Journal Issue

The Dynamics of Inflation: Forward Contracts and Money

International Monetary Fund. Research Dept.
Published Date:
January 1979
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One of The Stylized Facts that has been most important in shaping modern macroeconomic thought is the relatively sluggish response of wages and prices to changes in aggregate demand. This “stickiness” presents a major challenge to all theories that assume that prices adjust to equilibrate demand and supply. Indeed, its recognition led to the development of the Keynesian model and, more recently, to the nonmarket-clearing paradigm which is usually characterized by the assumption of some ad hoc gradual adjustment process. This assumption permits short-run deviations from “desired” equilibrium and is usually rationalized on the basis of adjustment costs. This approach has been widely applied in macroeconomic models (see Barro and Grossman (1976)) and is ubiquitous in the empirical literature on the demand for money. It has, nevertheless, been criticized for lacking a firm foundation of rational microeconomic behavior. 1 In its place a revival of equilibrium market models is occurring where contractual arrangements between the different economic agents give rise to the perceived sluggish price behavior. Previously, this effort has been directed to explaining unemployment. 2 The purpose of this paper is to extend this approach to the theory of money demand and the determination of the price level.

It is hoped that this paper will contribute to resolving a basic dilemma of modern monetarist theories of inflation, which can be described as follows. Money is one of many assets and as such its price (the inverse of the price level) should behave like any other asset price. Detailed observation of the markets for equity and other assets has led modern theoreticians to expect that asset prices adjust quickly to reflect all currently available information. Applying this theory to the demand for money, Sargent and Wallace (1973) show that the price level should change in the expectations of current and future monetary policy, implying that changes in the price level (the inflation rate) should be as unpredictable as changes in stock market prices. Even casual empiricism, however, yields the conclusion that the inflation rate is highly serially correlated and hence predictable, thereby contradicting predictions derived from the asset market theory.

Rather than rejecting this theory, economists have tended to regard it as a long-run proposition with little applicability to short-run behavior. For short-run analysis, two approaches have been taken. The more common one invokes the partial stock adjustment model for the demand for money. Unfortunately, this approach seems to be plagued with difficulties even on empirical grounds, since the estimated adjustment speeds are generally too slow to be credible and the money demand functions seem to be unstable over time. 3 The other approach, which is empirically untested, is to assume that goods market prices adjust slowly to changes in aggregate demand. 4 Although justified on quite different grounds, this approach yields very similar results to, and can be viewed as a special case of, the approach taken here.

This paper attempts to reconcile equilibrium theories of the demand for money with the empirical evidence. In particular, it will show that, even when all information is used efficiently and economic agents are always on their “desired” equilibrium demand schedules, the rate of inflation can exhibit strong serial correlation. Paralleling the recent literature on contractual arrangements in the labor market, this paper suggests that consumers enter into long-term contracts with producers. The upshot of this contractual arrangement is that some goods are sold in the spot market while others are sold at previously contracted prices. Since the latter are predetermined, only a portion of the price level (corresponding to the proportion of goods sold in the spot market) can adjust to reflect the new information; this process would then lead to an observed serial correlation of the inflation rate.

No attempt is made here to provide the microfoundations for such a market structure. Such a foundation would undoubtedly be based on risk diversification and information signaling services on the part of both consumers and producers. The market structure is left purposely vague in order to study the impact of money on price behavior under two theories of money demand. If one assumes that the proper deflator of money is the spot price index, this paper shows that the standard asset price behavior holds for an index of spot prices. The use of the general price index to test this theory then entails only a measurement error, and the observed serial correlation of the inflation rate cannot be used to contradict the asset view of money demand. On the other hand, if one assumes that money is deflated by the general price index, there is no measurement error, but then the spot price level tends to overshoot its long-run equilibrium position. 5

Section I describes the construction of a general price index when forward markets exist. Section II analyzes the impact on both the spot and general price levels under two assumptions about why money is held, Section III provides some empirically testable propositions that come out of this model, and Section IV presents a summary of conclusions.

I. Price Indices and Forward Contracts

Stripped to its bare bones, the idea of this paper is that time is an important element in the exchange of goods. In response to various economic factors, many, if not most, transactions are conducted such that prices are set sometime before the actual exchange takes place. For some commodities there is a formal futures market; for many goods and services there is an agreement to deliver in the future at a specified price. These examples are usually found in wholesale transactions, but even at the retail level there are implicit contracts. The simplest example is a catalog that publishes prices for a given period. As a less direct example, there is the retail store that does not change prices daily to reflect current supply and demand conditions.

