are irrational expectations rational? David Demery Nigel W. Duck1 ...

Are Irrational Expectations Rational? 14average error processesTXTX∆mc j,t = α i ε t−i + γ i u j,t−ii=0i=0where ε t is the shock that is common to all firms and u j,t is the idiosyncraticshock. The framework outlined in the first section of this paper then becomesrelevant.We assume, following Calvo (1983), that each firm resets its price withprobability 1 − θ each period, where θ is independent of the time elapsedsince the last adjustment. When resetting its price the firm aims to maximizeits expected discounted profits subject to the constraints imposed bytechnology, the wage rate and the possibility (defined by θ) thatitmayresetprice at some future date.We write the discounted value of the firm’s expected future profits fromsetting the price P ∗ j,1 as:E j,t DPF =∞X0(βθ) i E j,thP ∗ j,1 − g MC j,t+ii " P ∗ j,1P t+i# −φY t+iwhere MC g is the level (not the log) of the firm’s marginal costs. Therefore:∞X hE j,t DPF = (βθ) i E j,t Pj,1 ∗(1−φ) − MC g j,t+i P ∗−φ i " #Y t+ij,10P −φt+iAssume that Y t+i =[1+g] i Y 0 and P t+i =[1+π] i P 0 ; and normalize Y 0 sothat =1. Then maximizing E j,t DPF with respect to Pj,1 ∗ yields10Y 0P −ϕ0· ¸Pj,1 ∗ φ" Ã !#1+g∞" #Xi= 1 − βθφ − 1(1 + π) −φ (βθ) i 1+g0 (1 + π) −φ E g j,t MC j,t+i·So, writing (βθ) as z, we can write that for any expected stream¸1+g(1+π) −φof marginal costs E g j,1 MC j,t+i the optimal price is:· ¸Pj,1 ∗ φ∞X= [1 − z] z i E j,1 MCφ − 1g j,t+i0and the associated expected discounted stream of profits from this price is"· ¸#φ∞X1−φE j,1 DPF 1 =[1 − z] z i E j,1 MCφ − 1g 1j,t+i .1 − z0"· ¸#φ∞X−φ− [1 − z] z i E j,1 MCφ − 1g X ∞j,t+i z i E j,1 MC g j,t+i00h i 10 1+gAssuming that 0 < (βθ)< 1(1+π) −φ