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(a) Derive the optimality conditions for the household's problem and log-linearise them around the zero-inflation steady state. Pro- vide an intuitive interpretation for each of them. (b) Derive the first-order condition of the firm's problem, log-linearise it and provide an intuitive interpretation.

(a) Derive the optimality conditions for the household's problem and log-linearise them around the zero-inflation steady state. Pro- vide an intuitive interpretation for each of them. (b) Derive the first-order condition of the firm's problem, log-linearise it and provide an intuitive interpretation.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(b) Derive the first-order condition of the firm's problem, log-linearise it and provide an intuitive interpretation. (c) Show that in equilibrium: E1-1 {yt} = -y+E,-1 {az} where y = (d) Recall that: m - Pi = Y - ni Guess that i = p for all t. Show that under this guess the following conditions hold: Yt = -y+ a4-1+ E" nų = -y+ e" - %3D Remark: If you do not manage to show that the first equation holds, you can still use it and show that the other equations hold. (e) Assume that the guess i, = p in part (d) is correct. Explain the effects of a positive monetary policy shock ɛ" > 0 and com- pare them with the VAR evidence on monetary policy shocks discussed in the lectures.
Transcribed Image Text:(b) Derive the first-order condition of the firm's problem, log-linearise it and provide an intuitive interpretation. (c) Show that in equilibrium: E1-1 {yt} = -y+E,-1 {az} where y = (d) Recall that: m - Pi = Y - ni Guess that i = p for all t. Show that under this guess the following conditions hold: Yt = -y+ a4-1+ E" nų = -y+ e" - %3D Remark: If you do not manage to show that the first equation holds, you can still use it and show that the other equations hold. (e) Assume that the guess i, = p in part (d) is correct. Explain the effects of a positive monetary policy shock ɛ" > 0 and com- pare them with the VAR evidence on monetary policy shocks discussed in the lectures.
2. A model with price setting in advance Consider an economy where the representative household maximises EB' Eo{U (C, N,)} 0 < 3< 1 t-0 with C; denoting the CES aggregator of quantities consumed of the differentiated goods and N, standing for employment. Each house- hold maximises utility subject to a sequence of budget constraints of the form: PC+ Q,B = B-1 + W,N + D where P is the aggregate price index, W, denotes the nominal wage, BỊ stands for the stock of nominal bonds (which are in zero net sup- ply), Q. represents the price of the bond and D, are firm dividends. Assume that the per-period utility function is given by: U (C, N.) = log C -N+ 1+p Firms are monopolistically competitive, each producing a different good whose demand is given by Y,(i) = (30) Y,. Each firm has access to a linear production function: Y,(i) = A,N,(i) where productivity evolves according to a random walk, namely: a = a1-1 + e with a = log A,, e being i.i.d. and E{e1} = 0 for all t. In each period, firms set their prices in advance, i.e., before the realization of shocks. Under this assumption, they will choose a price P(i) in order to maximise the discounted profit: (P(i), Y;(i) P Et-1 P A subject to the demand schedule, where A-1,1 = B is the relevant stochastic discount factor. Finally, the money supply, M, is also a random walk which varies exogenously according to: m = m-1 + E" with m, = log M,, E" being i.i.d. and E,{ɛ"} = 0 for all t. (a) Derive the optimality conditions for the household's problem and log-linearise them around the zero-inflation steady state. Pro- vide an intuitive interpretation for each of them. (b) Derive the first-order condition of the firm's problem, log-linearise it and provide an intuitive interpretation.
Transcribed Image Text:2. A model with price setting in advance Consider an economy where the representative household maximises EB' Eo{U (C, N,)} 0 < 3< 1 t-0 with C; denoting the CES aggregator of quantities consumed of the differentiated goods and N, standing for employment. Each house- hold maximises utility subject to a sequence of budget constraints of the form: PC+ Q,B = B-1 + W,N + D where P is the aggregate price index, W, denotes the nominal wage, BỊ stands for the stock of nominal bonds (which are in zero net sup- ply), Q. represents the price of the bond and D, are firm dividends. Assume that the per-period utility function is given by: U (C, N.) = log C -N+ 1+p Firms are monopolistically competitive, each producing a different good whose demand is given by Y,(i) = (30) Y,. Each firm has access to a linear production function: Y,(i) = A,N,(i) where productivity evolves according to a random walk, namely: a = a1-1 + e with a = log A,, e being i.i.d. and E{e1} = 0 for all t. In each period, firms set their prices in advance, i.e., before the realization of shocks. Under this assumption, they will choose a price P(i) in order to maximise the discounted profit: (P(i), Y;(i) P Et-1 P A subject to the demand schedule, where A-1,1 = B is the relevant stochastic discount factor. Finally, the money supply, M, is also a random walk which varies exogenously according to: m = m-1 + E" with m, = log M,, E" being i.i.d. and E,{ɛ"} = 0 for all t. (a) Derive the optimality conditions for the household's problem and log-linearise them around the zero-inflation steady state. Pro- vide an intuitive interpretation for each of them. (b) Derive the first-order condition of the firm's problem, log-linearise it and provide an intuitive interpretation.
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