All these various market arrangements can be stylized by a model that considers that all goods are traded in both spot and forward markets. In this model every transaction is accompanied by a contract which describes (a) the price of the good, (b) the date at which the exchange and payment are to take place, and (c) the date on which the contract was agreed. A forward contract exists for each good for every future date, and all contracts are issued independently of each other. 6 In this one-good economy, the contract price (in logarithms) for the good to be delivered at time t, and agreed upon at time tn, as C(t,tn). It is evident that at any one time, barring perfect foresight, the same good is being exchanged at different prices. The price level of the economy is the weighted average of all prices at a point in time. For the economy under consideration, the price level (in logarithms) at time t, p(t), is defined to be the weighted average of all contracts maturing at time t.

where αi is the percentage of goods exchanged today under a contract agreed on in period ti and is assumed to be a constant.

Although the microeconomic setting is not specified here, any reasonable utility-maximizing framework will yield the property that the forward contracts will be priced so as to reflect expectations about future spot prices. In a more general formulation, there may be other factors, such as storage and financial costs and risk premiums, that interpose themselves between the forward and the expected spot prices. Since the inclusion of these other factors will not change the main results of the model, they are neglected here, and it is assumed that the contract price C(t,t-i) is equal to the price expected to prevail in the spot markets at time t.

where Et–i is the expectations operator. Substituting equation (2) in equation (1),

or, more conveniently,

where s (t, ti) = EtiC(t, t)

Equation (4) is the basic equation of the paper. It states that the price level is a weighted average of the expected spot price, where the expectations were formed at various periods in the past. The rest of the paper is devoted to deriving some of the implications of this formulation of the price index. For this purpose, it is convenient to assume some distribution for the weights in the price index. In. what follows, the weights are assumed to be declining geometrically with the maturity of the contract. 7

Substituting equation (5) in equation (4),

II. The Dynamics of Inflation

The previous section gives only a partial description of the economy considered in this paper and needs to be supplemented in order to arrive at a theory of the dynamics of inflation. For this purpose, Sargent and Wallace (1973) are followed, and a demand for money function in which real cash balances depend on the expected change in the price level is postulated. It is required that expectations be formed rationally and that markets clear so as to maintain economic agents on their demand schedules. The departure from previous models is that one must now distinguish between the spot and the general price levels. Here two experiments are performed. First, it is postulated that the proper deflator for balances is the spot price level. In the second experiment, the general price level is used as the deflator. In both cases, the argument in the real cash balance function is the change in the spot price level. For notational convenience, the first model is called the spot model and the second, the general model. 8 The spot model is written as

The general model is written as

Given that the nominal stock of money is exogenously determined, equations (7) and (8) become price level determination equations. To derive the rational expectations path of the price level, the following experiment is performed. Assume that the economy has been at equilibrium up to period t such that the spot price level, s, and the general price level, p, are equal to the nominal stock of money, m, in logarithms. At time t, the authorities increase unexpectedly and once and for all the nominal money supply by the proportion δ. Under these conditions, the paths of the spot and general price levels under the two money demand formulations can be derived.

Assume first the spot model (equation 7). Following Sargent and Wallace (1973), it is assumed that after a once-and-for-all increase in the supply of money, economic agents expect that eventually the price level will reach a stationary equilibrium; otherwise, either the price level or the real money stock will eventually be infinite. 9 To get the equilibrium path of the price levels, we will work backwards from this future equilibrium period to the current period, t. For any period in the future, equation (7) is generalized to yield

Assuming that after period t+k the economy is once again in a stationary equilibrium, then

To find the price level at period t+k–1, equation (10) is substituted in equation (9), where k = n + 1, and rearranged as follows:

Repeated substitutions in equation (9) yield that

Equation (12) shows that the spot price level jumps at period t from its previous level, m, to the new level m + δ such that money demand is immediately equilibrated. Since no new shocks are expected, the spot price level stays at the same level henceforth. This is the result described by Sargent and Wallace. Within the context of this model, it has a commonsense explanation. Suppose that the government issues new money to buy goods that are then redistributed among the population in such a way that redistribution effects can be neglected. The economic agents who exchange money for goods will do so only at the equilibrium price in this economy of perfectly distributed information and individuals with homogenous tastes. The equilibrium price is the one that eliminates the excess money demand; otherwise, the economic agents would be trading at a price that is expected to bring about a capital loss. Once the equilibrium is reached, there are no further shocks to disturb it.

