Appendix I: Derivation of demand for cash, demand deposits, and money
Clearly, all three functions are increasing in c. Denoting (I-i)/I by R, we have
Appendix II: Derivation of Φ(T)
To illustrate the idea that the equilibrium MPC can be thought of as the ratio of the average effective price to the current effective price, the function Φ(T) was used in equations (23). This function and its properties are now derived. Let the function Φ(T), 0<T<∞, be defined in implicit form, by
Clearly, Φ(T) is continuous for Tϵ(0,∞) since p2≠p1. Recalling that p2<p1, it follows from (II.2) that (a) Φ′(T)>0, (b) as T→0 lim[Φ(T)]=0, and (c) as T→∞ lim[Φ(T)]=1. Hence, 0<Φ(T)<1.
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Bloomfield, Arthur I., Monetary Policy under the International Gold Standard: 1880-1914 (New York: Federal Reserve Bank of New York, 1959).
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Brock, Philip L., “Reserve Requirements and the Inflation Tax,” Journal of Money, Credit, and Banking, Vol. 21 (1989), pp. 106–21.
Calvo, Guillermo A., “Staggered Prices in a Utility-Maximizing Framework,” Journal of Monetary Economics, Vol. 12 (1983), pp. 383–398.
Calvo, Guillermo A., “Temporary Stabilization: Predetermined Exchange Rates,” Journal of Political Economy, Vol. 94 (1986), pp. 1319–29.
Calvo, Guillermo A., (1989b), “Temporary Stabilization Policy: The Case of Flexible Prices and Exchange Rates” (unpublished), forthcoming in Journal of Economic Dynamics and Control.
Calvo, Guillermo A., and Carlos A. Végh, “Monetary and Interest Rate Policy in a New-Keynesian Framework,” unpublished manuscript (1989), International Monetary Fund.
Calvo, Guillermo A., and Carlos A. Végh, (1990a), “Credibility and Dynamics of Stabilization Policy: A Basic Framework,” unpublished manuscript, International Monetary Fund.
Calvo, Guillermo A., and Carlos A. Végh, (1990b), “Fighting Inflation with High Interest Rates: The Small Open Economy Case under Flexible Prices,” unpublished manuscript, International Monetary Fund.
Calvo, Guillermo A., and Carlos A. Végh, (1990c), “Interest Rate Policy in a Staggered-Prices Model,” unpublished manuscript, International Monetary Fund.
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Hume, David, “Of the Balance of Trade,” (1752), reprinted (in abridged form) in Barry Eichengreen, ed., The Gold Standard in Theory and History (London: Methuen, 1985), pp, 39–48.
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This research was initiated while Carlos Végh was in the European Department. An earlier version of this paper was presented at the IV Annual Meeting of the Central Bank of Uruguay (Montevideo, November 6-7, 1989) and at the XXIV Annual Meeting of the Argentine Association of Political Economy (Rosario, November 8-10, 1989). The authors are grateful to Aquiles Almansi, Julio de Brum, Mohsin Khan, and conference participants for helpful comments and suggestions.
Hume (1752) provided the classical exposition of the price-specie-flow mechanism. Governments’ frequent disregard for the “rules of the game” under the Gold Standard is the focus of the seminal work of Bloomfield (1959). At the time, observers were well aware of the active use of interest rates by Central Banks. Keynes (1924), for instance, argued that one of the main characteristics of the British monetary system under the Gold Standard was “the use of the bank rate for regulating the balance of immediate foreign indebtedness (and hence the flow, by import and export, of gold)” (p. 19).
Batten et al (1989) examine the implementation of monetary policy in the G5 countries and conclude that monetary authorities focus on influencing key short-term interest rates.
McCallum (1986) distinguishes between price level “indeterminacy” (the model does not determine the value of any nominal magnitude) and “non-uniqueness or multiplicity” of price level solutions (there are multiple paths of real money balances).
Barro (1989) suggests that interest rates should play a major role in a positive theory of monetary policy because “central bankers, including those at the Federal Reserve, seem to talk mainly in terms of controlling or targeting interest rates” (p.3).
