A Simple Model of the Private Gold Market, 1968–74: An Exploratory Econometric Exercise

The world’s private gold market has recently drawn an increasing amount of attention from central bank officials as well as academic economists. There is, however, as far as we know, no published paper that treats the subject in an integrated fashion. The main purpose of this paper is to propose a model of the private gold market, to estimate it, and to test it against historical data.

Abstract

The world’s private gold market has recently drawn an increasing amount of attention from central bank officials as well as academic economists. There is, however, as far as we know, no published paper that treats the subject in an integrated fashion. The main purpose of this paper is to propose a model of the private gold market, to estimate it, and to test it against historical data.

The world’s private gold market has recently drawn an increasing amount of attention from central bank officials as well as academic economists. There is, however, as far as we know, no published paper that treats the subject in an integrated fashion. The main purpose of this paper is to propose a model of the private gold market, to estimate it, and to test it against historical data.

The model is a nonlinear system consisting of: one supply function, two demand functions (industrial use and speculative demand), a generating function for price expectations, various definitional identities, and an equilibrium condition. The supply function is derived essentially from regulations governing the industry; the demand function for industrial use of gold is obtained from the behavior of firms that regard gold as an input for the production of such items as jewelry, electronic equipment, and dental services; and the speculative demand equation is derived from the conventional theory of portfolio choice.

The model is estimated by the two-stage least-squares method, using data for the period from the second quarter of 1968 to the fourth quarter of 1974. The starting date for the sample period was chosen because of structural change with the establishment of the two-tier system in March 1968. Before this time, the central banks had changed their stock of gold depending upon excess demand or supply in the market at prices fluctuating narrowly around $35.00 per fine ounce. The behavior of central banks was thus determined endogenously by market forces under the gold pool arrangement, while prices were set exogenously. Since the establishment of the two-tier system, however, changes in the stock of gold held by the central banks may be regarded as exogenous, while the price is determined by market forces. There is thus no reason to assume structural stability of behavioral equations over a period encompassing both of these systems. The closing date for the sample is constrained by data availability.

The estimated model is subjected to an in-sample dynamic simulation. The simulation exercise lends strong support to the model as a whole. For the reason already suggested, however, the validity of the model should not be taken for granted outside the sample period.

The paper is organized in four parts. Section I presents an overview of the problems in analyzing the world gold market. Section II briefly describes a quarterly econometric model. Section III discusses estimation and in-sample dynamic simulation results for the sample period—the second quarter of 1968 to the fourth quarter of 1974. Section IV summarizes major findings. Two appendices provide (1) some background notes on the supply curve of gold output and (2) data sources and derivations.

I. An Overview of the Problems

the nature of the market: stocks, flows, and speculation

An analysis of gold trading must begin by asking some fundamental questions about the nature of the market. Is gold essentially a commodity traded for its intrinsic attributes between producers and gold-using manufacturers, or is it primarily a financial asset, hoarded in the hope of speculative gains or avoidance of capital losses? In the first case, our concern is with the flow of mining output and the derived demand of manufacturers;1 in the second case, with the stock of gold in financial portfolios and adjustments to this stock. The question turns on relative magnitudes: if, on the average trading day, the stock adjustment is very large by comparison with producers’ supply and industries’ demand, the market is best characterized as a financial stock exchange; if small, we may treat the market like that for any other commodity flow.

It is far easier to formulate these criteria than to apply them: no data on either the size of private financial gold holdings or turnover on the major gold markets are available, and even production and industrial demand data are, at best, estimates for some countries. While it is plausible that the flow of newly mined gold supply and industrial gold demand is, on any market day, small relative to the stock of financial gold, it is not clear that it is infinitesimal or that the effects of flow disequilibria on price are completely swamped by the stock adjustment effects. A complete study of the market, therefore, requires explicit examination of the behavior of two, conceptually different, sets of transactors: the ordinary commodity trader, and the speculator.2

Traditionally, speculators are regarded as transactors with better than average foresight who, by anticipating the direction of price changes, diminish price fluctuations. In this light, speculators earn their gains in much the same fashion as merchants—by improving information transmission in the price system, their efforts hasten the equilibrating process and increase allocative and distributive efficiency. The traditional argument, however, is predicated on the assumption that “real” (by which is meant nonspeculative underlying demand and supply) forces dominate the market and that speculative turnover is only a small part of traded quantity, which can consequently influence only the magnitude, not the direction, of price changes. If this assumption is put aside and the alternative hypothesis (that real flow demand and supply on the market are infinitesimal relative to speculative turnover) is adopted, the analysis is much changed. The successful speculator under this condition still requires foresight, but rather about the future behavior of other speculators than about the real factors in the market. Even if, in the aggregate, speculation is unprofitable, the handsome rewards of a small group are sufficient to attract a shifting set of transients to the market. Keynes’s epigram (1936, p. 156) on the share market is relevant and succinct: “We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be.” Here, both the direction and magnitude of price changes may depend on the psychological investment climate taking little account of flow demand and supply; speculators can live off each other, independently of external events.3

Analysis of the gold market requires that speculation—at least in the aggregate—is not entirely whimsical, but may be well represented by a stable behavioral function. Furthermore, the speculator must be coupled with the more pedestrian market transactors in a consistent and stable set of interrelations.

requirements of the model

Beginning with the assumption that the stock of gold is large relative to net flow supply and demand, it is convenient for illustrative purposes to envision the price being set initially, each day, by financial transactors. This price then elicits demand and supply responses from the real side of the market. These responses probably necessitate further price adjustments, which in turn feed back to alter the demands of speculators. This seesawing process of price and quantity determination is in part illustrated in Figure 1.

