Investment and Interest Rate Policy in the Open Economy

where mc_t ≡ MMCt is real marginal cost.

PH,t

Firms set prices according to Calvo (1983), where in each period there is a constant
probability 1 — φ that a firm will be randomly selected to adjust its price, which is drawn
independently of past history. A domestic firm i, faced with changing its price at time t,
has to choose pH,t (i) to maximize its expected discounted value of profits, taking as given
the indexes P, PH, PF, Z and Z*:6

∞

max Et ]T(β^)^sXt,t+s {[pH,t(i) — MCt₊. (i)] [z∏,t₊. (i) + zH,t₊. (i)] } ,     (10)

pH,t(i)    _s=0

where

(H≠^-^-λ [ZH,+. + ZH,+,]

PH,t+s

and the firm’s discount factor is β^.Xt,t+. = [Uc(Ct+.)/Uc(Ct)][Pt/Pt+.].⁷ Firms that are
given the opportunity to change their price, at a particular time, all behave in an identical
manner. The first-order condition to the firm’s maximization problem yields

∞

λ

PH,t = λ - 1 Et    ^qt,t+sMCt+s.

The optimal price set by a domestic home firm P_H,_t is a mark-up ʌ--ɪ over a weighted
average of future nominal marginal costs, where the weight qt,t+. is given by

■. χt∙. PH.. ZH∙.    + /....

Et Σs=0(^β^sXt,t+sPH,t+s ZH+.S ⁺ ZH,t+s)

Since all prices have the same probability of being changed, with a large number of firms,
the evolution of the price sub-indexes is given by

r>1-λ   r>1-λ   I /-∣ ʌ nɪ-λ

PH,t = ΨPH,t-1 ^{+ (1 -} Ψ)PH,t

since the law of large numbers implies that 1 — φ is also the proportion of firms that adjust
their price each period.

⁶While the demand for a firm’s good is affected by its pricing decision pH_,t (i), each producer is small with
respect to the overall market.

⁷Under the assumption that all firms are owned by the representative agent, this implies that the firm’s
discount factor is equivalent to the individual’s intertemporal marginal rate of substitution.

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