Staggered prices in a utility-maximizing framework

Calvo analyzes the implications of assuming that individual nominal prices are not continuously revised and that price revisions across firms are non-synchronous, building a model along the lines of Phelps (1978) and Taylor (1979, 1980) but with a price-setting technology he describes as analytically more tractable and not dependent on the existence of explicit nominal contracts (Introduction, pp. 383-384). The key methodological improvement is enriching the demand side: rather than assuming aggregate demand is simply an increasing function of real money balances (as in earlier staggered-contract models, which mostly ignored interest-rate effects), Calvo derives demand from the optimizing behavior of Sidrauski-Brock infinitely-lived households under perfect foresight, making the model directly comparable to flexible-price optimizing frameworks such as Brock (1974), Fischer (1979), and his own earlier work (Section 1, p. 384).

Each of a continuum of identical firms can change its price only at the moment it receives a random “signal”; the probability (density) of receiving that signal h periods after the last one is δe^(-δh), independent of when the last signal arrived and independent (stochastically) across firms, which implies an expected length of a price quotation of 1/δ (Section 2, eq. 1, p. 385). Unlike earlier models that introduce an explicit cost of changing prices to determine optimal timing, Calvo – following Phelps and Taylor – treats the timing of price revisions as exogenous, an assumption he later notes (footnote 4) is not fully satisfactory in principle but is not very restrictive in practice, since all the paper’s results hold for any constant hazard rate δ > 0 (Section 2, p. 385).

A firm revising its price at time t sets a (log) price quotation V_t equal to a discounted integral, weighted by the probability the quotation survives to each future date, of the sum of the expected future average market price and a term proportional to expected future excess demand, so that V_t is increasing in both the average price charged by competitors and in expected future excess demand (Section 2, eq. 2, p. 386). Perfect foresight is a viable assumption despite the individual-level uncertainty about when a given firm’s next signal will arrive, because that timing uncertainty is assumed to wash out in the aggregate once there is a continuum of firms (footnote 2, p. 384), which is what allows V_t to be written as a function of actual (rather than merely expected) future values.

The aggregate (log) price level P_t is defined as the weighted average of all outstanding price quotations V_s, weighted by the surviving “number” (measure) of firms still charging each vintage, which Calvo shows – appealing to the law of large numbers over the continuum of firms – reduces to the integral formula P_t = δ∫ V_s e^(-δ(t-s)) ds (Section 2, eq. 3-4, pp. 386-387; proved in discrete time in the Appendix). Because P_t at any instant depends only on past price quotations, it is a predetermined variable that cannot jump discontinuously, in contrast to V_t, which is forward-looking and can jump whenever new information about the future path of P and excess demand arrives (Section 2, p. 386).

Differentiating the price-setting and price-level equations with respect to time, Calvo obtains Π̇_t = -bE_t (with b = δ²β > 0), meaning that the acceleration of inflation – not the inflation rate itself – is a decreasing function of excess demand E_t; he explicitly contrasts this with more naive Phillips-curve models in which the inflation rate itself is assumed to be an increasing function of excess demand (Section 2, eqs. 5-8, p. 387). Calvo notes, however, that this does not contradict the conventional relationship at the level of observed equilibrium outcomes: along an equilibrium path of the full model, inflation and excess demand can still end up positively associated, “in line with Taylor’s papers,” even though the deeper structural relationship implied by staggering runs through the second derivative of the price level (Section 2, p. 387).

Households are modeled as a Sidrauski-type representative family that maximizes the discounted integral of instantaneous utility from consumption and real money balances, subject to a flow budget constraint in which money holdings grow with income net of consumption, inflation-adjusted money balances, and lump-sum subsidies (Section 3, eqs. 9, 12-13, pp. 388-389). Applying optimal control methods, Calvo derives the standard conditions that the marginal utility of consumption equals the shadow value of wealth, and that this shadow value evolves according to the marginal utility of real balances and the sum of the discount rate and expected inflation; combining these yields a differential equation for consumption growth as a function of real money balances, the discount rate, and inflation (Section 3, eqs. 14-15, p. 390).

Because staggered pricing implies a non-degenerate distribution of prices across firms at any moment, the model needs a rule for how demand is allocated among differently priced firms; Calvo resolves this by assuming a costless “Price Regulating Mechanism” (PRM) that taxes firms whose price V exceeds the average P and subsidizes those whose price is below it (with penalties that grow stiffer the larger is excess demand), so that every consumer ends up paying the same uniform effective price, equal to the geometric mean of all outstanding quotations (Section 3, pp. 388-389). Calvo is explicit that the PRM is a simplifying device meant to isolate the aggregate analysis from microeconomic details of customer-firm matching that he judges not essential to the paper’s central questions (Section 3, p. 389).

Under a constant money growth rate and constant government spending (the “benchmark case”), Calvo shows the model reduces to a three-equation dynamic system in consumption, real money balances, and inflation, and proves in the Appendix that the linearized system around the steady state has exactly one negative and two positive-real-part characteristic roots, which guarantees existence of a unique saddle-path converging to the steady state and that convergence is monotonic (Section 3, eqs. 20-24; Appendix A.2, pp. 390-391, 396-397). This convergent path is depicted with a phase diagram in (m, c) space in which consumption is shown to be an increasing function of real balances along the equilibrium path, and the region of the diagram to the right of the steady-state money stock corresponds to demand exceeding capacity output, which the paper excludes from its main analysis by construction (Section 3, fig. 1, p. 392).

Starting from an excess-supply situation, Calvo shows that a once-and-for-all unanticipated increase in the money supply can move the economy directly to full employment, and that setting the rate of money growth so that the marginal utility of real balances is driven to zero at the steady state (the Friedman-rule-like condition v’(m) = 0) attains the socially optimal quantity of money (Section 4, eqs. 28-29, p. 393). By contrast, closing the same output gap through a permanent increase in government spending also restores full employment and constant inflation, but requires steady-state private consumption to fall permanently below what it would have been under the optimal monetary policy, so that fiscal expansion, while an effective demand-management tool, is Pareto-inferior to the monetary alternative in this model (Section 4, eqs. 30-38, pp. 393-394).

Calvo shows that if the monetary authority fixes the nominal interest rate at a constant target rate, the model’s dynamic system in consumption and inflation admits a whole continuum of paths (c_0, Π_0) consistent with convergence to a steady state, so that the initial values of consumption and inflation are indeterminate even though they are, in principle, free to jump at time zero (Section 4, eqs. 39-40, fig. 2, pp. 394-395). Calvo emphasizes that this result is notable because it extends the price-level indeterminacy Sargent and Wallace (1975) had found under an interest-rate peg in an ad hoc flexible-price model to a setting where the price level is by construction a predetermined, non-jumping variable – showing that the earlier indeterminacy result was not simply an artifact of assuming perfectly flexible prices (Section 4, p. 394-395).