In (a), STS and STd are, respectively, the stock supply of and demand for gold. The supply is not price responsive, since current flows involve small magnitudes relative to accumulated stock. The demand curve, drawn in price-quantity (P-Q) space, is extremely volatile, sensitive as it is to price expectations (Pe), wealth (w), interest rates (i), and inflation (π).4 The stock being virtually fixed in the short run, price adjusts instantaneously to clear the stock market. At every point in time the stock of gold is effectively demanded (held) by some transactor, and this stock can only be adjusted slowly. The emergent stock market-clearing price, then, elicits flow supply (Pr) and demand (D) responses as illustrated in (b).5 Notably, we have incorporated the well-established fact that the supply response is correlated negatively with price.6 Of course, while flow demand and supply (b) is conceptually separable from stock adjustment (a), separability in time is not possible. The process postulated—price set on the stock side of the model, resultant flow responses dependent on infinitesimally lagged price—is oversimplified for expositional purposes, and will not be evident from data over longer observation intervals.7 While we have drawn an arrow in (b) to indicate the directions of market responses to flow disequilibrium, these changes would themselves evoke stock adjustments and would probably be dominated by them.8 The market, of course, clears each day; excess demands of financial and industrial participants sum to zero. It is clearly possible that, over time, excess flow demand reduces the stock supply. We have abstracted from this in Figure 1 by assuming flows to be of insignificant magnitude relative to the accumulated stock. Finally, while stock-flow equilibrium appears to be guaranteed by price, P*, even this is a static oversimplification, as secular growth of wealth will, ceteris paribus, require growth in the stock demand for gold and, consequently, persistent excess of flow supply over flow demand.

The basic structure of the model that follows is illustrated by the description given here; certain concessions have, however, been made to empirical constraints. First, a dearth of shorter-run data has limited us to estimating the model with data over 27 quarterly observation periods. Over this length of interval, the model is fully simultaneous. Second, since data on neither the cumulated stock of private gold nor market turnover are available, we have considered speculative behavior with regard to quarterly net adjustments to the stock. The model, therefore, while rich in its ability to determine the effects of changes in net supply, has nothing to say about the redistribution of gold stocks among hoarders.

II. A Quarterly Model

supply of gold

The new supply of gold (Gs) in the private gold market consists of gold production (Pr), plus net sales by members of the Council for Mutual Economic Assistance (CMEA) and the People’s Republic of China (Z), plus central bank decumulation (B),9 as expressed in equation (1)

Gts=Prt+Z¯t+B¯t(1)

where the bar (—) on top of a variable indicates that the variable is exogenous to the model.10

The aggregate supply function of gold output in the noncommunist world is necessarily an amalgam of producers in various countries subject to different geological, technological, and institutional conditions. The aggregation problem, however, is not as serious as it appears; while Fells (1975) provides data on 18 gold producing countries, between 1968 and 1974 South Africa produced almost 80 per cent of total gold output in non-CMEA countries. With such proportions, it seems not too crude a simplification to consider the South African case as the norm and to assume that effects of deviations are randomly distributed about a zero mean.

The supply of gold from South Africa is not strictly the result of myriad profit-maximizing decisions of small mining companies, but rather the outcome of conscious national policy governing the mining industry.11

This policy—based presumably on a low rate of time discount and the expectation that future gold prices will be at least as high as those at present—seeks to ensure the longevity of the mining industry by the substitution of lower for higher grades of ore when gold prices rise. An agreement between mining companies and the authorities constrains the companies to mine to their average grade of ore reserves. Included in ore reserve calculations are only those that, at prevailing costs and prices, are marginally profitable. As a consequence of this agreement, an increase in the gold price (or decrease in mining costs) by lowering the minimum profitable grade of ore increases calculated ore reserves and decreases the average grade of ore mined. Writing g for grade, P for price of gold, and C for the costs 12 of ore extraction, this constraint may be specified linearly as

g=α+θC¯βPθ>0;β>0(2)

In the short run, the quantity of ore extracted (ϕ) is fixed by capacity constraints that, given the high capital intensity of gold mining, can be adjusted only very slowly. Writing the equation for output as

Pr=gφ¯(3)

it is immediately evident that, from equations (2) and (3),

Pr=φ(α+θC¯βP)(4)

This negative relation between output and price is further complicated by the tax-subsidy system where high profits are penalized, low returns subsidized.13

While price data are good and the negative correlation of price and output is well determined, cost data are poor and problematic. For various reasons,14 the effects of costs—measured by any general index—on output may be poorly determined and cannot be decided a priori.

demand for gold

The demand for gold (Gd) can be divided into two major components: the industrial use of gold (I), and (speculative) net hoarding demand (H).15 Therefore, we have

Gtd=It+Ht(5)

The industrial demand for gold is a derived demand and, as such, depends on both the output of those final goods for which gold is an input and the price of gold, as well as the prices of substitutes for gold in their production. The total industrial demand consists of three distinct categories and a residual: (a) demand in jewelry fabrication, (b) demand in electronics, and (c) demand in dentistry.16 It is, however, impossible to find disaggregated data for each of these categories on a global basis. Furthermore, the range of gold substitutes—broad and ever growing—varies across these different users.17 Clearly, indices of output and of substitute prices will have to be used. It is postulated, therefore, that the industrial use of gold is dependent on an income variable, together with the price of gold and that of alternative products. That is,

It=F2(Y¯,P,P¯a)(6)

where Y is world income, Pa represents the price of substitutes, and I and P are as defined before.18 It is expected that the industrial use of gold is correlated positively with Y and Pa, but negatively with P.

