ECON 451 Assignment 2 due April. 8th, 2021 Question 1: A Simplified Model (150 points) Note: Please derive the results step by step. Consider an economy where there is an infinite-lived representative consumer who has a preference (at an arbitrary time, denoted t = 0) given by U(c0, n0, m0, c1, n1, m1, c2, n2, m2, · · ·), where ct is the real consumption at time t, nt is the labor supply at time t, mt is the real money balance at time t. We assume that preference is additively separable as follows U(c0, n0, m0, c1, n1, m1, c2, n2, m2, · · ·) = u(c0, n0, m0) + βu(c1, n1, m1) + β 2u(c2, n2, m2) + · · · = X∞ t=0 β tu(ct , nt , mt), (1) where u(ct , nt , mt) is called the period utility function and satisfies standard properties. The discount factor β is a time preference parameter. Moreover, we assume the period utility function is u(ct , nt , mt) = log ct − n 1+ϕ t 1 + ϕ + log mt , where Uct > 0, Unt < 0,=”” umt=””> 0, and ϕ > 0. The budget constraint for the consumer is Ptct + Tt + Mt ≤ Wtnt + Mt−1 (2) Pt : price level Wt : nominal wage Tt : nominal lump-sum taxes Mt : nominal money balance & mt : real money balance. (a) Please derive the budget constraint in real terms. (25 points) 1(b) Please formulate the consumer’s problem and write down the Lagrangian function. (25 points) (c) Please derive the first order conditions which characterize the maximization of the utility. (25 points) (d) Consider there is a representative firm with production function yt = Atn 1−α t (3) where At is the exogenous technology and α ∈ (0, 1). Please write down the firm’s maximization problem and derive the condition which characterize the solution of the firm’s problem. (25 points) (e) Please solve for the equilibrium levels of employment, output, and wage. (25 points) (f) Does monetary policy determine any real variable? Please explain your answer. What is the policy implication? (25 points)

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