The general price level, however, will not jump to the new equilibrium, but rather will approach it asymptotically. To show this, equation (6) is rewritten so as to differentiate between two components, the expected spot prices before (i≤k) and after the shock occurs (i>k).

Consistency in forming expectations requires that economic agents should not expect to change their expectations of future variables unless new information is received. This implies that

Given that before the shock the price level was expected to be constant,

Substituting equations (14) and (15) in equation (13),

Substituting equation (12) in equation (16),

Figure 1 displays the expected equilibrium path of the spot and general price levels. Both of the paths were derived under the assumption that money is deflated by the spot price level. As can be seen in the above derivation, the general price level does not play any economic function in the model. It may be, however, that the spot price level is not measured by economic statistics and that the only information available is on the general price level. In such a case, the use of the general price level as a money demand deflator would lead to observation errors that are serially correlated.

Figure 1.Response of p and s to Jump in m, Assuming Money Deflated by Spot Price Level

Under the alternative formulation of the demand for money, the path of the price levels is more complex. As in the above derivation, the path of the spot price level will first be solved and then that of the general price index. For this purpose, equation (16) is substituted in equation (9) to get the general expression of the expected spot price level.

Again it is assumed that there is a period t+k such that there is no expected inflation henceforth and that γk+l is small enough to be neglected. The expected spot price level is, then, equal to the expected money supply.

Solving for the period t+k–1,

where ψk = l+βγk

Repeated substitutions in equation (18) give the path of the spot price level

where ωk=1ψk+1Σi=0βiΠj=0iψk+1+j

Equation (21) is similar to equation (12), the path of the spot price level under the alternative money demand function. The difference is that the increase in the money supply (δ) is now multiplied by a weight, ω, which depends in a complicated way on both the inflation elasticity of money, β, and the distribution parameter of forward contracts, γ. Since the expression giving the weight ω is rather complex, it cannot be worked with directly. Rather, it is convenient to derive the upper and lower bounds for the weights. In what follows it will be shown that the weight is always equal to or greater than one and that it has an upper bound, which is described by a much simpler expression.

To derive the lower bound, notice that

Thus, the weight ωk. is always equal to or greater than one. To derive the upper bound, notice that

The weight ωk is always less than or equal to the reciprocal of one minus γ raised to the power k+l. For a sufficiently large k, the upper bound converges on the lower bound, proving that the weights converge to one. This merely confirms the assumption that in the long run the spot price level is equal to the money supply. In the short run, however, the weight ωk, is greater than one, so that the spot price level overshoots the equilibrium price level. How much it overshoots depends on both the inflation elasticity of money demand and the distribution of forward contracts.

Given the upper and lower bounds of the path of the spot price level, similar bounds can be derived for the general price level’s path. Substituting equations (21) through (23) in equation (16) yields

The economic interpretation of these results is straightforward. Given that money demand is always in equilibrium, an incipient excess money supply must be eliminated by an increase in the general price level and/or an increase in money demand. The incipient excess money supply causes an initial rise in the general price level. Since most of the components of this price level are predetermined, the spot price level (the only free component) must increase much more to bring the price level high enough to eliminate the incipient excess money supply. This large spot price increase, however, causes expectations that the spot prices must decrease in the future to bring about long-run equilibrium. Since the money demand depends negatively on the expected inflation of spot prices, money demand increases in the short run. Thus, there are two effects bringing about equilibrium: the general price level is increasing and money demand is increasing. Which of the two effects is predominant depends on the inflation elasticity of money demand and on the distribution of forward contracts.