The speculative net (hoarding) demand—not the stock of demand for gold by private hoarders but rather their demand minus their supply, that is, their stock adjustment—depends upon portfolio choices.

Let ST be the stock (in metric tons) of private financial gold holdings. We assume that a conventional portfolio asset demand function is applicable to the speculative demand for gold,19 and write thus, in nominal terms,

PtSTt=F3(i¯,π¯,P˙e)w¯t(7)

Equation (7) states that some proportion of nominal wealth (w) is held in gold. This proportion depends upon the interest yield on competing financial assets (i), the rate of appreciation on competing real assets (π), and the expected rate of appreciation of gold {P˙e=100(Pt+1ePtPt)}.20

This may be rewritten in real terms, that is, deflated by the world consumer price index (Pc), as

PtSTtP¯ct=F3(i¯,π¯,P˙e)w¯tP¯ct

or,

ST=F3(i¯,π¯,P˙e)w¯tP¯ctP¯ctPt(8)

Letting w¯tP¯ct=W¯t and P¯ctPt=Rt and assuming a linear functional form for F3, we have

ST={γ0+γ1i¯+γ2P˙e+γ3π¯}W¯Rγ00;γ1,γ3<0;γ2>0(9)

By differentiating equation (9) with respect to time, and dividing through by WR, we arrive at a net hoarding function

(HW¯R)t=γ1Δi¯t+γ2ΔP˙te+γ2Δπ¯t+γ0(ΔW¯W+ΔRR)t+γ1it(ΔW¯W+ΔRR)t+γ2P˙te(ΔW¯W+ΔRR)t+γ3πt(ΔW¯W+ΔRR)t(10)

where H = ΔST and Δ is the first-difference operator. In this equation, there are both stock adjustment and flow effects. Stock adjustment is expected to dominate through the first three terms. The other terms take account of flow effects, that is, the growth of wealth over time and the assumption that some proportion of this wealth will be held in gold. This flow effect is a priori expected to be very small, since the flow of new savings is minute relative to the stock of wealth.

expectations formation function

The expected price of gold plays an important role in determining the net hoarding demand function. It is, therefore, essential to postulate the mechanism under which the expected price of gold is formed. For this purpose, let Pt+1e be the price of gold that wealth holders expect to prevail one period ahead (t + 1) of the current period (t).

It is hypothesized that portfolio goldholders see the gold market as primarily a market for a financial asset. They expect therefore that arbitrage will ensure a return on gold similar to that on alternative assets of similar risk. In the medium term, therefore, they see gold prices appreciating at a rate (r) equal to the rate of interest on alternative assets, which may be characterized as

Δlog(Pt+1e)=rt(11)

Insofar as these expectations are likely to be frequently incorrect, because of changes in the commodity value of gold or changes in the risk of gold-holding, they are corrected by a simple adaptive expectations mechanism. The expectations generating mechanism may thus be written:

Pt+1e=Pte(1+r)+η(PtPte)(12)

where 0 ≤ η ≤ 1.

equilibrium condition

The model is closed by the equilibrium condition for the demand and supply

Gts=Gtd

or,

Prt+Z¯t+B¯tItHt=0(13)

Thus, the model consists of five endogenous variables (Pr, I, H, P, Pe), which are uniquely determined by five equations (4), (6), (10), (12), and (13). The complete model is summarized in Table 1; a list of the variables is supplied in Table 2.

Table 1.

The Gold Market: Complete Model1

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The partial derivative of the ith function with respect to the jth argument is denoted, fi,j

Table 2.

List of Variables

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Various alternative proxies to those employed here are possible. Out of the large set of possible proxies, we have started here by using those most easily available as world aggregate variables. Clearly they are imperfect, but they may be as justifiable as most others. In any event, simulation results presented in Section III leave us skeptical of the value of increased sophistication.

It is implicitly assumed, by the use of an index of world real cash balances as a proxy for both Y and W, that the demand for money is a fixed proportion of wealth, and that wealth and income bear a constant relation to one another.

workings of the model

In order to elucidate the working of the model, one might consider the effects of changes in some of the exogenous variables. Consider, for example, a once and for all sale by the Union of Soviet Socialist Republics of part of its gold stock in the private market (i.e., Z¯>0). The stock supply schedule (STs; see Figure 1) would shift rightward, decreasing the stock market-clearing price. Various additional effects would follow immediately. First, as the price of gold declined, the price expected by the public to prevail in the following period would be revised downward.21 This would shift the stock demand curve leftward. Second, the decreased price would elicit a greater flow demand from industry, as they substituted gold for alternative inputs, and a greater production supply, owing to an improvement in the grade of ore mined. The increase in industrial flow demand would exceed the increase in production flow supply (as is clear from Figure 1), and flow disequilibrium would put upward pressure on the price.22 Full equilibrium would require either a stronger rightward shift in the stock demand curve or absorption of the additional quantity supplied over time by the excess flow demand.

III. Estimation Procedures and Empirical Results

This section discusses estimation procedures and presents estimation results for behavioral equations and in-sample dynamic simulation results for the model as a whole. The first subsection notes various important factors for estimation purposes. The second subsection examines each behavioral equation and pays special attention to the relative price elasticity of supply and demand functions. The third subsection considers whether the estimated model as a whole is reasonable in representing the actual gold market, and the fourth discusses problems in using the model for forecasting.

estimation procedures

The model developed in Section II has been modified slightly to take account of some special factors. The output function (equation (4)) and the equation for the industrial use of gold (equation (6)) may be specified in either simple linear or nonlinear form. Since there is no a priori information on the appropriate specifications of the functional forms, the choice is simply an empirical question. Log-linear specifications have been chosen on the basis of goodness-of-fit criteria.

