A Dynamic Model of the World Copper Industry

Richard Denis

doi:10.5089/9781451930443.024.A006

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A Dynamic Model of the World Copper Industry

Author: Richard Denis

Publication Date:: 01 Jan 1978

eISBN:: 9781451930443

Language:: English

Keywords:: SP; copper industry; price elasticity; refined copper; copper producer; utility function; Commodity markets; Stocks; Consumption; Global; fixed capital

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During the period February 1976 to February 1977, the commodity that contributed most to the cumulative total of shortfalls in world- wide export earnings was copper.1 Copper represents more than 20 per cent of total export earnings for five Fund members: Chile, Papua New Guinea, Peru, Zaïre, and Zambia. These countries accounted for 60 per cent of world copper exports in 1975. These facts are brought forward to show that difficulties affecting the copper industry are bound to create problems and hardships that will spill over into domestic economies of both producer and consumer countries. Paradoxically, attempts at identifying the sources of disequilibrium or just finding the major economic characteristics of the industry have been few. Although many models of the copper market are known to exist, they have usually been built by private industrial and consulting organizations and have not been published. The most commonly quoted published paper of the copper industry is the Fisher, Cootner, Baily model (1975), which is a yearly model.

Abstract

During the period February 1976 to February 1977, the commodity that contributed most to the cumulative total of shortfalls in world- wide export earnings was copper.¹ Copper represents more than 20 per cent of total export earnings for five Fund members: Chile, Papua New Guinea, Peru, Zaïre, and Zambia. These countries accounted for 60 per cent of world copper exports in 1975. These facts are brought forward to show that difficulties affecting the copper industry are bound to create problems and hardships that will spill over into domestic economies of both producer and consumer countries. Paradoxically, attempts at identifying the sources of disequilibrium or just finding the major economic characteristics of the industry have been few. Although many models of the copper market are known to exist, they have usually been built by private industrial and consulting organizations and have not been published. The most commonly quoted published paper of the copper industry is the Fisher, Cootner, Baily model (1975), which is a yearly model.

It is often argued that the general field of commodity model specification and estimation does not benefit from the higher standard usually applied to macroeconomic modeling, even though it is clear that commodity markets are ideally suited to hypothesis testing in such areas as non-tȃtonnement price theory, econometric estimation, growth model specification, and economic policy. The model of the world copper industry developed in this paper attempts to confirm this observation in a number of ways. This model is a quarterly one, specified in continuous time as a system of differential equations that represent the partial adjustment of a number of supply, demand, price, and stock functions. The specification of this model is obtained in two steps. Section I develops a general commodity model that is intended to show the relationships existing between supply, demand, price, and investment.

Two major aspects are formulated in this general commodity model. First, if the model is to be used for policy analysis, its behavior should be plausible in both the short term and the long term. This behavior implies an endogenous investment function that is regarded as a central part of the specification because of its effect in the form of feedbacks on both prices and quantities. In the particular case of copper, it is the slow adjustment of supply relative to demand that creates situations of excess supply, such as the one observed at the present time, and vice versa.

The second aspect is an attempt at specifying speculative activities in commodity markets. Speculative and trading demand for stocks depend on expected prices, and it is felt that a continuous time specification in the form of a differential equation system is helpful in embodying these forces in the price formation mechanism. Finally, this general commodity model is shown to be valid for both continuous production commodities (such as copper) and agricultural commodities, in which case production can be assumed to be exogenous in a first approximation. Section I emphasizes the nature of the dynamic adjustment and the long-run properties of the general commodity model.

Section II develops the model of the world copper industry, taking into account as many institutional factors that characterize the industry as possible. It is also shown that this model is basically an extension of the general model, and therefore the same properties concerning its specification and dynamic stability should hold. The approach is facilitated by the fact that copper is a homogeneous commodity, which has the additional characteristics of being noncorrodible and highly conductive. These characteristics imply that, at least theoretically, all the copper extracted in the past should still be available for reprocessing, thus giving rise to an important secondary industry. The conductivity aspect has made it particularly suitable for electrical use, and its non- corrodible properties for many other uses.

The validation of the model in terms of dynamic forecasting and some illustration of the use of the model for policy analysis are given in Section III. To bring to the fore the qualities of simultaneous estimation by full-information maximum likelihood (FIML), it has been kept in mind throughout this paper that the number of exogenous variables should be as small as possible. The conclusions in Section IV are followed by appendices describing the full steady-state solution of the copper model, a description of the estimation method, a set of charts depicting the predictive performance of the model, and data sources and definitions.

I. A General Commodity Model

the model

The purpose of introducing a general commodity model prior to specifying the model representing the world copper industry is mainly to illustrate the dynamic behavior and economic implications of such a simple model. The performance of such a dynamic commodity model is related directly to the existence and realistic character of its equilibrium solution and the implications attached thereto. It is assumed that the market organization is competitive. This is so because in many markets prices are determined by freely competing buyers and sellers dealing in a homogeneous commodity on the basis of organized exchanges. This form of market organization can be weakened when a few buyers or sellers dominate the market, as is true for the U. S. copper producers.

Following the exposition of the commodity model, the conditions for dynamic stability of this model are derived and commented upon. Assume a general commodity model as follows:

\begin{matrix} D \log p (t) = α_{1} \log [\hat{S} (t) / S (t)] - δ [D \log S (t) - μ] & (1) \end{matrix}

\begin{matrix} \hat{S} (t) = S_{0} {[\tilde{p} (t) / p (t)]}^{β_{1}} C {(t)}^{β_{2}} with β_{1,} β_{2} > 0 & (2) \end{matrix}

\begin{matrix} D log C (t) = α_{2} log [\hat{C} (t) / C (t)] & (3) \end{matrix}

\begin{matrix} \hat{C} (t) = C_{0} p {(t)}^{- b_{1}} Y {(t)}^{b_{2}} with b_{1}, b_{2} > 0 & (4) \end{matrix}

\begin{matrix} D \log Q (t) = α_{3} \log [\hat{Q} (t) / Q (t)] & (5) \end{matrix}

\hat{Q} (t) = F [p, Q (L (t), I (t))]

where

\begin{matrix} Q (t) = Q_{0} L {(t)}^{α} I {(t)}^{[1 - α]} e^{λ t} with 0 < α < 1 & (6) \end{matrix}

\begin{matrix} D m (t) = γ {α_{4} \log [Π (t) / p P k (t) I (t)] + γ_{0} - m (t)} & (7) \end{matrix}

\begin{matrix} D \log \tilde{p} (t) = α_{5} \log [p (t) / \tilde{p} (t)] & (8) \end{matrix}

\begin{matrix} D S (t) = Q (t) - C (t) & (9) \end{matrix}

\begin{matrix} D \log I (t) = m (t) & (10) \end{matrix}

where

D = d/dt = differential operator
$\hat{X} (t)$ denotes the desired value of a variable X(t)
$\hat{X} (t)$ denotes the expected value of a variable X(t)
C(t) = rate of consumption at time t
Q(t) rate of production at time t
S(t) = level of inventories at time t
1(t) = level of capital stock in fixed capital at time t
m(t) = proportional rate of increase in fixed capital at time t
Π(t) = nominal profits at time t
p(t) = P(t)/Ps(t) = relative price of the commodity at time t
ρ = rate of profit = (1 +r) where r is a rate of discount including risk
P(t) = nominal price of the commodity at time t
Ps(t) = nominal price of substitute commodities at time t
Pk(t) = nominal price of capital goods at time t
Y(t) = economic activity
L(t) = amount of variable inputs employed in the industry
λ = rate of growth of technical progress

This model assumes a continuous production process represented by a production function. For agricultural commodities, this production function may be substituted for an exogenous variable representing the quantities produced by a seasonal crop. The investment relationship represented by equation (7), however, would still be present in that case. A related prototype model was applied to the sugar market by Wymer (1975).

This model is essentially a representation of a non-tâtonnement process acknowledging the fact that, in commodity markets, trade takes place continuously, even though prices may not be in equilibrium. It also includes an investment feedback that represents the slow adjustment process of fixed capital formation. Disequilibrium situations in the latter are probably the major source of disturbances in the market. It can be observed from equations (2) and (4) that the global demand for the commodity is twofold: demand for the purpose of consumption represented by equation (4), and demand for the purpose of transfer or investment over time by (2). The demand for stock will usually be assumed to depend on two types of agent in the economy: speculators and traders. These demands introduce a mixed stock/flow element in the model, which will bear implications throughout the specification and estimation of the model.

The dynamics of price formation is usually assumed in general equilibrium models to be represented by a relationship of the form Dp = Hⁱ(Ei), where E_i is a vector of excess demands and where the function Hⁱ is assumed to be continuous, sign preserving, and not approaching zero as its argument becomes infinite. The adjustment postulated in equation (1) is found in Samuelson (1941) and represents a market in which equilibrium can be reached only when both stocks and flows are in equilibrium. The general character of this formulation should be acknowledged, because it holds more common price adjustment relationships as a special case. In the absence of stock factors, or under the equivalent assumption of instantaneous asset equilibrium, equation (1) would reduce to a flow/excess demand relationship where μ is the expected rate of growth of consumption. Conversely, in the absence of flow disequilibrium, price adjustment will be guided toward equilibrium through adjustment of portfolios by holders of stocks. The dominance of one or the other aspect in the price formation mechanism will be an empirical matter and will reflect characteristics of the commodity under consideration. The investment demand for a commodity depends, for instance, on such characteristics as value and storability. Perishable goods are worth transferring over time only if the cost of doing so, for example, storage, compares favorably with the value of these goods at the future time of consumption. The preceding formulation allows for the whole intermediate spectrum of possibilities whereby the single- equation solution of the differential equation (1) can range continuously between damped harmonic motion and exponential approach to equilibrium. If relationship (1) were considered in isolation, the necessary conditions for stability of such an equation would require that the real part of its characteristic roots (eigenvalues) be negative. Definitions of global and local stability are examined later in this section.

The demand for stocks defined in equation (2) is a joint representation of (a) the trading component of demand C(t) for these stocks with an income elasticity β₂ and (b) the speculative component that is influenced by the relative position of expected future price and actual price. The variables entering equation (2) generally would also include other factors affecting the demand for stocks, such as interest rates. Among other variables entering the decision set of speculators, one can find both the situation and attractiveness of other commodities suitable for speculative purposes that could induce speculators to enter or leave the commodity market under scrutiny and the cost of switching from one market to another. On the trading aspect, changes in efficiency of production or improvement in channels of distribution may imply that common rules of thumb attached to desired levels of stock/consumption ratios differ from constants, or similar simple relationships. The presence of uncertainty is taken into account via the specification of the expected price variable.

Under conditions of certainty, intertemporal price equilibrium was lucidly explained by Samuelson (1957) in the context of the literature on the supply of storage. The argument can be summarized as follows: Today’s future price must equal the current price for that future date so that $\tilde{p} (t, θ) = p (t + θ)$ where $\tilde{p} (t, θ)$ is the expected price at time (t) for period θ in the future and p(t+ θ) is the spot price prevailing at (t+ θ). The price spread between future price and present price is limited by the “carry-over” cost, and goods are stored only if it pays to do so. Or, $\tilde{p} (t, θ) = p (t) + r (t, θ)$ where r(t, θ) is the cost of transport of the goods from period t to t+ θ; over any period, the current price is determined so as to clear the market of the quantities produced in the current period or taken from storage.

The extension of the certain case to an Arrow-Debreu context of uncertainty implies that the foregoing points still have to hold. All the prices are now subject to the state of the world for which the expectation is formed; that is, $\tilde{p} [t, θ : i]$ is the expected price at time t for period θ in the future, contingent on the state i occurring. The market participants will maximize a utility function reflecting the global situation of uncertainty and the set of contingent prices. Elementary utility functions reflecting individual attitudes toward risk will be embodied in the process. The introduction of states of the world gives rise to differential- difference equations, which are beyond the capabilities of the estimation technique used in this paper. This three-dimensional problem will be solved by successive approximations. The state of the world dimension is reduced to a plane known as the certainty equivalent plane. The next approximation consists in representing the term structure of future prices by a straight line defined by the spot price and one future price that is conveniently chosen as the observation interval corresponding to one actual forward quotation of the commodity considered.

Price expectations are an important part of commodity markets, since buyers and sellers usually meet on exchanges where contracts as well as information are exchanged. The formation of expectations is defined by equation (8) and is postulated to be of the adaptive type whereby expected prices are determined by a first-order exponential distributed lag function of spot prices. The estimate of the speed of adjustment of expected prices to spot price α₅ will suggest whether these expectations are static or not. It is felt that adaptive expectation mechanisms are adequate in representing the formation of market expectations about spot prices, because the system estimation used to obtain numerical values of the parameters maintains a priori within and across equation restrictions in the estimation of α₅.

The supply side of the market is assumed to be represented by a production function (6). For simplicity, a Cobb-Douglas production function² is assumed but a constant elasticity of substitution (CES) production function could equally well be postulated without loss of generality. This would, however, complicate the model unnecessarily for the present purpose, since linearization of this production function would be required. In equation (6) two factors of production are assumed. Fixed factors, such as fixed capital, are represented by 1(t), and variable factors are represented by L(t). Production functions also allow for the existence of technological progress, which is represented in equation (6) by the trend term e^λt and is assumed to be neutral in Harrod’s sense. This last variable is found to be relevant to the longterm properties of the model, as is illustrated later in this section when the steady-state solution of the model is derived.

It is also assumed that some of the variables entering the production function cannot be formulated explicitly because no data are, in general, available in commodity markets about these variables. The supply equation (which follows) is derived to be only a function of fixed capital, technological progress, and the relative price of the commodity. Variable factors of production have been substituted using the assumption of profit-maximizing behavior.³ The supply equation used throughout the rest of this section is as follows:

\begin{matrix} \hat{Q} (t) = {Q_{0}}^{ʹ} I p^{m_{1}} e^{m_{2^{t}}} & (6^{'}) \end{matrix}

with $Q_{0}^{ʹ} = Q_{0}^{\frac{1}{1 - α}} α^{\frac{1}{1 - α}}, m_{1} = \frac{α}{1 - α}; m_{2} = \frac{λ + α ϵ}{1 - α}; 0 \leq m_{1} < 1$

where p is now the price of the commodity studied, relative to its substitutes. It has been assumed that the ratio of the nominal price of the commodity to the nominal price of variable inputs can be approximated by the relative price of the commodity multiplied by a time trend, corresponding to the ratio of the nominal price of substitutes to the nominal price of variable inputs (i.e., P/W = PPs/PsW with Ps/W = e^∊t). This constraint on the use of relative prices is due to the introduction of relative prices in the demand function, and homogeneity requires this relative price variable to be used throughout the model.

Investment in fixed capital, represented by equations (7) and (10), is the cornerstone of any commodity market specification. Situations of excess supply or excess demand create hardship for both producers and consumers. The speeds of adjustment characterizing the investment process in fixed capital of production are probably the major sources of disequilibrium in commodity markets, and the usually long time lag between the decisions to create new producing capacity and its availability for production is one of the least controllable sources of instability in these markets. It is assumed that for any given rate of profits there is a partial equilibrium proportional rate of increase in the stock of fixed capital, and that the actual proportional rate of increase in capital is increasing at a rate proportional to the excess of the partial equilibrium rate over the actual rate. The desired rate of fixed capital formation is assumed to depend on the rate of profit,⁴ which can be explained alternatively as the excess of the marginal product of capital (1 – ɑ)Q(t)/I(t) over the real marginal cost of capital ρPk(t)/P(t). The parameter γ₀ is the equilibrium rate of fixed capital formation associated with equality between the marginal product of capital and the marginal cost of capital. The existence of such a function is not affected by a possible exogenous production relationship, and it seems crucial that this investment function should be postulated because of its long-term effect on the dynamic properties of the system.

The demand function (4) is assumed to have the usual form of multiplicative demand functions derived from Cobb-Douglas utility functions, thus depending on relative prices and income.

The production and demand relationships postulated in this model are not myopic, as was shown in Bergstrom and Wymer (1976). For a given relationship such as

(a) D log C α₂ log [Ĉ/C] + γ with Ĉ = b_l Y where C represents consumption and γ income, and where γ is the expected rate of growth of income. It may appear that when income is following its long-run growth rate the ratio of consumption to income would always be less than the desired ratio. By redefining $b_{1} = b_{1}^{ʹ} e^{\frac{γ}{α^{2}}}$ the expression (b) $D \log C = α_{2} \log ({b^{'}}_{1} e^{\frac{γ}{α 2}} Y / C)$ will be seen to collapse to expression (a). Expression (a) implies that if consumption is at its desired level in relation to income, it is increasing at a proportional rate equal to the expected rate of growth of income.

Before proceeding toward the study of the dynamic properties of the preceding model, some comments are in order about the nature of the adjustment process taking place toward equilibrium. It is assumed that, in general, the actual level of a variable adjusts toward its desired or partial equilibrium value with an exponential lag. This does not mean, of course, that there is monotonic adjustment to long-term growth. The formal representation of that adjustment is $D y (t) = β [\hat{y} (t) - y (t)]$ and can be recognized in equations (1), (3), (5), (7), and (8). This relationship does not preclude rapid adjustment to equilibrium and the mean time lag |1/β|⁵ may be close to zero, thus implying that ŷ (t) ≃ y(t). Estimates of β can be obtained that are asymptotically unbiased even for relatively long observation periods. The next subsection deals with the stability and properties of the dynamic system specified earlier.

the dynamics of the model

The utility derived from the specification and estimation of commodity models is to provide some insight into the behavior of commodity markets for the purpose of decisionmaking. The main uses of commodity models are policy analysis and forecasting/simulation. The horizon for which the previously developed model is suitable can vary from a few months to several years. The usefulness of a commodity model bears a direct relationship to its ability to reproduce the observed dynamic behavior of the market under consideration. But one’s understanding is, of course, a direct consequence of the closeness of the specification to economic theory and the institutional structure of the market. The differential equation system, which is the formal representation of the model, must be dynamically stable in a global sense. This stability requirement will be a major check on the realistic specification of the model. Demand and supply in commodity markets are not observed, in reality, to diverge from equilibrium over a long period, although these variables may show substantial fluctuations. A correctly specified dynamic system should exhibit the same property in its long- run properties. The long-run behavior of the model should also be plausible. It can be inferred that implausible long-run behavior probably results from misspecifications in the postulated structure. Very few models are critically examined on these grounds, in particular in the area of estimated commodity models, and it should be pointed out that implausible long-run properties will lead to deriving meaningless conclusions from the model.

The myopic approach, consisting of dismissing long-run properties because the model is to be used for short-term applications, is totally misleading as the variables composing the specification bear fairly stable relationships to one another. An example of this is the fairly regular behavior of stock/consumption ratios or profit/investment ratios. The broad movements in these variables and their regularities suggest that steady-state growth paths exist toward which the markets tend in the long run. Hence, it becomes obvious that even short-run models should conform to plausible long-run properties. The existence of a steady- state solution to a system of differential equations may call for the imposition of restrictions on some of the structural parameters. The unicity of the steady-state solution is not strictly necessary, even though it may be useful. Finally, the investigation of steady-state properties of a commodity model will ensure that the estimated model is mathematically consistent. This may be a by-product of the plausibility of long-run behavior mentioned earlier.

The model previously expressed can be represented by DY(t) = F[Y(t), Z(t), θ] where Y(t) is a vector of endogenous variables, Z(t) a vector of exogenous variables, and θ the set of structural parameters. Assuming that all exogenous variables tend to exponential growth at a constant rate, that is, $Z_{i} (t) = Z_{i}^{*} e^{λ_{i^{t}}}$ for all i, the nonlinear system of differential equations has a steady-state solution, if a particular solution $Y_{i} (t) = Y_{i} {(t)}^{*} e^{ρ_{i^{t}}}$ exists to this system.

The stochastic components of the system introduce shocks that will disturb the system from its steady state, but, provided that the model is stable, the motion of the system will always converge toward the steady-state solution. The steady-state solution is obtained by substituting Y_i(t) and Z_i(t) and solving for the steady-state growth rates ρ_i and steady-state levels Y_i* [Y_i(t) for t 0] for all i.

