INTEREST-RATE RULES: Framework for Analysis 3

The source of the real effects of monetary policy in our model is that prices do not adjust immediately to shocks. Following Calvo (1983), we assume that prices are changed at exogenous random intervals.4 Specifically, a fraction (1 — a) of sellers get to choose а new price at the end of any given period, wThereas the others must continue using their old prices. Of those who get to choose a new price, a fraction 7 start charging the new price at the beginning of the next period. The remaining fraction (1 — 7) must wait until the following period to charge the new price or, put differently, they must post their prices one quarter in advance. These assumed delays explain why no prices respond in the quarter of the monetary disturbance and why the largest response of inflation to a monetary shock takes place only twro quarters after the shock so.

Let p\ denote the price set by sellers that decide at date t — 1 upon a new price to take effect at date t, and p’f the price set by sellers that decide at date t — 2 upon a new price to take effect only two periods later. These prices are chosen to maximize the contributions to expected utility resulting from sales revenues on the one hand, and the disutility of output supply on the other, at each of the future dates and in each of the future states in which the price commitment still applies. This means that pJ is chosen to maximize
over p. Here we have substituted the demand function (1.3) into the household’s objective function, written XT for the marginal utility (in units of period T utility flow) of additional nominal income during period T, and assumed that revenues each period are taxed at the constant rate r.5 The factor aT-t appears as the probability that the price that is charged beginning in period t is still in effect in period T > t (where we assume that this contingency is independent of all aggregate disturbances). Note that our assumption of complete contingent claims markets (including full opportunities for households to insure one another against idiosyncratic risk associated with different timing of their price changes) implies that the marginal utility of income process {A^} is the same for all households, and can be treated as an exogenous stochastic process by an individual household (whose pricing decisions will have only a negligible effect upon aggregate prices, aggregate incomes, and aggregate spending decisions). Similarly, an individual household treats the processes {Pt,Yt} as exogenous in choosing its desired price. The optimizing choice of p\ then must satisfy the first-order condition
As before, we wish to log-linearize this equilibrium condition around a steady state in which Yt = Y, Pt/Pt_i = 1, p\/Pt = 1, and AtPt = A at all times. (The requirement that these constant values satisfy (1.13) when = 0 at all times determines the steady-state value P.6 ) Percentage deviations of vy(yi]€t) from its steady-state value can be written as u(y{ — Yt), where uo = vyyY/vy> with partial derivatives evaluated at the steady state, yi = log(yl/Y), and Yt is a certain linear function of Using this notation, the log-linear approximation to (1.13) takes the form
is a composite exogenous disturbance