The hoarding demand equation (10) has been rearranged and expressed as a price equation. The inversion of the hoarding function was motivated mainly by pragmatic considerations. We are concerned primarily about prices; we would prefer to minimize the residual sum of squares in price space. The theoretical model does not argue against such an inversion. The model is an equilibrium model in that equilibrium in the private gold market as a whole is obtained in each observation period. What is important to note here is that whatever the supply of gold, someone will hold it of his own volition. Thus, price and expected price adjust so as to make the net absorption by hoarders equal to the excess supply of gold over industrial demand. We could estimate the motion of the price adjustment by a difference equation, Δe = f(H). However, since we have developed the hoarding function explicitly, we can invert it for the same effect. Therefore, we may write the equation as

RELP=1γ2SPECγ1γ2INTγ3γ2INFL+ARPL+100γ2γ0γ2WLTH(14)

by noting that

P˙te=100(P˙t+1ePtPt)(15)

and defining composite variables

RELP=100(P˙t+1e/Pt)(16)
SPEC=H/[(1+ΔW¯W¯+ΔRR)W¯R](17)
INT=[Δi¯+i¯(ΔW¯W¯+ΔRR)]/(1+ΔW¯W¯+ΔRR)(18)
INFL=[Δπ¯+π¯(ΔW¯W¯+ΔRR)]/(1+ΔW¯W¯+ΔRR)(19)
APRL=100(Pte/Pt1)/(1+ΔWW+ΔRR)(20)
WLTH=(ΔW¯W¯+ΔRR)/(1+ΔW¯W¯+ΔRR)(21)
R=P¯c/P(22)

Notably, all the structural parameters are uniquely identified.

In estimating the net hoarding demand function as a price equation (14), we have used several alternative expected price level series (Pe) that were generated by equation (12), with different values assigned to the parameter η. The series that minimized the residual sum of squares of estimates was that setting η = 0.2.23 The results of the estimated equations by the two-stage least-squares method are summarized in Table 3.24

Table 3.

Regression Results, Second Quarter 1968-Fourth Quarter 1974 1

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Figures in parentheses are t-ratios.

estimation results

The output equation for gold—(i) in Table 3—seems plausible. As expected, the price elasticity of this supply function is estimated to be negative and very inelastic. The high t-ratio (about 3) of the estimated coefficient (−0.1) indicates that it is significantly different from zero. The estimated coefficient of the variable (C) is negative but statistically insignificant, and may be regarded as zero.

The estimated equation for the industrial use of gold exhibits the expected signs for all the explanatory variables. The income elasticity is found to be about 0.6, while the price elasticity is estimated to be about 0.7 with a high statistical significance.

All the estimated coefficients in the hoarding demand function have the right signs, even though the statistical significance varies considerably among them. Since the structural parameters are uniquely identified, it is possible to derive these and their t-statistics. These are as follows:

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All parameters, except the constant term, are of expected sign and generally not implausible.25 Rewriting equation (iii) in Table 3, so that stock adjustment (H) is expressed in terms of other variables, yields

Ht=(0.93Δi¯t+0.25ΔP˙te0.70Δπ¯t)(w¯P)t*+(10.320.93i¯+0.25P˙te0.70π¯t)Δ(w¯P)t*

where (w¯P)t*=10,000(w¯P)t

Since the dependent variable (Ht) is a net stock adjustment, the coefficients given above should not be taken for stock demand elasticities. They do, however, clearly indicate that net stock adjustment can fluctuate widely with small changes in the independent variables.

On the basis of these empirical findings, we may consider the effects of a change in one of the exogenous variables on the market price and quantities. Suppose, starting from equilibrium, there is a sudden once and for all increase in stock supply. This is represented (in Figure 2) as a shift of ST from ST0s to ST1s A number of additional effects follow immediately: price falls from P* to P1, the new stock market equilibrium price; STd shifts slightly inward, as the impact effect of the price decline alters price expectations, accentuating the price decline slightly (to P2); and the market registers an excess flow demand with upward pressures on the price. In the interregnum, excess flow demand will drive up prices and, at any price between P* and P2, will absorb part of the stock increase. Stock-flow equilibrium can be re-established either by the increase in supply being whittled away by excess flow demand or by a shifting outward of the stock demand curve. Given the parameter values of the model, only the former can occur. The initial fall in price decreases price expectations slightly

dPt+1e=ηdPt=0.2dPt<0ifdPt<0

and shifts STd inward as described earlier.

Thus, abstracting from other changes in exogenous variables, equilibrium will be restored by the absorption of excess stock by excess flow demand over time. The price effects of the disturbance and rate of adjustment depend—as one might intuitively expect—upon the relative magnitudes of the disturbance and the excess flow demand. This question will be addressed in the simulation exercise of the following section.

simulation exercise

The importance of dynamic simulation in testing the validity of a model as a whole is well known. In order to examine how well the estimated behavioral equations, together with identities, describe the historical movements of the endogenous variables, the model has been subjected to dynamic simulation for the entire sample period.26 Some of the results of this simulation are presented in Charts 14.

Chart 1.
Chart 1.

Actual and Simulated Values: Net Hoarding Demand for Gold, 1968–74

(In metric tons)

Citation: IMF Staff Papers 1977, 001; 10.5089/9781451956443.024.A002

Chart 2.
Chart 2.

Actual and Simulated Values: Industrial Demand for Gold, 1968–74

(In metric tons)

Citation: IMF Staff Papers 1977, 001; 10.5089/9781451956443.024.A002

Chart 3.
Chart 3.

Actual and Simulated Values: Output of Gold, 1968–74

(In metric tons)

Citation: IMF Staff Papers 1977, 001; 10.5089/9781451956443.024.A002

Chart 4.
Chart 4.