The solution can be represented by ρ_i = g_i (λ, θ) and Y_i = G_i (Z*, λ, θ). This procedure applied to the general commodity model leads to the following solution. The system (1) to (10) using (6ʹ) and (7ʹ) has a particular solution given by

\begin{matrix} Q = Q^{*} e^{ρ_{1^{t}}} & with ρ_{t} = b_{2} λ_{\bar{y}} - b_{1} ρ_{2} \\ C = C^{*} e^{ρ_{1^{t}}} \\ S = S^{*} e^{ρ_{1^{t}}} \\ I = I^{*} e^{ρ_{1^{t}}} \\ p = p^{*} e^{ρ_{2^{t}}} & with ρ_{2} = - \frac{m_{2}}{m_{1}} = - (\frac{λ}{α} + ϵ) \end{matrix}

where the steady-state levels are

\begin{matrix} \log Q^{*} = \log [ρ_{1} S^{*} - C^{*}] \end{matrix}

\log C^{*} = \log C_{0} + b_{2} \bar{\log Y} - b_{1} \log p^{*} - ρ_{1} / α_{2}

\log S^{*} = \log C^{*} + \log S_{0} + \frac{δ μ}{α_{1}} + \frac{ρ_{1}}{α_{5}} - [\frac{1}{α_{1}} + \frac{β_{1}}{α_{5}}] ρ_{2} - [\frac{1}{α_{5}} + \frac{δ}{α_{1}}] ρ_{1}

log I^{*} = log Q^{*} + ψ / α_{4} - ρ_{1} / α_{4}

\log p^{*} = \frac{1}{m_{1}} [\frac{ρ_{1}}{α_{3}} - \frac{ψ}{α_{4}} + \frac{ρ_{1}}{α_{4}} - \log {Q^{'}}_{0}]

One restrictive assumption had to be made to obtain this solution, namely, β₂ should be equal to unity in order to meet the constraints imposed by equations (1) to (8).

The preceding steady-state growth rates are very similar to classical results obtained for one-sector growth models. Production, consumption, and stocks are constrained to grow at the same rate through the market clearing identity (9), and their rate of growth ρ₁ is the sum of the rate of growth of business activity outside the market, weighted by the income elasticity of demand of the commodity studied, and the rate of growth of relative price of the commodity, weighted by the price elasticity of demand. This is to say that in the absence of price effects the commodity market is allowed to grow at a rate that may differ from the rest of the economy and could possibly correspond to certain phases of the technological development cycle of the commodity. By using the steady-state solution, many results can be obtained regarding the comparative dynamic properties of the model.

The rate of growth of relative prices follows the familiar conclusion of one-sector growth models whereby this variable will vary as a function of the rates of technical progress between the commodity market and the rest of the economy. Very strong assumptions would be necessary in order to find a growth rate of relative prices equal to zero. It must be concluded that under normal conditions for the commodity market investigated, the steady-state rate of growth of relative prices is a nonzero quantity linked directly to technological progress outside the market.

Some basic theoretical features of the model can be derived by differentiating the foregoing steady-state levels with respect to the structural parameters. For example, the effect on the price level of an increase in the real cost of capital has a negative effect represented by

\frac{\partial \log p^{*}}{\partial ψ} = - \frac{1}{m_{1} α_{4}} = - [\frac{1 - α}{α α_{4}}]

This expression depends on the parameter m₁, which is a measure of the labor intensiveness of the industry. Hence, in a highly capital- intensive production process, one requires a small movement in the cost of capital in order to obtain sizable changes in the proportional rate of change of the price level. Changes in the cost of capital can be obtained by either changes in attitude toward risk, which will influence the rate of profit and therefore increase or decrease the cutoff point at which firms producing the commodity are willing to increase their fixed capital requirement, or by movements in interest rates in the money markets where firms find their capital. Similarly, structural changes in the production process could manifest themselves through shifts in the capital intensiveness of production. Numerous other cases could be investigated and similarly commented upon. The influence of the variable chosen in the previous example (ψ) is seen here to have a one-to-one impact on the level of relative prices.

The extension of the analysis to agricultural commodities is as follows. It is assumed that production fluctuates randomly, thus following the common assumption made about the weather. The rate of growth of production will, however, be postulated to be positive. This is a consequence of increases in yield per unit of output, which exists irrespective of the random character of production.

The system (1) to (10) with $\bar{Q} (t) = Q^{*} (t) e^{λ_{{\bar{Q}}^{t}}}$ has a particular solution given by

\begin{matrix} C = C^{*} e^{λ_{{\bar{Q}}^{t}}} & with λ_{\bar{Q}} exogenous \\ S = S^{*} e^{λ_{{\bar{Q}}^{t}}} \\ I = I^{*} e^{λ_{{\bar{Q}}^{t}}} \\ p = p^{*} e^{ρ_{1^{t}}} & with ρ_{1} = \frac{b_{2}}{b_{1}} λ_{\bar{Y}} - \frac{1}{b_{1}} λ_{\bar{Q}} \end{matrix}

where the steady-state levels are

\log C^{*} = \log Q^{*} + A - \log [1 + e^{A}]

with A = [\frac{1}{α_{1}} + \frac{β_{1}}{α_{5}}] ρ_{1} + \frac{δ}{α_{1}} λ_{\bar{Q}} - \log S_{0} - \frac{δ}{α_{1}} μ - λ_{\bar{Q}}

\log S^{*} = \log ({\bar{Q}}^{*} - C^{*}) - λ_{\bar{Q}}

\log I^{*} = \log {\bar{Q}}^{*} + ψ / α_{4} - λ_{\bar{Q}} / α_{4}

\log p^{*} = (b_{2} / b_{1}) \bar{\log Y} * - \frac{\log C^{*}}{b_{1}} + \frac{\log C_{0}}{b_{1}} - \frac{λ_{\bar{Q}}}{b_{1} α_{2}}

The endogenous rate of growth of the physical variables has become exogenously equal to $λ_{\bar{Q}}$ . The rate of growth of relative prices has, however, changed considerably. It is now equal to the rate of growth of business activity weighted by the ratio of the income elasticity of demand to the price elasticity of demand, minus the rate of growth of technological progress.

No comparative dynamic analysis similar to the previous type of model will be made here, as all the variables are influenced mostly by exogenous rates of growth and steady-state levels.

The models, complete with their steady-state solutions, must be shown to be asymptotically stable, at least in the neighborhood of the steady states. This analysis consists in showing that given the system with initial values in the neighborhood of the steady-state growth path, this system will converge toward the steady-state path if undisturbed. Because the models presented earlier are nonlinear systems with constant coefficients and because it is not the purpose of the present paper to discuss fully measures of stability, it is necessary to approximate the system by a linear system that does have constant coefficients. Any stability properties found in this way are necessarily local stability properties. A standard theorem used in stability analysis is the theorem of Perron. This theorem states that the solution y(t) = 0 of the differential equation system Dy(t) = Ay(t) + f [y,t] in which A is a matrix of constants and f a vector of functions, is asymptotically stable if f[y,t]/|y| tends to zero uniformly in t as |y| tends to zero and the eigenvalues of A have negative real parts.

The present system is convenient in the sense that the stability analysis can be conducted in terms of deviation from the steady-state solutions calculated earlier, these deviations being the logarithm of the ratios of the variables to their steady-state paths, that is, y_i = log [Y_i/Y_i*]. The nonlinear models expressed in this form do not involve t, and therefore the first part of the theorem requiring uniform convergence is by definition satisfied. The general model with endogenous production can conveniently be expanded in terms of deviations from its steady state as follows:

\begin{matrix} D^{2} y_{1} = & - [α_{5} + α_{1} β_{1}] D y_{1} - [α_{1} + δ α_{5}] D y_{5} + α_{1} D y_{2} \\ - δ D^{2} y_{5} + α_{1} α_{5} y_{2} - α_{1} α_{5} y_{5} \end{matrix}

{D y}_{2} = α_{2} [{- b}_{1} y_{1} + b_{2} x_{1} - y_{2}]

D y_{3} = α_{3} [m_{1} y_{1} + y_{4} - y_{3}]

D^{2} y_{4} = γ α_{4} [y_{3} - y_{4}] - γ D y_{4}

D y_{5} = Q^{*} / S^{*} [e^{y_{3} - y_{4}} - 1] - C^{*} / S^{*} [e^{y_{2} - y_{5}} - 1]

where

y_{1} = \log \frac{p}{p^{*} e^{ρ_{2^{t}}}}

y_{2} = \log \frac{C}{C^{*} e^{ρ_{1^{t}}}}

y_{3} = \log \frac{Q}{Q^{*} e^{ρ_{1^{t}}}}

y_{4} = \log \frac{I}{I^{*} e^{ρ_{1^{t}}}}

y_{5} = \log \frac{S}{S^{*} e^{ρ_{1^{t}}}}

x_{1} = \log \frac{Y}{Y e^{λ_{{\bar{y}}^{t}}}}

Taking a Taylor series expansion of equation (9) yields the approximate linear system (1) to (5), where equations (1) to (4) are identical and equation (5) becomes

D y_{5} = Q^{*} / S^{*} [y_{3} - y_{5}] - C^{*} / S^{*} [y_{2} - y_{5}]

The second condition can be shown to depend solely on the matrix of structural coefficients A of the system re-expressed as Dy(t) = Ay(t), which, provided that the eigenvalues of that matrix of constant coefficients have negative real parts only, implies that the steady state of the nonlinear model will be asymptotically stable in the sense that lim y(t) = 0 as t→+ ∞, given that its initial values are in the neighborhood of equilibrium. The eigenvalues already characterized by negative real parts may also be complex, and the afore-mentioned limit would therefore be approached with a harmonic motion. Since the elements of A are explicit combinations of the estimated parameters of the system, the asymptotic stability conditions can be expressed explicitly in terms of the structural parameters by using the Routh-Hurwicz conditions.

II. Application to the World Copper Industry

institutional background of the model

The model presented in this section is by necessity a simplified representation of an industry that is relatively complex in its institutional features. No attempt is made to provide a highly disaggregated description of either production or consumption on a geographical basis. Such a disaggregation would demand data resources that are just not available or too unreliable to be used, particularly on the supply side of the market, and would introduce additional exogenous variables that would then become an impediment to policy analysis related to the dynamic behavior of the structural model.

The model is estimated by FIML with quarterly data spanning the period beginning with the fourth quarter of 1960 to the fourth quarter of 1975. One helpful feature characterizing the industry that is used in specifying the model is the existence of a dual pricing system for the commodity. As a broad rule, the U. S. industry behaves with respect to the U. S. producer price, and the rest of the world deals on the basis of London Metal Exchange (LME) quotations.

Since few data are available on production, consumption, and pricing in the member countries of the Council for Mutual Economic Assistance (CMEA), the contribution of these countries to the working of the world industry has been partially excluded. Trade of refined copper with the Western world is being recorded, and exchanges take place on the basis of LME quotations. Hence, the model takes into account this trade and its relation to LME prices. Throughout this model, the form of copper under scrutiny is refined copper and all other forms, such as ore, concentrates, blister, or scrap, are expressed in metric tons of copper content.

On the supply side, production of refined copper is broken down between primary refined copper and secondary refined copper for both the United States and the rest of the world (referred to hereinafter as non-U. S.). In addition, for each primary production equation, an investment equation is specified to represent the behavior of mine capacity. Similarly, an equation representing the stock of copper available for reprocessing to the secondary industry is specified for both the United States and the non-U. S. The demand side is represented by two equations corresponding to non-U. S. and U. S. consumption of refined copper. The links between the U. S. and the non-U. S. markets can be classified as follows:

(a) Trade in both refined copper and ore and concentrates takes place between the two blocs. This trade has been affected by restrictions during the sample period chosen for estimation, a matter that is dealt with at greater length later on.

(b) Because trade takes place, arbitrage between the U. S. pricing system and the LME quotations tends to keep these prices in line with one another.

(c) The U. S. secondary industry deals at LME prices, thus ensuring the working of the arbitrage mentioned earlier. This institutional feature is due to the fact that international trade in copper scrap is fairly sophisticated and takes place with non-U. S. partners following the LME.

Copper can be traded at various stages of the production process, and the general classes most often used are ores and concentrates, blister copper, and the various types of refined copper. Most copper is sold under bilateral contracts between a producer and a fabricator or a merchant, but some sizable quantities are sold on a spot basis or through a terminal market. Many contracts exhibit similar clauses, but there is no standard contract; the content of these contracts tends to remain confidential. The most common clauses are force majeure, currency of payment, quotational period, and exchange rate variations. Concentrates are sold on long-term contracts where pricing is variable but usually based on LME quotations in the non-U. S. or on the U. S. producer price for some transactions with the United States. Blister sales can be either sold or treated on “toll” by refiners, and the prices used will be the same as above. Contracts for sale of refined copper tend to be annual contracts, where the price chosen for international transactions is the LME cash settlement price for wirebars either on a particular day or averaged over a calendar month. Except in the United States and Canada, some domestic prices exist for domestic trading; these prices will usually reflect currency fluctuations but will be close to LME prices. Owing to the importance of LME quotations, it may be opportune to expand on the role of the LME.

The LME provides the services of a free commodity market for six nonferrous metals: copper, lead, silver, tin, zinc, and recently aluminum. The main purpose of the LME is to ensure the functioning of spot prices and futures quotations as well as providing warehousing facilities for the efficient working of their operations. A low proportion of non-U. S. consumption is actually traded through the LME, but the justification for the existence of the LME should not be looked at from the point of view of the quantities traded. One of its major justifications is that it provides a futures market that facilitates equilibrium across time. The fact that metals are storable goods provides a potential choice between present and future output and consumption. The one-directional (or irreversible) character of this intertemporal adjustment is important insofar as currently produced goods can be held for future instead of current consumption, but in general it is not possible to consume future output. It is the quotation of both cash and forward prices by the LME that highlights its role as a main indicator for the exchange of contracts between producers and consumers in the greater part of the world, even though the physical exchange might not take place on the LME. Another, but no less important, aspect of the function of the LME is to reflect the views of traders and speculators about directions in the price of refined copper, which information is in effect used by the U. S. producers in fixing their own domestic price.

But probably the major justification of the LME is to provide some ground for speculative demand and a market for risks under conditions of uncertainty. In the previous section, the concept of demand for deferred consumption over time was introduced for storable commodities, and this concept is re-examined here in connection with the LME. As mentioned then, the total demand for stocks is composed of a hedging aspect as well as a speculative aspect. Traders can equally be speculators or hedgers, but the existence of speculators facilitates the exchange of contingent claims and therefore helps the hedging function, which is the major usefulness of the commodity market. The reasons why commodities, and, in particular, nonferrous metals, on the LME are subject to speculative demand can be stated as follows. (a) Metals can be dealt in at organized exchanges. The organization implies standardized contingent claims in the form of contracts that are perfectly durable, fairly liquid, and defined by the exchange rules. (b) As shown in Section I, under conditions of certainty, speculative activity adjusts the spot price in such a way as to equate the difference between expected price and current price with the sum of interest cost and carrying cost. Uncertainty implies inequality between current price and expected price because it pays speculators to enlarge or reduce their commitments. As consequences of the foregoing points, one can see that the organization of the market facilitates entry and exit by speculators wishing to switch asset composition within their portfolio; copper, for instance, has been found to be attractive to speculators when other assets, such as gold or currencies, do not offer sufficient attraction.

Hedging is typically described as the action whereby a purchaser under a forward transaction at a fixed price protects himself against an adverse price movement by, at the same time, effecting a forward sale at a comparable fixed price and due at the same future period. Similarly, a seller under a forward transaction at a fixed price will protect himself, by a hedging purchase, from loss that could result from a price rise before the maturity of the principal contract. The hedging function has always been a natural one on the LME and is intended to provide insurance against uncertainty about deliveries of metal on schedule. This uncertainty is the underlying motive for holding some stocks. The necessity to maintain the production process is also a major factor. The level of stocks held is related directly to the likelihood of a breakdown in the production stream. This implies that even if stocks are not expected to show profits from price appreciation, they will still exhibit advantages from the standpoint of minimizing a production loss function. Thus, stocks will still be held when prices are expected to decline substantially, because a valuation of these stocks would include the opportunity cost associated with the necessity of maintaining continuous production.

In the U. S. market, the producers set the price to reflect what they believe to be a sustainable medium-term level of copper prices, taking into account their own resulting supply decisions. This price is fairly stable in terms of variance, particularly when compared with the LME spot price. The group of U. S. copper producers forms a relatively weak oligopoly whose decision set is bound by the necessity to remain relatively close to the LME price, fairly strict antitrust regulations, powerful labor unions, the necessity to operate the mining process as near capacity as possible, etc. The U. S. copper industry is highly integrated, and the mining companies often hold the whole of the production process from the extraction of ore to the production of semifinished products. The latter are defined as consumption for the purpose of statistical recording.

Most exports of copper from the United States to the rest of the world are refined copper; most U. S. imports are in the form of ore and concentrates, which are processed by U. S. smelters and refineries.

There also exists a commodity exchange in New York known as comex, used mostly by merchants and consumers; this exchange is a terminal market that theoretically can perform functions similar to the LME. comex is a clearinghouse market where clearing members settle their trade balance daily with the Central Clearing Association. The trade taking place is in a range of seven delivery months, and price limits are in operation to prevent distortions owing to this trading. The quality of copper acceptable for delivery is lower than is permitted on the LME.

Some comments may be appropriate about the data used in the estimation. A full set of sources and definitions is provided in Appendix D. The collection and building of data series describing the various components of supply and demand in the world copper industry do not yet benefit from the technique of double-entry accounting, which characterizes the formation of other macroeconomic variables, such as balance of payments figures or national accounts. A number of defects have been observed in the areas of consumption, stock, and, to a lesser extent, production. U. S. data, in general, tend to be more reliable than non-U. S. data.

Consumption has been affected by some amount of underrecording. Quantification of that underrecording is the subject of many arguments among the statisticians establishing or using this consumption data. The major sources of underrecording are new areas of consumption of refined copper in developing countries. The quantities consumed that are not recorded may stem from national smelting that is not being accounted for by known smelting and refining companies. The view taken here is that the underrecording that may have been quite substantial in the 1950s and earlier has diminished considerably, and the data compiled today benefit from better statistical methods. The reasons just given are far from exhausting potential sources of bias in the data series.

The recording of stocks has always been done on a reported basis that has also been found to embody many forms of bias. If, as is true here, price formation is to be a function of stocks, the only reliable way of dealing with the problem is to assume virtual stocks formed from identities. The consistency of the series is thus ensured and is not affected by leads and lags in recording. By this method, any other inconsistencies in the series are reconciled, and therefore closing the model becomes possible. Mention should also be made of the character of LME stocks. These stocks have been observed to be relatively low throughout the 1960s. This was due mostly to the partial loss of importance of the London Metal Exchange during the 1959-63 attempt by some producers to stabilize prices. The level of these stocks has risen during the present decade with the exception of the price peak in 1973–74, and this may indicate a shift in the economic importance of these stocks. The present situation of excess supply has provided some grounds for such a shift, and some producers and traders have found it worthwhile to hold and maintain some of their stocks in LME warehouses. Among the reasons quoted are the ease of supply in time of transport uncertainties, concerning such areas as producing countries in Southern Africa, and also some considerations of quality for demanding consumers who require premium grades for electrical purposes. For the latter, obtaining premium grade refined copper from LME-held sources would imply additional refining of the metal. This procedure can be avoided by contracting with refiners that will produce the required grade in one step, thus saving one set of refining and transport charges. Such patterns have been observed in some European countries and would imply that standard grade copper acceptable under the LME rules is kept outside the main channels of consumption until the supply/demand situation makes it worthwhile to use it.

The major source of data problems on the production side is the secondary industry, which, outside the United States, is very difficult to monitor and leads to bias in the data series on secondary refined copper. There are three types of copper scrap: new scrap, old scrap, and ashes and residues. These three types are usually assumed, for data recording purposes, to compose two classes: new scrap and old scrap, where the latter is now the aggregation of old scrap and ashes and residues. This results from the definitions of the different types of scrap. New scrap consists mainly of trimmings, offcuts, grindings, and turnings produced by the transformation of basic shapes into semifinished goods. Most new scrap arises in the plants manufacturing these semifinished products, and this scrap is usually recovered within the plants or via the refiners providing the original shapes. For the purpose of this paper, new scrap is ignored, since it has been found that these quantities revolve in the system practically without losses until they are effectively consumed.

