Answer Happy

2 New-Keynesian Phillips curve Derive the standard New-Keynesian Phillips curve in the following steps. Here we assume Calvo (1983)- type price stickiness. The final-good firm produces output Y, by choosing a combination of intermediate inputs {Yf,t} so as to maximize profit Passio subject to the production technology PtYt - 1+X x₁ = { [ ' v , ^ } ¹*^. Yt f,t (3)

Similarly, we can show that the price index is given by = {/' Pitar}^^. Pt = Facing the consumption-good firm's demand, each intermediate-good firm sets the price of its product on a staggered basis à la Calvo (1983). In each period, a fraction 1 - 0 € (0, 1) of intermediate-good firms reoptimizes prices, while the remaining fraction remains prices unchanged. Then, firms which reoptimize prices in the current period maximize the expected profit, Et Σi pi Pf,t – Wt+jYf,t+j; Pt+j j=0 subject to the demand function obtained in Problem 1. Wt is the real wage at t. (5)

5. Show that we can obtain equation (6) in the following steps. First, write the equation of Problem 4 for the period of t + 1 instead of t. Second, calculate (the equation of Problem 2.4 for t) minus {(the equation of Problem 4 for t + 1) times 03}. Third, simply the equation without using Et Σj=0· ………. where t = 03 0 = Pt - Pt-Wt+ Et (Pt - Pt+1). 1- ᎾᏰ 6. Log-linearize equation (4) and show that we can obtain equation (7). 0 1 0 Pt/Pt-1 represents the inflation rate and represents its log-linearized form. √πt = Pt - Pt, 7. Derive the New-Keynesian Phillips curve using equations (6) and (7). (7)