Article

A Macroeconomic Model of the United Kingdom

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1978
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Few industrial economies have been as much analyzed and commented upon in recent years as that of the United Kingdom. For much of the period since 1945, Britain has had rather slow economic growth, low investment, and recurrent balance of payments problems. Furthermore, after the introduction of a more flexible exchange rate in 1972 the U. K. economy experienced a period of rapid monetary expansion and inflation. In recommending policies for stabilizing the economy, some economists have emphasized the causal linkages running from the rate of domestic monetary expansion to price inflation, while others have stressed the effects of wage demands on the domestic price level. Each of these approaches, at least in its extreme form, emphasizes one part of the inflationary process to the exclusion of others and tends to obscure crucial elements of an advanced industrial economy. Although inflation cannot continue without monetary expansion, the money supply itself is not necessarily determined independently of the behavior of the private sector, because the authorities respond to market developments in determining the stance of their policies. A most important case in point occurs if, over time, the authorities want to maintain a target level of unemployment. In this case, attempts by workers to push the real wage above its equilibrium level may induce policymakers to raise government spending in order to maintain employment. If the monetary authorities are attempting to stabilize nominal interest rates, however, the increased fiscal deficit of the public sector will induce monetary expansion and inflation.

To understand more fully how the government and the private sector interact during the inflationary process, it is necessary to construct models that have two features. In the first place, they must include both the real and financial sectors of the economy and explain how they are linked with each other and with markets in the rest of the world. Second, these models must try to explain how the actions of the private sector lead to predictable reactions by the authorities. Only when such reactions are specified as an integral part of the system can structural parameters, even of the private sector’s behavior, be estimated consistently. Furthermore, only when the dynamic properties of a system with given reaction functions have been analyzed is it possible to suggest changes in the behavior of policymakers that would enhance the achievement of major economic goals.

The econometric model that is presented in this paper attempts to meet these requirements. Its structural equations describe the interactions between domestic residents and foreigners in markets for goods and financial assets. The real sector consists of the markets for output and labor services, and these markets are linked by an explicit production function that both constrains the long-run relation between domestic output and the supply of 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 behavior of home and foreign residents in the markets for domestic money balances, advances, government and private sector securities, and net Eurocurrency liabilities of banks resident in the United Kingdom is also specified.

In the model, the excess demands that arise in real and financial markets may lead to disequilibrium in the balance of payments at the current exchange rate. If the domestic currency is not floating freely, these changes will induce accommodating movements of international reserves that feed back into the domestic economy in many ways, particularly via changes in the domestic money supply. The model takes explicit account of the financial implications of changes in the government borrowing requirement, and it includes reaction functions for four policy instruments: taxes, government spending, Bank rate, and base money. In the United Kingdom, where it is convenient for certain purposes to aggregate the Treasury and the Bank of England into a single official sector, the latter policy instrument is best specified as the rate of acceleration in nonofficial holdings of U. K. Government debt.

While the behavior of real and financial markets has been studied intensively in previous empirical work, the last two relationships—the budget constraint of the authorities and their reaction functions—have received less attention in macroeconometric models. Yet, it is important to emphasize at the outset that these relationships are fundamental to a thorough understanding of recent developments in the U. K. economy.

While the model described in this paper is not wedded to either the neo-Keynesian or monetarist schools of economic analysis, it is grounded on extensive previous theoretical and empirical research. In particular, it builds on the work of Bergstrom (1966), Bergstrom and Wymer (1976), and Knight and Wymer (1975). Section I sets out the equations of the model and provides a condensed discussion of its basic structure. The analytical properties of the model are considered in Section II, where the steady-state solution of the model is discussed. Section III describes the estimation of the model and presents the empirical results, including point estimates of the behavioral parameters and in-sample forecasts. The stability of the model is considered in Section IV, and the implications for the authorities’ policy reaction functions are analyzed. Some conclusions of the study are summarized in Section V.

I. The Structure of the Model

The model specifies the activities of four sectors: the private sector, the banking system, the monetary and fiscal authorities, and foreign residents. The private sector is assumed to produce a homogeneous domestic good and to use the income from production together with funds from foreign borrowing to purchase domestic or foreign goods, domestic securities, or money balances. Banks resident in the United Kingdom finance their domestic advances and security holdings by accepting deposits or by increasing their net Eurocurrency liabilities. The Central Government finances its spending by taxation or by issuing securities, while the monetary authorities set the Bank rate and determine the proportion of government stock that is to be held outside the official sector. Foreigners purchase U. K. exports and provide its imports as well as holding government bonds, claims on the U. K. private sector, and Eurocurrency deposits.

The interest rate and price level prevailing in world markets are assumed to be exogenous to the model. This implies that in long-run equilibrium U. K. interest rates and prices are determined by their levels in world markets. But sectors and markets are not constrained to adjust to changes instantaneously. The model is a partial adjustment system composed of demand and supply functions, price and quantity adjustment functions, and wealth and market identities. Since the structure of markets is often such that a few sectors have a predominant effect on prices and quantities, adjustment functions are assumed to respond to only one or two excess demands. This implies that the rates at which domestic output, prices, expenditure, and money balances converge to long-run equilibrium depend crucially on the speed with which various sectors and markets respond to change. In contrast to the properties of the steady-state solution of the model, which are consistent with macroeconomic theories of full equilibrium and should hold for virtually any developed economy, the dynamic structure of the model is specific to the particular country under consideration. And since governments are usually concerned with short-term or medium-term targets, it is of the utmost importance for them to understand how the economy responds to various changes if they are to operate their policies in a stabilizing manner. The model described in this paper has been designed with this purpose in mind, and the empirical results in Section III provide a great deal of information about the short-run behavior of the U. K. economy.

Before describing the model in detail, it is important to discuss one last, but crucial, assumption. In the model, the U. K. exchange rate is taken as an exogenous policy variable in that the authorities are assumed to determine a target level of sterling on the foreign exchange market, and to use intervention policy to achieve this target. There is no problem with this assumption for most of the period over which the model is estimated, when the U. K. exchange rate was fixed. 1 In June 1972, however, the U. K. authorities ceased to declare a par value for sterling. The scale and direction of intervention since that time indicate that the U. K. authorities have generally sought to maintain sterling at some target level. Although the model could be modified to take account of managed floating, it was decided not to attempt to make the exchange rate endogenous at this stage because of the difficulties in estimating a model of this complexity using data from both fixed and flexible exchange rate periods. 2 Given the target exchange rate, however, the model developed here could be used for analysis in the period since June 1972, provided that the private sector’s behavior has not been altered by the elimination of the par value for sterling, an assumption that seems reasonable for such a model.

The variables of the model are defined in Table 1, and its equations are presented in Table 2. Unless otherwise noted, all output and expenditure variables are defined in real terms, while variables that represent financial holdings are valued in money terms. Many of the behavioral equations are linear in the natural logarithms of the variables, so that they have constant elasticities, but expressions derived from the production function are nonlinear. Some functions in the stochastic equations and most identities are linear in variables. The model is specified in continuous time as a system of stochastic differential equations. For economy of notation, the error terms are omitted and the model is described in deterministic form.

Table 1.United Kingdom: Variables of the Model
endogenous
Output and expenditure variables
C = real consumption expenditure of the private sector
E = real exports of goods and services
K = stock of fixed capital
k = proportional rate of increase of fixed capital
G = central government expenditure on current goods and services
I = real imports of goods and services
Y = net domestic product and income
V = inventories of goods and work in progress
L = employment
p = prices
w = money wage rate
Financial variables
R = official reserves of gold and foreign exchange
M = domestic money stock (commercial bank deposits)
B = total stock of government bonds excluding official holdings
b = proportional rate of increase in volume of government bonds
H = liquid assets held by commercial banks
Bb = commercial bank holdings of government bonds (market value)
Bp = government bonds held by the private sector (market value)
Bf = foreign holdings of domestic government bonds
N = domestic banks’ Eurocurrency loans to foreign residents
A = domestic banks’ advances to private sector
T = tax receipts less transfers (valued in money terms)
S = net liabilities of private sector to foreign residents
r = market yield on long-term government bonds
r0 = central bank discount rate
exogenous
F = domestic banks’ Eurocurrency deposits
rf = foreign interest rate
є = exchange rate (price of foreign currency in terms of domestic currency)
q = price of foreign output in terms of the foreign monetary unit
t = time
Table 2.United Kingdom: Equations of the Model1
DlogC=α1log(Cd/C)+α2log(Md/M)(1)
where
Cd=γ1eβ5(rDlogp)(YT/P)(1’)
and
Md=pγ2rβ6rfYβ7(1”)
DlogE=α3log(Ed/E)(2)
where
Ed=γ3(pϵq)β1eλ3t(2ʹ)
Dk=α4{γ4[β3(Y/K)1+β4r+Dlogp]+γ5k}(3)
DlogG=α18log(Gγ14Y)+α19log(Lγ15eλzt)+α20Dlog(Lγ15eλzt)(4)
DI=α5{γ6(pϵq)β8(C+DK+E+G)I}+α6{γ7(C+DK+E+G)V}(5)
DY=α7{[1γ6(pϵq)β8](C+DK+E+G)Y}+α8{γ7(C+DK+E+G)V}(6)
Dlogp=α9log{γ8β2weλ1t[1β3(Y/K)β4](1+β4)/β4p}(7)
DlogL=α10log{β2eλ1t(Yβ4β3Kβ4)1/β4L}(8)
Dlogw=α11log{β2eλ1t(Yβ4β3Kβ4)1/β4γ9eλzt(wpeλ1t)β9}(9)
Dlogr=α12log(Md/M)(10)
DlogS=α13log(Md/M)+α14log(Bpd/Bp)(11)
where
Bpd=pγ10rβ10rfβ11Y(11ʹ)
DlogBb=α15log(Hd/H)(12)
where
Hd=γ11r0β12M(12ʹ)
Dlog(N/F)=α16log{rβ13rfβ14(Nγ12F)}(13)
DlogA=α17log(A8/A)(14)
where
A8=γ13r0β15rβ16M(14ʹ)
Db=α21log(Rγ16M)+α22Dlog(Rγ16M)+α23log(Lγ15eλ2t)+α24Dlog(Lγ15eλ2t)(15)
Dlogr0=α23log(rγ19rf)+α26log(Rγ16M)+α27Dlog(Rγ16M)(16)
DlogT=α28log(Tγ17pY)(17)
DlogBf=α29log(Bfd/Bf)(18)
where
Bfd=ϵqγ18(r/rf)β17eλ3t(18ʹ)
DlogK=k(19)
DlogB=b(20)
DV=Y+1CDKEG(21)
DR=pEpI+DFDN+DBf+DS(22)
DM=pG+pEpIT+DADBp+DS(23)
DB=pGTDH+DR(24)
H=M+FBbNA(25)