Old scrap originates through product obsolescence of both capital and consumer goods throughout the economies. Ashes and residues included in that category arise from manufacturing processes. In cases where these residues have a high copper content, it is usually worth recovering the metal, and this is done by the factories; otherwise, these residues are disposed of. Copper scrap is graded internationally, based mostly on the copper content of scrap and its ease of recovery. The main motivation of scrap metal industries is to realize revenues and profits in the upgrading process. Ultimately, if it pays to do so, all scrap can be recovered and transformed into refined copper of a quality that is identical to copper produced from primary sources. This is a consequence of the qualities and homogeneity of the metal; it also explains the relatively large size of the secondary sector in copper compared with other metals.

The stock of recoverable copper from scrap depends on the life of copper-containing products. Decisions to scrap are influenced by the business cycle, fixed capital formation in the economy, and the price obtainable for the copper and other recoverable metals contained in these products. Scrap processing is not a feature of domestic economies alone. The international trading of copper-containing goods is very active and is influenced by the price incentive that can be found, for instance, in moving to Europe or Japan U. S. scrap that the major U. S. producers are not willing or equipped to process.

Finally, in this subsection it may be worthwhile to comment on the long-term price of copper. Chart 1 is taken from a paper by Radetzki (1977) and is used here as an illustration of the real price of copper since the turn of the century. The major comment is the absence of a long-term trend in the data over the whole sample period. The period since 1963 has been a time of high copper prices, which probably reflects the situation of excess demand that prevailed during these years. The period 1940-46 is probably affected by price controls that distort the series. It can be argued that the real price of copper is of a cyclical nature where the length of the cycle would be about 50 to 60 years and where the period 1920-50 was a period of gross excess supply. It is not the purpose of the present explanation to document these hypotheses, but the purpose of Chart 1 is to show that the period over which the model is estimated is a time of relatively high real prices that is probably not expected to continue over the long run.

a model of the world copper market

The notation and variables composing the world copper model are as follows:

$\tilde{p} (t)$ = expected LME spot price of copper relative to the price of aluminum
p(t) = LME spot price of copper relative to the price of aluminum
q(t) = LME 3-month forward price of copper relative to the price of aluminum
pu(t) = U. S. producer price of copper relative to the price of aluminum
Sr(t) = non-U. S. stocks of refined copper
Cr(t) = non-U. S. consumption of refined copper
Qtr(t) = non-U. S. total production of refined copper
Qp(t) = non-U. S. primary production of refined copper
Qs(t) = non-U. S. secondary production of refined copper
NMu(t) = U. S. imports of refined copper and ore and concentrates
MCr(t) = non-U. S. mine capacity
Kr(t) = non-U. S. stock of recoverable copper
Qup(t) = U. S. primary production of refined copper
Qus(t) = U. S. secondary production of refined copper
Cu(t) = U. S. consumption of refined copper
MCu(t) = U. S. mine capacity
Ku(t) = U. S. stock of recoverable copper
mr(t) = proportional rate of increase in non-U. S. mine capacity
mu(t) = proportional rate of increase in U. S. mine capacity
Su(t) = U. S. producer stocks of refined copper

The exogenous variables are as follows:

Yr(t) = index of manufacturing output for non-U. S.
Yu(t) = index of manufacturing output for the United States
SOCd(t) – SOCx(t) = rate of import minus rate of export of refined copper from CMEA countries
Suc(t) = U. S. consumer stocks of refined copper
USSL(t) = U. S. strike losses

Finally,

D = d/dt = differential operator
$\hat{X}$ denotes the desired value of a variable X(t)
$\tilde{X}$ denotes the expected value of a variable X(t)

The set of relationships describing the model are as follows:

\begin{matrix} D \log p (t) = α_{1} \log [\hat{S} r (t) / S r (t)] - δ [D \log S r (t) - μ] & (1) \end{matrix}

\begin{matrix} \hat{S} r (t) = S_{0} {[\tilde{p} (t) / p (t)]}^{β_{2}} C r (t) & (2) \end{matrix}

\begin{matrix} D \log q (t) = α_{2} \log [\tilde{p} (t) / q (t)] & (3) \end{matrix}

\begin{matrix} D \log C r (t) = α_{3} \log [\hat{C} r (t) / C r (t)] & (4) \end{matrix}

\begin{matrix} \hat{C} r (t) = C_{0} p {(t)}^{{- b}_{1}} Y r {(t)}^{b_{2}} & (5) \end{matrix}

\begin{matrix} D \log Q p (t) = α_{4} \log [\hat{Q} p (t) / Q p (t)] & (6) \end{matrix}

\begin{matrix} \hat{Q} p (t) = Q p_{0} p {(t)}^{m_{1}} M C r (t) e^{m_{2^{t}}} & (7) \end{matrix}

\begin{matrix} D \log Q s (t) = α_{5} \log [\hat{Q} s (t) / Q s (t)] & (8) \end{matrix}

\begin{matrix} \hat{Q} s (t) = Q s_{0} p {(t)}^{n_{1}} K r (t) e^{n_{2^{t}}} & (9) \end{matrix}

\begin{matrix} D m r (t) = γ_{1} α_{6} \log [Q p (t) / M C r (t)] - γ_{1} m r (t) - γ_{1} ψ_{1} & (10) \end{matrix}

\begin{matrix} D \log p u (t) = β_{3} \log [C u (t) / Q u p (t)] & (11) \end{matrix}

\begin{matrix} D \log Q u p (t) = α_{7} \log [\hat{Q} u p (t) / Q u p (t)] & (12) \end{matrix}

\begin{matrix} \hat{Q} u p (t) = Q u p_{0} p {(t)}^{{m^{'}}_{1}} M C u (t) e^{{m^{'}}_{2^{t}}} & (13) \end{matrix}

\begin{matrix} D \log Q u s (t) = α_{8} \log [\hat{Q} u s (t) / Q u s (t)] & (14) \end{matrix}

\begin{matrix} \hat{Q} u s (t) = Q u s_{0} p {(t)}^{{n^{'}}_{1}} K u (t) e^{{n^{'}}_{2^{t}}} & (\begin{matrix} 15 \end{matrix}) \end{matrix}

\begin{matrix} D \log C u (t) = α_{9} \log [\hat{C} u (t) / C u (t)] & (16) \end{matrix}

\begin{matrix} \hat{C} u (t) = C u_{0} p u {(t)}^{- {b^{'}}_{1}} Y u {(t)}^{{b^{'}}_{2}} & (17) \end{matrix}

\begin{matrix} D m u (t) = γ_{2} α_{10} \log [Q u p (t) / M C u (t)] - γ_{2} m u (t) - γ_{2} ψ_{2} & (18) \end{matrix}

\begin{matrix} D \log N M u (t) = α_{11} \log [\hat{N} M u (t) / N M u (t)] & (19) \end{matrix}

\begin{matrix} \hat{N} M u (t) = N M u_{0} {[p (t) / p u (t)]}^{- ζ} C u (t) & (20) \end{matrix}

\begin{matrix} D S r (t) = Q t r (t) - C r (t) - N M u (t) - [\bar{S O C d (t) - S O C x (t)}] & (21) \end{matrix}

\begin{array}{l} D K u (t) & = Q u p (t) + N M u (t) - D S u (t) - \bar{D S u c} (t) \\ = C u (t) - Q u s (t) & (22) \end{array}

\begin{array}{l} D K r (t) & = Q p (t) - N M u (t) - D S r (t) \\ = C r (t) - Q s (t) + [\bar{S O C d (t) - S O C x (t)}] & (23) \end{array}

\begin{array}{l} D S u (t) & = Q u p (t) + Q u s (t) - C u (t) + N M u (t) - \\ \bar{D S u c} (t) + U S S L (t) & (24) \end{array}

\begin{matrix} D Q t r (t) = D Q p (t) + D Q s (t) & (25) \end{matrix}

\begin{matrix} D \log M C r (t) = m r (t) & (26) \end{matrix}

\begin{matrix} D \log M C u (t) = m u (t) & (27) \end{matrix}

The expectation formation mechanism is as follows:

\begin{matrix} D \log \tilde{p} (t) = β_{1} \log [p (t) / \tilde{p} (t)] & (\begin{matrix} 28 \end{matrix}) \end{matrix}

This model can be shown to be closely related to the general commodity model investigated in the previous section. Equations (1), (2), (4), (5), (6), (7), (10), (21), (26), and (28) represent the non-U. S. market specification of the general model, except that total production of refined copper has been divided into (a) a primary production—equation (7)—and its adjustment—equation (6)—and (b) a secondary production—equation (9)—and its adjustment—equation (8). These two components of total non-U. S. production of refined copper are linked by the definitional relationship (25).

For estimation purposes, expected spot prices $\tilde{p}$ formed by the adaptive mechanism (28) are not observable and are substituted in equations (1) and (3), thus yielding second-order differential equations independent of $\tilde{p}$ but still identifying the parameter β₁

The forward price equation (3) is an attempt to use information that is readily available and that can be used in connection with the explanation of the spot price variable. It can be interpreted as showing the adjustment of three-month forward prices to expected spot prices $\tilde{p} (t)$ It is recognized that this adjustment mechanism may be unrealistic as a representation of the formation of three-month forward prices. Except for the price series itself, however, very little else is available in the way of data on holdings of contracts and their movements in the forward market. This fact has always been a major impediment to research in this area, and the LME is among the many commodity markets that neither publish nor even collect data on transactions taking place on the exchange. The only other series published by the LME are its daily total turnover and the level of the stocks held on warrants in its warehouses.

Equation (9) representing non-U. S. secondary production is a production function similar to that postulated in the general model and in equation (7). The fixed capital variable is now the stock of recoverable copper that has been accumulated over the years and that is defined by equation (23). This equation represents the accumulation and transfer of refined copper from underground to overground. The cumulative stock of overground copper represented by equations (22) and (23) does not represent the actual quantities available to the secondary producers. The amount actually recovered is a very small proportion of the total and consists of only the most obvious quantities to be recovered, that is, the higher grades of scrap. The production functions used for the secondary sectors do not contain any technological progress trend variables, since it is assumed that the technology moves in line with the primary sector. This is certainly true of the upgrading and refining stages, which are identical in character to the production of refined copper from primary sources. The major potential area for technological improvements is the collection and sorting out of scrap; however, it is that part which is postulated to remain the same. Within the time span of the model, that is not a loss of generality.

The fixed capital variable entering equation (7) is mine capacity. The production process is characterized by inflexibilities, implying some technical constraints on output. Some technical and economic factors entering the decisionmaking process of the primary copper producers are mentioned later. A mine is determined to operate on a long-term schedule that optimizes the technical characteristics of the mine, such as ore grade, operating rate, and lifetime of the mine, among other factors. The viability of a mine is determined at the investment stage, and this rigidity in technical characteristics will have to be carried throughout the economic life of the mine, unless the long-term prospects or financial support become so unfavorable that the mine has to be shut down.

A variety of explanations can be given in connection with the investment equations (10) and (18). The first possible specification is identical to the previous formulation put forward in the general commodity model. The firms in the industry are assumed to maximize profit, thus leading to the following mine capacity equation:

D \log \hat{M} C (t) = α_{6} \log [\frac{Π}{ρ P k (t) M C (t)}]

where MC(t) is mine capacity and Pk(t), the nominal price of capital goods employed in the formation of fixed capital. This expression has been shown in footnote 4 to reduce to

\begin{matrix} D \log \hat{M} C (t) = & α_{6} \log [Q (t) / I (t)] + ψ \\ with ψ = α_{6} \log [\frac{(1 - α) P (t)}{ρ P k (t)}] + ψ_{0} \end{matrix}

Because of both the scarcity and the bad quality of data series, the preceding equation may be the convergence of several alternative specifications.

Another interpretation is to postulate that the rate of change of fixed capital is a linear function of the gap between actual and potential output, where potential output is represented by the total amount of fixed capital, that is,

$D \hat{M} C (t) = α_{6} \log [λ Q (t) - M C (t)]$ where λ is the long-run capital/output ratio. This formulation, more akin to macroeconomic concepts, can be shown to be equivalent to the profit-maximizing approach expressed earlier by assuming that

λ = α_{0} P (t) / α P k (t), thus D \log \hat{M} C (t) = α_{6} \log [\frac{α_{0} P (t) Q (t)}{ρ P k (t) M C (t)} - 1]

Whether the approximation using λ is correct may be arguable, but in the absence of relevant series for ρ or Pk, hypothesis testing is not feasible.

It has, however, been argued that the non-U. S. primary copper industry does not behave like a profit maximizer. Since the nationalization of many mining companies in some major primary-copper producing countries during the 1960s, the management of mineral extraction has radically changed in many commodities and countries. Prior to nationalization, mining was in the hands of a few large international mining companies. After nationalization, the respective countries have implemented their respective views on the management of the mining sector. The main impact of these measures has been an attempt to integrate the mining sector into the respective economies of the producing countries. In particular, the pure profit motive has often been substituted for an export revenue maximization approach, and some of the efficiency in the extraction process has been traded for a higher level of employment of the local labor force. It is certainly true that in many instances the transition has not been smooth, but it must be accepted that this change of objective still has to be rational if the industry is to continue to operate. The present formulation for equations (10) and (18) has been designed to be ambiguous, but it also exploits the property that if the production process is to be represented by a Cobb-Douglas production function, then profit maximization will also coincide with the maximization of export earnings as was shown in footnote 4.

Another major issue in investment decisions has been the fact that new mines coming on stream should be mines with low operating costs that will displace those with high production costs within the total population of active mines. More and more, there is a tendency to open in developing countries mining ventures that will also contribute to the industrialization of those countries. These projects often mean that the investment meets the cooperation of the local authorities, and they are often less risky to the international investor. Other factors should be taken into consideration to support the similarity of specification between the two investment equations. Over the two periods, the nature of ownership of producing facilities throughout the world exhibits some continuity of complete private ownership and partial state ownership. The majority of investment projects undertaken today are initiated by the same international companies as before or during the early 1960s. Nationalization of producing capacities in some instances has only meant a change of control in shareholding that leaves the previous owners as the second largest group of shareholders. Many changes have taken place in mining laws in the non-U. S. that have transferred the role of participating countries and governments toward a larger share of total risks to be borne. This has resulted in larger investment projects contributing to the downstream processing of minerals as an integrated part of industrialization programs. The exploitation of copper deposits in many developing countries now has to include some infrastructure building and processing plants for the treatment of copper to obtain refined copper as the exportable product. Cooperation and risk sharing between international investment syndicates and local authorities is enhanced in this process, and sovereign risk remains about the same as it always has been.

The consumption of copper is spread over five major end-uses: the electrical, construction, transport, mechanical, and consumer goods sectors. An adequate study of consumption would disaggregate total consumption between these different end-uses in the respective countries. The reasons for doing so are that some of these sectors may behave in a countercyclical manner to other sectors (e.g., construction and consumer goods) and that the respective elasticities of each sector may differ from one sector to the next. These comments describe the familiar problems associated with derived demand. The functions for copper consumption (5) and (17) in the two geographical areas defined in the model are assumed to be of the same form. The income variables in these functions are explained in Appendix D, and the non-U. S. variable consists in a purposely built index of manufacturing activity, where component indices are weighted geometrically by the yearly share of total demand of each major non-U. S. consumer country.

The relative price of copper is formed by deflating the nominal price of copper in each geographical area by the free market price of aluminum. The proper deflator would be a weighted variable of the prices of stainless steel, plastic, and aluminum, because these three materials are the most common competing substitutes for copper. The substitute for copper with the biggest share is aluminum because the physical and electrical properties of aluminum are the nearest to those of copper.

The major aspect of substitution is its irreversible character. Very few areas of copper usage allow for capital equipment to be ambivalent between copper and its substitutes. This technical rigidity implies that a switch to a substitute material should be accompanied by new investment in machinery that would consequently incur sizable capital expenditure. It follows that the decision to substitute copper for other materials arises at the product development stage when decisions on characteristics and production costs of the product are made. Even further, copper is substituted for other materials through technological innovations that are not even primarily concerned with the wish to include or eliminate copper as a component material. The net substitution taking place is the algebraic sum of new products using copper as a component material and products where copper is phased out.

It has been a policy of the highly integrated aluminum industry to maintain stable prices in order to encourage the growth of the market for the metal. The technology related to aluminum has evolved rapidly since the turn of the century, and this opens many more areas of application for that metal. The substitution of other materials for copper is to be expected as new areas of use come to light for these materials. An example would be the potential impact of plastics in telecommunications.

No special attention has yet been given to the U. S. sector. The preceding comments about investment and consumption are mostly valid for both sectors but primary production differs between the United States and elsewhere.

An important factor in the U. S. primary production industry is strikes. Labor unions in the industry negotiate contracts with the mining companies for a period of three years. At renewal time, and depending on a variety of factors affecting the bargaining position of the parties entering the contracts, strikes have happened. Their effect on the industry is more or less pronounced, but the 1967-68 strikes, for instance, were estimated to have led to a loss of more than 900 thousand tons of copper over the period beginning with the third quarter of 1967 through the first quarter of 1968. Instead of using dummy variables to account for strikes, it was decided to add strike losses to actual production. Strikes have little impact on other variables, since they are usually well anticipated and discounted. It cannot be denied, however, that imports of copper in all forms by the United States from the rest of the world can have a sizable impact on the LME quotations, and this series has been left intact.

The formation of the U. S. producer price as represented by equation (11) is a difficult area of specification in the U. S. sector. Even though some earlier references have been made to the oligopolistic nature of the primary producing industry in the United States, the approach adopted for specification is one of coalitions. The fundamental approach taken is that examination of economic structure, such as bilateral monopoly or duopoly, leads to conclusions that are usually valid for structures involving a small number of participants. In general, it is doubtful that stable equilibrium can be reached within a noncooperative framework. The specification to be adopted must be one that, irrespective of implicit or explicit arguments made and whatever the individual action of producers, reflects the possible combinations of gains representing the outcome of a game. The general structure used for the formation of the U. S. producer price is the concept of arbitrage between producers, merchants, and consumers. A better description of the arbitrage at work is the operation whereby a coalition moves from a given allocation to another and finds it profitable to do so. This phenomenon is also more in line with the general context of LME price formation as a non-tâtonnement process; the variables affected by this game-theoretic approach can be both quantities and prices that can be derived without particular reference to the initial actions of participants in the coalition.

For the U. S. producer price, it is argued that the U. S. producers are forced to adopt the price changes fixed implicitly by the consumers, and this is so through the following set of developments. Secondary producers and merchants that draw their production from both domestic sources and international trade enforce a price/quantity arbitrage between the non-U. S. price/quantity position and the U. S. consumer. The U. S. consumers use these channels to fill the gap between their actual demand and their basic supplies. The U. S. producers therefore observe constant fluctuations in their share of total demand, which therefore enforces the latter arbitrage relationship defining the U. S. producer price. This is the basic argument leading to equation (11); the steady-state price in the United States is defined by the LME price if trade between the two sectors takes place, and it does. U. S. consumption is subject to demand-related oscillations that will be met by both primary and secondary supplies. The U. S. producers have to reconcile all these factors by setting a reasonable price that maintains their relative position in the market and therefore solves the arbitrage game.