The upper and lower limits of the path correspond to extreme values in the parameters β and γ. It is evident that, if the price index is composed only of spot prices (γ=0), the lower limit of the spot price level’s path will coincide with the upper limit of the general price level’s path. The result is a once-and-for-all jump in the general price level. Divergences exist between the paths of the two price levels when forward prices enter into the general price level. Figure 2 shows the results for the two polar values of the inflation elasticity: zero and infinity. First, assume that the inflation elasticity is zero. This implies that all adjustment must come through an increase in the general price level. As could be expected, the general price level jumps to the new equilibrium level and remains there henceforth. On the other hand, the spot price level increases by an amount proportional to the reciprocal of its weight in the general price level. Over time, more components of the price index reflect the new expectations, and the spot price level approaches the new equilibrium asymptotically. Thus, in the case of zero inflation elasticity, all adjustment must come through a change in the general price level and the paths of the two price levels are given by the upper limits in Figure 2.

Figure 2.Response of p and s to Jump in m, Assuming Money Deflated by General Price Level

Not surprisingly, the lower limit of the paths corresponds to the case when the money demand is infinitely elastic with respect to spot price inflation. Now spot prices must jump to the new long-run equilibrium and stay there henceforth; otherwise, explosive inflation or deflation will occur. For example, assume that the spot price level overshoots the long-run equilibrium: then a deflation is expected and the demand for money becomes infinitely large, forcing a further deflation of both the price levels. A similar argument in the opposite direction can be made when the spot prices undershoot the new equilibrium. Thus, in this case, the spot price level jumps to the new equilibrium and the general price level approaches the new equilibrium asymptotically from below. It is evident that this case exhibits the same paths for the two price levels as that of the alternative money demand function. The reason for this similarity is that in both cases the only stable path for spot prices is that of an immediate jump to long-run equilibrium.

One could envisage a hybrid model in which both the spot and general price levels played a role in determining money demands. The simplest version of such a model would deflate cash balances by a weighted average of spot and general price levels. This would be equivalent to postulating a different distribution in the maturity weight of the price index such that the spot price level has a proportionally greater weight than the forward prices. The dynamic paths of the price levels, however, remain qualitatively the same as those shown in Figure 2. 10

III. Application to Some Empirical Issues

In this section the type of empirical evidence that is compatible with the above hypothesis, as well as some implications of the model that are subject to empirical testing, is briefly suggested. The hypothesis has implications on closed as well as open economy models. In the closed economy, it can explain why some tests of the Fisher hypothesis of interest behavior have failed. It can also be used to explain, by means of the general model, why an increase in the rate of change of the money supply can bring about a decrease in the nominal interest rate. Furthermore, formulations of the demand for money that use the models set out above, rather than nonmarket-clearing type models, can be derived, and their relative merits can be subject to empirical testing. In the open economy, the hypothesis can be used to describe the purchasing power parity (PPP) relationship, as well as to provide the underpinning for models that assume that exchange rates adjust much faster than prices to excess demands.

Evidence in the Closed Economy

The observation that the long-term bond rate is related to the inflation rate has engendered a long and prolific debate among economists. The most famous explanation of this observation was proposed by Fisher, who suggested that the interest rate was equal to a constant real rate and the expected rate of inflation. Within the context of efficient capital markets, Fama (1977) proposed the following test: a regression of the inflation rate on the lagged short-term interest rate should yield (a) a slope term equal to unity and (b) additional variables that do not add to the explanatory power of the regression. This test has been shown to fail. 11 Fama retorts that this failure could be attributed to “systematic measurement errors in the data.” The above analysis suggests, however, that the measurement error is not in the collection of data but rather is inherent in the price concept used. It is suggested here that the interest rate forecasts the inflation rate in the spot markets rather than the general inflation rate. The general inflation rate was shown above to yield an unbiased but serially correlated estimate of the spot price level, and Fama’s regression should (as it does) reflect this serial correlation. Given this measurement problem, Fama’s hypothesis should be amended to propose that the addition of other variables to a regression that has past inflation rates and the lagged interest rate would not increase the explanatory power of the regression.