Actual and Simulated Values: Average Price of Gold, 1968–74

(In U. S. dollars per fine ounce)

Citation: IMF Staff Papers 1977, 001; 10.5089/9781451956443.024.A002

A casual look at Chart 1 indicates quite clearly that almost every turning point in the net hoarding demand for gold has been captured by the model, and differences between the actual and simulated values are very small, compared with the variance in historical values.

From the second quarter of 1968 until the end of 1972, industrial demand fluctuated around a mean value of 330 metric tons per quarter. While the model tracked these fluctuations quite well, it tended to lead or lag actual movements over different subperiods. The major turning point at the beginning of 1973 was picked up after a lag of one quarter. From the second quarter of 1973 until the end of 1974, actual and simulated values fluctuated around a mean of 195 metric tons. Despite leading or lagging actual values and missing some minor turning points, reference to Chart 2 indicates a fairly good tracking performance with a root-mean-squared-error equal to 5.53 per cent of the mean value.

Tracking performance of output—as illustrated in Chart 3—tells much the same story. Despite some mistiming of turning points and missing some of the minor turning points, the overall performance is excellent, with the root-mean-squared-error only 1.56 per cent of the mean of observed values.

The model simulates price developments over the period surprisingly well, considering some of the extraordinary influences on the market during the period that could not be incorporated explicitly. From the second quarter of 1968 until the fourth quarter of 1971, minor fluctuations in observed values are accentuated by the model. From 1972 until the end of 1974, despite missing a minor turning point, the tracking performance is excellent. Over all, the root-mean-squared-error of the simulation is 7.12 per cent of the mean of observed values, that is, less than US$5.00.

The preceding examination of the results obtained by the dynamic simulation clearly shows that the model as a whole is quite reliable in representing the gold market in the real world over the period considered.

problems of projection

While the model presented here could, of course, be used for projection, there are a host of reasons why such an exercise might be hazardous. First, econometric forecasting is always based on an assumption of behavioral similarity between the estimation and the forecast periods. Current debate as to the role of gold in the monetary system, however, forebodes structural change in the market.

Second, while the model was estimated over a sample period characterized by a rising gold price, events in 1975 halted the trend. For forecasting purposes, it would have to be assumed that all estimated coefficients in the model are symmetrical for negative and positive changes in the variables to which they are attached. It may be argued, however, that given resources devoted to developing gold substitutes over a long period of rising gold prices, the price elasticity of industrial demand might be lower in a subsequent downward price trend. Our sample period data allows no testing of this asymmetry hypothesis.

Finally, prediction requires—and depends critically upon—informal forecasts of exogenous variables. These rough estimates are impossible to make with much confidence.

For all these reasons, the model is better regarded as a simple formalization of our current stock of knowledge than as a device for looking into the future.

IV. Summary and Conclusions

Two prevalent views of the world gold market are discussed in the paper: that of an ordinary commodity flow market in which gold is traded for its intrinsic commodity value, and that of a financial stock market where gold is traded as a financial asset. This paper integrates these views by separating industrial demand for gold from portfolio adjustment and specifying behavioral equations for each. A price-expectations generating mechanism, required by the portfolio adjustment equation, is added. On the supply side, mine production is endogenized by a simple supply function. The model is closed by a market-clearing condition.

The econometric model estimated is not much modified from the suggested specification. Five endogenous variables are determined: price, output, industrial use, and speculative net hoarding demand for gold, as well as the price expected in the following period. The model is estimated by the two-stage least-squares method for the period from the second quarter of 1968 through the fourth quarter of 1974, and the estimated model subjected to a dynamic simulation for the sample period in order to test its tracking performance. The simulation results indicate that it is quite plausible in representing historical developments in the gold market.

The model presented here is very simple. It is, we believe, the simplest possible model capable of explaining a market consisting of both commodity flows and financial stock adjustments. A justification for further elaboration would, of course, depend on the incremental richness of results.

APPENDICES

I. A Note on the Supply Curve

A typical analysis of a firm’s supply schedule for an exhaustible resource, such as gold, starts with the behavior of a firm trying to maximize the present value of its net income (profit) stream over time.27 Hotelling (1931), Goldsmith (1974), and Weinstein and Zeckhauser (1975) all have shown that a firm, whether it is a monopoly or a competitive firm, will produce at a rate such that the marginal profit (the marginal revenue minus the marginal cost) is growing at the same rate as the firm’s discount rate.28 The same conclusion is obtained for a cartel by Salant (1976, p. 1080).

As a simplification, every … “plant” is assumed to have the same oil stock and cost function. In the absence of a cartel, each plant is owned by a different firm. A cartel is said to form when more than one plant comes under control of the same firm. The cartel is then in the distinctive position of owning larger reserves than any other firm. Moreover, if the marginal cost function of each plant is upward sloping, the cartel can extract at the same rate as another firm for a smaller cost.

These two distinguishing characteristics give the cartel market power over every other firm. If it is assumed that the rest of the world’s oil stock is divided equally among a sufficiently large number of other extractors, a dominant firm model results. In such a case, each small firm acts like a competitor which takes as given the price path set by the cartel and chooses a sales path to maximize the sum of discounted profits. The cartel takes the sales path of the “competitive fringe” as given and chooses a price path—supported by cartel sales—to maximize its discounted profits.

The quotation from Salant’s work seems applicable, in a highly idealized way, to an analysis of the world gold mining industry. Output of South African gold mines is determined not by considerations of individual mining companies but by an overall national policy. The group of South African companies, acting in unison under regulation of the Government, may be regarded as a cartel. Numerous small gold mining companies in various other countries constitute the “competitive fringe.” Thus, one would suppose, a dominant firm model emerges.