The specification of imports and exports of copper in all forms between the United States and the rest of the world is a difficult area for a variety of reasons. The major problem is the quality of the data. Imports are composed of both ore and concentrates to be treated by U. S. refineries, and refined copper. The quantities can vary greatly from one quarter to the next. Exports are mostly of refined copper and are quite variable, reflecting only a few contracts with other industrialized countries. The net import equation (20) therefore depends on the relative price variable, which ensures the price arbitrage between the U. S. producer price and the LME spot price. The sign of the elasticity cannot be rigorously determined a priori but can be expected to be negative on the assumption of a greater proportion of imports with respect to exports; the latter also justifies U. S. consumption as the income variable. Some export controls were in operation during the period 1966-68, and this fact has also been an incentive in defining a joint equation. No mention has yet been made throughout the model of government-held stocks of copper, such as the General Services Administration (GSA) stockpile in the United States. The U. S. stockpile is probably the most documented, but it is not unique in the world. France and Japan are also known to hold small quantities of copper as stockpiles, so as to offset potential variations in demand. The role of the GSA stockpile is mostly to ensure complete self-sufficiency of the country for strategic purposes. But it has also acted as a regulator for the industry: the highest level of stocks was held in 1960 when 1,040 thousand tons were held by the agency; the level was above one million tons throughout the period 1958-63, but its present level has dropped to about 20,000 tons. The impact of this stockpile may have been quite important in helping the U. S. copper industry to go through the difficult period of the late 1950s and early 1960s.

The system is completed by a set of definitional identities (21) to (27), all of which have been explained. Equation (24) as such, which does not occur explicitly anywhere else in the model, has been substituted in equation (22).

the steady-state properties of the world copper model

The steady state of the model can be derived as for the prototype model in Section I. The growth rates for the variables in the copper model are given in Table 1.

Table 1.

World Copper Model: Steady-State Growth Rates of Variables

The steady state is derived on the assumption that the rate of growth of the world economy is $λ_{\bar{y}}$ . This is a standard assumption if trade is taking place and if other restrictions do not completely isolate some part of the world. The position of CMEA countries has not been given any particular status concerning a recognition of the different economic structure prevailing in these countries; the only assumption is that the trade of these countries with the rest of the world follows the growth rate of the world economy. The preceding type of argument concurs to show that a steady state is only a particular solution that is not a priori unique. This point is commented on later in this subsection.

Similarly, it is assumed that the rate of growth of technological progress in the copper industry is the same throughout the world. This is plausible because that variable will reflect the differences in technological progress that may exist between the copper industry and other materials. Any technological innovations occurring either within the industry or in the industries that produce substitutes for copper are assumed to be spread throughout the industries. The steady-state growth rates for the endogenous variables are observed to be identically equal to λ₁ for all the physical variables. This is required for homogeneity, since different rates of growth for subsets of these variables could lead to implausible results. For instance, if imports and exports between the U. S. and the non-U. S. sectors were allowed to grow at different rates, a situation would develop whereby one or the other sector would sooner or later disappear. Similarly, different rates of growth between production and consumption variables would result in unacceptable situations of market imbalances. The steady-state rate of growth of relative prices allows for changes in the relative position of costs of production between the copper industry and industries producing substitutes for copper as well as for changes in the relative capital intensiveness of these industries. The latter changes may be due to technological advances in either sector. The rate of growth λ₁ reflects changes in the rate of growth of relative prices but also allows the physical variables to grow at a different rate from the world economy.

It can be observed that cases where the rate of growth of prices is equal either to zero or to λ₁ would be obtained only by imposing strong assumptions representing a great loss of generality. This steady-state solution can be seen as incomplete in the sense that most of the properties related to endogenous variables depend on crucial exogenous growth rates and levels. Relative prices can be made endogenous only by the addition of a second sector representing the behavior of substitutes for copper. Thus, a true description of commodity markets and the properties attached to them can be realized only by a multisector growth model representation. As shown in the previous section, the asymptotic stability properties of the copper model depend only on the matrix of structural coefficients.

estimation method

In recent years, satisfactory econometric techniques and computer programs have become available that enable models specified in continuous time to be estimated. Since the nature of commodity markets is that of a non-tâtonnement process, a simultaneous specification and estimation approach is required. Historically, the simultaneity aspect of the specification was usually arrived at through a two-step approach consisting in estimating the recursive relationships in the model and then using these results in the remaining simultaneous block, the usual estimator being two-stage least squares. FIML estimation of the parameters of a commodity model is only recent; this enables the functions to be estimated simultaneously, eliminating the need for any sort of stepwise approach.

The attraction of an FIML estimator is the qualities associated with this estimator. The FIML estimation makes use of all a priori restrictions imposed on the structural model, including across-equation restrictions that may have to be imposed directly from economic theory. The statistical properties of the FIML estimator are probably its main attraction—it is consistent and asymptotically efficient. This efficiency enables a more stringent test of the hypothesis embodied in the model. Adequately specified models usually give robust estimates, but sensitivity to specification errors is still often held as a liability to FIML estimation. A more proper interpretation would be that this sensitivity to specification imposes more stringent standards on the task of the economist. Tests have been advanced to cope with specification problems such as the chi-square test on the ratio of the restricted to unrestricted likelihoods, and even though this test is valid only for relatively large samples, it is a useful tool for comparing alternative specifications.

The case for differential equation systems arises from two factors. First, although many individual economic decisions are made at regular intervals, the variables usually observed are the outcome of a large number of decisions taken by different individuals at different points in time. Second, the intervals between observations are usually larger than the intervals between the decisions that they reflect.

From the point of view of estimation, the present model is estimated by FIML in its continuous time specification using the set of computer programs written by Clifford R. Wymer. A description of the technique and programs used is given in Appendix B.

results

This section presents estimates⁶ of the parameters of the world copper model for a sample of quarterly observations spanning the period from the fourth quarter of 1960 to the fourth quarter of 1975 (Table 2). Estimates of the eigenvalues of the continuous model are also given at the end of this section. A set of consistent estimates of the parameters of a stochastic differential equation system may be obtained from a sample in which the variables are observed at discrete intervals, as long as the model is linear in variables.

The model specified earlier has been linearized about its steady-state solution by means of a Taylor series expansion, truncated after the first order. The linear model obtained is composed of second-order equations in stock variables and first-order equations in flow variables. This case and details about the estimation procedure used are presented in Appendix B. The second-order system of differential equations is further reduced to a first-order system of 20 equations. After removal of the first-order moving average in the data, a procedure required by the estimation method, 57 observations were used.

During the estimation, the constraints embodied in the present model, including the across-equation restrictions and the constraints inherent in the linearization of the model, were maintained. These restrictions include those that were derived from the analysis of the steady-state solution of the model.

The consumption equations for the United States and the non-U. S. are characterized by similar income elasticity of demand, but the estimate for the non-U. S. is much more precisely defined than the U. S. estimate. This result is probably a consequence of the use of the specially built indicator of non-U. S. manufacturing activity. A similar index should perhaps be built for the United States, using published indices of component industries weighted by the shares of these industries in total U. S. copper consumption in order to take into account possible countercyclical behavior among the component sectors, such as construction and consumer goods, that may have a relatively large impact on copper consumption. A comparison of the U. S. results can be made with the model by Fisher and others (1975); it is found that the income elasticity of demand for the United States is similar to the estimate obtained in that study using the index of U. S. industrial production published by the United Nations. That model also illustrates the sensitivity of income elasticity estimates to alternative indices that may be used as proxies for economic activity in general. In the United States, the structure of the industry may perhaps have warranted a specification for consumption that would recognize the importance of fluctuations in stocks held by consumers. Because consumption as defined in the present model refers to a derived demand for copper, the variable representing economic activity should perhaps take into account the cyclical behavior of stocks. The non-U. S. income elasticity of demand cannot be compared with the estimates obtained by Fisher and others (1975), since in the present model non-U. S. demand is not disaggregated.

Table 2.

World Copper Model: Full-Information Maximum Likelihood (FIML) Estimates of Structural Parameters, Fourth Quarter 1961-Fourth Quarter 1975

(Quarterly data)

Table 2.

World Copper Model: Full-Information Maximum Likelihood (FIML) Estimates of Structural Parameters, Fourth Quarter 1961-Fourth Quarter 1975

(Quarterly data)

Parameters	Estimate	Asymptotic Standard Error	t-value
α₁	0.0814	0.0387	2.10
α₂	0.7727	0.2916	2.65
α₃	0.6896	0.1697	4.06
α₄	0.9998	0.2409	4.15
α₅	0.4697	0.1501	3.13
γ₁	1.2444	0.3062	4.06
α₇	0.8776	0.2110	4.16
α₈	1.4444	0.2772	5.21
α₉	0.6253	0.1405	4.45
γ₂	0.3212	0.1026	3.13
α₁₁	1.1969	0.2777	4.31
β₁	3.2061	0.8015	4.00
β₂	10.8691	1.8176	5.98
β₃	0.1606	0.0381	4.21
b₁	0.0523	0.0594	0.88
b₁	0.7444	0.0489	15.22
b’₁	0.3764	0.3452	1.09
b’₂	0.6918	0.3127	2.21
m₁	0.0808	0.0274	2.95
m₂	-0.0008	0.0004	2.04
m’₁	0.5974	0.1766	3.38
m’₂	-0.0077	0.0019	4.01
n₁	0.3438	0.1302	2.64
n₂	-0.0144	0.0026	5.60
n’₁	0.4076	0.0536	7.60
n’₂	-0.0154	0.0011	14.50
α₆	0.0757	0.0386	1.96
α₁₀	0.0407	0.0191	2.13
ζ	0.6436	0.3877	1.66
δ	0.2485	0.0955	2.60

Table 2.

World Copper Model: Full-Information Maximum Likelihood (FIML) Estimates of Structural Parameters, Fourth Quarter 1961-Fourth Quarter 1975

(Quarterly data)

Parameters	Estimate	Asymptotic Standard Error	t-value
α₁	0.0814	0.0387	2.10
α₂	0.7727	0.2916	2.65
α₃	0.6896	0.1697	4.06
α₄	0.9998	0.2409	4.15
α₅	0.4697	0.1501	3.13
γ₁	1.2444	0.3062	4.06
α₇	0.8776	0.2110	4.16
α₈	1.4444	0.2772	5.21
α₉	0.6253	0.1405	4.45
γ₂	0.3212	0.1026	3.13
α₁₁	1.1969	0.2777	4.31
β₁	3.2061	0.8015	4.00
β₂	10.8691	1.8176	5.98
β₃	0.1606	0.0381	4.21
b₁	0.0523	0.0594	0.88
b₁	0.7444	0.0489	15.22
b’₁	0.3764	0.3452	1.09
b’₂	0.6918	0.3127	2.21
m₁	0.0808	0.0274	2.95
m₂	-0.0008	0.0004	2.04
m’₁	0.5974	0.1766	3.38
m’₂	-0.0077	0.0019	4.01
n₁	0.3438	0.1302	2.64
n₂	-0.0144	0.0026	5.60
n’₁	0.4076	0.0536	7.60
n’₂	-0.0154	0.0011	14.50
α₆	0.0757	0.0386	1.96
α₁₀	0.0407	0.0191	2.13
ζ	0.6436	0.3877	1.66
δ	0.2485	0.0955	2.60

The elasticity of demand with respect to the price of copper relative to that of aluminum for both the U. S. and non-U. S. sectors is estimated to be not significantly different from zero. The demand functions postulated throughout this study assume that prices of substitute commodities are relevant to commodity markets, and consequently the analysis is carried out in terms of prices of the commodity studied relative to its substitutes. The LME quotation of the price of aluminum has been assumed to be a proxy for the price of substitutes for copper, but the prices of plastics and some qualities of steel should perhaps also have been included, although the larger part of substituted copper has gone to aluminum. The lack of significance of the price elasticities of demand is probably caused by irreversibility of substitution, which may be not price induced but technologically induced. The mean time lags are similar for the non-U. S. and the United States at about five months and are comparable to judgmental industry estimates for these lags.

The supply equations for secondary copper yield price elasticities of supply that are similar in magnitude and are significantly different from zero for both the United States and the non-U. S. The U. S. price elasticity of secondary production is equal to 0.41 and is marginally higher than the elasticity of 0.32 derived by Fisher and others (1975). The non-U. S. price elasticity is equal to 0.34 and can be compared with an elasticity of 0.16 in the earlier-mentioned study. The assumptions made about the importance of LME price quotations in markets for secondary copper are confirmed. The mean time lags, however, are quite different—two quarters for the non-U. S. and two months for the United States. The main explanation for this difference is probably to be found in the construction of the non-U. S. secondary data series, which is formed as a residual from total world production minus non- U. S. primary production, minus U. S. total production. The two trend parameters n₂ and n’₂, representing the rates of growth in the cost of variable factors of production, are significant, and both negative signs indicate that relative costs have increased over the estimation period. The constant α defined in the Cobb-Douglas production function introduced in Section I can be calculated from the two price elasticities of supply, equal to 0.229 for the non-U. S. and 0.289 for the United States. Because constant returns to scale have been assumed, the corresponding parameters for the fixed capital variable are equal to 0.771 for the non-U. S. and 0.711 for the United States. It can be concluded that the two secondary production functions are not perceived to be fundamentally different.

The mean time lags of primary production are of the order of one quarter for both the United States and the non-U. S. The price elasticities of primary production for the United States (0.60) and the non- U. S. (0.08) are quite different. The U. S. price elasticity estimate is much different from the estimate obtained by Fisher and others (1975). The latter, however, do not fit production functions including mine capacity variables to the primary supply side of the market, and therefore the price variable is their only explanatory variable in the behavioral relationships describing primary production. The non-U. S. price elasticity of demand is very low, albeit significantly different from zero. This low price elasticity, along with a relatively fast speed of adjustment, seems to suggest that non-U. S. primary producers can absorb small disturbances through changes in technical efficiency of extraction. This possibility is not thought to exist in the United States. The variable input parameters of the primary production functions are 0.074 for the non-U. S. and 0.375 for the United States. The size of the latter indicates that variable factors of production, such as energy and labor, have a much larger impact on U. S. primary production than the same inputs in the non-U. S. The time trend m₂, representing the joint effect of technical progress and costs of variable factors of production in the non-U. S., suggests that gains in price competitiveness from the substitute industries do not offset the relative increase in costs of variable factors of production. The same conclusion can be drawn for the United States, where the time trend mʹ₂ is significantly different from zero, a result that seems to indicate that the U. S. primary industry is as competitive as the non-U. S. industry with respect to the production of substitutes for copper. The rate of growth of technological progress λ postulated in the primary production function has been assumed to be the same throughout the world industry and can be calculated from the set of parameters estimated in both the U. S. and non-U. S. primary and secondary supply equations. The value of λ is 0.002 and does not represent so much the increase in efficiency of capital as a whole as the increase in efficiency of the copper industry with respect to the substitute industries. The increase in efficiency of capital is effectively embodied in the fixed capital formation process and the investment function.

At this point, comments can be made about the relative position of the different sectors in terms of speed of adjustment. In the United States, a change in demand is met more rapidly by changes in secondary supplies than in primary supplies. This point is further confirmed by the net import results, which react with a mean time lag of about two months. Therefore, changes in demand can be met equally by changes in net imports or secondary production, and these two sectors act faster than do primary supply sources. The price elasticity of net imports has the right sign but is not significantly different from zero. This result is not totally unexpected in the face of the quality of the data series. The position of these relative adjustments in the non-U. S. is markedly different. Changes in demand are met by changes in primary supplies, which are then followed by changes in secondary production.

Estimates of the mixed stock/flow formulation of price formation are found to be significant, thus verifying this hypothesis. Both stocks and flows are relevant to price determination, and a specification of price determination that is mutually exclusive between the two possible determinations is bound to omit one major aspect of price determination. The adjustment parameter of stocks to prices α₁, is relatively large, thus emphasizing the importance in determining price of the underlying level of copper stocks in the market. The relationship representing the desired demand for stocks yields a significant price elasticity, which implies that stocks are practically infinitely price elastic. Expected spot prices are reflected in the actual spot price with a mean time lag of about four weeks; the forward price adjusts in about four months. The expected spot price three months ahead is not found to coincide with the three-month forward price by a difference of about one month. This may be so because the expected price defined and estimated does not necessarily imply that all expectations are realized, and similarly the three-month forward price is only a measure of realized expectations attached to contracts that have actually been exchanged. The adjustment parameter attached to the U. S. producer price β₃ is also relatively large and suggests that U. S. producers react quite rapidly to changes in their market shares.

The investment equations yield mean time lags in the levels of investment of the order of 3.3 years in the non-U. S. and 6 years in the United States. These adjustments are significantly different from zero. These parameters represent the global movement of investment decisions and imply that investment in the non-U. S. is much more active than in the United States. The parameters α₁ and α₂, which can be assumed to depend on costs associated with variations in the rate of fixed capital formation, show some large differences between the two sectors, suggesting that the non-U. S. is more capable of reacting to changes in the rate of investment than is the U. S. primary industry. It may also be that U. S. mining companies investing internationally choose to be more active outside the United States.

Table 3 gives the eigenvalues of the equivalent system defined in terms of deviations about the steady-state solution (Appendix A). The system is dynamically stable, since all the eigenvalues have negative real parts. The largest damping period, which is the mean time lag of the system in reaching its steady state if undisturbed, is about 35 years. The standard errors of the two pairs of imaginary parts are relatively large and therefore do not allow for particular comments on whether or not a significant cycle can be traced for the copper industry. The length of the damping period is a reflection of the slow speed of adjustment affecting the industry. Some comments on the major parameters influencing the cyclical behavior of the industry are given in the next section, which deals with the potential use of the model for policy analysis.

Table 3.

World Copper Model: Estimate of Eigenvalues of Continuous System

Table 3.

World Copper Model: Estimate of Eigenvalues of Continuous System

	Eigenvalue		Asymptotic Standard Errors		Damping Period	Period of Cycle
No.	Real part	Imaginary part	Real part	Imaginary part	(In quarters)
1.	-4.057		0.679		0.246
2.	-3.206		0.801		0.312
3.	-2.166		0.364		0.462
4.	-1.439		0.277		0.695
5.	-1.201		0.277		0.833
6.	-1.050		0.305		0.952
7.	-1.021		0.203		0.979
8.	-0.803		0.193		1.245
9.	-0.793		0.270		1.261
10.	-0.507		0.188		1.972
11.	-0.462		0.147		2.164
12.	-0.086		0.063		11.628
13.	-0.048		0.033		20.833
14.	-0.023		0.000		43.478
15.	-0.017		0.007		58.823
16.	-0.007		-0.000		142.857
17.	-0.233	0.073	0.076	0.058	4.292	86.071
18.	-0.026	0.035	0.003	0.033	38.461	179.519

Table 3.

World Copper Model: Estimate of Eigenvalues of Continuous System

	Eigenvalue		Asymptotic Standard Errors		Damping Period	Period of Cycle
No.	Real part	Imaginary part	Real part	Imaginary part	(In quarters)
1.	-4.057		0.679		0.246
2.	-3.206		0.801		0.312
3.	-2.166		0.364		0.462
4.	-1.439		0.277		0.695
5.	-1.201		0.277		0.833
6.	-1.050		0.305		0.952
7.	-1.021		0.203		0.979
8.	-0.803		0.193		1.245
9.	-0.793		0.270		1.261
10.	-0.507		0.188		1.972
11.	-0.462		0.147		2.164
12.	-0.086		0.063		11.628
13.	-0.048		0.033		20.833
14.	-0.023		0.000		43.478
15.	-0.017		0.007		58.823
16.	-0.007		-0.000		142.857
17.	-0.233	0.073	0.076	0.058	4.292	86.071
18.	-0.026	0.035	0.003	0.033	38.461	179.519

III. Policy Analysis and Simulation

This section briefly discusses some policy issues on which the model has a bearing. It is, however, mostly descriptive and simple, because the actual use of the model for policy purposes along with the methodology used is the object of further research. Forecasting and simulation of the model conclude this section.

The major aim of the construction of dynamic models, such as the one presented in the previous section, concerns their use for policy analysis. Since the dynamic adjustments in the system can extend over a fairly long spectrum, it is important that the effect of current policy decisions on the future course of the commodity market be examined. As already mentioned, the theoretical nature of the model and the dynamic structure specified are designed to capture the most important feedback in the world copper industry so that they should be capable of reproducing the broad movements of the major endogenous variables corresponding to changes in exogenous variables of policy parameters.

It is shown later in this section that qualitative information can be obtained from the specified copper model to answer questions on the choice of potential policy instruments. But the present state of thinking on the problems associated with commodity markets and the solutions that should be sought tend to evolve around the modification of existing institutional structures that would result in “better” working of these markets.