Xd refers to the partial equilibrium demand for variable X, and r refers to its desired supply. All variables are defined at time t, and D is the differential operator d/dt.

Xd refers to the partial equilibrium demand for variable X, and r refers to its desired supply. All variables are defined at time t, and D is the differential operator d/dt.

Adjustment parameters are denoted by α, long-run elasticities by β marginal propensities or other desired or partial equilibrium ratios of variables by γ, and exogenous growth rates by λ, where:

  • λ1 = rate of technical progress

  • λ2 = rate of growth of the labor force

  • λ3 = rate of growth of foreign real income

  • λ4 = rate of growth of foreign volume of money

The a priori restrictions are 0 < γ1 <1, β4 >– 1, 0 < γ6 < 1, β8 < 1 and all other parameters are positive except the following adjustment coefficients: α2, α15, α16, α18, α19, α20, α25, α26, α27, α28, and possibly α21 and α22. The value of λ2 might be negative for the United Kingdom during the sample period, but the other λs are expected to be small positive numbers. Since the model set out in Table 2 is described in detail in Knight and Wymer (1975), the discussion that follows will be somewhat more impressionistic, and will stress several changes to the specification of the earlier paper that are the result of interaction between theoretical analysis and the empirical research.

expenditure, output, and the labor market

The first part of the model describes the markets for domestic and foreign output and for labor services. Following Bergstrom and Wymer (1976), the marginal products of labor and capital are derived from a production function that has constant elasticity of substitution (CES). 3 These marginal productivity terms are determinants of the demands for labor services and capital goods, so that they enter the adjustment functions for employment, wages, prices, and capital investment. The model does not require that total output be constrained by the aggregate production function at every moment, but factor utilization rates will be adjusted in response to changes in wages, prices, and the real interest rate. In the long run, therefore, the production function does constrain the rate of growth in output relative to increases in labor and capital. This way of modeling the productive sector reflects the important assumption that the aggregate production function, which assumes homogeneous units of labor and capital, is not a purely technological relationship. Rather, it indicates the amounts of labor and capital needed to sustain output at a particular level in the long run on the basis of normal rates of capacity utilization and standard working hours. More (or less) output may be obtained temporarily by putting workers on overtime (or part time), and such changes in utilization rates would not be reflected in the available time series for labor and capital. Production can, in this sense, be “off” the production function in the short run. In the longer term, however, the model assumes that real wages and employment will gradually tend to fall whenever the rate of labor utilization exceeds that given by the production function, and similarly for capital.

The model assumes that U. K. residents and foreigners transact in the markets for two composite goods: domestic output (with price p) and foreign output (with foreign currency price q). The first four equations specify the various expenditure components: consumption (1), exports (2), fixed capital formation (3), and government spending (4).

A common feature of each of these equations is that the level of spending on each component is assumed to adjust to the excess demand for it.

In equation (1) consumption is assumed to rise in response to either an excess demand for consumer goods or an excess supply of money, the latter term being included on the argument that the private sector eliminates an excess demand for money balances either by forgoing current consumption or by inducing an increase in the interest rate as in equation (10). Because the excess demand for money is homogeneous of degree zero in prices and the model is continuous, the demand for money balances and the level of these balances may be defined in either real or nominal terms. Since the model does not distinguish between the corporate and personal segments of the private sector, a decrease in consumption would not affect the excess demand for money directly, but the subsequent decrease in output and imports would decrease the demand for money and simultaneously increase the domestic volume of money. Estimates of this relationship showed, however, that disequilibrium in the money market did not have a significant effect on consumption in Britain during the sample period, and to obtain more efficient parameter estimates, this term was eliminated in the model whose estimates are given in Table 3 by setting α2 to zero. This equation was also modified by replacing real taxes in the consumption function. Since the partial equilibrium level of real taxation from equation (17) is

desired consumption in equation (1) may be approximated by

where

The export equation (2) assumes that the level of exports responds to foreigners’ excess demand for goods produced in the United Kingdom. Since short-run movements in total output are determined by equation (6), the export equation involves the implicit assumption that a rise in foreign demand for U. K. output increases the profitability of exports relative to production for the domestic market. It might seem that the specification of (2) implies that exports are determined entirely by foreign demand and that domestic supply considerations exert no influence on their level. But this interpretation is incorrect for two reasons. First, at given output the rate at which domestic producers will increase exports depends on the adjustment parameter α3. Second, in contrast to simple neo-Keynesian models in which total output is demand determined, the production function in the model constrains the level of output in the long run. Thus, any increase in exports that increases domestic output will induce a rise in wages and prices, and so limit the growth in foreign demand for home output.

Equation (3), which determines the rate of change in fixed capital formation, follows Bergstrom and Wymer (1976) in assuming that investment depends on the difference between firms’ desired proportional rate of increase in fixed capital and the actual rate, k, at which net investment is currently taking place. The desired rate of growth of the capital stock depends on a linear function of the difference between the marginal productivity of capital derived from the production function β3 (Y/K)1+β and the real rate of interest (r–D log p). The real rate of interest is the nominal interest rate minus the expected rate of inflation. As the model is not concerned with the investigation of alternative expectations behavior, the expected inflation rate is proxied by the actual current rate of change of prices, which implies that expectations are adjusted rapidly to changes in the inflation rate. This simplifying assumption is made primarily for convenience, since the introduction of even a single adaptive process raises the order of all equations where expectations appear. The final component of domestic expenditure—government spending—is determined by the policy reaction function (4) and is discussed later.

Equations (5) and (6), which determine the levels of imports and total output, are most conveniently described together. Total sales of goods and services to home and foreign residents (C + DK + E + G) are supplied from either domestic output Y or imports I with inventories acting as a buffer to absorb short-term discrepancies between demand and supply. Both output and imports are assumed to adjust to their partial equilibrium level, which for imports is some proportion φ of total sales and for output, the proportion (1–φ) of sales. This proportion, which is defined to be φ = γ6 (β ∊q)β8, varies in response to the terms of trade. Since real imports, as defined in the model, are nominal imports divided by the domestic price level, the price elasticity of demand for imports, in volume terms, is –(1 + β8). During the earlier stages of estimation, plausible values of β8 that were significantly different from zero could not be found, so β8 was set to zero. The level of inventories is a residual in the goods market, given by equation (21), but it is assumed that producers have a desired ratio γ7 of inventories to total sales and that they will increase both output and imports whenever inventories are below their desired level. Thus, the role of inventories as a buffer between supply and demand in the goods market is directly analogous to that of money balances as the residual of the private sector’s budget constraint.

The determination of domestic prices in equation (7) is a disequilibrium version of the standard short-run markup function, which says that firms tend to raise prices whenever marginal labor costs (as derived from the production function) rise relative to the current price level. The numerator of that expression, excluding γ8, is the marginal cost of producing current output with the current stock of capital. The parameter γ8 allows for imperfect competition if it is greater than unity; perfect competition implies that γ8 will be unity.

The market for labor services is given in equations (8) and (9), which describe the determinants of employment and money wages. Firms in the domestic economy are assumed to operate in conditions of monopolistic competition. At the current price level each firm regards its sales as exogenously given and attempts to maximize profits by minimizing unit labor cost, which is obtained by calculating labor input per unit of output from the production function. Thus, equation (8) states that firms will hire workers whenever actual employment is less than the amount of labor, derived from the production function, that is required to sustain output at the current level using the current capital stock.

