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Explain the following Question 2 (50 points) This question considers the macroeconomic effects of a collapse in consumer demand in the New Keynesian model. There

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Question 2 (50 points) This question considers the macroeconomic effects of a collapse in consumer demand in the New Keynesian model. There are a continuum of identical households. The representative household makes consumption (C) and labor supply (N) decisions to maximize lifetime expected utility: (1) 1=0 subject to their budget constraint: C+ + Be = we Ne + (1 + 2-1) P. (2) where we is the real wage, N, is hours worked, B, are real bond holdings at the end of period t, it-1 is the nominal interest rate paid between t - 1 and t, P is the price of the final consumption good and D, are real profits from firms that are distributed lump sum. As usual, 0 0 and o > 0. Z, is a household preference shock, which is a way of generating shocks to demand. The production side of the model is the standard New Keynesian environment. Monopolistically competitive intermediate goods firms produce an intermediate good using labor. Intermediate goods firms face a probability that they cannot adjust their price each period (the Calvo pricing mechanism). Intermediate goods are aggregated into a final (homogeneous) consumption good by final goods firms. The production side of the economy, when aggregated and linearized, can be described by the following set of linearized equilibrium conditions (the production function, the optimal hiring condition for labor and the dynamic evolution of prices): ye = mu (3) wit = met (4 ) At = BE,(14+1) + Amic, (5) The resource constraint is: yt = C (6) Monetary policy follows a simple Taylor Rule: 4 = 0.#: (7) The (linearized) preference shock follows an AR(1) process: 2 = pz-1 + et (8) is i.i.d. In percentage deviations from steady state: me, is real marginal cost, & is consumption, w, is the real wage, it is hours worked, y, is output. In deviations from 3 steady state: 4, is the nominal interest rate, if, is inflation. A is a function of model parameters, including the degree of price stickiness.' Assume that o, > 1, 0 0. For simplicity, we assume that consumption of farmers can be negative, that is c E R. Moreover, farmers can save (or borrow) in a non-state contingent and non-defaultable asset a, which has a rate of return r. Farmers face the following "No- Ponzi" condition on assets lim 1-+o (1 + r)= 2 0. Moreover, farmers can produce and hold capital, k, which is rented to the representative firm in competitive markets (1 unit of final consumption good produces 1 unit of capital). Let r* be the rental rate of capital and o the depreciation rate. Unlike farmers, workers don't own land, and they use their available time to work in the representative firm. Assume each worker is endowed with one unit of time. They cannot trade the asset a, but they can produce and hold capital, k (with the same technology as farmers). Their per-period utility is given by u(c), with u'(c) > 0, u"(c)

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