where y; (j) is the output of good j, N (j) is the amount of labor employed by firm j, and Z; is exogenous labor productivity. Given this production function, firm j's marginal cost is w;/Z;, where w; is the real wage (Wi/B). e Why is w;/Z; the expression for marginal cost? Suppose that firms set prices subject to a cost of price adjustment equal to: ; 2 v ( pe(d) 1) 2 \\pt-1(7) ' with / > 0. This cost is measured in units of the consumption basket. It implies that the price of product j will be set as a time-varying markup over (nominal) marginal cost: ; : Wy pe (3) = 1t (5) Pt?, t where g, (j) > 1. If prices are flexible, the markup is constant and equal to /(60 1). Imposing symmetry across firms in equilibrium, p; (j) = P, and y, (7) = p,. It follows that: 1 wr = Zt. Hy There is a wedge between the real wage and labor productivity equal to the reciprocal of the markup. e Use the last equation and the first-order condition for the optimal choice of labor supply to find the level of the labor income tax rate 7; that removes the impact of monopoly power and nominal rigidity on the labor market (i.e., the labor income tax rate that delivers labor market efficiency). What is the expression of this tax rate if prices are flexible? e If you did things right, you found that the labor income tax rate that removes the effects of distortions on the labor market is negative. This means that, in order to restore labor market efficiency, the government must actually subsidize labor (instead of taxing it). In other words, the government must tax leisure. 7nfR The budget constraint in period is: PCi+ Biy1 =1 1) WeNy + (1 +4) By + Ty + I, where P, is the consumer price level, W; is the nominal wage, B; denotes nominal bonds at the beginning of period , i; is the nominal interest rate, 7, is the rate of taxation of labor income, and T; is a lump-sum transfer with which the government rebates the revenue from taxation back to the consumer. The tax rate 7, and the transfer T; are the only instruments of government policy in this exercise. Note that 74 can be negative, in which case 7; is a subsidy rate and T} is a lump-sum tax that finances the subsidy. Both the tax (or subsidy) rate 74 and the lump-sum transfer (or tax) T; are taken as given by the consumer (i.e., the consumer does not recognize that the amount of the transfer T; depends on the amount of labor supplied). Finally, IT; denotes the profits made by firms with monopoly power. The firms are owned by the consumers, so their profits are transferred to the representative consumer as the lump-sum amount II; (note: I1; is different from the inflation rate mp = (P P;1) /Pi1). There is one budget constraint like the one above in each period. e Write the Euler equation for the consumer's choice between consumption and bond accumulation and the optimality condition for labor supply. Explain the reasoning that led you to write these equations as you did. Assume that consumption consists of a bundle of differentiated goods produced by firms that operate under monopolistic competition: 1 ., 1o - U i) dj] , 0 where ; (7) is consumption of the differentiated product j (produced by firm 7), there is a continuum between 0 and 1 of such firms, and 6 > 1 is the elasticity of substitution between their products. e Let p;(j) denote the price of product j. Given the amount of consumption C; determined by the Euler equation, what is the expression for the consumer's demand of product j in period t? Explain this expression intuitively. Suppose that firm j's production function is: 6 of 8 vt (3) = ZeNy (4) Problem 3: Labor Tax Policy in the New Keynesian Model (30 Points) This problem asks you to think about labor tax policy in the New Keynesian model. We have not talked about this topic, but the material we covered should be sufficient for you to answer the questions in the problem. Suppose the representative consumer maximizes the intertemporal utility function: oo E. Y 8 u(Cs1-Ny), s=t where E; denotes the expectation conditional on information available at time , Cy is consumption in period s, Ny is labor effort supplied in the same period, the duration of the period has been normalized to 1, and [ is a discount factor strictly between 0 and 1. What is your intuition for this result? (This question is harder. The penalty if you do not answer it correctly will be small.)