In the wage equation (9), the numerator of the term on the right-hand side is the partial equilibrium demand for labor, while the denominator is the desired supply of labor services. Labor supply is assumed to depend on the rate of growth of the population of employable age and on the real wage rate. Thus the equation states that, at given prices, nominal wages will tend to rise whenever there is an excess demand for labor in the economy. 4 An alternative formulation would be to set β9 = 0, and to add an extra term α(D log p –ρ) to (9), where ρ is the expected long-term rate of inflation. This would imply that nominal wages rise pari passu with the rate of price inflation, but without adjustment for previous divergences in wage and price movements. The specification used here, however, is superior for a country where nominal earnings are not adjusted automatically in response to changes in the cost of living.

The U. K. financial system

The monetary and financial system of the United Kingdom is described by equations (10)–(16) and (18) and the identities (20) and (23)–(25). Banks resident in Britain 5 issue money in the form of deposit liabilities to domestic residents M; and they hold liquid asset reserves H, government bonds Bb, and domestic advances A. Their behavior as major intermediaries in the Eurocurrency market will be described later. Since the interest rate on deposit accounts in the United Kingdom is closely related to Bank rate r0, the commercial banks cannot increase their sources of funds by raising the rates they offer, so the banks’ total domestic deposits are predetermined with respect to their behavior. Given M, the size of their net Eurocurrency liabilities (as given by equation (13)) and their wealth constraint (25), the banks can freely adjust only two of the three remaining assets in their portfolio. The banking system must maintain a legal minimum ratio of liquid asset reserves (high-powered money) to domestic liabilities, and thus banks have a demand for reserve assets that depends on both the level of their total deposits and the opportunity cost of holding reserves. Equation (12) posits that they buy or sell gilt-edged stock in order to keep their reserve assets at the desired level.

The domestic private sector holds bank deposits and government securities, and it borrows from the banking system by means of advances. In Knight and Wymer (1975), it was assumed that when banks lend on overdraft, as in the United Kingdom, the level of bank advances responds to the private sector’s excess demand for loans. However, the sample period of the model covers several successive intervals of credit restraint and such conditions indicate a supply-determined specification, The empirical work provided evidence that advances responded to the excess supply of the U. K. banks during the sample period, and accordingly the specification of equation (14) has been revised to incorporate this finding.

Transactions between the U. K. private sector and the other sectors in the model are settled by transfers of money balances, and, since money balances are the most liquid assets held by the private sector, the model incorporates the assumption that these balances act as a buffer for all receipts and payments. Thus the actual rate of change in the money holdings of the private sector at any moment equals the money value of its purchases and sales of goods and securities, as given by the private sector budget constraint (23). Equation (10) assumes a partial adjustment version of the liquidity preference theory with the nominal rate of interest adjusting in response to the private sector’s excess demand or supply of real money balances. Changes in the yields in financial markets then affect fixed capital formation in equation (3) and the consumption-saving choice (1). The excess demand for money also enters the adjustment function for private sector liabilities to foreigners, thereby affecting the capital account of the balance of payments. In this way, the private sector’s excess demand for money plays a crucial role in the linkages between the real and financial sectors in the domestic economy, and also in the connections between financial markets in the United Kingdom and abroad.

the balance of payments

Transactions between domestic residents and foreigners in the markets for current output and financial assets give rise to excess demands and supplies in the foreign exchange market at the current exchange rate, and hence to movements in the above-the-line items of the balance of payments. The authorities must accommodate the excess demand or supply of foreign exchange on the part of private transactors in order to hold the exchange rate at their target level. Equation (22) states that the rate of change of official reserves DR must always equal the current account surplus (pE – pl) plus the net capital inflow (DF – DN + DBf + DS) that results from portfolio adjustments by home and foreign residents. Furthermore, the private sector’s wealth constraint (23) says that an excess demand for cash has as its counterpart an excess supply of goods, securities, or both. Such an excess supply of commodities or securities by domestic residents implies a balance of payments surplus. A current account surplus (pE – pl) > 0 occurs whenever domestic output exceeds total expenditure by U. K. residents, and the determinants of domestic output and expenditure have already been discussed. The capital account of the balance of payments is the resultant of the portfolio behavior of all four sectors in the model (authorities, banks, and domestic and foreign residents). Since the model incorporates a stock-adjustment approach to portfolio behavior, capital flows occur either as a result of growth in the size of domestic or foreign portfolios or in response to changes in the level of U. K. interest rates relative to yields abroad. In the latter case, capital movements involve a once-for-all change in the composition of asset portfolios, so that they endure only during the time that the stock adjustment is being made. Equation (11) specifies the determinants of capital flows between the U. K. private sector and the rest of the world on the assumption that home residents tend to increase their liabilities to foreigners whenever they have an excess demand for either money or domestic bonds.

The behavior of banks resident in Britain in the market for Eurocurrency deposits is specified in (13). The acceptance rate that the banks offer on Eurocurrency deposits is assumed to be determined in the world market, so that the level of U. K. banks’ Eurocurrency deposits F is also exogenous. The domestic banks adjust their Eurocurrency loans in order to secure their desired level of net Eurocurrency liabilities (FIN), and changes in the net liability position induce capital movements between the U. K. economy and the rest of the world. In the initial version of the model, foreign holdings of U. K. Government securities, B, were taken as exogenous. The addition of equation (18) allows them to be determined within the system as a response to foreigners’ excess demand for U. K. Government securities.

policy functions of the U. K. authorities

The introductory portion of this paper emphasized the importance of direct feedback from the behavior of domestic and overseas residents onto economic policy in the United Kingdom. In the model, six equations describe the policy reactions of the Central Government and the monetary authorities. Real government expenditure on goods and services G adjusts according to equation (4), while nominal tax receipts (net of subsidies and other transfers) depend on money income (17). In contrast to many macroeconomic models, the model presented here explicitly incorporates the authorities’ budget constraint. The difference between the Government’s expenditure and its tax receipts in nominal terms (its budget deficit) may be financed either by issuing government debt B or by creating high-powered money H. Thus, the public sector borrowing requirement equals spending less taxes plus the difference between the accumulation of international reserves and increases in high-powered money. This is the amount that the U. K. Government must finance by selling government debt B in the open market as given by (24). In countries where the central bank controls the rate of growth of high-powered money according to some rule, the monetary policy function would determine H. In the United Kingdom, however, the Treasury and the Bank of England work closely together to control b, the rate at which securities are issued on the open market, and thus monetary policy should be described in terms of their reactions with respect to b according to the function (15). For a given government budget deficit, an increase in b implies that less of the deficit is being financed by the creation of high-powered money. Thus, a high level of b implies, ceteris paribus, a low rate of monetary expansion, and vice versa. In addition to this “open market” variable, the model also includes the discount policy of the authorities (16).

Given the broad framework of the government sector in the United Kingdom, the determinants of the policy reaction functions may now be described. The stance of government policy depends on the authorities’ objectives with regard to such economic goals as the level of employment, the price level, the rate of growth of output, and the state of the balance of payments. First consider the determinants of fiscal policy as given in (4) and (17). Since taxes adjust merely in response to changes in nominal income, it has been assumed, in effect, that during the sample period the authorities adjusted government spending rather than making any significant use of taxing policy in operating discretionary fiscal policy. Equation (4) assumes that besides having a desired ratio of real government spending to real income the authorities have a target level of employment (γ15eλ2t), and they try to stimulate the economy by spending more whenever employment falls below the target level. In addition, however, the rate of change of the ratio of employment to its target level is also assumed to affect government spending. This reflects the hypothesis that if actual employment is below its target level and falling, government spending should be more stimulative than it would be if employment were steady at the low level.

According to this specification, government spending is devoted entirely to internal balance, whereas the reaction functions (15) and (16) for the two monetary policy instruments respond to considerations of both internal and external balance. In addition to the employment targets already mentioned, the open market reaction function responds to deviations of the ratio of foreign reserves to money balances from the level desired by the authorities. It is, however, important to understand the exact nature of this response. As equation (23) indicates, an increase in the trade surplus or capital inflows leads directly to a rise in private sector holdings of money. If the ratio of reserves to money balances rises above the level that the authorities desire, they will pursue a more expansionary policy, financing a larger proportion of their budget deficit by monetary creation. On these arguments, α21 and α22 should be negative. 6 Similarly, when actual employment is above the authorities’ target level they will tend to sell gilt-edged stock at a more rapid rate in an attempt to damp down domestic spending, so that α23 and α24 are greater than zero. The final policy equation in the model specifies that Bank rate is set in response both to movements in the ratio of reserves to money balances and to changes in the level of nominal interest rates abroad.

In Section III, the parameters of the authorities’ reaction functions are estimated along with those of other sectors. If the authorities’ targets are consistent, there will be restrictions across the policy reaction functions; for example, the target level of employment implied by the estimates of the government spending equation (4) should be the same as that in the open market equation (15). These restrictions were imposed during estimation.

