Problem (30 points) Prof. Robert King Consider an economy in which there is one period and there are many, identical households. Each household derives utility from consumption (c), leisure (l) and a public good (g) according to u(c; l; g) = [cc l l g g ] where c + g + l = and all parameters are positive. that each household s budget constraint is c = ( n )wn Suppose further where n = l is the amount of work e ort, w is the wage rate, and n is the labor income tax rate. Firms produce output y according to y = an where n is labor input and a is labor productivity. They sell their output to households as consumption (c) and to the government as the public good. Competition requires that w = a. The government has a budget constraint of the form g = n wn (a) [5 points] Consider the household s optimal consumption and labor/leisure decision plan, given the wage rate and the tax rate. Show that it takes the form c = c a( n ) l = l Derive the coe cients c and l. Discuss why a, n, and g do not a ect leisure, even though each of these enters into the budget constraint or the objective. (b) [5 points] Determine the optimal level of public good provision in this economy. You can set this up as a Ramsey problem, it is easier to use the results of part (a) directly. Comment on the nature of the trade-o s faced by the government in this economy.
Problem 2 (60 points) Consider a household that has an intertemporal objective of the form X t u(c t ; l t ; g t ) with the momentary objective being the same as in the previous question (u(c; l; g) = [cc l l g g ] ). Suppose further that the household has a present value budget constraint, with t p t being the price of future goods, of the form X a 0 + t p t [( n t )w t n t + ( k t )q t k t + T t c t i t ] 0 where a 0 is an initial stock of nancial assets, k t is the capital income tax rate, q t is the market rental rate on capital at date t, k t is the household s holdings of capital at date t, and T t is the level of transfer payments (positive or negative) that the household receives at date t. The level of household investment, i t, alters the future capital stock according to k t+ = ( )k t +i t, where is the depreciation rate. (a) [0 points] Determine the e ciency condition that must be satis ed for investment if the household is to be willing to (i) hold positive amounts; and (ii) not be in nitely wealthy. (b) [0 points] Under the condition from part (a), show that the household budget constraint is equivalent to fa 0 + [( k 0)q 0 + ]k 0 g + X t p t [( n t )w t n t + T t c t ] 0 and discuss the economic interpretation of the { } term in this expression. (c) [5 points] Show that the rst order conditions for the consumer in this intertemporal model are t u c (c t ; l t ; g t ) t p t = 0 t u l (c; l; g) t p t ( n t )w t = 0 2
where is a multiplier on the intetemporal budget constraint. (d) [0 points] Implications of these rst-order conditions are: u c (c t+ ; l t+ ; g t+ ) u c (c t ; l t ; g t ) u l (c t ; l t ; g t ) u c (c t ; l t ; g t ) t+ l l( ) g( ) t+ gt+ ] t l l( ) g( ) t g = c( ) c[c c( ) c [c = lc t c l t = ( n t )w t t ] = p t+ p t so that log( c t+ c t ) = log( p t+ p t ) + 2 log( ( n t+)w t+ ( n t )w t ) + 3 log( g t+ g t ) with = ; ( c+ l )( ) 2 = l ( ) ;and ( c+ l )( ) 3 = g( ). Concavity of utility in the combination of consumption and leisure implies that the ( c+ l )( ) denominator is negative, even if <. Comment on how this expression for consumption growth di ers from that developed when public goods do not a ect utility, explaining the economic mechanisms involved. Include an explanation of why the value of ( > or < ) is important for the nature of the results. (e) [5 points] Suppose that there are two di erent consumers, one rich and one poor in terms of wealth, but otherwise facing the same prices (p t ; w t ) and tax rates ( n t ; k t ). How would the rich consumer and the poor consumer di er in terms of consumption and leisure levels? in terms of growth rates? 3
Pollution in the neoclassical growth model Consider a version of the neoclassical growth model in which production produces pollution (x t ) which reduces utility: max U = β t [log(c t ηx t )] K t+ = I t + ( δ)k t zkt α N α = C t + I t The variable K t denotes the stock of capital, C t the level of consumption, and N is the fixed number of hours worked. Production generates γ units of pollution per unit of output: x t = γzkt α N α a) Compute the first-order conditions for the planner s problem for this economy. b) Compute the steady-state value of the stock of capital implied by the planner s problem. c) Suppose that the private sector takes the level of pollution in the economy as exogenous. The government imposes a tax on output at rate τ. The proceeds from the tax are rebated to private agents as lump sum subsidies, s t. s t = τzk α t N α. The private sector solves the following problem: max U = β t [log(c t ηx t )] K t+ = I t + ( δ)k t ( τ)zkt α N α = C t + I t + S t where both X t and S t are treated as exogenous by each private agent. In equilibrium X t = x t and S t = s t. What is the value of the tax rate on output, τ, that makes the solution to the private sector problem coincide with the solution to the planner s problem?
SZG Doctoral Program, 2009 Macroeconomics, Week 4 Jordi Galí Question Week 4 (70 points) Consider an economy with Calvo-type staggered price setting, where a continuum of monopolistically competitive firms have access to a technology Y t = A t N t, where Y t is output, N t denotes hours of work and A t is an exogenous technology parameter. The representative consumer has a period utility U(C t, N t ) = log C t N +ϕ t +ϕ., where C t is a CES function of the quantities consumed of the different types of goods. The technology parameter a t log A t is assumed to follow a random walk process, i.e. a t = a t + ε t, where {ε t } is white noise. Firms desired markups are constant. All output is consumed. The labor market is perfectly competitive. The implied equilibrium conditions take the form ỹ t = E t {ỹ t+ } (i t E t {π t+ } ρ) () π t = β E t {π t+ } + κ ỹ t (2) where ỹ t y t y n t is the output gap, y t is (log) output, y n t is the (log) natural level of output, π t p t p t is the rate of inflation, and i t is the short-term nominal rate. (a) Derive equilibrium condition (), using the consumer s Euler equation (b) Derive equilibrium condition (2), given inflation equation π t = β E t {π t+ }+ κ mc t, where mc t mc t mc is the (log) deviation of real marginal cost from its steady state level. (c) Determine the natural level of output yt n as a function of the technology parameter. (d) Suppose that the monetary authority adopts a simple interest rate rule of the form i t = ρ + φ π π t (3) where φ π >. Determine the equilibrium path of inflation, the output gap, and output as a function of exogenous technology.shock. (e) How would your answer to (d) change if the technology process was instead given by a t = ρ a a t +ε t with ρ a (0, )? Derive your result analytically and explain it intuitively. (f) Explain how your answers to (d) and (e) would change if the central bank followed the price level targeting rule i t = ρ + φ p (p t p ) (4) (g) How would you go about determining the relative desirability of (3) and (4)? (Make any additional assumptions explicit).