A related presumption is that an unanticipated increase in the rate of monetary expansion must drive the interest rate down, at least temporarily. This presumption, if correct, creates a puzzle in terms of some monetary theories that predict that the same monetary shock will cause an increase in the expected rate of inflation. The conflict is usually resolved by creating a mechanism which drives the real interest rate down. The general model postulated above provides an explanation of this counterintuitive movement of the interest rate while maintaining a constant real rate. Since the monetary shock causes the spot price level to overshoot, there is, after the shock, an expected deflation in spot markets. Given that it reflects the latter and not the general inflation rate, the interest rate decreases rather than increases. Indeed, in the extreme case, which assumes an infinite inflation rate elasticity of money, the nominal interest rate remains constant while the ex post “real” interest rate (measured by the difference between the nominal interest and general inflation rates) may be negative until equilibrium is regained. Thus, the phenomenon of “liquidity effects” on the short-term interest rate, as well as persistent ex post negative real interest rates, can be explained within the context of equilibrium models, even in the case where ex ante real interest rates, when properly measured, remain constant.

As mentioned in the introduction, one alternative to the models presented here are models that use partial stock adjustment. In what follows, an alternative formulation of the demand for money that can be empirically tested is derived. For this purpose, the backward expectations operator, Bi, is defined such that Bix(t,t) = x(t,ti). Using this operator, equation (6) can be rewritten as

or, assuming the weak expectation property that expectations of the current variable are equal to its actual value (Etx(t) = x(t,t) = x(t)) and

or, adding and subtracting (l–γ) p(t–1) and rearranging,

where Δp(t) = p(t) – p(t–l) ≈ actual rate of inflation

Δ*p(t) = p(t,t–l) – p(t–1) ≈ expected rate of inflation

Equation (27) is the discrete time representation of the behavior of the inflation rate. It states that the inflation rate is a weighted average of the expected inflation rate and the discrepancy between today’s spot price and the previous period’s price level, both in logarithms. To arrive at a continuous time representation, Turnovsky and Burmeister (1977) are followed, and equation (26) is rewritten as

rearranging, adding γp(t+h), and dividing by h on both sides,

where θ=1γγh

To get to the continuous time formulation, the observation interval h is made smaller and smaller, approaching zero at the limit. If the inflation rate ((t)) is now defined as the difference in the next moment’s price level (p(t+h)) and the current price level (p(t)) and averaged over the period h

and the expected rate of inflation (π(t)) is defined as the difference between current expectations of the next moment’s price level (p(t+h,t)) and the current price level (p(t)) and averaged over the period h

then the limit of the left-hand side of equation (29) is the difference between actual and expected inflation rates:

Taking the limit of equation (29) and substituting equation (32) gives the continuous time version of the behavior of the rate of inflation

Thus, the actual rate of inflation is equal to the expected rate of inflation plus a proportion of the discrepancy between the spot price and the general price level. 12 Thus, if the spot and general price levels are equal, the rate of inflation follows the expected path. Deviations from this path will occur when there is a discrepancy between the spot and general price levels. To arrive at an estimation equation, a money demand function similar to the asset approach formulation used in Section II is postulated, so that

where F(.) is an unspecified function, and is substituted in equation (27) or (33), depending on the estimation approach taken. The resulting model can then be estimated and compared against other models of the demand for money, if one can find an appropriate proxy for the lagged expected rate of inflation. 13 Usually, the latter can be made to depend on lagged inflation rates and perhaps other variables.

Khan (1979) argues that an often observed phenomenon is that real cash balances (measured as money deflated by the general price index) increase rather than decrease after an increase in the nominal money supply, and proposes a modified partial adjustment model that can exhibit such behavior. Such a phenomenon is a natural outcome of the models proposed here, since, except in an extreme case, the general price level does not increase proportionally after the shock.

Evidence in the Open Economy

The purchasing power parity relationship is one of the most useful and yet controversial relationships in international economics. It states that the ratio of two countries’ price levels (or the difference between their inflation rates) is equal to the exchange rate (or the rate of change of the exchange rate). There have been numerous empirical tests of this proposition, usually resulting in the finding that at best PPP is a long-run relationship. 14Magee (1978) warns that much of the evidence may be invalid because the studies neglect the contractual nature of international trade. Because it takes time to import goods, the price of imports reflects the contracted, rather than the actual, price. The implication is that PPP holds only if spot price levels are compared. The hypothesis presented in this paper suggests this argument and extends it. Contracting will influence the results even when domestic price indices are used, causing spurious deviations from PPP and making direct tests of the hypothesis difficult because of the autocorrelation in the deviations from PPP thereby induced. One possible test of PPP is to derive the PPP relationship within the context of efficient capital markets and test for market efficiency in a way similar to Fama’s test of Fisher’s hypothesis. 15 In this context, it can be shown that, given the past price history, the use of the past exchange rate history should not improve the forecasts of prices.

