Interest Rate Pegging and Price Level Indeterminacy

In the representative household models with money in the utility function and cash-in-advance constraints (such as the ones we have analyzed so far), one can show that if monetary policy pegs the nominal interest rate, then the price level and the money supply cannot be determined uniquely.

This result is known as price level indeterminacy and was first alluded to by Wicksell [1898] in the context of a static monetary model, and, more recently, by Sargent and Wallace [1975], in the context of an IS-LM macro model with rational expectations. The same indeterminacy applies in the representative household dynamic general equilibrium models with money that we have examined so far. For simplicity, we will assume that there is no uncertainty and that the nominal interest rate is pegged by the central bank to a constant level i₀.

12.8.1 Interest Rate Pegging and Price Level Indeterminacy in Representative Household Models

From the money demand function of the representative household model with money in the utility function, the money demand equation implies that

In (12.71) we have imposed the equilibrium condition in the goods market: C_t = Y_t. Given that real income Y_t is exogenous, this condition is satisfied for an infinite number of combinations of M_t and P_t. For given Y_t, if it is satisfied for M₀ and P₀, it is also satisfied for λM₀ and λP₀, for any λ. Thus, both the money supply and the price level are indeterminate.

The reason for the indeterminacy is that, under interest rate pegging, no monetary anchor can determine the price level, as in the case where the central bank determines the money supply.

The same problem arises in the cash in advance representative household model. From the Euler equation for consumption (12.42) and the goods market equilibrium condition C_t = Y_t for every t, it follows that

Because income is exogenous and the nominal interest rate is fixed, (12.72) is satisfied for any price level. Multiplying P_t and P_t+1 by a coefficient λ means that (12.72) continues to be satisfied, because it is linearly homogeneous in prices. The price level is thus indeterminate.

Again, the reason for the indeterminacy is that, under interest rate pegging, there is no monetary anchor that can determine the price level, as in the case where the central bank determines the money supply. Neither the price level nor the money supply can be identified uniquely under interest rate pegging. The equilibrium condition for consumption can be satisfied with both high prices and a consequent high stock of money, and with low prices and a consequent low stock of money (i.e., virtually for any level of prices).

This, indeterminacy is especially problematic, because the key monetary instrument used by most central banks has not been the money supply, but nominal interest rates. How is it possible for the price level to be determinate if interest rates are the key monetary instrument of central banks?

12.8.2 The Wicksell Solution to the Problem of Price Level Indeterminacy

The main solution of this problem was provided by the monetary economist who first realized its existence, namely Wicksell [1898, p.

So long as prices remain unaltered, the banks’ rate of interest is to remain unaltered. If prices rise, the rate of interest is to be raised; and if prices fall, the rate of interest is to be lowered; and the rate of interest is henceforth to be maintained at its new level until a further movement of prices calls for a further change in one direction or the other.

Wicksell’s interest rate rule can be written as

where P₀ is the target price level of the central bank, and ϕ a positive coefficient denoting the response of the central bank to deviations of the actual price level from the target price level.¹²

Substituting (12.73) for i₀ in (12.72), assuming that output is constant and taking logs, we get

If ϕ is positive, as Wicksell proposed, then (12.74) is a stable difference equation that fully determines the price level. The price level is uniquely defined, as it adjusts immediately to P₀, the target price level of the central bank. If ϕ is equal to zero, then we have price level indeterminacy, as in equation (12.72).¹³

Wicksell’s rule is a good example of a stabilizing interest rate rule that makes the nominal interest rate a function of endogenous variables, such as the price level, about which the central bank is concerned. The nominal interest rate is not pegged but is a positive function of the price level. Such an interest rate rule does not result is price level indeterminacy. Instead, it stabilizes the price level around the target price level of the central bank.

Alternative ways to solve the problem of price level indeterminacy when the policy instrument of the central bank is the nominal interest rate have subsequently been proposed: inflation targeting rules; nominal income rules (McCallum [1988]); and more recently, the Taylor [1993] rule, which is a generalization of the Wicksell rule.

12.8.3 The Fiscal Theory of the Price Level

One theoretical development worth mentioning in this context is the fiscal theory of the price level, proposed in Leeper [1991], Sims [1994], and Woodford [1994, 1995]. This theory argues that even if monetary policy rules do not suffice to determine the price level, the price level can be determined as the level ensuring that nominal government debt does not follow an explosive path. This requirement is sufficient in such models to determine the price level. The path for the price level ensures a path of nominal government debt that satisfies the intertemporal budget constraint of the government.

12.8.4 The Pigou Effect and Price Level Determinacy in OLG Models

Note that the problem of price level indeterminacy under interest rate pegging does not arise in OLG models. Unlike the representative household model, where both the current and the future price level are non-predetermined variables, in the OLG models, the price level is determined through the predetermined nominal financial assets of old households. These function as a monetary anchor and help determine the price level.

For example, in the cash in advance version of the Samuelson OLG model, the equilibrium condition in the goods and services market is given by (12.55). Assuming that the nominal interest rate is pegged at i₀ by the central bank, one can solve (12.55) for the price level:

The price level is uniquely defined. Because consumption depends on the real value of financial assets of the old households and these assets are positive, the price level is determined and positive, regardless of the interest rate policy of the central bank.

In static ad hoc monetary models, such dependence of consumption on the financial wealth of households has been called the Pigou effect (see Pigou [1943]), or the real balance effect (see Patinkin [1956]). As Sargent and Wallace [1975] indicated in their original analysis of price level indeterminacy, in the presence of a Pigou or real balance effect, the problem of price level indeterminacy does not arise, even if the central bank pegs the nominal interest rate at a constant level, because nominal financial assets anchor the price level.

12.9