While this characterization of the South African gold mining industry seems theoretically plausible, it proves entirely incapable of explaining observed price and output patterns. This implies that the objective function is somehow incorrectly specified: first, perhaps, because the Government does not consider that it exercises monopolistic price-setting power, because of the hoarded gold stock and its potential effect on price; and second, because the Government puts a special emphasis on the stability of its foreign exchange earnings from gold. This does not necessarily imply a conflict of interest between mining companies and the authorities. While such a conflict might arise from divergent intertemporal rates of discount or different expectations regarding future prices, the evidence is that the industry itself puts a special premium on stability of profits and dividends.

The robust negative correlation between price and output can, however, easily be explained by explicit reference to the constraints imposed by the authorities on the mining companies. Indeed, as is shown later, in the short run the constraints are so severe that, given costs and prices, the output of each mining company is determined by these constraints and supported by the structure of taxation.

A long-standing agreement between mining companies and the authorities constrains the companies to mine to their average grade of ore reserves.29 Ore reserves considered are only those that, at prevailing costs and prices, are marginally profitable—that is, the marginal cost of gold extraction is below the marginal revenue derived. As a result of this agreement, any increase in the gold price (or decrease in mining costs), by lowering the minimum profitable grade of ore, increases calculated ore reserves and decreases the average grade of ore mined.30 Writing g for grade, P for price of gold, and C for costs of extracting ore (not gold), we may specify this constraint as:

g=f1(P,C)wheref11<0;f12>0(1)

or, linearly, as,

g=θCβP(1)

The tax rate on profits of gold mines is progressive and depends positively on the ratio of taxable profits to total revenue.31 Denoting J as the tax rate and Q as units of gold output, we may write:

J=f2(1C/QP)=f2(C,P,Q)(2)

where f21 < 0; f22 f23 > 0

or, linearly, as,

J=δQ+ηPγC(2)

The quantity of ore extracted (ϕ) is fixed in the short run, by capacity constraints that, given the high capital intensity of gold mining, can be adjusted only very slowly. This leads to a simple relationship between quantity and grade, where

Q=φg(3)

In the case under consideration, output of ore is fixed in the short run, and gold production is determined entirely by the agreement between mining companies and the Government (equation (1)). A simple, static, profit-maximization problem will serve to illustrate.

Combining the after-tax profit function:

Π=(1J)(QPC)(4)

with constraints (1′), (2′), and (3), we get a Lagrange equation

L=(1J)(QPC)+λ1(J+γCδQηP)+λ2(Qφg)+λ3(g+βPθC)(5)

Clearly, any attempt to find profit-maximizing Q results in an overdetermined system of equations.32 Output (Q) is determined uniquely from the latter two constraints.

Q=φ(θCβP)(6)

From equation (6) we would thus expect a supply negatively related to price and positively related to cost. Costs, however, are a somewhat problematic concept in the context of South African gold mines. While the large majority of mine employees are recruited nonunionized labor, “skilled” employment is monopolized by domestic unionized labor.33 Differential pay scales are entrenched by custom and “job reservation” law.34 Any increase in the employment ratio, resulting from a relaxation of job reservation, would lower wage costs (independent of the general wage level in the economy) and, by equation (6), decrease output.35 Furthermore, since most mine labor is not unionized, it is often argued that wage adjustment depends upon the marginal revenue product and the discretion of a monopsonistic employer. On this argument, wage costs would be correlated positively with gold prices and negatively with output.36 Finally, reductions in costs that are due to improved techniques for extraction and treatment of ores (for example, the cyanide process) are unlikely to be captured in any economy-wide index of costs,37 but would decrease output. For these reasons, the effects of costs—measured by any general index—on output cannot be determined a priori.

II. Data Sources and Derivations

The data for the variables listed in Table 2 were obtained from the following sources:

A. Bank for International Settlements, Annual Report, various issues.

B. Consolidated Gold Fields, Limited, Annual Report, various issues.

C. Deutsche Bundesbank, Monthly Report of the Deutsche Bundesbank.

D. International Monetary Fund, Data Fund.

E. Wall Street Journal.

Endogenous variables

(1) Pr: Output of gold (in metric tons).

Quarterly data for a given year are derived from the total free world annual data by the following formula:

Pri=Pr×(Σj=110PRij/Σj=110Pri)

where i = the ith quarter; j = the jth country. The countries are Australia, Canada, Colombia, Ghana, India, Japan, Mexico, Nicaragua, the Philippines, and South Africa. The aggregate output of these countries represents about 90 per cent of the total free world output for the period 1968–74. Sources: Pr, B; Prij D.

(2) I; Industrial use of gold (in metric tons).

Quarterly data are estimated from the annual data by the following formula:

Ii=I×EXP(Σj=115wilog(IPij))/EXP(Σj=115wilog(IPi))

where the subscripts are the same as in (1), I = industrial use of gold, and IP = industrial production indices. The countries represent 15 major industrial users,38 including Belgium, Canada, France, the Federal Republic of Germany, Greece, India, Italy, Japan, Mexico, the Netherlands, South Africa, Spain, Switzerland, the United Kingdom, and the United States. The jth country’s weight, represented by wj, is based upon its average share (1968–74) in the sum of industrial use over these countries. The weights are as follows:

article image
Sources: I, B; IP, D

(3) H: (Speculative) hoarding demand for gold (in metric tons). Quarterly data are obtained from equation (13).

(4) P: Quarterly average spot price of gold in London market; U. S. dollars per fine ounce. Quarterly average price is obtained by taking the mean of the daily prices for the quarter. Source: E

(5) Pe: Expected price of gold one period ahead; U. S. dollars per fine ounce. It is generated by equation (12).