The general concept of stabilization and the design of policy instruments to achieve this objective is a much-debated subject, at both the political and economic levels. The view taken in the present work is that stabilization is essentially a dynamic adjustment phenomenon that must be investigated at the level of the whole commodity market or industry. It has been shown earlier that steady-state solutions that are representations of equilibria exist for the models that have been investigated. It seems therefore a natural task to describe stabilization as the attempt to bring the market toward its intrinsic equilibrium, this solution optimizing the utility of its participants, such as producers, consumers, and speculators. The dynamic adjustments for each sector of the market have been shown to differ, and small as well as large deviations from equilibrium may be attributable to the inability of the market participants to react to changes in the relative adjustment of the different sectors. It has been shown that investment in fixed capital adjusts slowly to changes in output and the real marginal cost of capital. This low speed amounts to inflexibility that creates large movements in the relative adjustment of investment to demand. The design of policy measures should perhaps attempt to modify these relative adjustments of the different sectors of the market in order to reduce the deviations of the respective variables from their steady-state paths. By examining the long-run solution of the general commodity model given in Section I, one can obtain some pointers on the consequences of stabilization policies for some variables or combinations of variables.

The underlying movement in the relative price level of the commodity to its substitutes is seen to depend only on a combination of structural parameters and growth rates. These parameters include the price elasticity of primary supply, the real marginal cost of capital, the rate of growth of the physical variables in the model, the intercept of the supply function, and two speeds of adjustment: the adjustment parameter of primary supply and the adjustment parameter of the level of fixed capital. Changes in the relative price level can result only from shifts in these structural parameters, but the path of adjustment toward a new level and its consequences on the dynamic behavior of the system cannot be assessed a priori. Some instances of comparative dynamics have been illustrated following the exposition of the steady-state solution. Global relative revenues are much more important than the relative price level as a variable to focus on, because this variable represents the aggregate level of many countries’ export earnings derived from the commodity considered, and as such is an important component of national income for each country. Total relative revenues, defined as the product of the relative price of the commodity and the total consumption of it, depend crucially on the level of economic activity, which also happens to be an exogenous variable. The size of the price elasticity of demand is seen to be relevant to the transmission of shocks occurring in the level of economic activity. For values of that elasticity below unity, the impact of such shocks is dampened, whereas values of the income elasticity above unity amplify the impact of these shocks on the level of relative revenues. Again, here a comparative dynamics approach fails to pinpoint the nature of the adjustment process following shifts in adjustment parameters or disturbances in exogenous factors. It can therefore be tentatively advanced that shifts in structural parameters should be part of the policy objectives but will not contribute very much to an understanding of the impact of the policies on market stability. Differing degrees of stability, however measured, represent in effect the ability of the commodity market to adjust to short-term changes. The view expressed in this section is that the relevant area for development of stabilization policies is the study of a market approach to stabilization.

The market approach to stabilization can be summarized as follows: A utility function can be formed to measure the satisfaction of all the market participants, and this function is in effect an objective function, such as the ones postulated in control theory. This utility function can be postulated to measure levels of dynamic stability, different degrees of aversion toward risk by market participants, the differences in importance to the participants in reaching expected levels of utility, and many other factors. Such a measure of market stabilization takes care of the controversial aspects of partial versus total stabilization because the utility function is a measure of the total net gain or loss to the market of the stabilization policies investigated. Because total stabilization of a single objective is generally held as unfeasible, the measuring of the net market gain becomes an important aspect of the problem that is covered by the market objective-function approach. Another way of approaching the same concept is to introduce the concept of mathematical distance, which measures the position of the system in an n-dimensional space in terms of deviations from its equilibrium solution. This distance is usually a scalar that can be defined in a variety of ways but, over all, will allow a measurement of different states of the model brought about by the modification of the specification to allow different values of the structural parameters or additional equations to represent the workings of the agencies implementing the policies, such as a buffer stock agency. These distances can be shown to be closely related to utility functions and also, under certain assumptions, to behave like Lyapunov functions—the latter being directly related to measures of global system stability.

It has also been shown in the last two sections that the eigenvalues characterizing the model intrinsically contain all the dynamic information about this system. A system of differential equations Dy(t) = Ay(t) can be equally represented by y(t) = He^Δtc, where A = H^-1A H is the matrix of eigenvalues and H, the matrix of eigenvectors. A method of partial differentiation of the right-hand side of y(t) can be applied to obtain matrices of coefficients reflecting the relative importance of parameters, eigenvectors, or variables on the dynamics of the system of differential equations representing the commodity market. These recent developments by Wymer (1978) are usually referred to as sensitivity analysis.

Sensitivity analysis performed on the eigenvalues of the model linearized about the steady state yields a matrix of partial derivatives showing the absolute influence of each parameter on each eigenvalue. These derivatives can be used in a variety of ways, two of them being the tracing of misspecifications associated with an unstable system and the discovery of policy instruments insofar as some parameters, and there-fore the equations associated with these parameters, can be used to influence the dynamic properties of the system as a whole. Each partial derivative measures the absolute efficiency of a given parameter for a given eigenvalue, but the possible existence of several nonzero derivatives for such an eigenvalue leads to the concept of relative efficiency of the corresponding parameters. This relative efficiency is measurable by taking the ratio of the inverse of these derivatives. For example, given an eigenvalue, two potential policy instruments are found relevant (nonzero derivatives) with respect to this eigenvalue:

\frac{\partial λ}{\partial α_{1}} = a and \frac{\partial λ}{\partial α_{2}} = b for - \infty < {a, b} < + \infty

implies that

\frac{\partial α_{1}}{\partial α_{2}} = - b / a

this quantity being a measure of the relative influence of α₁ compared with α₂ with respect to λ.⁷ A subset of the sensitivity matrix of eigenvalues with respect to the parameters of the system is given in Table 4. The parameters that appear crucial in changing the stability of the model toward dynamic instability are α₁, b₁, and m₁ on λ₁₇. Dynamic instability would result if the real part of the eigenvalue became positive as a result of shifts in parameters. It is difficult to quantify likely impacts, but it can be observed that it would take changes of the order of 0.5 in the absolute value of the price elasticity of both demand and supply to bring a possible change of sign in the real part of xλ₁₇. Similarly, the impact of parameters γ₂, β₃, and α₁₀ on λ₁₈ could also be investigated. For a given stable system, the examination of the sensitivity matrix may help in tracing parameters that are potential policy instruments. If it is thought that the intrinsic cyclical behavior of the system is more disturbing than an exponential convergence to equilibrium, then the parameters most relevant in influencing the imaginary part of the complex eigenvalues should be considered. It can equally be checked that some parameters that may be the object of policy use will not disturb the existing stability of the system if the value of these parameters is changed radically. An interesting result to be found in Table 4 is the little effect of the parameters attached to the formation and impact of expected prices on eigenvalue λ₁₇. This suggests that expectations about prices are not a major source of instability in the commodity market studied.

Table 4.

World Copper Model: Subset of Sensitivity Matrix of Eigenvalues

Table 4.

World Copper Model: Subset of Sensitivity Matrix of Eigenvalues

	λ₁₇		λ₁₈
	Real	Imaginary	Real	Imaginary
Parameter	–0.0261	±0.0350	–0.2330	±0.0730
α₁	0.0504	0.3373
α₂	–0.0086	–0.0057
α₇			0.0117	–0.0619
α₈			0.0355	–0.0574
γ₂			–0.4077	0.0966
ζ	–0.0036	–0.0382
δ	–0.0132	–0.0074
β₁	0.0003	0.0077
β₂	–0.0002	–0.0061
β₃			–1.0070	0.7594
b₁	0.0360	–1.0080
b’,₁			–0.1805	0.1046
α₁₀			1.1194	1.0844
α₆	0.0013	–0.0177
m₁	0.0240	0.8898
m’₁			–0.1496	0.1281
n₁	0.0057	0.1634

Table 4.

World Copper Model: Subset of Sensitivity Matrix of Eigenvalues

	λ₁₇		λ₁₈
	Real	Imaginary	Real	Imaginary
Parameter	–0.0261	±0.0350	–0.2330	±0.0730
α₁	0.0504	0.3373
α₂	–0.0086	–0.0057
α₇			0.0117	–0.0619
α₈			0.0355	–0.0574
γ₂			–0.4077	0.0966
ζ	–0.0036	–0.0382
δ	–0.0132	–0.0074
β₁	0.0003	0.0077
β₂	–0.0002	–0.0061
β₃			–1.0070	0.7594
b₁	0.0360	–1.0080
b’,₁			–0.1805	0.1046
α₁₀			1.1194	1.0844
α₆	0.0013	–0.0177
m₁	0.0240	0.8898
m’₁			–0.1496	0.1281
n₁	0.0057	0.1634

The tendency to modify the institutional features of the commodity market mentioned at the beginning of this section can take the form of agencies that are designed to implement the policy objectives chosen. These may include buffer stock schemes, compensatory financing facilities, and supply management measures. The design of these schemes is bound by certain constraints and usually is accompanied by some restriction on the number and combinations of policy objectives and instruments. The operations of these schemes must also be consistent with the structure and behavior of the commodity market, and the introduction of these new objectives and instruments needs to be reconciled with the steady-state properties of the model. It therefore follows that the specification of these agencies and their compatibility with the existing market may lead to restrictions on the behavioral parameters of the agency. In the context of buffer stock intervention, the agency may choose to follow a mixed integral/proportional stabilization policy of the type

D \log B = - α β_{1} \log p^{*} - α β_{2} \int_{0}^{t} \log p^{*} d t - α \log B

w i t h p^{*} = \frac{p_{0} e^{λ t}}{p}

where B is the level of the agency’s holdings and p*, the price objective pursued. There will then exist restrictions on λ and P_o such that the newly defined system still has a long-run equilibrium solution. These restrictions also ensure that the above-mentioned scheme is compatible with other schemes that may be implemented. A failure to enforce restrictions necessary for the existence of a long-run equilibrium solution may lead to the impossibility of other instruments reaching their targets. The mixed integral/proportional stabilization policy just given as an example of possible buffer stock agency behavioral rule is a straightforward application of the policies put forward by Phillips (1954), whereby the desired level of the buffer stock agency holdings are proportional and of opposite sign to the deviations of price from its target level and also to the integral of all the deviations from this target. The coefficient α represents the speed of adjustment of the transaction process to reflect the state of uncertain knowledge, owing to lack of data or other factors, in which the agency operates. The foregoing rule, however, is for illustration purposes only, and its simplicity should not be taken as a limiting factor on the capabilities of the model.

Although the purpose of the estimation of the copper model is directed toward policy analysis, the model can be used for medium-term predictions. The relative simplicity of the model makes it unlikely that this model would provide accurate short-term forecasts (e.g., quarter on quarter) compared with “guesstimates” that could be obtained from brokerage companies, but if theory is to be of any use for forecasting purposes, the more stringent theoretical basis of the model and the emphasis put on plausible long-run behavior does suggest that this model may be preferable for medium-term forecasting.⁸

Table 5 contains the mean and root-mean-square errors (RMSE) for both static and dynamic forecasts, starting in the fourth quarter of 1961 and finishing in the last quarter of 1976. The charts for a selected number of the dynamic forecasts are given in Appendix C. The dynamic forecast, or multiperiod forecast, uses the actual values of the appropriately transformed variables in the last quarter of 1961 as starting values for the endogenous and exogenous variables and uses the predicted values of the endogenous variables thereafter until the fourth quarter of 1976. The last four quarters are ex post forecasts, and their asymptotic standard errors are given with the point estimates in Table 6 in Appendix C.

Table 5.

World Copper Model: Mean and Root-Mean-Square errors (RMSE) in Single Period and Multiperiod Forecasts, Fourth Quarter 1961-Fourth Quarter 1976

Table 5.

World Copper Model: Mean and Root-Mean-Square errors (RMSE) in Single Period and Multiperiod Forecasts, Fourth Quarter 1961-Fourth Quarter 1976

	Single Period		Multiperiod
Variables	Mean error	RMSE	Mean error	RMSE
Cr	0.000271	0.011375	0.000348	0.017639
Qp	0.003657	0.029369	0.004582	0.033463
Qs	–0.002278	0.089568	–0.000856	0.092443
MCr	–0.000330	0.005217	0.005113	0.021025
pu	0.002095	0.037254	–0.004827	0.066932
Qup	–0.002587	0.063861	–0.000456	0.076702
Qus	0.000993	0.088961	0.002233	0.090199
Cu	0.002467	0.104367	0.008432	0.116982
MCu	–0.000039	0.001669	–0.005255	0.019731
NMu	–0.023900	0.536448	–0.146480	0.577152
Sr	0.001634	0.058366	0.014211	0.190533
Ku	–0.000012	0.001002	0.000808	0.004266
Kr	0.000009	0.001048	0.000594	0.002695
Qtr	0.002723	0.034464	0.002476	0.032005
P	–0.001420	0.072557	–0.004163	0.093769
q	–0.000763	0.072557	–0.007155	0.084256

Table 5.

World Copper Model: Mean and Root-Mean-Square errors (RMSE) in Single Period and Multiperiod Forecasts, Fourth Quarter 1961-Fourth Quarter 1976

	Single Period		Multiperiod
Variables	Mean error	RMSE	Mean error	RMSE
Cr	0.000271	0.011375	0.000348	0.017639
Qp	0.003657	0.029369	0.004582	0.033463
Qs	–0.002278	0.089568	–0.000856	0.092443
MCr	–0.000330	0.005217	0.005113	0.021025
pu	0.002095	0.037254	–0.004827	0.066932
Qup	–0.002587	0.063861	–0.000456	0.076702
Qus	0.000993	0.088961	0.002233	0.090199
Cu	0.002467	0.104367	0.008432	0.116982
MCu	–0.000039	0.001669	–0.005255	0.019731
NMu	–0.023900	0.536448	–0.146480	0.577152
Sr	0.001634	0.058366	0.014211	0.190533
Ku	–0.000012	0.001002	0.000808	0.004266
Kr	0.000009	0.001048	0.000594	0.002695
Qtr	0.002723	0.034464	0.002476	0.032005
P	–0.001420	0.072557	–0.004163	0.093769
q	–0.000763	0.072557	–0.007155	0.084256

The properties of the FIML estimator can again be observed. The asymptotic efficiency of this estimator owing to a greater use of a priori information is shown in Table 5 where mean errors are always below 1 per cent with the exception of net U. S. imports and non-U. S. stocks. In general, RMSE are small by commodity model standards with, again, the exception of net U. S. imports. This deficiency was known at the specification stage, and it is thought that little can be done at present to improve that equation using the data series now available. It should perhaps also be pointed out that such a dynamic forecast depends on the initial period chosen and the deviations from the steady state of the endogenous variables at that time. However, it is one property of the FIML estimator in dynamic forecasts that the system is more likely to correct itself if properly specified than are other methods of estimation.

IV. Conclusions

This paper develops a dynamic simultaneous model of the world copper industry that gives plausible estimates of its structural parameters and exhibits a realistic dynamic behavior. The properties of the FIML estimation procedure used were illustrated in Section III, where such qualities as consistency truly come to light in the dynamic simulation exercise.

The mixed stock/flow hypothesis of price formation representing the non-tâtonnement process prevailing in commodity markets has been found to be verified. The introduction of price expectations and their importance in specifying the speculative demand for copper have proved to be invaluable in obtaining a reasonable price relationship. It has also been shown that the estimation of an investment relationship is possible and yields credible results. The investment feedbacks that are cornerstones of the models specified in this paper are crucial to the long-run properties of the models as well as their dynamic stability.

The model developed in this paper could be used to study the impact of decentralized economic policy on the copper industry. Three schemes that are worth investigating for stabilizing the copper industry are a buffer stock agency, some supply management measures, and the already existing compensatory financing facility operated by the Fund. The design of policy reaction functions as well as the allocation of targets and instruments to the separate agencies involves considerable further research, which relies heavily on the dynamic properties of the estimated copper model along with many issues of measurement of the impact of stabilization measures.

APPENDICES

A. The Steady-State Solution of the World Copper Model

The model described in Section II has a particular solution given by Y = Y^* e^λit where for the following variables (Cr, Cu, Qp, Qs, Qup, Qus, Sr, Kr, Ku, Qtr, MCr, MCu, NMu) X, = X, and for the remaining variables $(p, p u, q, \bar{p}) λ_{i} = λ_{2}$ the corresponding steady-state levels are

\log {C r}^{*} = \log C_{0} + b_{2} \bar{\log Y r} * - b_{1} \log p^{*} - \frac{λ_{1}}{α_{3}}

\log {C u}^{*} = \log {C u}_{0} + {b^{'}}_{2} \bar{\log Y u} * - {b^{'}}_{1} \log {p u}^{*} - \frac{λ_{1}}{α_{9}}

{Q p}^{*} = λ_{1} {Q u}^{*} - Q_{s}^{*}

Q_{u}^{*} = λ_{1} S_{r}^{*} + C_{r}^{*} + N M u^{*} + [\bar{S O C d - S O C x}]

Q_{s}^{*} = {[λ_{1} e^{A_{1}} p^{* - n_{1}} + 1]}^{- 1} [{C r}^{*} - [\bar{S O C d - S O C x}]] w i t h A_{1} = \frac{λ_{1}}{α_{5}} - \log Q_{0}

\log Q u p^{*} = \log {C u}^{*} - \frac{λ_{2}}{β_{3}}

Q u s^{*} = {[λ_{1} e^{A_{2}} p^{* - {n^{'}}_{1}} + 1]}^{- 1} {C u}^{*} w i t h A_{2} = \frac{λ 1}{α_{8}} - \log Q u s_{0}

\log {S r}^{*} = \log C^{*} + \log S_{0} - A_{3} w i t h A_{3} = \frac{λ_{1} δ β_{1} + [β_{1} + α_{1} β_{2}] λ_{2} + δ β_{1} μ}{α_{1} β_{1}}

{K r}^{*} = \frac{{C r}^{*} - {Q s}^{*} - [\bar{S O C d - S O C x}]}{λ_{1}}

{K u}^{*} = \frac{{C u}^{*} - {Q u s}^{*}}{λ_{1}}

\log {N M u}^{*} = \log {N M u}_{0} + \log {C u}^{*} - ζ \log p^{*} + ζ \log {p u}^{*} - \frac{λ_{1}}{α_{11}}

\log {M C r}^{*} = \log {Q p}^{*} + \frac{ψ}{α_{6}} - \frac{λ_{1}}{α_{6}}

\log {M C u}^{*} = n {Q u p}^{*} + \frac{ψ_{2}}{α_{6}} - \frac{λ_{1}}{α_{10}}

\log p^{*} = m_{1}^{- 1} [\frac{λ_{1}}{α_{4}} + \frac{λ_{1}}{α_{6}} - \frac{ψ_{1}}{α_{6}} - \log Q_{0}]

\log {p u}^{*} = {m^{'}}_{1}^{- 1} [\frac{λ_{1}}{α_{7}} + \frac{λ_{1}}{α_{10}} - \frac{ψ_{2}}{α_{10}} - \log {Q u p}_{0}]

\log q^{*} = \log p^{*} - [\frac{α_{2} + β_{1}}{α_{2} β_{1}}] λ_{2}

Calculations similar to the comparative dynamic examples carried out in Section I could be made here. Shifts in structural parameters as well as changes in initial conditions or in exogenous variables could be investigated. However, it is felt that such an analysis can be left to the reader, owing to the vast number of comments that would be necessary if a systematic investigation of these properties were to be made. The foregoing solution shows that a particular relation can be found and that a totally qualitative analysis of the model can be carried out.

