# A note on the specification of the demand for money in open economies.

I. Introduction

Open economy models which make a distinction between traded and non-traded goods have been widely used to analyze a variety of issues. For simplicity, in many cases the real money stock is defined by using as the relevant deflator the price of either of those goods. Moreover, this is true both for the case of models in which the demand for money is derived via explicit utility maximization, as in [6; 7] and for models with an "ad-hoc" demand for money [3; 4; 5].

The purpose of this brief note is to show the extent to which the choice of the deflator can alter the adjustment of a monetary economy to various shocks in a qualitative, and hence important, way. As it is shown below, while the qualitative response to monetary innovations will in general not depend on the choice of the numeraire, the same is not true for the response to changes that modify the relative price between traded and non-traded goods (i.e., the "real exchange rate"). For the latter type of changes, different choices of the deflator can mean the difference between monetary accumulation and a temporary balance of trade surplus, and a fall in the real money stock and a temporary balance of trade deficit, both as a response to the same exogenous change.

In sections II and III we present a very simple model and provide examples. In section IV we discuss the implications and justify the importance of the issue.

II. The Model

The main points can be demonstrated in a small open economy model with traded and non-traded goods.(1) Normalizing the exogenous foreign price of the traded good to unity, the domestic traded good price is equal to the nominal exchange rate, E. The deflator used in defining the demand for real money balances is given by the index

[Mathematical Expression Omitted],

where 0 [less than or equal to] [Sigma] [less than or equal to] 1 and [P.sub.H] is the domestic currency price of the non-traded good. This index can also be expressed as P = [[Epsilon].sup.[Sigma]][P.sub.H], where [Epsilon] is the relative price of the traded good in terms of the non-traded good and is defined as the real exchange rate. The price of the non-traded good is perfectly flexible so the market clearing condition,

[x.sub.H] = [c.sub.H], (2)

is always satisfied, where [x.sub.H] is non-traded good output and [c.sub.H] is non-traded good consumption.

Individuals are identical and infinitely lived, and the representative individual maximizes the functional

[integral of] U([c.sub.T],[c.sub.H],m)[e.sup.-[Delta]t] dt between limits of [infinity] and 0, (3)

where [Delta] is the constant rate of discount, [c.sub.T] is traded good consumption, and m is the stock of real money balances, defined as the nominal money stock deflated by the price index (1). It is assumed that the utility function is strictly concave and, for simplicity, separable in real money balances and both consumption goods.

The individual receives fixed flows of the traded good and the non-traded good,(2) holds domestic real money balances as the only asset, and receives lump sum government transfers, [Tau]. The individual's flow budget constraint, expressed in terms of the traded good, is

[Mathematical Expression Omitted],

where [x.sub.T] is traded good output and [Mathematical Expression Omitted]. It is assumed that the government controls the path of the nominal exchange rate and rebates the cost of maintaining real cash balances through the lump sum transfers:

[Tau] = [[Epsilon].sup.[Sigma]-1]m[Pi].(3) (5)

The maximization of (3) by choice of [c.sub.T], [c.sub.H], and m, subject to (4) and the initial value of m, along with (2), (4), and (5), yields the following:

[Epsilon] = [U.sub.T]/[U.sub.H] (6)

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the rate of revaluation set by the government. Equation (6) determines the real exchange rate where, for normal consumption goods, [Delta][Epsilon]/[Delta][c.sub.T] [less than] 0 and [Delta][Epsilon]/[Delta][c.sub.H] [greater than] 0. Equation (7) is the balance of payments equation. Since there is no capital mobility, the real money stock varies with the trade account. The steady state conditions from (7) and (8) are as follows:

[Mathematical Expression Omitted]

[Mathematical Expression Omitted],

where * denotes steady state values. Equation (9) indicates that in the steady state the trade account is balanced. Equation (10) implicitly defines a reduced form, steady state demand for real money balances of the form

[Mathematical Expression Omitted].

Differentiation of (10) with respect to [[Pi].sub.T], [Mathematical Expression Omitted], and [Mathematical Expression Omitted] yields

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted].

The sign of (11) is negative, while the signs of (12) and (13) are ambiguous for [U.sub.TH], [U.sub.HT] [greater than] 0. An increase in the consumption of one good affects the real exchange rate (recall [Delta][Epsilon]/[Delta][c.sub.T] [less than] 0 and [Delta][Epsilon]/[Delta][c.sub.H] [greater than] 0), causing a substitution effect away from consumption of the other good. This implies opposing effects on the demand for money. For long run consumption of either commodity and the long run real money stock to be positively related, these substitution effects must be of smaller magnitude than the direct effects of consumption on money demand, which will be the case if

[Sigma][U.sub.TH] [less than] -(1 - [Sigma])[Epsilon][U.sub.HH] (14)

and

(1 - [Sigma])[U.sub.HT] [less than] -[Sigma][U.sub.TT]/[Epsilon]. (15)

With the simple model presented above, the implications of different "definitions" of the real money stock (i.e., of different deflators or "numeraires") can be seen by considering different values of [Sigma]. The cases of [Sigma] = 0 and [Sigma] = 1 are of particular interest because they are common simplifying assumptions employed in open economy models. An important characteristic of the money demand function derived above is that [Sigma] = 0 implies that [Mathematical Expression Omitted] and [Mathematical Expression Omitted], while [Sigma] = 1 implies that [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Since the effects of shocks to the economy on important variables (the trade account, the real exchange rate, etc.) may in part depend on the effects on real money demand, the choice of numeraire in the definition of real money demand is important. Examples presented below serve to clarify and illustrate this point.