Exogenous variables

(6) Z: Net sales of gold by the People’s Republic of China and CMEA countries (in metric tons).

Annual data on Z is available from annual reports of the BIS and Consolidated Gold Fields, Limited. Quarterly interpolations were done on the basis of information in Pick’s Currency Yearbook, and reports of London bullion brokers. Sources: A, B

(7) B: Decumulation of the stock of gold held by central banks and official international organizations.

It is obtained by the negative of the first differences in the stock of gold held by the organizations at the end of the period (B*). It is obtained as follows:

B*=B^/(ER×35×32,150.75)

where B^ is the stock of gold in U.S. dollars; ER is the exchange rate (expressed in U. S. dollars per SDR); 35 represents the SDR value per fine ounce; 32,150.75 is the number of ounces per metric ton. Source: B^,D

(8) Pa: Prices of alternative industrial inputs to gold (proxied by an index of consumer prices, expressed in U. S. dollars, in the major gold consuming countries), 1970 = 1.00.

Pa=Exp(Σj=118ajlog(PcjEj))

where aj, Pc, and Ej are, respectively, the jth country’s weight, the consumer price index in local currency (1970 = 1.00), and an index of the exchange rate (the jth country’s currency per U.S. dollar, 1970 = 100). The following weights (aj) are obtained from the share in the aggregate gross national product in 1970 (expressed in U. S. dollars) for the 18 countries:

article image

(9) C: Index of costs of inputs (proxied by an index of consumer prices, expressed in U. S. dollars, in the major gold producing countries). It is derived by the following formula:

C=Exp(Σj=17cjlog(PcjEj))

where cj is the jth country’s weight, and Pc and E are the same as defined in (8). The weights are obtained by the share of output in 1970 among the following 7 countries:

article image

(10) Y: Index of world income, proxied by an index of world real cash balances (Mr).

Mr = Mw/Pc, where Mw and Pc are indices of world nominal money supply (narrow) and consumer prices, respectively. Sources: Mw, D; Pc, D

(11) W: Index of real wealth in the world, proxied by an index of world real cash balances (Mr).

(12) w: Index of the world nominal money supply (narrow) = Mw. Source: Mw, D

(13) Pc: Index of the world consumer prices. Source: Pc, D

(14) i: Equilibrium interest rate on three-month Euro-deutsche mark deposits. Use of the uncovered Euro-dollar rate as the opportunity cost for gold holders may be erroneous for various reasons. Crises of confidence in the dollar lead, at once, to a flight into gold and to high “overnight” Euro-dollar rates in the scramble for credit to buy other assets. In addition, of course, any expected depreciation of the dollar raises, simultaneously, the Euro-dollar rate and the forward discount on dollars. The rate, i, is calculated from the interest parity theorem:

1+i=FRSR(1+r)

where r is the three-month Euro-dollar deposit rate; FR and SR are the forward and spot exchange rates (quoted in terms of deutsche mark per U. S. dollar), respectively. The equilibrium deutsche mark rate was chosen as the most reliable proxy for opportunity cost. The actual Euro-deutsche mark rate was, however, found to be only negligibly different from the calculated equilibrium on quarterly averages, except in times of crisis. Sources: i, C; FR, C; SR, C; and r, E

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*

Mr. Lipschitz, economist in the Asian Department, is a graduate of the London School of Economics and Political Science.

Mr. Otani, Senior Economist in the Asian Department, is a graduate of the University of California at Berkeley and the University of Minnesota.

The authors wish to thank, without implication, colleagues in the Fund for useful comments on earlier drafts of the paper. An earlier version was presented at the meetings of the Econometric Society held in Atlantic City, New Jersey, in September 1976.

1

Some work has been done on analyzing the commodity demand for gold; see Michalopoulos and Van Tassel (1971).

2

The two types of transaction are separated by motive, not by agent. Clearly, the same agent may act in both capacities, each motivated by a separate set of considerations. “Speculator,” in this paper, is used to refer to any transactor who buys gold not intending to use it as an input in some final good but only to hold as a financial asset in the hope of appreciation or as a hedge against capital loss. Clearly, our speculators might include arbitrageurs who buy spot while simultaneously selling forward. We have ignored this distinction because forward markets in Europe are reputedly very thin, and futures markets in North America are very recent phenomena.

3

The literature on speculation and stability is vast and often repetitive. See, for example, Kaldor (1939); Friedman (1953); Baumol (1957); and Johnson (1976).

4

The explicit functional form is given in equation (9).

5

The model suggested here is similar to the orthodox stock-flow models of investment theory, in particular those of Keynes (1936) and Lerner (1944), and the extension and correction of these models by Witte (1963). Tobin (1965; 1969) has expanded the model within an overall portfolio balance framework; Foley and Sidrauski (1971) have included a two-sector production model where the flow supply of capital may be determined by supply considerations. Both elaborations maintain the familiar dichotomy of stock and flow sectors. Our portfolio balance equation—derived later in the paper—is essentially a simple application of the Tobin-Markowitz general theory of portfolio selection, abstracting, however, from variances and covariances of rates of return on different assets. The precise response functions are derived in Section II.

6

This correlation is corroborated by many studies besides that which follows here. See, for example, Hirsch (1968); Brown (1971); Barattieri and Brown (1973).

7

Step-by-step cause and effect relations allow simple exposition. The relations here will, of course, be simultaneous, although the feedback from flows to stock adjustment may be negligibly weak. On a more metatheoretical level, however, it has been argued that causal relations are necessarily recursive and that the appearance of simultaneity arises out of excessively long observation intervals. See, for example, Wold (1953), pp. 43–53 and 64–71.