At any point in time, the steady-state levels can be calculated and used to measure the actual deviation of the commodity market from equilibrium. The nonlinear system developed in terms of deviations from its steady state is as follows:

\begin{matrix} D^{2} y_{1} = & - [β_{1} + α_{1} β_{2}] {D y}_{1} + α_{1} {D y}_{2} - [α_{1} + δ β_{1}] {D y}_{13} \\ - δ D^{2} y_{13} - α_{1} β_{1} y_{13} + α_{1} β_{1} y_{3} & (1) \end{matrix}

\begin{matrix} D^{2} y_{2} = - [β_{1} + α_{2}] {D y}_{2} + α_{1} β_{1} y_{1} - α_{2} β_{1} y_{2} & (2) \end{matrix}

\begin{matrix} {D y}_{3} = α_{3} [{- b}_{1} y_{1} + b_{2} x_{1} - y_{3}] & (3) \end{matrix}

\begin{matrix} {D y}_{4} = α_{4} [m_{1} y_{1} + y_{6} - y_{4}] & (4) \end{matrix}

\begin{matrix} {D y}_{5} = α_{5} [n_{1} y_{1} + y_{16} - y_{5}] & (5) \end{matrix}

\begin{matrix} D^{2} y_{6} = γ_{1} α_{6} [y_{4} - y_{6}] - γ_{1} {D y}_{6} & (6) \end{matrix}

{D y}_{7} = β_{3} [y_{10} - y_{8}]

\begin{matrix} {D y}_{8} = α_{7} ({m^{'}}_{1} y_{1} + y_{11} - y_{8}) & (8) \end{matrix}

\begin{matrix} {D y}_{9} = α_{8} ({n^{'}}_{1} y_{1} + y_{15} - y_{9}) & (9) \end{matrix}

\begin{matrix} {D y}_{10} = α_{9} [{- b}^{'}_{1} y_{1} + {b^{'}}_{2} x_{2} - y_{10}] & (10) \end{matrix}

\begin{matrix} D^{2} y_{11} = γ_{2} α_{10} [y_{8} - y_{11}] - γ_{2} {D y}_{11} & (11) \end{matrix}

\begin{matrix} {D y}_{12} = α_{11} [{λ y}_{1} - {λ y}_{7} + y_{10} - y_{12}] & (12) \end{matrix}

\begin{matrix} {D y}_{13} = \frac{Q_{t r}^{*}}{S_{r}^{*}} [e^{y 16 - y 13} - 1] - \frac{C_{r}^{*}}{S_{r}^{*}} [e^{y 3 - y 13} - 1] + \frac{{N M}_{u}^{*}}{S_{r}^{*}} [e^{y 12 - y 13} - 1] - \frac{[\bar{{S O C}_{d} - {S O C}_{x}}] *}{{S r}^{*}} [e^{x 3 - y 13}] & (13) \end{matrix}

\begin{matrix} {D y}_{14} = \frac{C_{u}^{*}}{K_{u}^{*}} [e^{y 10 - y 14} - 1] - \frac{Q_{u s}^{*}}{K_{u}^{*}} [e^{y 9 - y 14} - 1] & (14) \end{matrix}

\begin{matrix} {D y}_{15} = \frac{C_{r}^{*}}{K_{r}^{*}} [e^{y 3 - y 15} - 1] - \frac{Q_{s}^{*}}{K_{r}^{*}} [e^{y 5 - y 15} - 1] - \frac{[\bar{{S O C}_{X}^{*} - {S O C}_{D}^{*}}]}{K_{r}^{*}} [e^{x 3 - y 15} - 1] & (\begin{matrix} 15 \end{matrix}) \end{matrix}

\begin{matrix} {D y}_{16} = \frac{Q_{p}^{*}}{Q_{t r}^{*}} [e^{y 4 - y 16} - 1] + \frac{Q_{s}^{*}}{Q_{t r}^{*}} [e^{y 5 - y 16} - 1] & (16) \end{matrix}

Taking a Taylor series expansion of equations (13) to (16) yields the approximate linear system (1) to (16) where equations (1) to (12) are identical. The remaining equations become

{D y}_{13} = \frac{Q_{t r}^{*}}{S_{r}^{*}} [y_{16} - y_{13}] - \frac{C_{r}^{*}}{S_{r}^{*}} [y_{3} - y_{13}] + \frac{{N M}_{u}^{*}}{S_{r}^{*}} [y_{12} - y_{13}] - \frac{[\bar{{S O C}_{d}^{*} - {S O C}_{x}^{*}}]}{S_{r}^{*}} [x_{3} - y_{13}]

{D y}_{14} = \frac{C_{u}^{*}}{K_{r}^{*}} [y_{10} - y_{14}] - \frac{Q_{u s}^{*}}{K_{u}^{*}} [y_{9} - y_{14}]

{D y}_{15} = \frac{C_{r}^{*}}{K_{r}^{*}} [y_{3} - y_{15}] - \frac{Q_{r}^{*}}{K_{r}^{*}} [y_{9} - y_{15}] - \frac{[\bar{{S O C}_{x}^{*} - {S O C}_{d}^{*}}]}{K_{r}^{*}} [x_{3} - y_{15}]

{D y}_{16} = \frac{Q_{p}^{*}}{Q_{t r}^{*}} [y_{4} - y_{16}] + \frac{Q_{s}^{*}}{Q_{t r}^{*}} [y_{5} - y_{16}]

where the vector of ys is the vector of deviations from the steady-state solution y = log [Y/Y^*]

B. Estimation of Linear Stochastic Differential Equation Systems

This Appendix presents a short summary of the technique used in estimating the model in this paper, essentially reproducing the text in Wymer (1975), and it also refers to the computer programs used. The presentation relies almost exclusively on the work by Sargan (1974) and Wymer (1972; 1978).

A recursive model of rth order differential equations can be represented as follows:

\begin{matrix} D^{r} y (t) = Σ_{k = 1}^{r} A_{k} D^{k - 1} y (t) + B z (t) + u (t) & (1) \end{matrix}

where D is an operator equivalent to stochastic differentiation, y(t) is a vector of endogenous variables, z(t) a vector of exogenous variables, u(t) a vector of disturbances, and A and B matrices of coefficients.

The disturbances u(t) are assumed to be generated by a stationary process with constant spectral density, so that the integral

ϵ (t) = \int_{0}^{t} u (s) d s

is a homogeneous random process with uncorrelated increments. Since ∊(t) is “white noise” and nondifferentiable, u(t) cannot be rigorously defined. But as Wymer (1972) has shown, (1) may be considered as

\begin{matrix} d D^{r - 1} y (t) = Σ_{k = 1}^{r} A_{k} D^{k - 1} y (t) d t + B z (t) d t + d \bar{ϵ} (t) & (2) \end{matrix}

where u(t) is replaced by the mean-square differential of e(t), which can be defined rigorously. This may be written as the first-order system

\begin{matrix} d y^{*} (t) = A^{*} y^{*} (t) d t + B^{*} z (t) d t + d ξ^{*} (t) & (3) \end{matrix}

The exact discrete model derived from the solution to (3) is as follows:

\begin{matrix} y^{*} (t) = e^{A *} y_{t - 1} + \int_{t - 1}^{t} e^{(t - s) A *} B * z (s) d s + w * (t) & (4) \end{matrix}

The derivation of (4) and the definition of w^*(t) are given in Wymer (1978).

The observations generated by (2) will satisfy (4), irrespective of the length of the interval between successive observations. Hence, the properties of (2) can be approached by studying the properties of (4). Generally, the variables z(t) will not be analytic functions of time, and the integral

\int_{t - 1}^{t} e^{(t - s) A *} B^{*} z (s) d s

cannot be evaluated directly. This integral can be evaluated directly, however, if z(s) is a linear or quadratic function of s.

Since it is computationally expensive to estimate (4) subject to general a priori restrictions on A^* and B^*, Wymer formulated an approximation that maintains the structural form but requires less computing time and that is generally used to find the satisfactory specifications of the model prior to the estimation of the exact discrete model (2).

This approximation to (2) is a nonrecursive discrete model derived by integrating over the interval (t –1, t) and using the approximations

\int_{t - 1}^{t} D y (s) d s = Δ y (t); \int_{t - 1}^{t} y (s) d s = M y (t)

where Δ = 1 – L, M = 0.5 (1 + L), and L is the lag operator. The approximate discrete model is

\begin{matrix} Δ y^{*} (t) = A^{*} M y (t) + B^{*} M z (t) + v^{*} (t) & (5) \end{matrix}

where v^*(t) is a vector of disturbances depending on the errors in (2) and the errors of approximation. The approximate estimating equation is

\begin{matrix} Δ^{r} y_{t} = Σ_{i = 1}^{r} A_{i} (M^{r - i + 1} Δ^{i - 1}) y_{1} + M^{r} B_{t} + v_{t} & (6) \end{matrix}

where v_t is a moving-average process of order r –1 and depends on the coefficients of (5) and v^*_t. This, in turn, can be approximated by a moving-average system of order r –1 that is independent of the parameters of the model. The estimates obtained for A^* from (6) are biased, but this bias is known and will generally be small for small eigenvalues of A^*.

The estimation technique can be extended to a system of mixed-order linear differential equations, where the structural equations are of any order including zero, by redefining A^* in (3) to take this into effect. Although the approximation to the continuous system has been derived under the assumption that the variables in the model are either all instantaneous (that is, variables measurable at a point in time) or all flows, the mixed- order system allows that analysis to be extended to a mixed stock/flow model.

Consider a simple model

\begin{matrix} d y (t) = a y (t) d t + b x (t) d t & (7) \end{matrix}

where x is a stock and y a flow. Integrate twice over the interval (t –1, t) to obtain

\begin{matrix} \int_{t - 1}^{t} d \int_{s - 1}^{s} x (θ) d θ d s = a \int_{t - 1}^{t} \int_{s - 1}^{s} y (θ) d θ d s + b \int_{t - 1}^{t} \int_{s - 1}^{s} x (θ) d θ d s & (8) \end{matrix}

By defining

\int_{s - 1}^{s} x (θ) d θ = x^{*} (s)

\int_{s - 1}^{s} y (θ) d θ = y^{*} (s)

equation (8) becomes

\begin{matrix} \int_{t - 1}^{t} d x^{*} (s) = a \int_{t - 1}^{t} y^{*} (s) d s + b \int_{t - 1}^{t} x^{*} (s) d s & (9) \end{matrix}

and (9) is approximated as follows:

Δ x_{t}^{*} = a M y_{t}^{*} + b M x_{t}^{*}

Δ M x_{t} = a M y_{t} + b M^{2} x_{t}

where y^*(t) is measured by y(t) over the observation period. Thus, y^*_t refers to quarterly or yearly observations of some flow variable, for instance, and x_t refers to stocks at end of period; x^*_t refers approximately to stocks at midperiod. The double integration does lead to a first-order moving-average process in the disturbances, which can be approximated by using a moving-average process

ϵ_t= u_t – 0.268u_t–1

and truncating after a few terms.

The order of the moving-average process is determined by the order of systems in its flow variables.

The program used to estimate A^* and B ^*in (5) was resimul, providing FIML estimates and allowing within and across nonlinear restrictions on the equations of the system.

Data transformations prior to estimation were handled by transf. The exact discrete model (4) was derived from the output of RESIMUL using Discox; the ex post forecast and simulation of the continuous system (by PREDIC) used equation (5); and, finally, the dynamic properties of the system were calculated using CONTINEST. All programs were written by Clifford R. Wymer and were operated on the Fund’s Burroughs 7700 computer.

C. Selected Dynamic Forecasts

Table 6.

World Copper Model: Selected Dynamic Forecasts, 1976

Table 6.

World Copper Model: Selected Dynamic Forecasts, 1976

Variables		Point Estimate	Asymptotic Standard Error	Actual Value
First quarter
	Cr	5.47065	0.03693	5.55222
	Qp	5.49760	0.03798	5.46325
	Qs	3.91584	0.09459	3.99280
	pu	0.42285	0.03299	0.45871
	Qup	4.49677	0.06890	4.58216
	Qus	3.40903	0.09421	3.44317
	Cu	4.80238	0.10794	4.76085
	NMu	2.61850	0.54016	3.31872
	Sr	5.82810	0.02615	5.78990
	Ku	8.13067	0.00112	8.13076
	Kr	8.70138	0.00047	8.70259
	Qtr	5.59321	0.03510	5.57479
	P	0.25082	0.07818	0.33523
	q	0.29322	0.07179	0.36131
	MCr	5.59833	0.00554	5.61058
	MCu	4.76734	0.00183	4.76902
Second quarter
	Cr	5.53505	0.03664	5.57074
	Qp	5.52761	0.03792	5.45593
	Qs	3.95578	0.09412	3.95576
	pu	0.46053	0.03290	0.50373
	Qup	4.56534	0.06625	4.54898
	Qus	3.43396	0.09383	3.37239
	Cu	4.82533	0.10743	4.82299
	NMu	2.81455	0.54261	3.55493
	Sr	5.79585	0.02623	5.77204
	Ku	8.13931	0.00112	8.13957
	Kr	8.71444	0.00047	8.71489
	Qtr	5.62425	0.03501	5.56388
	P	0.35275	0.07789	0.49415
	q	0.38714	0.07178	0.52090
	MCr	5.62919	0.00567	5.63152
	MCu	4.77261	0.00174	4.77273
Third quarter
	Cr	5.55637	0.03629	5.55669
	Qp	5.54475	0.03875	5.50765
	Qs	3.95398	0.09363	4.12372
	pu	0.51535	0.03292	0.43829
	Qup	4.56657	0.06616	4.57411
	Qus	3.47791	0.09413	3.43079
	Cu	4.85118	0.10660	4.83646
	NMu	2.97576	0.54065	3.13382
	Sr	5.73816	0.02648	5.73337
	Ku	8.14901	0.00111	8.14956
	Kr	8.72703	0.00046	8.72689
	Qtr	5.63938	0.03564	5.63286
	P	0.54827	0.07792	0.41440
	q	0.57279	0.07196	0.44353
	MCr	5.63962	0.00559	5.64444
	MCu	4.77548	0.00176	4.77563
Fourth quarter
	Cr	5.55471	0.03663	5.51111
	Qp	5.56068	0.03762	5.48619
	Qs	4.03939	0.09496	3.93803
	pu	0.45169	0.03298	0.33653
	Qup	4.55327	0.06708	4.63932
	Qus	3.42592	0.09427	3.41880
	Cu	4.87137	0.10774	4.72651
	N Mu	2.82706	0.53993	2.62973
	Sr	5.77364	0.02594	5.75307
	Ku	8.15918	0.00112	8.15879
	Kr	8.73815	0.00047	8.73826
	Qtr	5.66571	0.03481	5.58767
	P	0.26445	0.07940	0.25900
	q	0.30599	0.07310	0.29088
	MCr	5.65341	0.00543	5.65866
	MCu	4.77769	0.00175	4.77882

Table 6.

World Copper Model: Selected Dynamic Forecasts, 1976

Variables		Point Estimate	Asymptotic Standard Error	Actual Value
First quarter
	Cr	5.47065	0.03693	5.55222
	Qp	5.49760	0.03798	5.46325
	Qs	3.91584	0.09459	3.99280
	pu	0.42285	0.03299	0.45871
	Qup	4.49677	0.06890	4.58216
	Qus	3.40903	0.09421	3.44317
	Cu	4.80238	0.10794	4.76085
	NMu	2.61850	0.54016	3.31872
	Sr	5.82810	0.02615	5.78990
	Ku	8.13067	0.00112	8.13076
	Kr	8.70138	0.00047	8.70259
	Qtr	5.59321	0.03510	5.57479
	P	0.25082	0.07818	0.33523
	q	0.29322	0.07179	0.36131
	MCr	5.59833	0.00554	5.61058
	MCu	4.76734	0.00183	4.76902
Second quarter
	Cr	5.53505	0.03664	5.57074
	Qp	5.52761	0.03792	5.45593
	Qs	3.95578	0.09412	3.95576
	pu	0.46053	0.03290	0.50373
	Qup	4.56534	0.06625	4.54898
	Qus	3.43396	0.09383	3.37239
	Cu	4.82533	0.10743	4.82299
	NMu	2.81455	0.54261	3.55493
	Sr	5.79585	0.02623	5.77204
	Ku	8.13931	0.00112	8.13957
	Kr	8.71444	0.00047	8.71489
	Qtr	5.62425	0.03501	5.56388
	P	0.35275	0.07789	0.49415
	q	0.38714	0.07178	0.52090
	MCr	5.62919	0.00567	5.63152
	MCu	4.77261	0.00174	4.77273
Third quarter
	Cr	5.55637	0.03629	5.55669
	Qp	5.54475	0.03875	5.50765
	Qs	3.95398	0.09363	4.12372
	pu	0.51535	0.03292	0.43829
	Qup	4.56657	0.06616	4.57411
	Qus	3.47791	0.09413	3.43079
	Cu	4.85118	0.10660	4.83646
	NMu	2.97576	0.54065	3.13382
	Sr	5.73816	0.02648	5.73337
	Ku	8.14901	0.00111	8.14956
	Kr	8.72703	0.00046	8.72689
	Qtr	5.63938	0.03564	5.63286
	P	0.54827	0.07792	0.41440
	q	0.57279	0.07196	0.44353
	MCr	5.63962	0.00559	5.64444
	MCu	4.77548	0.00176	4.77563
Fourth quarter
	Cr	5.55471	0.03663	5.51111
	Qp	5.56068	0.03762	5.48619
	Qs	4.03939	0.09496	3.93803
	pu	0.45169	0.03298	0.33653
	Qup	4.55327	0.06708	4.63932
	Qus	3.42592	0.09427	3.41880
	Cu	4.87137	0.10774	4.72651
	N Mu	2.82706	0.53993	2.62973
	Sr	5.77364	0.02594	5.75307
	Ku	8.15918	0.00112	8.15879
	Kr	8.73815	0.00047	8.73826
	Qtr	5.66571	0.03481	5.58767
	P	0.26445	0.07940	0.25900
	q	0.30599	0.07310	0.29088
	MCr	5.65341	0.00543	5.65866
	MCu	4.77769	0.00175	4.77882

D. Data Definitions

The data used consist of 65 observations from the fourth quarter of 1960 through the fourth quarter of 1976. The first four observations are eliminated at the data preparation stage by the construction of lags and the removal of the first-order moving average described in Appendix B. The last four observations are reserved for ex post forecasting.

All the prices used in the world copper model are end-of-period prices, converted wherever necessary into U. S. cents per pound using the corresponding appropriate end-of-period exchange rate. All the variables related to quantities of copper are expressed in thousand metric tons of copper content.

The non-U. S. index of manufacturing activity has been constructed using ten components of manufacturing activity in the ten biggest copper-consuming countries. These components are indices of manufacturing activity (seasonally unadjusted) extracted from the Organization for Economic Cooperation and Development, Main Economic Indicators. (The U. S. index of manufacturing activity is also taken from the same source.) The manufacturing activity indices are then geometrically weighted by the share of each country out of the total amount of copper consumed by the ten countries on a yearly basis. The ten countries are Belgium, France, the Federal Republic of Germany, Italy, Spain, Sweden, the United Kingdom, Japan, Canada, and Australia.

The series describing physical quantities of copper have been compiled from: World Bureau of Metal Statistics Bulletin, Metals Week, Metallgesellshaft Aktiengesellschaft Metal Statistics, U. S. Department of Commerce publications, U. S. Bureau of Mines publications, and various bulletins published by the United Nations Conference on Tariffs and Trade. The stock of recoverable copper in the United States was cumulated on the base figure of 10 million metric tons in December 1960. The same variable in the non-U. S. was cumulated on the base figure of 20 million tons. Stock series of refined copper were cumulated on the values published for December 1960. The non-U. S. secondary production series, which is not directly available, are obtained by subtracting the U. S. total production and the non-U. S. primary production from the world total of copper produced, taking into account the net quantities of ore and concentrates imported by the United States.

BIBLIOGRAPHY

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SUMMARIES

The Demand for International Reserves Under Fixed and Floating Exchange Rates—h. robert heller and mohsin s. khan (pages 623–49)

The purpose of this paper was to examine if there was a fundamental shift in the demand for international reserves of countries in 1973 because of the change in the international monetary system from one of generally fixed exchange rates to one of greater exchange rate flexibility. Particular attention was also paid to the question whether the relationship between reserves and certain important variables remained stable during the period 1973-76. By answering this question, we were able to ascertain whether the move to greater flexibility in exchange rates changed the basic behavior pattern underlying countries’ desires to hold resources in the form of international reserves.