II. Steady-State Properties of the Model

The model described in the previous section was designed for policy analysis in the short to medium term (that is, for periods of one or two quarters to several years) as well as for forecasting and simulation. For such analysis to be valid, the properties of the model must be consistent with the actual dynamic behavior of the economy. For example, implausible long-run behavior could indicate that an important feedback has been submitted from the model, thus limiting its usefulness.

It might seem that if the model is to be used to analyze short-term to medium-term policy problems, the question of the existence of a steady state is of little interest. But it is a well-documented fact that, whatever the short-run behavior of certain variables, they bear fairly stable relationships to one another in the long run. In virtually all industrial societies the long-run rate of real economic growth does not fluctuate widely, and the same observation holds for the ratio of consumption to income and the capital/output ratio. These regularities suggest that in developed economies variables tend toward steady growth paths in the long run. If this is so, then a properly specified model should have a plausible steady-state solution, even if the major purpose of the model is an analysis of short-term behavior. Further, analysis of the steady-state properties of the system often suggests restrictions on behavioral parameters that would not otherwise be apparent. For this reason it is useful to examine the steady state in some detail. 7

The model specified in equations (1) to (25) may be written

where Y(t) is the vector of endogenous variables, Z(t) the vector of exogenous variables, and θ the vector of parameters α, β, γ. Assuming that all exogenous variables tend to grow at a constant exponential rate, so that Zi(t)=Zi*eλit the system has a steady state if there exists a particular solution to the set of nonlinear differential equations (26) such that Yi(t)=Yi*eρit for all i. The Yi* are the steady-state levels of the endogenous variables Yi(t) when t = 0, and the ρi are the corresponding growth rates. Thus, the steady state is the cantus firmus, relative to which the variables are moving according to the laws of harmonic motion described earlier. Since the model is stochastic, it can always be displaced from the steady state by random events, but, provided that the model is stable, it will tend to return to the steady state.

Analysis of the steady state has two aspects. First, the system must allow a solution for the set of growth rates ρ, which are usually simple functions of the rates of growth of the exogenous variables λ. The model (1) to (25) incorporates assumptions 8 sufficient to yield such a solution as follows: 9

VariableSteady State Growth Rate
Lλ2
C, E, K, G, I, Y, Vλ1 + λ2
pλ4λ3
wλ4λ3 + λ1
M, T, R, B, B, B, H, A, S, Nλ4
r,ro0

The second part of the steady-state solution involves expressing the initial levels Y* of the steady-state paths as explicit functions of the parameters of the model, as well as the exogenous growth rates A and the initial levels of the exogenous variables Z*. It is then possible to derive the partial derivatives of the Y* with respect to the values of the exogenous variables and the parameters. 10

To obtain a solution for the steady-state levels, it is necessary to assume that the parameters β5, β8, and β9 are zero, so that the steady-state level of consumption does not depend on the real interest rate and the steady-state supply of labor is independent of the real wage rate. These assumptions arise from the fact that the model is not designed to explain the long-run response of the labor force to move- ments in its real income. The assumption that β9 = 0 means that in the steady state the level of employment is independent of government policy. 11 In deriving the steady state of the model, no solution could be found for values of these parameters other than zero. It may be that some nontrivial particular solution of the set of differential equations does exist where β5 and β9, are nonzero and this would be another steady state of the system, but it seems doubtful that this is true. The assumption that β8 = 0 constrains domestic demand for imports in volume terms to equal unity, so that growth in demand, whether at home or abroad, is not biased toward either the home good or its foreign substitute. Since, as noted earlier, home and foreign output grow at the same rate in the steady state, these assumptions imply that the equilibrium barter terms of trade will remain constant in the long run, an implication that is consistent with purchasing power parity. However, although the preceding assumptions are useful in developing the theoretical analysis of the steady state, it is not necessary to impose all of them as restrictions on the model used for estimation during any particular sample period. 12

III. Estimation Results

This section presents estimates of the parameters of the U. K. model for a sample of quarterly observations extending from the first quarter of 1955 to the fourth quarter of 1972. For estimation, the model has been approximated by taking a first-order Taylor series expansion about the sample means of the logarithms of the variables. The method of estimation and the properties of the estimator are discussed in Wymer (1976).

The constraints embodied in the theoretical model specified earlier, including the across-equation restrictions and the constraints inherent in the linearization of the model, were imposed during estimation. Furthermore, certain restrictions derived from analysis of the steady- state solution to the model were also imposed. The implications of these restrictions will be considered later. Since a full-information maximum-likelihood (FIML) procedure was used, the estimated parameters have an asymptotic normal distribution. 13

On the whole, the empirical results for the model are very satisfactory. All except three of the estimated parameters have the sign expected a priori, and none of the parameters with incorrect signs are significantly different from zero. These parameters have been retained in the final model to allow more detailed analysis of its dynamic properties. Of 57 estimated values, 43 are significantly different from zero at the 5 per cent level. Twelve parameters from the full theoretical model have been set equal to unity or zero, and several pairs of interest rate elasticities were restricted to be equal. These restrictions were imposed because there were good theoretical reasons for doing so, and because in the initial stages of estimation their values were found not to differ significantly from the values to which they were eventually restricted. The additional restrictions increase the efficiency of the estimates of the remaining parameters.

Since the adjustment parameters are fundamental to the dynamic behavior of the model, their estimated values will be considered in some detail. Many adjustment functions are based on the implicit assumption that economic agents adjust the level of a variable (say, X) that is under their control to its desired or partial equilibrium level (X^) with an exponentially distributed lag:

This relationship may be written as

where X^(t)/X(t) is a measure of the excess demand or supply of X. Thus, the inverse of each adjustment parameter indicates the time required to adjust X to eliminate 63 per cent of any excess demand. These mean time lags, and their standard errors, are presented in columns 4 and 5 of Table 3. Adjustment functions of the form

where X and Y are different cannot be given an interpretation similar to (27). In equations for prices, wages, and interest rates, the adjustment parameter itself gives the elasticity of response in the market, that is, the percentage rate of change in the price variable that results from excess demand of 1 per cent.

Of the 27 adjustment parameters in the estimated model, 14 21 are significant at the 5 per cent level. The mean time lags and response elasticities give an interesting picture of the relative speeds of adjustment in various markets. The excess demand for money balances does not appear in the consumption equation (α2 = 0) because during the earlier stages of estimation this adjustment parameter was not significantly different from zero and was usually positive. Although this restriction does not affect the steady state of the model, it could have a substantial effect on the overall dynamics of the system, and on its stability. The partial derivatives of some of the largest eigenvalues of the model with respect to this parameter are sufficiently large and negative that a relatively small negative value of α2 could make the model unstable. Under fixed exchange rates, an excess supply of money balances can be eliminated directly through capital flows, as in equation (11), and indirectly by a decrease in interest rates, as in (10).

Table 3.United Kingdom: Estimated Adjustment Parameters of the Model
ParameterEntering

Equation

Number
EstimateAsymptotic

Standard

Error
Mean Time

Lag
Standard Error

of Mean

Time Lag
output and expenditure sector
α1(1)0.2880.1263.471.52
α2(1)0.0*
α3(2)1.4410.2340.700.11
α4(3)0.6890.1271.450.26
α5(5)1.3610.2030.740.11
α6(5)0.0410.025
α7(6)2.1750.3820.460.08
α8(6)0.0*
α9(7)0.5400.2151.850.74
labor market
α10(8)0.1190.0168.401.13
α11(9)0.2840.103
financial sector
α12(10)0.1320.049
α13(11)0.4200.043
α14(11)0.2660.094
α15(12)0.0220.017
α16(13)–0.1260.038
α17(14)0.2300.0484.350.91
α29(18)0.0250.00640.009.60
government policy reactions1
α18(4)–0.1490.0746.713.33
α19(4)–0.7500.167
α20(4)–3.9101.074
α21(15)–0.1360.084
α22(15)–2.9610.729
α23(15)–0.1490.175
α24(15)1.6971.173
α25(16)–0.1530.138
α26(16)–0.0810.035
α27(16)–1.2670.632
α28(17)–4.5080.7680.220.03

Parameters α21andα22 are defined in footnote 1 to Table 4.

Parameters α21andα22 are defined in footnote 1 to Table 4.

In the output and expenditure equations, aggregate supply has a shorter mean time lag than does aggregate demand. For example, the estimates imply that if inventories are in partial equilibrium, 60 per cent of the discrepancy between actual imports and their desired level will be eliminated by a change in imports in about two months, while the mean time lag for domestic output is only about five weeks. The response lag for exports is about two months, while that for consumer spending is more than ten months.

In the labor market, the mean lag in the response of employment to a change in the demand for labor is about eight quarters, a figure that seems reasonable for the United Kingdom, where employment levels were generally high during the sample period, and where reductions in employment take place by not replacing retiring or resigning staff rather than by layoffs. Turning to the market adjustment functions for prices and wages, the gross domestic product (GDP) deflator adjusts to its partial equilibrium level with a mean time lag of just under two quarters, while the elasticity in the response of nominal wages to an excess demand for labor is 0.3.