8

In Figure 1, the arrow represents the direction of response to excess demand in the flow side of the model. Abstracting from the stock side, stability in (b) depends upon the relative slopes of the demand and supply functions, according to the solution of the differential equation.

dp/dt = k{∂D/∂P (P − P*) − ∂Pr/∂P (P − P*)}

such that,

P(t) = [P(0) − P*] EXP k(∂D/∂P − ∂Pr/∂p)t + P*

Thus, stability requires that the flow demand curve be flatter than the flow supply curve. This is subsumed in Figure 1 and confirmed by the empirical results in estimation results in Section III.

9

B includes decumulation of gold stocks of official international organizations.

10

All the variables are expressed in volume terms (e.g., metric tons).

11

A more detailed account of the determinants of the supply of gold is given in Appendix I. In addition, Houghton (1964), Hirsch (1968), and Brown (1971) contain interesting information. Wilson (1972) provides, useful insights on wage costs.

12

C is merely an index of consumer prices in the major gold producing countries. It is assumed to be exogenous, and may be regarded as a proxy for the costs of capital and labor services to gold producers. These costs are determined independently of the price and quantity of gold. This is not, however, to deny the endogeneity of the cost of gold production. In fact, as can be seen very easily from equations (2) and (3), the cost of producing gold is endogenous. Alternatively, equation (4) may be regarded as a supply curve with the relative price of gold (P/C) as one of its arguments. As will be seen later, however, it is estimated without the parameter constraints implied by this view.

13

This fiscal arrangement seems to be prevalent, too, in various other producing countries. For more detail on the tax structure, see Appendix I.

15

H can take either negative or positive value. As will be shown later, when it takes a negative value, the industrial use of gold is greater than Gs, and vice versa.

16

The term “industrial” is, of course, used loosely as a catchall. It includes the manufacturing and service sectors.

17

Substitution of various alloys for gold among the latter two categories of user has received great impetus from price increases since 1971–72.

18

Pa in equation (6) is proxied by an index of consumer prices in the major gold consuming countries.

19

The functional form postulated is a standard simplification of the Tobin-Markowitz portfolio selection theory. See footnote 5 and Learner and Stern (1970), pp. 102–103.

20

The function is linearly homogeneous in wealth. This homogeneity attribute follows from the proportionality assumption given earlier. F3 (…) is homogeneous of zero degree: that is, portfolio selection is independent of whether one considers the real or the nominal return on competing assets, since all returns would be deflated by the same inflation rate.

21

This may be clarified by the following equations:

Pt+1e=Pte(1+rη)+ηPtdPt+1edPte=η

where 0 ≤ η ≤ 1

22

See footnote 8. The relative slopes of the curves ensure equilibrating price responses from the flow side of the model.

23

The procedure adopted here is analogous to the maximum likelihood estimation method. See, for example, Cagan (1956, pp. 92–97).

24

The model is nonlinear in variables, but linear in parameters. All the behavioral equations satisfy the order condition for identifiability. Hence, the two-stage least-squares method will give consistent estimators for the parameters of the system. See Kelejian (1971), Edgerton (1972), and Amemiya (1974).

25

Inference from the estimated stock-adjustment equation to the actual stock demand should not be made. The data set is too small and too unusual to be able to infer past accumulation.

26

The simulation procedure used reduced forms of the full model, consisting of behavioral and definitional equations (4), (6), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21), and (22).

27

The firm will maximize the present value of the expression

t1t2Π(Q(t))eγtdtλ{t1t2Q(t)dtK}

where π = the profit, Q(t) = the output per period, λ = Lagrangian multiplier, γ = interest rate for the purpose of discounting the future profit, K = the fixed amount of a resource whose reserves are exhausted by a certain period, say, t2.

28
That is,
MΠ(t)=MR(t)MC(t)=λertor,(ΔMΠMΠ)=γ
29

This agreement covers all mines operating under state leases. See South African Government (1960, p. 502); Transvaal and Orange Free State Chamber of Mines (c. 1968, pp. 7–8); and Hirsch (1968, p. 445).

30

Costs are defined as in footnote 12 as some general index of the costs of capital and labor services.

31

Hirsch (1968, p. 448). Tax credits to submarginal gold mines work on a similar formula, except that the ratio (QP-C)/QP may be close to zero or negative. See Government Gazette (1968) and Department of Mines (1968). Mine leasing costs are set up similarly to ensure the longevity of mines.

32

Differentiating with respect to Q, subject to the constraints, yields the over-determined system of four equations in only three endogenous variables: J, Q, and g.

33

“Recruited labor” refers to the majority of mine employees, who are recruited from the Bantustans or from outside South Africa.

34

See Houghton (1964, p. 142); Wilson (1972, pp. 10–13) and appendices.

35

See Wilson (1972), Appendix 24: “Relative Wages and Employment Ratios,” and Appendix 27: “The Colour Bar as a Tax.”

36

Legislation limiting mobility for Bantu labor distorts free market determination of wage rates in the economy as a whole. See Wilson (1972), Appendix 25: “Monopsonistic Collusion.”

37

See Hirsch (1968, p. 445).

38

Industrial use by these 15 countries represented on average (1968–74) 78 per cent of total free world industrial use. “Geometric means are used since they assure that if all of the countries of an area have constant, but different, rates of increase, the area average will have a constant rate of increase.” (Source: D; specifically, International Financial Statistics, “Introduction—Section 4. Consumer Prices and Money Supply.”)

IMF Staff papers: Volume 24 No. 1
Author: International Monetary Fund. Research Dept.