Question 2. (110 points) Consider the following closed economy. Households. The economy is populate ...
Question 2. (110 points) Consider the following closed economy. Households. The economy is populated by a representative household with ? identical individuals. Each individual is endowed with one unit of time. The household maximizes lifetime utility U(0) = Z 1 0 e??t?[log c (t) + ? log (1 ? l (t))] dt, ?, ? > 0 2 where ? is the individual discount rate, ? is preference for leisure and c is consumption per capita. The household faces the flow budget constraint ?a = ra + (1 ? ?) wl + G/? ? c, where a is assets holding, r is the rate of return on assets (the interest rate), w is the wage rate, l is the fraction of time that each individual allocates to work, ? is the tax rate on labor income, and G/? is a lump sum transfer from the government. Firms. The economy is also populated by N production firms that sell a homogenous good whose price is normalized to one. The good can be either consumed or invested. Firms have access to the production technology Yi = (Ki)? (TiLi)1?? , 0 < ? < 1, i = 1, ...,N where Ki is capital, Li is labor and Ti is labor-augmenting technology described by the relation Ti = Zi + ? XN j6=i Zj , 0 < ? < 1 where Zj is knowledge accumulated by firm j and ? is a parameter governing spillovers across firms. Capital accumulates according to ?K i = Ii. For simplicity we assume that capital does not depreciate and that knowledge accumulates as a by-product of investment, ?Z i = ?Ki, so that after suitably normalizing terms we can write Zi = Ki. Finally, we assume that N is su¢ciently large that each firm in equilibrium commands a negligible market share so that we can focus on a competitive equilibrium where increasing returns are treated as external to the firm (i.e., firms take Ti as given). Aggregation. Sincewehave N firms, we need to specify the following aggregation rules: Y = XN i=1 Yi; K = XN i=1 Ki; L = XN i=1 Li; I = XN i=1 Ii; V = XN i=1 Vi, 3 that define aggregate output, capital, employment, investment and stock market value (think of the household as holding shares of a fully diversified equity fund whose price per share is V ). Government. Thegovernmentcannotborrowandsatisfies thebudget constraint ?wL = G. In other words, we assume that the government sets tax rates and rebates in a lump-sum fashion the revenues to the household. Answer the following questions. 1. Write down the Hamiltonian for the household’s problem and derive the Euler equation. Interpret the Euler equation as an equation characterizing the reservation after-tax rate of return on assets demanded by savers. What are the e§ects of tax rates on this rate of return? 2. Write down the Hamiltonian for the firm and derive the equations characterizing the behavior of the firm. Interpret carefully the equation characterizing the after-tax rate of return on capital generated by firms. What are the e§ects of tax rates on this rate of return? 3. Define the variable x ? C/K, where C = c? is aggregate consumption. Show that the equilibrium of the labor market yields aggregate employment L as a downward sloping function of x. Explain why the relation is downward sloping. Next, show that the Euler equation for saving and the resource constraint of the economy yield the di§erential equation x? x = x ? (1 ? ?) ? 1 + ? (N ? 1) N L(x) ?1?? ? ?. Use this equation to discuss the equilibrium dynamics of this economy. Is the equilibrium trajectory unique? What is the equilibrium value of aggregate employment? 4. Show that the equilibrium just discussed yields log c (t) + ? log (1 ? l (t)) = u0 + gt, where u0 and g are expressions that depend only on the fundamentals. Next, show that U(0) = 1 ? ? u0 + g ? ? . 4 This is a handy expression for evaluating the welfare e§ects of policies. Interpret it carefully. In particular, observe that dU (0) du0 = 1 ? and dU (0) dg = 1 ?2 , so that dU (0) dg = 1 ? dU (0) du0 . Notice that we typically think of ? = .02 so that in practice the welfare gain from a given dg is 200 times (!) the welfare gain from an equal du0. 5. Assume an unanticipated, immediate, permanent increase in ?. What happens to welfare in this economy? Note that according to the characterization above, you need to show what happens to both u0 and g.