In the financial sector, the estimated responses seem plausible for the interest rate and private sector capital flows. However, the time lags implied by the adjustment parameters in the equations for domestic advances and foreign holdings of U. K. Government securities are long. For example, the estimates imply that the lag in the adjustment of bank advances to a change in the money stock is over one year. These relatively small estimated adjustment parameters are at least partly a result of the fact that the U. K. authorities imposed ceilings on the supply of advances at various times during the sample period, and of the slow adjustment of overseas holdings of official sterling assets. Nevertheless, the results for this sector seem to indicate that, at least for the United Kingdom, there might be advantages to modeling the financial sector at a higher level of aggregation. This is suggested also by the positive but very small and insignificant parameter in the adjustment of banks’ holdings of government securities to the excess demand for liquid assets.

Finally, consider the adjustment speeds in the reaction functions for government spending and taxes. Assuming that employment is at the level desired by the authorities, the mean time lag in the response of government spending to a change in income is 20 months while that for taxes is about one month. Thus the initial effect of a fall in home income is an increase in the government budget deficit.

The estimated proportions, demand elasticities, and marginal propensities of the partial equilibrium supply and demand functions and policy reaction functions, together with the estimated long-run growth rates of the model, are presented in Table 4. As already noted, several parameters in the demand and supply equations were restricted for theoretical reasons. Two such restrictions result from the assumption that the model has a steady-state solution that requires that the real interest rate elasticity in the consumption function β5, and the elasticity of labor supply with respect to the adjusted real wage, β8, be zero. Estimates of these parameters and their asymptotic standard errors when they were not restricted were β, = 0.228 (1.048) and β9 = 2.905 (1.135). Thus, the restriction of β5 can be justified on empirical grounds, but this is not true of β9 at the 5 per cent level of significance.

The basic model was specified such that the partial equilibrium level of income in the consumption function, excluding the effect of real interest rates, was γ1(Y–T/p). This expression was linearized in terms of the logarithms of output and real taxes. But because of difficulties in estimating the parameters of the equation, the expression was reduced to γ1Y, and it is the estimate of γ1 rather than γ1 that is given in Table 4. The partial equilibrium level of real taxes is γ17, so that the marginal propensity to consume out of real disposable income is γ1=γ1(1γ17)=0.88 where γ17=e(λ1+λ2)/α28.

The estimated price elasticity of demand for exports, β1 in equation (2), was not significantly different from unity and was eventually restricted to this value in order to increase the efficiency of the estimates of the other parameters. In addition, the exponent of the relative price term in the import adjustment function β8 was set to zero, so that the relative price elasticity of demand for imports was restricted to –1. It was also found that the excess demand for stocks was not a significant determinant of domestic output, so that α8 = 0 in equation (6). The mean time lag of the adjustment of stocks to a change in aggregate sales is about six years, which is reflected in the relatively long cycles inherent in the model.

In the CES production function, estimates of β4 were not significantly different from unity. Thus, β4 was set equal to one, so that the elasticity of substitution between labor and capital is 0.5. This additional over-identifying restriction increased the significance of other parameters in the investment function (3), the employment function (7), and the wage and price functions (8) and (9). The estimates of the other parameters in the production function were not altered significantly when this restriction was imposed. The long-run elasticity of output with respect to capital β3 (Y/K) had an average value over the sample period of 0.259.

Table 4.United Kingdom: Estimated Proportions, Demandelasticities, and Growth Rates
Parameter 1Entering

Equation

Number
Point

Estimate
Asymptotic

Standard

Error
β1(2)1.0*
log β2(7), (8), (9)–6.1100.166
β3(3), (7), (8), (9)2.3930.816
β4(3), (7), (8), (9)1.0*
β5(1)0.0*
β6(1), (10), (11)0.8760.237
β7(1), (10), (11)0.0*
β8(5), (6)0.0*
β9(9)0.0*
β10(11)0.4730.249
β11 = β10(11)
β12(12)2.8960.600
β13(13)1.8291.154
β14 = β13(13)
β15(14)1.0861.205
β16 = β15(14)
β17(18)–0.8700.904
log γ1(1)–0.3260.031
log γ2(1), (10), (11)–3.0151.060
log γ3(2)7.4670.005
γ4(3)0.0800.011
γ5(3)0.00710.0060
γ6(5), (6)0.2340.000
γ7(5), (6)1.0621.133
γ8(7)0.0*
log γ9(9)3.1280.576
log γ10(11)–0.2190.232
log γ11(12)–2.3791.261
log γ12(13)0.0100.006
log γ13(14)–0.2160.167
log γ14(4)–1.6300.663
log γ15(4), (15)3.1560.212
log γ16(15), (16)0.0*
log γ17(17)–1.7220.463
log γ18(18)8.0940.385
γ1(7), (8), (9)0.006790.00051
γ2(4), (9), (15)0.001020.00149
γ3(2), (18)0.011060.00036

The model for which estimates are given includes the ratio of exports to imports rather than the ratio of reserves to money balances in the monetary policy equations. Equation (15) becomes

Db=α21log(Eγ16I)+α22DlogEγ16I+α23log1γ15eλ2t+α24DlogLγ15eλ2t

The model for which estimates are given includes the ratio of exports to imports rather than the ratio of reserves to money balances in the monetary policy equations. Equation (15) becomes

Db=α21log(Eγ16I)+α22DlogEγ16I+α23log1γ15eλ2t+α24DlogLγ15eλ2t

Portfolio theory suggests that a sector’s asset demand and supply functions should include both the interest rates on all securities in its portfolio and the level of its total assets. Under the assumption of utility maximization subject to a wealth constraint, these functions are homogeneous of degree one in total assets and of degree zero in all interest rates. Because of the degree of aggregation in the model, it is not possible to include all relevant interest rates in each asset demand function, so that these properties need not hold. The assumption that the economy has a steady state, however, leads to results similar to those of portfolio theory in that the demand functions are unit elastic with respect to total assets, and the sum of the interest elasticities is zero. These restrictions were imposed during estimation in order to increase the efficiency of the estimator. Thus the asset demand and supply functions in this model have the form X^=γr1β1r2β2W, where W is some scale variable and β1 = β2. 15

When these restrictions are imposed, all parameters in Table 4 except) β17 have the correct sign, and 19 of the 29 estimated parameters of the partial equilibrium equations are significant at the 5 per cent level. The long-run growth rate of foreign income that is obtained from the model is about 4.4 per cent per annum, and is significantly different from zero, while the rate of technical progress is 2.7 per cent. The growth rate of the U. K. labor force is only 0.4 per cent, a result that reflects the fact that the labor force first fell and then rose again during the sample period.

The policy responses of the authorities are of major interest in this study. Initially the model was estimated with only the ratio of international reserves to domestic money balances, the ratio of desired to actual employment, and the rates of change of these variables included in equation (15) for the authorities’ open market operations. But if the system is to have an analytical steady-state solution, it appears to be necessary that the authorities should have a target ratio of exports to imports. 16 The ratio of exports to imports and the rate of change of this ratio were included in equations (15) and (16), in effect replacing the international reserves target, that is, the ratio of reserves to money, in these functions. Since the estimate of the partial equilibrium ratio of imports to exports had a very small standard error and was not significantly different from one, this parameter was set equal to one, implying that the authorities’ target is balanced trade.

The estimates indicate that during the sample period the U. K. authorities used government expenditure to pursue both integral and proportional policies with respect to the employment target but that open market operations were used only as a proportional policy. That is, the aim of open market operations was to increase or reduce the current rate of change of employment, but not to eliminate discrepancies between actual and desired employment that might have developed in the past. Changes in Bank rate seem to have been used mainly to alter the ratio of international reserves to money balances and as a defensive mechanism to reduce the effect of changes in foreign interest rates. The estimates indicate that the authorities placed little weight on long-term discrepancies between the actual ratio of international reserves to money and their desired ratio, despite the fact that the desired ratio is well determined.

It is difficult to provide simple statistics for testing the validity of the overall structure of a simultaneous equation model. The likelihood ratio statistic, which is distributed asymptotically as a Chi-square distribution, does allow a test of whether the overidentifying restrictions are consistent with the data for the FIML estimator, but this test may not be valid for a sample of the size normally available in macroeconomics. The estimates of the parameters are asymptotically normally distributed, but recent theoretical work on small-sample approximations to the asymptotic distribution of parameters in dynamic models and Monte Carlo studies by Phillips (1978) suggest that for sample sizes of 20 to 30 observations the accuracy of the approximation depends heavily on the stability of the model. Convergence of the small-sample parameter distribution to the asymptotic distribution as the number of observations increases can be slow if the model is unstable. It should be noted, however, that the sample used here, with 64 observations, is much larger than that used in the Monte Carlo studies. The estimated standard errors of the parameters given earlier are an approximation to the asymptotic standard errors, so that the parameter estimates can be tested using a normal distribution with standard deviation given by the estimated standard error.