By real effects, we mean effects on consumption—which determines welfare. Permanent policies, however, will affect the real money supply.
Another independent, and more familiar, reason for the counterproductive nature of an interest rate hike in high-inflation countries is that in many cases the government is one of the main borrowers. Thus, the policy tends to worsen the fiscal situation even further.
The use of the cash-in-advance constraint in continuous-time models is discussed in Feenstra (1985).
The sum of cash and demand deposits will be referred to as “money” (M); that is, M=H+Z.
The incorporation of demand deposits into the cash-in-advance constraint can be found in Walsh (1984). Brock (1989) assumes that both assets reduce transaction costs or “shopping” time. In both cases, the two assets are posited to be imperfect substitutes. Englund and Svensson (1988) distinguish between “cash” goods and “check” goods, both of which are subject to liquidity constraints. Demands for both cash and demand deposits have been derived in a Baumol-Tobin context by Santomero (1979) and Whitesell (1989). Given the different costs of using cash versus debitable accounts, cash is used for small transactions while debitable accounts are used for large transactions (see Whitesell (1989)).
A subscript i on a function denotes the partial derivative with respect to its ith argument.
The assumption lhz>0 equivalent to ruling out perfect substitutability between h and z. (Notice that if lhz=0, then, by Euler’s theorem, lhh=lzz=0, in which case l(h,z) is a linear function.) As discussed below, the analysis still applies to the case of perfect substitution but it involves a corner solution in which only demand deposits are used.
It will be referred to as the pure nominal interest rate to distinguish it from it, which will be referred to simply as the nominal interest rate.
It has been assumed that β=r to ensure the existence of a steady-state. This implies that there are no intrinsic dynamics in the model so that all dynamics will result from the implementation of temporary policies.
The constraint (2) holds with equality at an optimum given that it has been assumed that both I and (I-i) are positive.
To see how this increased demand for money comes about, consider the following example. Suppose that the liquidity-services production function is Cobb-Douglas with equal weights; that is, l(h,z)=(hz)½. Then, c=(hz)½ and z/h=I/(I-i) so that z>h. An increase in i results in z increasing by the same proportion than h falls because consumption is given. Because z>h, the demand for z rises by more than the demand for h falls thus increasing the demand for m.
Notice that if lhz=0 equation (13) does not apply because h and z would be perfect substitutes, as indicated earlier. In that case, the consumer uses only demand deposits because they have a lower opportunity cost than cash. The effective price of consumption is 1+I-i (assuming l(h,z)=h+z) so that an increase in i decreases the effective price of consumption.
Alternatively (as shown below), the authorities can be viewed as issuing only high-powered money and determining the interest rate paid on demand deposits by controlling reserves requirements, given that I is exogenously given.
For notational simplicity, no superscripts are introduced to denote equilibrium values.
Note that a constant path of the effective price constitutes the only case in which the equilibrium MPC is unity for all t. If the effective price varies over time, the equilibrium MPC may be unity for some t but it cannot be unity for all t.
If cash and demand deposits were modeled as reducing transaction costs, as in Brock (1988), then changes in i would affect available resources. In this paper, by adopting the liquidity-in-advance specification, we abstract from such effects in order to concentrate exclusively on intertemporal price speculation effects.
It should be clear that, given that permanent changes in i have no real effects, an anticipated decrease in i would have the same real effects as a temporary increase in i.
In the context of anticipated devaluations, an anticipated discontinuity is possible only if there is no capital mobility (see Calvo (1989a)).
In the Cobb-Douglas case (c=hαz1-α, where z/h=[(1-α)/α][I/(I-i)], it can be shown that the fall in the effective price of consumption for a given increase in i is greater the larger is 1-α until 1-α reaches a value of 0.684, at which point the ratio z/h equals 4.3 (assuming I/(I-i)=2)). (The liquidity-services production function needs to be specified for this exercise because third derivatives are involved.)
Naturally, the discontinuity of c(T) at T=0 does not imply that the welfare loss is discontinuous at that point. In technical terms, the integral is continuous if the integrand is piecewise continuous.
This equivalence would not hold under flexible exchange rates, however, because the pure nominal rate is not exogenously given.