To test these hypotheses, we first estimated demand equations for six country groupings over the period 1964–76, and then applied statistical tests of parameter stability. The results indicated that there was a shift in the demand for reserves by industrial countries in response to the move to floating, but that this shift occurred toward the end of 1973 rather than at the beginning of the year. Obviously, there was some lag in the response of these countries to the change in the system; however, the behavior of non-oil developing countries did not appear to be affected by the change. This can perhaps be attributed to the fact that most of these countries continued to peg their currencies to another currency, and thus there was no real change in the exchange rate regime relevant to them. We further observed that, after the structural change, the function explaining reserves for both groups of countries continued to be stable in the floating rate period. Because the relatively short period precluded formal statistical testing, however, this latter conclusion must be regarded as preliminary.

Exchange Rate Policies for Developing Countries: Some Simple Arguments for Intervention—leslie lipschitz (pages 650–75)

This paper examines stabilization policy options in an open, less developed economy that is subject to transitory shocks. Particular emphasis is given to exchange rate policy as an instrument of stabilization.

It is shown by means of a simple model, in which the authorities seek to minimize fluctuations in absorption around some sustainable level, that shocks originating in aggregate demand should be dealt with by flexible exchange rates. Shocks originating in supply fluctuations are best dealt with by intervention in the exchange market to maintain a fixed exchange rate. Where the two sorts of shock occur, a managed floating system is the superior strategy, with the extent of intervention positively related to the predominance of supply shocks. The particular terms-of-trade shock modeled in this paper under extremely restrictive assumptions argues for a flexible exchange rate regime.

When active monetary policy is introduced, it is shown that, for demand shocks, pure flexible exchange rates automatically achieve the same result as would an active monetary authority with perfect foresight. For output shocks, appropriate domestic credit policy may be required not only to offset the depletion of external reserves but also to increase the overall money supply. This policy response to a balance of payments deficit, which seems quite straightforward in the present context, appears perverse if considered in the context of an orthodox demand-determined model.

A consideration of the dynamic properties of the model makes it clear that, unless the authorities use credit policies, the use of reserves cannot indefinitely stave off a need for real adjustment, but can only alter the time path of adjustment. Where active credit policy is used to offset the effects of real supply shocks on absorption, the enhanced stability comes only at the cost of greater fluctuations in external reserves.

Modeling the Demand for Liquid Assets: An Application to Canada—donal j. donovan (pages 676–704)

This paper develops and applies a model of the demand for liquid assets by Canadian households, based on the “direct utility of money” approach suggested by Milton Friedman. It is shown that previous models adopting this approach neglected to take into account important microeconomic and econometric considerations implied by the use of a utility maximization framework. Utilizing recent developments in duality theory, the model derives from a generalized representation of individual household preferences a set of constrained liquid asset demand functions, each of which contains as variables the rental prices of all assets, together with total expenditure on the services of money and near-money. Aggregation over households then provides aggregate liquid asset demand equations for estimation purposes. An important feature of the model is that it permits an explicit test of the underlying hypothesis of utility maximization.

The model, which is applied to annual data for the period 1952-74, includes four assets: chartered bank personal savings deposits; trust and mortgage loan company (TML) savings deposits; TML deposits for periods of more than one year; and Canada Savings Bonds. The results indicate, first, that for almost two thirds of the estimation period the data were consistent with the maximization of an underlying well-behaved preference ordering, defined over the services of money and money substitutes. Second, there is only slight evidence of gross substitutability between money and near-money--most assets are gross complements. Third, the results indicate that money is a normal good, while near-money is a luxury good. Thus, the rapid growth in near-bank liabilities during the period can be attributed to higher elasticities of expenditure, rather than to relative price changes in favor of financial institutions other than chartered banks.

Some Factors Influencing the United Kingdom’s Economic Growth Performance—leslie g. manison (pages 705–41)

This paper examines some factors accounting for the relatively poor economic growth performance of the United Kingdom since 1960. The examination is approached from the side of capital, using international comparisons. The United Kingdom’s slow growth rate, relative to that of other major industrial countries, is attributable to the interrelated phenomena of relatively low investment and gross incremental output/capital ratios. The deceleration in the growth of output in the 1970s in the United Kingdom, as in most other major industrial countries, is ascribed to the marked decline in the apparent productivity of new capital formation, especially of that from the manufacturing sector.

It is argued that a poor domestic saving performance, which has limited the capacity to invest, and a low and falling rate of return on capital employed, which has diminished the willingness to invest, have contributed to the United Kingdom’s relatively low rate of capital formation. These factors, in turn, have reflected a high and rising proportion of corporate value added being absorbed by labor costs. The decline in the physical productivity of capital and, more recently, rises in fuel and material costs have also accounted for the fall in company profits.

It is contended that an important factor contributing to the United Kingdom’s relatively low apparent productivity on new capital formation has been the relatively low level of investment allocated to the more dynamic export-oriented manufacturing industries. As a result, U. K. industry was unable to take full advantage of the marked upswing in world trade in manufactures in the 1960s and the early 1970s. This tendency manifested itself in the loss of export market shares and sharp increases in import penetration of domestic markets, which in turn led to periodic balance of payments constraints on the growth of output.

A Macroeconomic Model of the United Kingdom—malcolm d. knight and clifford r. wymer (pages 742–78)

The model that is specified and estimated in this study describes the dynamic behavior of major economic aggregates in the economy of the United Kingdom. In the real sector of the model, the markets for output and labor services are linked by a production function that both constrains the long-run relation between domestic output and inputs of labor and capital and influences the short-run behavior of wages, prices, and investment in fixed capital. A detailed financial sector that specifies the determinants of domestic money balances, advances, Eurocurrency deposits, and government debt is also included. A basic feature of the model is that it allows government monetary and fiscal policies (taxes, government spending, open market operations, and Bank rate) to respond to changes in target variables, such as income, the balance of payments, and employment. The budget constraints of the government sector are incorporated explicitly, so that the government borrowing requirement is financed by changes in the volume of base money and government securities. Thus, the model is well-suited to detailed policy analysis. The model is also designed both to capture the short-run adjustment processes of the U. K. economy and to have steady-state properties consistent with macroeconomic theory and with empirical observation over periods longer than the sample used for estimation. Thus, the steady-state solution of the model and the implications for both estimation and policy analysis are considered in detail. The model is estimated subject to the overidentifying restrictions of the theoretical structure. The parameter estimates and other statistics, including in- sample forecasts, indicate that the model is a plausible representation of the U. K. economy. A general analysis of the dynamic properties of the model is provided and is integrated with a discussion of the effects of the authorities’ policy reactions on the stability of the system.

A Dynamic Model of the World Copper Industry—den is richard (pages 779–833)

A general commodity model represented by a system of 10 first-order differential equations is specified to include hypotheses derived from economic theory, such as a mixed stock/flow formulation to represent the dynamics of price formation; an asset approach to the demand for the commodity that allows an implicit distinction between hedging and speculative demand, the latter including expected prices as an argument; and a production function where the fixed factors of production are endogenously determined by investment functions. Plausibility of long-run behavior and dynamic stability are examined, and it is shown that seasonal commodities can be accommodated by the model, thereby preserving the endogenous investment feedback.

This model is then extended and, using quarterly data, is applied to the world copper industry to include as many institutional features and as few exogenous variables as possible. Estimation of the simultaneous system of 20 first-order differential equations is carried out by full information maximum likelihood method, and the dynamic stability properties of the estimated structure are enhanced. The potential use for policy analysis with some designs for stabilization schemes form the latter part of the paper, along with some illustrations of the forecasting capabilities of the model.

RESUMES

La demande de réserves internationales dans un système de taux de change fixes et en régime de taux de change flottants — h. robert heller et mohsins. khan (pages 623–49)

Le présent document a pour objet d’examiner s’il y a eu, en 1973, un changement fondamental dans la demande de réserves internationales des pays par suite du passage du systéme monétaire international d’un régime de taux de change en général fixes à un régime caractérisé par une plus grande flexibilité des taux de change. On a également porté une attention particulière à la question de savoir si les relations entre les réserves et certaines variables importantes sont demeurées stables au cours de la période 1973-76. La réponse à cette question nous a permis d’établir si la plus grande flexibilite dans les taux de change vers laquelle on s’est orienté a modifié le comportement fondamental qui sous-tend le désir des pays de détenir des ressources sous forme de réserves internationales.

Pour vérifier ces hypothèses, nous avons d’abord estimé des équations de demande pour six groupes de pays au cours de la période 1964-76, et nous avons ensuite testé par voie statistique la stabilité des paramétres. Les résultats ont montré qu’un changement est intervenu dans la demande de réserves des pays industrialisés sous l’influence du passage au flottement généralisé, mais que ce changement s’est produit vers la fin de 1973 plutôt qu’au début de l’année. Evidemment, ces pays ont mis un certain temps à réagir au changement intervenu dans le système; toutefois, le comportement des pays en développement non producteurs de pétrole ne parait pas avoir été affecté par ce changement. Cette absence de réaction peut sans doute être attribuée au fait qu’ils ont toujours, pour la plupart, rattaché leur monnaie à une autre monnaie et qu’en conséquence le régime de taux de change qui les intéresse n’a subi aucune véritable modification. Nous avons en outre observé que, à la suite du changement structures, la fonction explicative de la demande de réserves pour les deux groupes de pays est demeurée stable au cours de la période de flottement des taux. Toutefois, it y a lieu de considérer cette dernière conclusion comme provisoire puisque la briàveté relative de la période en question n’a pas permis de recourir à des tests statistiques formels.

Les politiques dé taux de change et les pays en voie de développement : quelques arguments simples en faveur de l’intervention — leslie lipschitz (pages 650–75)

Le présent document examine les options qui s’offrent à un pays en matière de politique de stabilisation dans une économie ouverte, moins développée, sujette à des chocs transitoires. L’accent est mis particulièrement sur la politique de taux de change en tant qu’instrument de stabilisation.

L’analyse montre au moyen d’un modèle simple, dans lequel les autorités cherchent à réduire au minimum les variations de l’absorption intérieure autour d’un certain niveau soutenable, qu’il faudrait appliquer des taux de change flexibles pour connaître les chocs émanant de la demande globale. L’intervention sur le marché des changes, pour maintenir un taux de change fixe, est le meilleur moyen de connaitre les chocs émanant des fluctuations de l’offre. Mais, lorsqu’on se trouve en présence de ces deux sortes de choc, un système de flottement dirigé représente une stratégie supérieure, le degré d’intervention étant positivement corrélé à la prédominance des chocs de l’offre. Le choc particulier causé par les termes de l’échange, dont le modèle été établi dans le présent document en se fondant sur des hypothèses extrêmement restrictives, fournit des arguments en faveur d’un régime de taux de change flexibles.

Avec I’introduction d’une politique monétaire active, I’analyse montre que, dans le cas des chocs causés par la demande, des taux de change flottant sans intervention produisent automatiquement le même résultat qu’obtiendrait une autorité monétaire active dotée d’une perspicacite parfaite. Pour ce qui est des chocs émanant de la production, it se peut qu’il faille recourir à une politique du crédit intérieur appropridée non seulement pour compenser l’épuisement des réserves extérieures, mais aussi pour accroître l’offre de monnaie globale. L’application d’une telle politique pour faire face à un déficit de la balance des paiements, qui paraît tout à fait normale dans le contexte actuel, semble contreindiquée si on l’examine dans le contexte d’un modèle orthodoxe deéterminé par la demande.

Lorsqu’on examine les propriétés dynamiques du modèle, on s’aperçoit clairement que, en l’absence d’une politique du crédit, l’utilisation des réserves ne réussira pas indefiniment à écarter la nécessité d’un ajustement réel mais seulement à modifier la trajectoire de I’ajustement. En présence d’une politique du crédit active pour compenser les effets des chocs réels de l’offre sur l’absorption, une plus grande stabilité ne s’obtient qu’au prix de variations plus fortes des réserves extérieures.

Etablissement d’un modèle de demande d’actifs liquides : application an Canada — donal j. donovan (pages 676–704)

Le présent document développe et applique un modèle de la demande d’actifs liquides des ménages canadiens, foné sur le concept de I’“utilité directe de la monnaie” avancé par Milton Friedman. L’auteur montre que, dans les modèles antérieurs reposant sur ce concept, on a négligé de tenir compte des importantes considérations microéconomiques et économétriques qu’implique l’utilisation d’un cadre de maximisation de l’utilité. S’appuyant sur I’évolution récente de la théorie de la dualité, le modèle établit, à partir d’une représentation généralisée des préférences individuelles de ménages, une série de fonctions sous contrainte de la demande d’actifs liquides, contenant chacune comme variables le loyer de tous les actifs, de même que la dépense totale au titre des services rendus par la monnaie et la quasi-monnaie. L’agrégation sur l’ensemble des menages fournit alors, aux fins d’estimation, des equations globales de la demande d’actifs liquides. Une importante caractéristique du modèle est qu’il permet de vérifier explicitement I’hypothese sous-jacènte de la maximisation de l’utilité.

Le modèle, qui est appliqué aux données annuelles pour la période 1952-74, comporte quatre actifs : les dépôts d’épargne personnelle des banques à charte; les dépôts d’épargne des sociétés de fiducie et des sociétés de prêt hypothécaire; les dépôts de ces sociétés pendant plus d’un an et les obligations d’épargne du Canada. Les résultats indiquent, tout d’abord, que pour près des deux tiers de la période d’estimation, les données étaient compatibles avec la maximisation d’un ordre sous-jacent de préfラrences ayant les propriétés habituelles defini d’après les services rendus par Ia monnaie et ses substituts. Ensuite, it n’existe que peu d’éléments de fait sur la substituabilite brute entre la monnaie et la quasi-monnaie — la plupart des actifs sont des complements bruts. Enfin, les résultats indiquent que la monnaie est un bien ordinaire, tandis que la quasimonnaie est un bien de luxe. On peut ainsi attribuer la croissance rapide des quasi-engagements des banques durant la période à des éasticités plus grandes de la dépense, plutót qu’aux variations relatives des prix en faveur d’institutions financières autres que les banques à charte.

Facteurs influant sur la croissance économique du Royaume-Uni — leslie g. manison (pages 705–41)

Dans cette étude, l’auteur examine certains facteurs expliquant la croissance économique relativement faible du Royaume-Uni depuis 1960. 11 se place sur le plan de la formation de capital et se base sur des comparaisons internationales. La lenteur relative du taux de croissance du Royaume-Uni, par rapport à ceux des autres principaux pays industriels, est imputable à deux phénomèlnes étroitement liés : un taux d’investissement relativement bas et un coefficient marginal brut de capital relativement faible. Le ralentissement de la progression de la production pendant les années 70 au Royaume-Uni comme dans la plupart des autres principaux pays industriels est atrribué à la baisse marquée de la productivité apparente du capital nouvellement investi, surtout dans le secteur manufacturier.

Selon l’auteur, le taux peu élevé de l’épargne intérieure, qui à limité la capacité d’investissement, et le rendement faible et en baisse du capital utilisé, qui à découragé I’investissement, sont deux facteurs qui ont contribué au taux relativement faible de la formation de capital au Royaume-Uni. Ces facteurs ont, à leur tour, reflété (’absorption par les charges salariales d’une proportion élevée et croissante de la valeur ajoutée par les sociétés. La baisse de la productivité matérielle du capital et, plus récemment, la hausse des coûts des carburants et des matières premieres ont également contribué à Ia diminution de rentabilite des sociétés.

L’auteur soutient que l’un des facteurs importants expliquant la productivité apparente relativement faible du capital nouvellement investi au Royaume-Uni à été le niveau lui aussi relativement bas des investissements affectés aux industries manufacturières plus dynamiques tournées vers (’exportation. De ce fait, l’industrie britannique n’a pas été en mesure de tirer pleinement parti de l’augmentation sensible du commerce mondial des produits manufactures au cours des années 60 et au début des années 70. Cette tendance s’est traduite par la réduction de parts de marchés et par la forte progression de la pénétration des importations sur les marchés interieurs, qui, à leur tour, ont limité la croissance en imposant périodiquement des contraintes de balance des paiements.

Un modèle macroéconomique du Royaume-Uni — malcolm d. knight et clifford r. wymer (pages 742–78)

Le modèle qui est spécifié et estimé dans la prèsente étude décrit le cornportement dynamique des principaux agrégats économiques dans l’économie du Royaume-Uni. Dans le secteur réel du modèle, les marchés pour la production finale et les services de la main-d’œuvre sont liés par une fonction de production qui exerce une contrainte sur la relation à long terme entre la production intérieure et les inputs main-d’œuvre et capital et influence le cornportement à court terme des salaires, des prix et de la formation de capital fixe. Les auteurs ont également inclus un secteur financier détaillé qui spécifié les déterminants des encaisses monétaires intérieures, du crédit au secteur privé, des dépôts en euro-monnaies et du volume des emprunts publics. L’une des caracteristiques essentielles du modèle est qu’il permet aux politiques monétaire et budgetaire de l’Etat (impôts, dépenses publiques, opérations d’ open market et taux d’escompte) de réagir aux variations de variables-objectifs telles que le revenu, la balance des paiements et l’emploi. Les contraintes budgétaires du secteur public sont explicitement incorporées, de sorte que les besoins de financement de l’Etat sont couverts par la variation du volume de la base monétaire et des valeurs d’Etat. Le modèle se prête donc bien à une étude détaillée de la politique économique. 11 est également concu pour à la fois appréhender les processus d’ajustement à court terme de l’économie du Royaume-Uni et avoir des propriétés de croissance équilibrée compatibles avec la théorie macroéconomique et l’observation empirique sur des périodes plus longues que l’échantillon utilisé pour l’estimation. Ainsi, la solution de croissance équilibrée du modèle et les implications qui en découlent tant pour l’estimation que pour l’analyse de la politique économique sont examinées en détail. Le modèle est estimé sous réserve des restrictions suridentificatrices imposées par la structure théorique. Les estimations des paramètres et autres statistiques, y compris les prévisions effectuees dans l’échantillon, indiquent que le modèle constitue une reprèsentation plausible de l’économie du Royaume-Uni. La prèsente étude comporte une analyse générale des propriétés dynamiques du modèle, qui est intégrée dans le cadre d’une discussion de (’impact des réactions des mesures économiques adoptées par les autorités sur la stabilité du systéme.

Un modèle dynamique de l’industrie mondiale du cuivre — denis richard (pages 779–833)

Cet article est consacré à ]’étude d’un modèle général applicable aux produits de base. Ce modèle comporte un système de dix équations différentielles du premier ordre; it est spécifié de manière à inclure des hypothèses dérivées directement de la théorie économique, en particulier: 1) une reprèsentation de la dynamique de formation des prix caractérisée par un phénomène de non-tâtonnement, 2) une approche de la demande pour les produits de base par le biais des actifs et 3) une fonction de production. La formation des prix tette qu’elle est spécifiée dans ce modèle consiste en une formulation mixte des stocks et des flux. L’approche de la demande, adoptée dans cette étude, permet de distinguer implicitement les caractères transactionnels des caractères spéculatifs de cette demande, ces derniers contenant les espérances de prix comme variable explicative. La fonction de production comporte des facteurs fixes de production qui sont déterminés d’une manière endogène par des fonctions d’investissement. L’auteur examine la vraisemblance du comportement à long terme et la stabilité dynamique du modèle; it démontre également que ce modèle peut être étendu pour, par exemple, prendre en compte les produits de base saisonniers tout en maintenant les relations d’investissement endogènes.

Dans un deuxième temps, l’auteur élabore ce modèle et, utilisant des données trimestrielles, l’appliqué à l’industrie mondiale du cuivre; ainsi est-il en mesure d’inclure autant de facteurs institutionnels que possible tout en maintenant un minimum de variables exogènes. L’estimation du modèle est réalisée en utilisant la méthode du maximum de vraisemblance à information complète; ainsi l’auteur parvient-il à procéder à une estimation simultanée des vingt équations différentielles de premier ordre reprèsentant le modèle afin d’élaborer les propriétés relatives à la stabilité dynamique du système estimé. La dernière section de cette étude montre l’utilisation potentielle du modèle à des fins de politique économique, en particulier (’étude de programmes de stabilisation, et elle prèsente quelques illustrations des possibilités prévisionnelles de ce modèle.