Even if formal tests of the overall structure of the model are not available, it may be of interest to consider the mean square errors of the endogenous variables in the estimated model. The two sets of mean square errors are given in Table 5. The first is the mean square of the residuals calculated from the restricted reduced form for the sample period. This is equivalent to calculating “forecasts” of the endogenous variables for each observation in the sample period in which all predetermined variables take on their actual values. The second set of mean square errors is that for a dynamic “forecast” in which the values of the lagged endogenous variables for the first period are the observed values but for all later forecasts the lagged endogenous variables have the values forecast by the model for the previous period. Thus, these errors will tend to be larger than those of the single-period case. 17 In evaluating the model, the only exogenous variables are the foreign interest rate, the foreign (import) price level, and a time trend. In particular, this means that the forecasts are based on the implicit assumption that foreign real income grows steadily. This neglect of fluctuations in foreign output around its trend growth path tends to increase the errors in both the single period and dynamic forecasts.

Table 5.United Kingdom: Expost Root-Mean-Square Errors, Second Quarter 1956-Fourth Quarter 1972
VariableRoot-Mean-Square Error

of Single-Period Forecasts
Root-Mean-Square Error

of Dynamic Forecasts
C0.0150.046
E0.0420.094
k0.0010.001
G0.0210.060
I0.0430.048
Y0.0180.040
P0.0310.092
L0.0040.036
w0.0290.072
r0.0300.164
S0.0830.350
Bb0.0500.147
N0.0520.211
A0.0221.051
b0.0410.032
ro0.0770.539
T0.1680.177
Bf0.0120.048
K0.0010.015
B0.0200.214
V0.0050.042
R0.1120.502
M0.0360.271
Bp0.0290.484
H0.1202.292

Since the model is in logarithms, the root-mean-square error gives the average error as a proportion of the actual level of the endogenous variable. The second column of Table 5 indicates that the errors in the one-period “forecasts” are very small. Of the 25 endogenous variables, only 3 (those for reserves, high-powered money, and taxes) have root-mean-square errors of more than 10 per cent. For 18, the errors are less than 5 per cent. The larger errors on international reserves and high-powered money in the United Kingdom are not surprising, given the volatile nature of these variables.

With a few exceptions, the in-sample dynamic simulations are also satisfactory. As expected, the root-mean-square errors are somewhat larger than in the one-period case. Nevertheless, 13 variables still have errors of less than 10 per cent. It is clear, however, that the model is much more successful in estimating the levels of real sector variables than it is for those of the financial sector. Furthermore, a number of individual variables have prediction errors that are rather large, most noticeably, the levels of high-powered money and advances. However, the errors in liquid assets are offset by compensating errors in advances, so that the errors in the predicted volume of money are much smaller (27 per cent). This is due to the inclusion of an explicit portfolio identity for the banking sector in which the volume of money is determined by the private sector and advances and liquid assets are set by the banks.

IV. Stability Analysis

The stability properties of the model can be analyzed given estimates of the parameters, and these properties are useful in formulating economic policy. Since the basic model is nonlinear, it could be argued that these properties should be evaluated by simulation. It is often more informative, however, to linearize the model about some suitable path or point and to determine its dynamic properties using the eigenvalues of the linear model. Generally this can be used only to determine local stability, that is, the dynamic behavior of the model in the neighborhood of the point about which the system is linearized, but models such as (1) to (25) may be transformed to allow an analysis of global stability.

The model may be written, as in (26),

where it is assumed, as in the analysis of the steady state in Section II, that the exogenous variables are on long-run paths given by Zi(t)=Zi*eλit for all i. Such a model will be stable if, for some transformation of the variables—say, y(t)—equation (30) can be written

where lim lim|y|0h(y,Z*,θ,λ,t)|y| is uniformly convergent in t, and all eigen- values of A have negative real parts. A property of the model (1) to (25) is that it can be represented by another system of nonlinear differential equations defined in terms of logarithmic deviations of the variables about their steady state, and which is independent of t. This model 18 may be written

where, in general, yi(t)=log[Yi(t)/Yi*eρit] with Y* and p defined as in Section II. Equation (32) can be linearized by a Taylor series expansion about the steady state to give a model of the form (31) in which h is independent of t. Since the remainder terms are of order greater than one in |y| 1, the stability properties of the model (1) to (25) may be examined by evaluating the eigenvalues of the linearized model.

The estimates of eigenvalues of the model that includes the ratio of exports to imports rather than the ratio of reserves to money balances in the monetary policy function are given in Table 6. Since the wealth identity of the banking sector (25) is a zero-order equation, the system has only 24 eigenvalues.

All the real eigenvalues are negative, and three of the four complex conjugate eigenvalues have negative real parts. These correspond to cycles of about 11 years, 17 years, and a very long cycle that is not significant. The damping period of the 11-year cycle is relatively short, with its amplitude declining by 63 per cent in about 18 months. The damping period of the 17-year cycle is about 5 years. As all the real eigenvalues are negative, the system will converge to the limit cycle with a period of 33 years given by the pair of complex conjugate eigenvalues with a positive real part rather than converging to a continuing state of inflation or depression.

Since the transformed model (32) is defined in terms of deviations of the endogenous variables about their steady-state paths, and since the elements of the matrix of coefficients of the linearized model are functions of the set of parameters θ, it is possible to evaluate the effect of changes in particular parameters on the stability of the system. Partial derivatives of the eigenvalues with respect to the parameters of the model may be calculated and used to provide a sensitivity analysis of the dynamic behavior of the system. Given a set of values of the parameters of the model, these results show the effect on the behavior of the model of a small change in the value of any parameter. This analysis is discussed in Wymer (1976).

Table 6.United Kingdom: Eigenvalues of the Model
EigenvaluesAsymptotic

Standard Error
Damping Period

(Quarters)
Period of Cycle

(Quarters)
–4.5080.7830.2
–2.8110.4020.4
–1.3420.8730.7
–1.0410.2151.0
–0.8960.1861.1
–0.6510.2391.5
–0.4500.1742.2
–0.2750.2953.6
–0.1350.0747.4
–0.0900.09111.2
–0.0400.02224.7
–0.0280.01435.4
–0.0110.00090.9
–0.0110.00091.4
–0.0100.00097.4
–0.0060.000164.0
–0.182± 0.146i0.091, 0.0985.543.1
–0.048 ± 0.116i0.016, 0.06320.654.3
0.044 ± 0.048i0.043, 0.029133.0
–0.020± 0.006i0.017, 0.01050.81,093.3

Such an analysis has two particular uses. First, it allows the effect of a change in the structure of the model (such as the introduction or elimination of some feedback) to be determined, which is often useful during the formulation of a model. Second, the implications of certain policy measures, such as a change in the weight that the authorities place on certain targets, or the choice of alternative instruments to achieve a given target, may be evaluated.

An example of the use of this analysis in indicating the effect of changing the structure of the model can be seen by considering the introduction of the excess demand for money in the consumption function. In the model for which estimates are given, this term has been omitted, so that α2 is restricted to zero. Table 7 gives the partial derivatives of the relevant eigenvalues with respect to this parameter at the point α2 = 0.0.

Since α2 should be negative and probably greater than -0.1, the inclusion of the excess demand for money term in the consumption function would increase the real part of the eigenvalues in Table 7, thus making the model less stable. The cycle corresponding to the complex eigenvalue would have a longer period. The partial derivatives of the other eigenvalues with respect to α2 have positive real parts or are sufficiently small that the overall stability of the model would not be affected significantly by any plausible value of α2. This result can be explained by considering an increase in the rate of growth in foreign demand for goods, which would lead to an increase in exports and, hence, of money balances. The resultant fall in the interest rate and simultaneous increase in output causes investment to increase, so that eventually the capital stock necessary to satisfy the increased demand for domestic output will be accumulated. But, if the increase in money balances directly causes consumption and, hence, imports to rise, the estimates show that in this model the interest rate will fall by less than it would otherwise, and so the adjustment to the higher rate of growth of exports will take longer.

Table 7.United Kingdom: Sensitivity Coefficients of Critical Eigenvalues with Respect toα2
Eigenvalue (μ)–0.27–0.090.04± 0.04i
μα–19.31–3.84–(1.47 ± 1.76i)

Within this model, where the export/import ratio is a target of the authorities, the effect of using monetary policy, and particularly open market operations, to achieve some target level of unemployment can be compared with the use of fiscal policy. If the ratio of the level of employment to the target level desired by the authorities affects neither government expenditure nor open market operations, so that α19, and α23 are zero, the partial derivatives of the critical eigenvalues are as shown in Table 8.