For the purposes here, the distinguishing feature of the open economy is that the exchange rate can act as a proxy for the ratio of two domestic spot market price levels. If one takes the monetary approach to the exchange rate, then the dynamics between excess demands and that of the exchange rate will depend on which deflator is believed to be appropriate for money demands. The path that the exchange rate takes will be the same as that of the spot price level in Figures 1 and 2. If the spot price level is the appropriate deflator, then the exchange rate moves only enough to equilibrate the relative excess money demands. If the general price level is more appropriate, the exchange rate must overshoot its long-run equilibrium value. It may be possible to determine by empirical tests which of the two views explain actual behavior better. In either case, the hypothesis presented here would predict that, as shown by recent experience, exchange rates will be more volatile than relative price levels.

IV. Conclusion

There is an increasing awareness in the economic literature that much of the observed “stickiness” of prices reflects contractual arrangements rather than some inherent market failure. By explicitly considering these contractual arrangements, macroeconomic theories have been able to explain many phenomena which could not otherwise be explained without assuming that markets were slow to react to new information. This paper shows that one can explain the sluggish behavior of prices within the context of an equilibrium theory of money demand. The essential ingredient is that one must distinguish between prices set in the spot markets and prices previously contracted. Since the general price index is a weighted average of both types of prices, it will only partly reflect new information, which may be erroneously interpreted as a lack of market clearing. Furthermore, this distinction between spot and general price levels has important implications for the expected behavior of the dynamics of inflation. Under the model in which money is deflated by the spot price level, the latter jumps to clear an excess supply of money while the general price index approaches the equilibrium level asymptotically. In this case, the usual money demand functions are misspecified because of the use of a proxy (the general price level) that exhibits systematic divergences from the true variable (the spot price level). On the other hand, if the general price level is the appropriate deflator in the money demand function, then the spot price level will overshoot its equilibrium level. In this case, the demand for money temporarily increases, since the spot prices are expected to deflate although the general price level is inflating.

This approach has wide empirical applications. It can explain the serial correlation of the inflation rate. In the closed economy it has implications for the behavior of the interest rate as well as money velocity. In the open economy, it can explain the observed “divergences” from purchasing power parity as well as the possible overshooting of the exchange rate. Empirical tests of these implications can be formulated, and further research may show the general validity of the hypothesis.


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Mr. Brillembourg, economist in the Special Studies Division of the Research Department, is a graduate of Harvard University and of the University of Chicago.

See the section, “Macroeconomics: An Appraisal of the Non-Market-Clearing Paradigm,” American Economic Review, Papers and Proceedings (May 1979), pp. 54–69.

This is the counterpart of the overshooting hypothesis of the exchange rate in the open economy with sticky prices; see Dornbusch (1976).

This assumption rules out average cost pricing of contracts, that is, the issuing of a contract that allows the transaction to take place at a fixed price and at any time from now to, say, six months hence. (The reader will recognize this to be akin to an option rather than a forward contract.) While allowing for this possibility may be more realistic in that it models sticky prices and recognizes the cost of issuing forward contracts, it does not change the substance of the argument but does complicate it greatly.

A more general formulation would recognize that these weights are determined by the behavior of the agents in the economy and, thus, cannot be taken as given. Indeed, it is likely that their distribution depends on the variability of inflation, among other things. The determination of these weights is left for future research.

Note that in both cases the alternative of holding money is holding goods; hence, the opportunity cost of holding money is the expected appreciation of goods in terms of money. The fact that individuals can issue future contracts does not change this trade-off in the aggregate.

If the deflator is equal to (1 – λ)s (t,t)+λp(t,t), the paths of the price levels are given by equations (21) and (24), where ψi = 1+β–λγi.

This equation is similar in structure to one proposed by Mussa (1976), derived under the assumption of sticky prices. Because of the different approach taken, however, the interpretation and, consequently, the use which can be made of it are quite different.

As in Section II, one can assume that the appropriate deflator for cash balances is a weighted average of both the spot and general price levels. Unfortunately, the resulting models will not be empirically distinguishable from those that deflate only by the spot price level.

See Journal of International Economics, Vol. 8 (May 1978), which is devoted to a discussion of PPP-related issues.

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