III. Examples

The system (7) and (8) will be used to evaluate the effects of two exogenous changes: an increase in the rate of devaluation and an increase in the flow of non-traded good output. It is straightforward to show that this system is saddle path stable around the steady state. The phase diagram of this system is shown in Figure 1.(5)

Assume the system has attained the steady state defined in (9) and (10) and consider an increase in the rate of devaluation, [[Pi].sub.T]. By (2), (6), and (9), the steady state real exchange rate is unaffected, so the opposing effects on steady state money demand discussed above to not occur. By (11), [Delta]m*/[Delta][[Pi].sub.T] [less than] 0 for any value of [Sigma], and the adjustment of the system will be qualitatively similar, regardless of the price used to deflate nominal money balances in the utility function: there is a temporary trade deficit and, by (6), a temporary reduction in the real exchange rate. The adjustment is shown in Figure 1: if the initial steady state equilibrium was at a point such as point b, the result is an initial increase in [c.sub.T] to the indicated saddle path, and a gradual fall of both [c.sub.T] and the real money stock.

IV. Conclusions

Why is this point relevant? In the first place, it conveys a general-message of caution about the use of the deflator used for the real money stock, in particular in the specification of models not derived explicitly from utility maximization: for certain changes, the effects can be qualitatively different. Secondly, one should notice that there are two different questions involved. The first is the typical index number problem, i.e., which is the correct specification of the general price level. This is not strictly a question in monetary theory. In the case of form (1), which is the value of the coefficient [Sigma]. The second is a very different question which pertains to monetary theory, and this is which should the deflator for the money stock be. The inclusion of the real money stock in the utility function, per se, does not solve the problem, unless there is a further specification on the nature of the "utility" provided by money, which would then indicate the correct deflator to be used. One rationalization for such an inclusion is that money saves in transaction costs; on this account, there are conceivable scenarios which would allow to argue that most transactions take place in either home or traded goods, and this could make it plausible to specify a coefficient [Sigma] equal to zero or to unity. Another possible rationale for money providing utility is related to the "store of value" function of money; on this account, it would seem logical to use the general price level as the appropriate deflator. In other words, the choice of the deflator implies, in a fundamental sense, the choice of a theory of why money is held.

Leonardo Auernheimer Texas A&M University College Station, Texas

Michael A. Ellis Kent State University Kent, Ohio

1. This model is a simplified version of the model presented in Auernheimer [1].

2. Assuming fixed outputs reflects the assumption that there are significant lags in adjusting outputs to changes in relative prices.

3. This is one of the simplest among other possible assumptions concerning the uses of seigniorage and the inflation tax. See, for example, Calvo [2].

4. The terms [U.sub.T] and [U.sub.H] denote partial derivatives of the utility function with respect to traded good consumption and non-traded good consumption, respectively.

5. To avoid clutter, in the "experiments" that follow the locus [Mathematical Expression Omitted] indicated in Figure 1 is assumed to be, in each case, the locus relevant after the change, with point a or point b being the initial equilibrium point.

6. The initial increase in both traded and non-traded good consumption has opposing effects on the real exchange rate, but the real exchange rate must increase to clear the non-traded good market.

References

1. Auernheimer, Leonardo. "On the Significance of Foreign Debt: Some Fundamentals," in International Indebtedness, edited by M. Borchert and R. Schinke. London: Routledge, 1990, pp. 35-50.

2. Calvo, Guillermo, "Devaluation: Levels Versus Rates." Journal of International Economics, May 1981, 165-72.

3. ----- and Carlos A. Rodriguez, "A Model of Exchange Rate Determination Under Currency Substitution and Rational Expectations." Journal of political Economy, June 1977, 617-25.

4. Dornbusch, Rudiger, "Expectations and Exchange Rate Dynamics." Journal of Political Economy, December 1976, 1161-76.

5. -----, "Devaluation, Money and Nontraded Goods." American Economic Review, December 1973, 871-80.

6. Engel, Charles, "The Trade Balance and Real Exchange Rate under Currency Substitution." Journal of International Money and Finance, March 1989, 47-58.

7. Livitan, Nissan, "Monetary Expansion and Real Exchange Rate Dynamics." Journal of Political Economy, December 1981, 1218-27.