Table 8.United Kingdom: Partial Derivatives of Critical EigenValues with Respect toα19 AND α23
Eigenvalue (μ)–0.27–0.18± 0.15i–0.05± 0.1i0.04± 0.04i
μα190.09–0.09± 0.02i0.07± 0.02i–(0.01± 0.00i)
μα23773.615.42± 22.85i12.69±0.54i–(2.14± 6.67i)

It is expected that the coefficient of the employment term in the government expenditure function α19 would be negative and greater than –1, and the corresponding term in the open market operations function would be positive and less than 1. The partial derivatives in Table 8 indicate that the use of monetary policy to achieve some target level of employment, and hence to bring about internal balance, will have a destabilizing effect on the economy in that an increase in the value of α23 will increase the real parts of the eigenvalues and, for sufficiently large but plausible values of α23, could make some positive. On the other hand, the real part of the eigenvalue that is positive will be reduced. The effect of α13 on the stability of the system is negligible. Apart from a and α23, these partial derivatives are evaluated at the point given by the estimates of the parameters of the model, and their values will vary for other values of the parameters. The partial derivatives with respect to α19 and α23 of the other eigenvalues of the system are all sufficiently small or the signs of their real parts are such that these eigenvalues can be neglected when considering the stabilizing effect of the employment term in the policy functions. The conclusion that can be drawn from Table 8 is that according to this model the overall effect of using open market operations to achieve some target level of employment, when the export/import ratio is another target, will be more destabilizing than using fiscal policy to achieve the same result.

The stability of a system in which the authorities have a balance of payments target rather than a balance of trade target was also investigated, with results that contrast with those just discussed. Specifically, it was assumed that in determining open market operations the authorities have a desired ratio of international reserves to money balances, rather than a desired ratio of exports to imports. The eigenvalues of this model are given in Table 9.

The eigenvalues show that having a target for the balance of payments rather than the balance of trade results in a more stable system. The small positive real eigenvalue means, however, that there will be gradual convergence to a permanent state of depression or inflation. The partial derivatives of the coefficients on the employment term in the government expenditure and monetary policy functions are all sufficiently small that any plausible values of these coefficients will have little effect on the stability of the model. 19

Table 9.United Kingdom: Eigenvalues of the Model with Reserves Target
EigenvalueDamping Period (Quarters)(Period of Cycle (Quarters))
–4.5080.2
–2.7950.4
–2.010.5
–1.3360.7
–0.9081.1
–0.7161.4
0.4082.4
0.3053.3
0.1337.5
–0.04621.7
0.03826.0
–0.02638.6
0.01281.2
0.01189.3
0.01190.9
–0.01190.0
0.0002,206.2
0.002
–0.157 ± 0.127i6.349.6
–0.093 ± 0.055110.8114.9
–0.047 ± 0.085121.373.7

V. Conclusion

The model specified and estimated in this study describes the dynamic behavior of the major economic aggregates in the real and financial sectors of the U. K. economy. As well as making extensive use of economic theory, the specification incorporates major features of the institutional structure of the U. K. economy. The model was designed to capture the short-run adjustment processes that characterize an industrial economy and to have long-run properties consistent with macroeconomic theory and empirical observation over a much longer period than the sample period used for estimation. The parameter estimates and other statistics presented earlier indicate that the overall structure of the model is a plausible representation of the U. K. economy.

A basic feature of the model is that it allows government monetary and fiscal policy to respond to major economic variables such as income, the balance of payments, and employment. The budget constraints of the government sector are specified explicitly within the model, so that it is well suited for policy analysis.

Analysis of the model suggests that if, under a fixed exchange rate regime, the authorities use open market operations to pursue a balance of trade target and simultaneously attempt to achieve some desired employment level, they will tend to destabilize the system. If they pursue a reserve target, as defined in the model, instead of a balance of trade target, the destabilizing effect may still exist but will be much less. This effect holds even if the desired employment level of the authorities is the “natural” or steady-state level inherent in the model. Thus, if employment is below the level that the authorities wish to achieve, open market operations will be used to increase the volume of money and hence to decrease the interest rate and to increase investment. But the relatively long lags in the system, especially in capital formation, and the feedback through increased sales and output to employment mean that the effect of monetary policy on employment is very slow, and the tendency for destabilization suggests it is not an effective instrument for short-term stimulation of employment.

The estimates of the model have some general implications for the specification of macroeconomic models. Although the estimates of the real sector are very satisfactory, the estimates of the equations representing the domestic financial sector indicate that a higher degree of aggregation of some assets would not fundamentally alter the properties of the model. They suggest that liquid assets and banks’ holdings of government securities could be aggregated so that liquid assets would be included in the definition of total bonds issued by the authorities. Thus, the equation for banks’ holdings of bonds would be deleted, as would the behavior function for liquid assets, but the supply of advances by the banks would remain. The aggregate of liquid assets and bonds held by banks would be determined by the wealth identity of the banking sector. Thus, the model could be simplified without reducing its usefulness.

Although the model has been specified and estimated for a fixed exchange rate regime, it can easily be extended and used to study the implications of certain policies under the assumption of a flexible rate. For example, it may be assumed that the authorities have a target exchange rate and are willing to intervene in the market to achieve this target. The policy function for the exchange rate would contain such variables as the level of reserves and their rate of change in order to capture the feedback from current reserve changes to the exchange rate, so that if the authorities attempted to achieve a rate not consistent with that inherent in the economy, reserves would rise or fall and induce the authorities to intervene differently in the market. It would be necessary to modify some behavior functions by replacing nominal interest rates with the corresponding covered rates and perhaps to change some parameter values. In full equilibrium, this model would exhibit long-run properties of both interest rate parity and purchasing power parity.

DATA APPENDIX

Sources of data

AA

Annual Abstract of Statistics (U. K. Central Statistical Office)

BE

Quarterly Bulletin (Bank of England)

BLS

British Labour Statistics: Yearbook (U. K. Department of Employment)

ET

Economic Trends (U. K. Central Statistical Office)

FRB

Federal Reserve Bulletin (Board of Governors of the Federal Reserve System)

FS

Financial Statistics (U. K. Central Statistical Office)

MDS

Monthly Digest of Statistics (U. K. Central Statistical Office)

NIE

National Income and Expenditure (U. K. Central Statistical Office)

Definition of series

The time series used in this study consist of quarterly observations for the period 1955–72. They are defined as follows:

C Real consumption

Consumer expenditure at current prices deflated by the gross domestic product implicit price deflator, p, as defined later in this Appendix. Source: ET.

E Real exports

Exports of goods and services at current prices deflated by the implicit price deflator, p. Source: ET.

K Real fixed capital formation

Gross domestic fixed capital formation at current market prices less interpolated quarterly capital consumption (as in Bergstrom and Wymer (1976)) deflated by the implicit price deflator, p, and cumulated on a base stock of £ 120,200 million in the fourth quarter of 1970. Sources: Gross capital formation—ET; Capital consumption—AA; Fixed capital stock—NIE.

G Real government expenditure

Public authorities’ current expenditure at current prices divided by the implicit price deflator, p. Source: ET.

I Real imports

Imports of goods and services at current prices deflated by the implicit price deflator, p. Source: ET.

Y Real output

Gross domestic product at current factor cost plus taxes on expenditure less subsidies less interpolated quarterly capital consumption as defined earlier in this Appendix, deflated by the implicit price deflator, p. Source: ET.

V Inventories

Value of physical increase in stocks at current prices deflated by the implicit price deflator, p, and cumulated on a base stock of £ 14,203 million in the fourth quarter of 1970. Sources: ET and MDS.

L Employment

Employees in employment. Sources: BLS and MDS.

p Price level

Gross domestic product at current market prices divided by gross domestic product at 1970 market prices. Source: ET.

w Wage rate

Income from employment divided by employment, L, as defined earlier in this Appendix. Sources: BLS and ET.

R Foreign currency reserves

Total currency flow in pounds sterling cumulated on a base figure of 4,033 million in the first quarter of 1955. An inaccurate base stock figure will affect the level of reserves and of foreign holdings of domestic assets, but changes in these variables will be correct. Although changes in the logarithm of these variables will be incorrect, this error would be affected by compensating errors in the coefficients of the log-linear approximation to the budget constraint and by the intercept term in the identity. Source: FS.

M Volume of money

Total deposits held by nonbanks with the U. K. banking sector (defined as the London clearing banks, the Scottish and Northern Ireland banks, British overseas and foreign banks, other banks, Accepting Houses, and Discount Houses) less Eurocurrency liabilities of the banking sector, F. An aggregate series on the first component is available from the first quarter of 1963 but before that date the series had to be produced from disaggregated data. Sources: BE and FS.

B Total government securities (excluding holdings of the authorities)

Total British Government and government-guaranteed securities. Sources: BE and FS.

H Liquid assets

Cash and balances with the Bank of England of the London clearing banks and Scottish and Northern Ireland banks, plus Treasury bills held by the U. K. banking sector as defined in M plus special deposits of the London clearing banks with the Bank of England. Sources: BE and FS.

Bb, Government securities held by the banking sector

This series is the sum of government securities and government-guaranteed securities excluding Treasury bills held by the groups that comprise the U. K. banking sector, namely, the London clearing banks, the Scottish and Northern Ireland banks, British overseas and foreign banks, other banks, Accepting Houses, and Discount Houses. Sources: BE and FS.

Bp Government securities held by the private sector.

Total British Government and government-guaranteed securities excluding holdings of the authorities, B, less government securities held by the banking sector, B6, less government securities held by the foreign sector, Bf. Sources: BE and FS.