RESUMENES

La demanda de reservas internacionales con tipos de cambio fijos y flotantes h. robert heller y mohsin s. khan (páginas 623–49)

Lá finalidad del prèsente trabajo es examinar si se produjo un desplazamiento fundamental en la demanda de reservas internacionales de los países en 1973 à causa de la modificación del sistema monetario internacional, que pasó de ser un sistema de tipos de cambio généralmente fijos à un sistema de mayor flexibilidad de los tipos de cambio. También se prèstó atención especialmente à la cuestión de si había permanecido estable durante el período 1973–76 la relación entre las reservas y ciertas variables importantes. Hallando la respuesta à esta cuestión, pudimos averiguar si el movimiento hacia una mayor flexibilidad de los tipos de cambio modificaba la modalidad bica de comportamiento en la que se funda el deseo de los países de mantener recursos en forma de reservas internacionales.

Para comprobar estas hipótesis, estimamos primero las ecuaciones de demanda de seis grupos de países durante el período 1964–76, y luego aplicamos pruebas estadisticas de la estabilidad de los parámetros. Los resultados indicaron que se produjo un desplazamiento en la demanda de reservas por parte de los países industriales, como reacción ante el movimiento hacia la flotación, pero que dicho desplazamiento ocurrió hacia finales de 1973 y no à comienzos de año. Evidentemente, hubo un desfase en la reacción de estos países à la modificación del sistema; sin embargo, el comportamiento de los países en desarrollo no productores de petróleo no parece haber sido afectado por la modificación. Esto quizϡ pueda atribuirse al hecho de que la mayorϭa de dichos países continuaron vinculando su moneda à otra moneda, y por lo tanto no se produjo una modificación real en el regimen cambiario que les afectaba à ellos. Observamos también que, después de la modificación estructural, la función explicativa de las reservas para ambos grupos de países siguió siendo estable en el perϭodo de flotación de los tipos de cambio. No obstante, esta última conclusiþn debe considerarse preliminar, dado que el período relativamente corto impidió la realización de pruebas estadísticas adecuadas.

Políticas de tipo de cambio para los países en desarrollo: Algunos argumentos sencillos à favor de la intervención— leslie lipschitz (páginas 650–75)

En el prèsente trabajo se examinan las opciones de la poíitica de estabilización en una economía abierta y menos desarrollada, que esté sujeta à conmociones transitorias. Se hace hincapié en la política de tipos de cambio como instrumento de estabilizacióm.

Con un modelo simple, en el que las autoridades tratan de reducir al mínimo las fluctuaciones de la absorción en torno à cierto nivel sostenible, se demuestra que debe hacerse frente à las conmociones derivadas de la demanda agregada con tipos de cambio flexibles. La mejor manera de hacer frente à las conmociones derivadas de las fluctuaciones de la oferta consiste en intervenir en el mercado de divisas para mantener un tipo de cambio fijo. De producirse ambos tipos de conmociones, la mejor estrategia consiste en un sistema de flotatión controlada, y el grado de interventión estará positivamente relacionado con el predominio de las conmociones de oferta. La conmoción concreta de relación de intercambio analizada mediante un modelo en el prèsente trabajo, en supuestos sumamente limitadores, aboga en favor de un régimen de tipos de cambio flexibles.

Incorporando una poíitica monetaria activa, se demuestra que con tipos de cambio flexibles puros se logran automáticamente, por lo que se refiere à las conmociones de demanda, el mismo resultado que el obtenido por una autoridad monetaria cuya previsión fuera perfecta. En cuanto à las conmociones de producto, quizá se necesite una poíitica créditicia interna apropiada, no sólo para contrarrestar el agotamiento de las reservas externas sino también para incrementar la oferta monetaria global. La mencionada reactión de poíitica ante un déficit de balanza de pagos, que parece perfectamente diáfana en el contexto actual, parece perversa si se Ia considera en el marco de un modelo ortodoxo determinado por la demanda.

Al estudiar las propiedades dinamicas del modelo resulta evidente que, à menos que las autoridades recurran à medidas créditicias, el recurso à las reservas no permite aplazar indefinidamente la necesitad de un ajuste real y no puede más que alterar Ia senda del ajuste. Cuando se utiliza una poíitica créditicia activa para contrarrestar los efectos producidos en Ia absorción por las conmociones de la oferta real, la mayor estabilidad se consigue sólo à costa de mayores fluctuaciones en las reservas externas.

Construcción de un modelo de demanda de activos líquidos: Una aplicación à Canadá—donal j. donovan (páginas 676–704)

En este documento se desarrolla y aplica un modelo de la demanda de activos líquidos de los hogares canadienses, basado en el método de la directa del dinero” propuesto por Milton Friedman. Se demuestra que en modelos anteriores en que se adoptó este método, no se tuvieron en cuenta importantes consideraciones microeconómicas y econométricas inherentes al use de un marco de maximización de la utilidad. Empleando recientes mejoras de la teoría de la dualidad, el modelo deriva de una representación généralizada de las preferencias de hogares individuales, un conjunto de funciones de demanda de limitados activos líquidos, cada una de las cuales contiene como variables el precio de arriendo de todos los activos y el gasto total por servicio del dinero y el cuasidinero. Luego, con la suma de_, todos los hogares, se obtienen las ecuaciones de la demanda agregada de activos líquidos con fines de estimación. Una caracteristica importante del modelo es que permite probar explícitamente las hipótesis básicas de maximización de la utilidad.

El modelo, que se aplica à los datos anuales del período 1952–74, incluye cuatro activos: depósitos de ahorro personal en bancos autorizados; depósitos de ahorro en sociedades de prèstamos hipotecarios (TML, por trust and mortgage loan company); depósitos à más de un año en TML, y Canada Savings Bonds. Los resultados indican, primero, que durante casi dos terceras partes del período de estimación, los datos fueron consistentes con la maximización de una ordenación de preferencias básica de comportamiento racional, definida en relación con el servicio del dinero y los sustitutos del dinero. Segundo, sólo hay pruebas insignificantes de sustituibilidad entre dinero y cuasidinero en général; casi todos los activos son mayormente complementos. Tercero, los resultados indican que el dinero es un bien ordinario y el cuasidinero, un bien suntuario. Por lo tanto, el rápido crecimiento de los pasivos cuasibancarios durante el período puede atribuirse à mayores elasticidades de gasto, más que à variaciones de precios relativos à favor de instituciones financieras distintas de los bancos autorizados.

Algunos factores que influyen en el comportamiento del crecimiento económico del Reino Unido—leslie g. manison (pdginas 705–41)

En el prèsente documento se analizan algunos factores à que cabe atribuir los resultados relativamente modestos obtenidos por el Reino Unido en materia de crecimiento económico desde 1960. Se aborda este análisis desde el punto de vista del capital, utilizando comparaciones internacionales. El ritmo lento de crecimiento del Reino Unido, comparado con el de otros países industriales importantes, es atribuible à dos fenómenos relacionados entre sí, à saber, la inversión y las relaciones incrementales brutas de capital–producto relativamente pequeñas. La desaceleración del crecimiento del producto registrada en el Reino Unido durante el decenio de 1970, como en casi todos los países industriales de primera importancia, se atribuye à la acusada disminución de la productividad aparente de la formación de capital nuevo, especialmente en el sector manufacturero.

Se alega que los escasos resultados obtenidos en materia de ahorro interno, con la consiguiente limitation de la capacidad de inversión, y la baja y decreciente tasa de rendimiento del capital empleado, que ha frenado el estímulo para invertir, han contribuido à la tasa relativamente baja de formación de capital en el Reino Unido. à su vez, esos factores han obedecido à la parte elevada y creciente del valor agregado emprèsarial que absorben los costos de la mano de obra. La baja de la productividad material del capital y, más recientemente, la subida registrada por los costos del combustible y de los materiales también explican la disminución de los beneficios obtenidos por las empresas.

Se sostiene que el nivel relativamente bajo de la inversión asignada à las industrias manufactureras más dinámicas orientadas hacia la exportación constituye un factor importante que ha contribuido à la productividad aparente relativamente baja de la formacion de capital nuevo en el Reino Unido. Como consecuencia de ello, la industria britáica no pudo beneficiarse plenamente del notable auge registrado por el comercio mundial de productos manufacturados en el decenio de 1960 y primeros años del de 1970. Esta tendencia se manifestó con la pérdida de participación en los mercados exteriores y con la acusada y creciente penetración de los productos importados en los mercados internos, lo cual à su vez dio Lugar à que el crecimiento del producto se viera sometido à prèsiones periódicas de balanza de pagos.

Modelo macroeconómico del Reino Unido—malcolm d. knight y clifford r. wymer (páginas 742–78)

El modelo especificado y estimado en este estudio describe el comportamiento dinámico de importantes agregados económicos del Reino Unido. En el sector real del modelo, los mercados del producto y de los servicios laborales están vinculados por una función de producción que limits la relación à largo plazo entre el producto y los insumos internos del trabajo y el capital e influye en el comportamiento à corto plazo de los salarios, los precios y la inversión en capital fijo. Se incluye también un sector financiero detallado en el cual se especifican las determinantes de la liquidez monetaria interna, créditos bancarios, depósitos en euromonedas y deuda del gobierno. Una característica básica del modelo es que permite que las poíitical monetarias y fiscales del gobierno (impuestos, gasto público, operaciones de mercado abierto y tasa de descuento) respondan ante los cambios de variables metas tales como el ingreso, la balanza de pagos y el empleo. Se incorporan explícitamente las limitaciones prèsupuestarias del sector gubernamental, de modo que la necesidad de endeudamiento del gobierno se financia mediante modificaciónes del volumen de la base monetaria y las obligaciones del gobierno. Por lo tanto, el modelo se prèsta muy Bien para el análisis de poíitica detallado. El modelo se ha concebido de tal manera que, además de captar los procesos de ajuste à corto plazo de la economía del Reino Unido, sus características respecto al estado estacionario (“steady state”) son compatibles con la teoría macroeconómica y con la observación empírica durante períodos más largos que los de la muestra usada para fines de estimación. De este modo, se consideran en detalle la solución de marco estable del modelo y las repercusiones desde el punto de vista tanto de la estimación como del análisis de poíitica. Se ha estimado el modelo con sujeción à restrictiones que superidentifican la estructura teórica. Las estimaciones de los parámetros y otras estadisticas, inclusive los pronósticos obtenidos en la muestra, indican que el modelo constituye una reprèsentación plausible de la economía del Reino Unido. Se prèsenta un análisis général de las propiedades dinámicas del modelo y dicho análisis se integra en el examen de los efectos que tienen en la estabilidad del sistema las decisiones de las autoridades en materia de poíitica.

Un modelo dinámico de la industria mundial del cobre —denis richard (páginas 779–833)

Se especifica un modelo général de productos primarios reprèsentado por un sistema de diez ecuaciones diferenciales de primer orden, que incluye hipótesis derivadas de la teoría económica, tales como una formulación mixta de masag y flujos para reprèsentar la dindmica de la formación de precios; un mdtodo basado en los activos en cuanto à la demanda de productos primarios, que permite una distinción implícita entre demanda con fines de cobertura y demanda especulativa, incluyendo esta última como argumento los precios previstos, y una función de producción en la que los factores fijos de producción se determinan de forma endógena mediante las funciones de inversión. Se examina la verosimilitud del comportamiento de largo plazo y la estabilidad dinámica, y se demuestra que los productos primarios estacionales pueden tener cabida en este modelo, conservandose así la retroacción de la inversión endógena.

Se extiende luego este modelo y, utilizando datos trimestrales, se le aplica à la industria mundial del cobre incluyendo el mayor número posible de aspectos institucionales y el menor número posible de variables exógenas. Se lleva a cabo la estimación del sistema simultáeo de 20 ecuaciones diferenciales de primer orden, mediante el mdtodo de máxima verosimilitud con plena información, con to que aumentan las propiedades de estabilidad dinámica de la estructura estimada. La última parte del prèsente estudio se refiere al use potencial para análisis de medidas de poíitica con algunas aplicaciones à programas de estabilización, junto con algunas ilustraciones de la capacidad de pronóstico que tiene el modelo.

In statistical matter (except in the résumés and resfiúenes) throughout this issue,

Dots (…) indicate that data are not available;

A dash (—) indicates that the figure is zero or less than half the final digit shown, or that the item does not exist;

A single dot (.) indicates decimals;

A comma (,) separates thousands and millions;

“Billion” means à thousand million;

A short dash (–) is used between years or months (e.g., 1971–74 or Januaryóctober) to indicate a total of the years or months inclusive of the beginning and ending years or months;

A stroke (/) is used between years (e.g., 1973/74) to indicate a fiscal year or a crop year;

Components of tables may not add to totals shown because of rounding.

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Balance of Payments Yearbook

Volume 29

Volume 29 of the Fund’s Balance of Payments Yearbook was complèted with the publication in December of the comprehensive annual issue, which supersedes the 11 booklets issued monthly throughout 1978. This annual book contains definitive tables and notes for 110 countries, with annual figures for most of them for 1969–77 and quarterly summaries continuing into 1978 for some countries. à Supplement to Volume 29 contains about 60 series of standard categories organized by topic instead of by country.

In the Balance of Payments Yearbook, the Fund publishes figures that are comparable for each country included. About 30 of the 110 countries provide quarterly or half-yearly figures as well as annual data.

Volume 29 was the last to be based on the Third Edition (1961) of the Balance of Payments Manual. Henceforth, figures will be published according to the concepts and definitions of the Fourth Edition of the Balance of Payments Manual (Washington, 1977), beginning with the monthly booklets in Volume 30 of the Yearbook.

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Mr. Richard, economist in the Commodities Division of the Research Department when this paper was prepared, is currently economist in the Central Banking Service. He is a graduate of both the Faculties of Science and of Law and Economics at the University of Montpellier and of the University of London.

The author extends grateful acknowledgments to Dr. Leo Drollas of British Petroleum and to his former colleagues at the Commodities Research Unit and Forex Research Ltd.

International Monetary Fund, IMF Survey, Vol. 6 (March 7, 1977), p. 68.

A general Cobb-Douglas production function can be represented as follows:

$Q = Q_{0} \underset{i = 1, n}{Π} {L i}^{α i}$

with 0 < α < 1 for i = 1 … n and Σα_i = 1

which can accommodate n factors of production. This function is homogeneous of the first degree and exhibits constant returns to scale if Σα_i = 1; its elasticity of substitution is equal to unity.

Profits can be expressed as

$\begin{array}{l} Π = P Q - W L = P Q \\ - W {[Q / Q_{0}]}^{\frac{1}{α}} I^{\frac{α - 1}{α}} e^{\frac{- λ}{α} t} \end{array}$ where W is the nominal price of variable factors Although each firm can vary the amount of its fixed capital, only the rate of change of this variable can be varied instantaneously. At any time, firms must regard the amount of fixed capital as part of the factors affecting their decisions. The supply equation is therefore derived from profit maximization as

$Q = Q_{0}^{\frac{1}{1 - α}} I {[α P / W]}^{\frac{α}{1 - α}} e^{\frac{λ}{1 - α} t}$

Equation (7) can be re-expressed by eliminating W in Π = PQ – WL

$Q = Q_{0}^{\frac{1}{1 - α}} I {[α P / W]}^{\frac{α}{1 - α}} e^{\frac{λ}{1 - α} t}$

one obtains Π (1 – α)P Q, that is, profits are a constant proportion of the total revenue. Then (7) becomes

$D m (t) = γ {α_{4} \log [\frac{(1 - α) p (t) Q (t)}{ρ P K (t) I (t)}] + γ_{0} - m (t)}$

$\begin{matrix} D m (t) = γ {α_{4} \log [Q (t) / I (t)] + ψ - m (t)} with ψ = α_{4} \log [\frac{(1 - α) P (t)}{ρ P k (t)}] + γ_{0} & (7^{'}) \end{matrix}$

This proportional relationship between profits and revenues is a feature of the Cobb- Douglas production function and will turn out to be most helpful in the explanation of alternative behaviors in the copper industry. The quantity ψ is assumed to be constant because of lack of data on p and Pk(t). The steady-state rate of growth of the quantity comparable to a real rate of interest should, however, be zero.

The mean time lag |1/β| is the time taken for 63 per cent of the discrepancy between y(t) and ŷ(t) to be eliminated by changes in y(t) following a change in ŷ(t), since:

$\int_{0}^{| 1 / β |} β e^{β s} d s = 0.63$

This adjustment process can be represented in operator form as $y (t) = \frac{β}{(D + β)} \hat{y} (t)$ which is equivalent to the first-order exponential distributed lag function:

$y (t) = \int_{0}^{\infty} {β e}^{- β s} \hat{y} [t - s] d s$

The attractiveness of such an adjustment is that it lends itself readily to the concept of excess supply and demand. Higher-order exponential distributed lag functions f(s) with, say, a humped time profile can be specified. Such higher-order functions often result from eliminating variables, such as expectations, by replacing them with distributed lag functions of other variables. These functions are always strictly positive, the area under the curve being equal to unity, and they tend to zero asymptotically as β increases:

$f (s) > 0 f o r a l l s; \int_{0}^{\infty} f (s) = 1; \lim f (s) = 0 a s \to + x$

It is thought difficult to find economic foundations to higher-order lag functions unless they are the result of the combination of elementary adjustment functions such as the one just explained.

The estimates of parameters have an asymptotic normal distribution and the t-value referred to hereafter is merely the absolute value of the ratio of the parameter estimate to an estimate of the standard deviation of that asymptotic normal distribution. The use of the term “t-value” is for convenience only and does not imply that this value has a Student’s t-distribution. At the 5 per cent level, the ratio of the parameter estimate to its standard error should exceed 1.96 for the estimate to be significantly different from zero.

From $d λ = [\frac{\partial λ}{\partial α_{1}}] d α_{1} + [\frac{\partial λ}{\partial α_{2}}] d α_{2} = > {[\frac{\partial α_{1}}{\partial α_{2}}]}_{λ} = - [\frac{\partial λ \partial α_{1}}{\partial λ \partial α_{2}}] \frac{α_{1}}{α_{2}}$

One feature of the forecasts of the model perhaps should be noted here. The transformation necessary to eliminate the moving average inherent in a stock/flow model does not “smooth” the data, and hence the forecasts derived from the estimated model are not “smoothed.” On the contrary, the moving-average transformation is a prewhitening procedure required by the nature of the observations, and it could be argued that the usual discrete model, estimated by using observations on flow variables that are implicitly integrated over the observation period, use “smoothed” data and hence will produce forecasts that are “smoothed.”

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IMF Staff papers: Volume 25 No. 4

Author: International Monetary Fund. Research Dept.

Volume/Issue:: Volume 1978: Issue 004

Publisher:: International Monetary Fund

ISBN:: 9781451930443

ISSN:: 1020-7635

Pages:: 234

DOI:: https://doi.org/10.5089/9781451930443.024

Variables	Steady-State Growth Rate
Cr, Cu, Qp, Qup, Qus, Sr, Kr, Qtr, NMu, MCr, MCu	$λ_{1} = b_{2} λ_{\bar{y}} - b_{1} λ_{2}$
p, pu, q	$λ_{2} = \frac{{m^{'}}_{2}}{{m^{'}}_{1}} - \frac{m_{2}}{m_{1}}$

Variables	Steady-State Growth Rate
Cr, Cu, Qp, Qup, Qus, Sr, Kr, Qtr, NMu, MCr, MCu	$λ_{1} = b_{2} λ_{\bar{y}} - b_{1} λ_{2}$
p, pu, q	$λ_{2} = \frac{{m^{'}}_{2}}{{m^{'}}_{1}} - \frac{m_{2}}{m_{1}}$

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Abstract

I. A General Commodity Model

the model

the dynamics of the model

II. Application to the World Copper Industry

institutional background of the model

a model of the world copper market

the steady-state properties of the world copper model

estimation method

results

III. Policy Analysis and Simulation

IV. Conclusions

APPENDICES

A. The Steady-State Solution of the World Copper Model

B. Estimation of Linear Stochastic Differential Equation Systems

C. Selected Dynamic Forecasts

D. Data Definitions

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Balance of Payments Yearbook

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