Bf Government securities held by the foreign sector

British Government and government-guaranteed securities held by overseas residents. Sources: BE and FS.

N Eurocurrency assets of the banking sector

Eurocurrency assets of the U. K. banking sector as defined in M. Sources: BE and FS.

A Advances

The volume of money, M, plus Eurocurrency liabilities, F, less Eurocurrency assets, N, of the banking sector less liquid assets, H, less government securities held by the banking sector. This series is constructed to satisfy the wealth constraint of the banking sector.

T Taxes less transfers to the private sector

Government expenditure at current prices, pG, less increase in total government securities, B, less increase in net liquid assets of the banking sector, H, plus increase in foreign currency reserves, R. This series is constructed to satisfy the wealth constraint of the authorities.

S Foreign holdings of domestic assets

Increase in foreign currency reserves, R, plus imports of goods and services at current prices less exports of goods and services at current prices less increase in non-sterling deposits of the banking sector, F, plus increases in non-sterling deposits of the banking sector, N, less increase in foreign holdings of government securities, Bf, cumulated on a base figure of 1,875 million in the first quarter of 1955. This series is constructed to satisfy the wealth constraints of the three sectors of the domestic economy. Leads and lags, for example, are subsumed in S.

r Interest rate

Gross redemption yield on long-term government bonds. Sources: BE and FS.

r0 Bank rate

Sources: BE and FS.

F Eurocurrency liabilities of the banking sector

Eurocurrency liabilities of the U. K. banking sector as defined in M. Sources: BE and FS.

rf Foreign interest rate

Interest rate on long-term U. S. Government securities. Source: FRB.

∊.q Foreign prices in terms of domestic currency

Imports of goods and services at current market prices divided by imports of goods and services at 1970 market prices. Since the model was defined for a fixed exchange rate regime, it was not necessary to include separate variables for the exchange ratio, , defined as the price of foreign currency in terms of domestic currency, and the price of foreign output, q, defined in terms of the foreign monetary unit. Source: ET.

t Time trend

Time trend where the first quarter of 1955 is –35.5, the second quarter of 1955 is –34.5, … , the fourth quarter of 1972 is 35.5, so that the sample mean is zero.

In general, all stock series and price and interest rate series were measured at end of period. The logarithms of all variables were deseasonalized by removing the mean seasonal variation about the sample mean with trend adjustment as suggested by Durbin (1963). All series measured at end of period were averaged to produce series consistent with flow data as discussed in Wymer (1976). and first differences produced to give the proportional rate of change of government securities b(= D log B) and the proportional rate of change of the fixed capital stock k (= D log K). All data were transformed according to the procedure discussed in Wymer (1976) in order to eliminate the moving average inherent in a model containing flow data.

REFERENCES

    BergstromA. R.“Nonrecursive Models as Discrete Approximations to Systems of Stochastic Differential Equations,”Econometrica Vol. 34 (January1966) pp. 17382.

    BergstromA. R. and C. RWymer“A Model of Disequilibrium Neoclassical Growth and Its Application to the United Kingdom” in Statistical Inference in Continuous Time Economic Modelsed. byAbram R.Bergstrom (Amsterdam1976) 267327.

    DurbinJ.“Trend Elimination for the Purpose of Estimating Seasonal and Periodic Components in Time Series” in Proceedings (Symposium on Time Series Analysis held at Brown University1962) ed. byMurrayRosenblatt (New York1963).

    JonsonP. D.E. R.Moses and C. RWymer“A Minimal Model of the Australian Economy,”Reserve Bank of Australia Research Discussion Paper No. 7601 (November1976).

    KnightM. D. and C. RWymer“A Monetary Model of an Open Economy with Particular Reference to the United Kingdom,”Ch. 8 in Essays in Economic Analysis: The Proceedings of the Association of University Teachers of Economics Sheffield1975ed. byM. J.Artis and A. RNobay (Cambridge University Press1976) 15371.

    PhillipsP. C. B. (1973) “A Sampling Experiment with a Dynamic Model of the Product Market” University of Essex Discussion Paper in Economics.

    PhillipsP. C. B. (1978) “Edgeworth and Saddlepoint Approximations in the First-Order Noncircular Autoregression” Biometrika Vol. 65 (April1978) 9198.

    WymerCliffordR.“Continuous Time Models in Macro-Economics: Specification and Estimation,”paper presented at SSRC-Ford Foundation Conference on Macroeconomic Policy and Adjustment in Open Economies Ware England (April28—May011976).

Mr. Knight, economist in the Financial Studies Division of the Research Department, holds degrees from the University of Toronto and from the London School of Economics and Political Science, where he also served as a member of the Economics Department from 1972 to 1975. He is the author of articles in the fields of macroeconomics and international finance.

Mr. Wymer, Senior Economist in the Financial Studies Division of the Research Department, is a graduate of the University of Auckland and of the London School of Economics and Political Science, where he taught from 1966 to 1976. During the last three years of that period, he directed the U. K. Social Science Research Council’s International Monetary Research Program.

This paper developed from work begun within the International Monetary Research Program at the London School of Economics, financed by the U. K. Social Science Research Council.

Actually, the sample period extends from the first quarter of 1955 to the fourth quarter of 1972, so that it includes two quarters of managed floating.

Such estimation is feasible, but it was considered preferable to obtain a satisfactory theoretical and empirical specification of the major relationships within the model using data from the fixed rate period before extending the model.

The production function is

Y={(1β2eλ1tL)β4+β2Kβ4}1/β4

The elasticity of substitution is 1/(1 + β4), and the elasticity of output with respect to capital is β3(Y/K)β4. Technical progress is assumed to be Harrod neutral.

The labor supply function in the model differs slightly from that given by Knight and Wymer (1975) because of a misspecification in the earlier model that was brought to the attention of the authors by Professor John Williamson. In the earlier model the demand for labor was expressed in efficiency units, while the supply was in actual units. This led to the conclusion (p. 164) that a consistent set of long-run growth rates could exist only if the supply of labor was inelastic with respect to the real wage. In the current version, this error in the dimensionality of the excess demand for labor has been corrected by adjusting the real wage rate for changes in the efficiency of the labor force. The new specification allows price changes to have an immediate effect on nominal wages through their impact on real earnings and on labor supply. Nevertheless, while the conclusion as stated in Knight and Wymer (1975) is incorrect, a similar restriction on the elasticity of the supply of labor still seems necessary in order that a steady state should exist.

The banking system in the model is highly aggregated and includes the clearing banks, overseas and foreign banks, and the accepting houses.

A policy of strict sterilization of the monetary consequences of payments flows would imply α21, α22 > 0, but such a reaction would tend to destabilize the system.

However, the estimation results that are reported in the next section do not incorporate the restrictive assumption that the model was near to its steady state during the particular sample period.

The most important assumptions are that wealth and income elasticities equal unity.

This solution was derived on the assumption that the steady-state paths of the exogenous variables are

F(t)=F*eλ4t,q(t)=q*e(λ4λ3)t,r(t)=rf*,andϵ(t)=ϵ*

Moreover, for a full international steady state, the rates of growth of domestic and foreign output must be equal, that is, Alλ1λ2 = λ3.

The steady-state solution of the system is not presented in this paper but is available upon request from the authors, whose address is Research Department, International Monetary Fund, Washington, D. C. 20431.

That is, it is independent of those government policies that are incorporated in the model; it would not be independent of structural labor market policies.

The estimation results in the next section make use of the assumptions β8 = 0 and β4 = 1, but these restrictions were imposed only after research indicated that the estimates did not differ significantly from these values.

In the discussion of the results, the term “t-ratio” simply denotes the ratio of a parameter estimate to the estimate of its asymptotic standard error, and does not imply that this ratio has a Student’s t-distribution. In a sufficiently large sample, this ratio is significantly different from zero at the 5 per cent level if it lies outside the interval ± 1.96 and significantly different from zero at the 1 per cent level if it is outside the interval ± 2.58.

In the initial stages of estimation, 3 adjustment parameters in the full theoretical model were found to be insignificant and to have a sign different from that expected a priori. These parameters were set to zero.

These restrictions have the additional advantage that they eliminate the estimation problems that would result from multicollinearity between the different interest rates in the model.

That is, the authors have been unable to find a steady-state solution of a model that does not include this ratio.

The use of dynamic simulation, even through the sample period, provides a reasonable test of the dynamic structure of the model and of the feedbacks within the system. It is, of course, preferable to produce ex post forecasts for a postmodel-building period. Ex ante forecasts are less useful because of their dependence on some forecast of the values of the exogenous variables.

A copy of this model may be obtained upon request from the authors, whose address is Research Department, International Monetary Fund, Washington, D. C. 20431.

In this model, the introduction of the excess demand for money into the consumption function will increase the stability of the model. This result is consistent with the model of the Australian economy described in Jonson, Moses, and Wymer (1976), which incorporates an interest rate policy function and assumes a flexible exchange rate. In that model, the effect of the excess demand for money balances on consumption has a stabilizing influence.

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