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UW-Madison ECON 312 - Problem Set 1

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Economics 312MacroeconomicsNoah WilliamsProblem Set 1Due in class on February 12.1. Suppose that instead of simply being a waste of output, government purchases areused for infrastructure which increases current productivity. In particular, supposethat the government increases its current purchases (G increases) and that this inturn increases total factor productivity (z increases). The government funds thehigher expenditure by lump sum taxes. Using the static general equilibrium modelfrom class (and diagrams when possible), discuss the effects this policy change willhave on the equilibrium levels of output, employment, and consumption.2. Suppose that there is a progressive tax on labor income. We model this by supposingthat for a given wage if N ≤ N∗then the household faces a tax rate t, but if N > N∗the tax rate increases to t0> t for hours worked above N∗. That is, with N > N∗after-tax labor income is (1 − t)wN∗+ (1 − t0)w(N − N∗).(a) Suppose a worker has unearned income π and faces this tax schedule. Drawthe worker’s budget constraint. Suppose that preferences (MRS) differ acrossworkers, and illustrate the optimal choice of consumption and leisure for differ-ent individuals. Show that there is likely to be a mass of workers who chooseN = N∗.(b) Now suppose that there is an increase in the top labor tax rate t0, but t isunchanged. What happens to the labor supply choices of different workers?3. Consider the optimal growth model from class, but add government spending. Thatis, there is now a specified amount of government spending G which must be fundedevery period via lump sum taxes.(a) Compared to the case of no government spending, how does having G > 0affect the optimal steady state levels of consumption and capital c∗and k∗?(b) How are the dynamics affected? That is, suppose the economy is initially inthe steady state (k∗0, c∗0) associated with G = 0. Then there is announcementthat there will be government spending G > 0 for all future dates. How doconsumption and capital respond, both immediately upon the announcementand then in the succeeding periods?14. Suppose that instead of labor being supplied inelastically, households value leisure.That is, in the optimal growth model we now have preferences:∞Xt=0βt[U(ct) + v(1 − Nt)],where households have 1 unit of time each period, and Ntis labor, so 1−Ntis leisure.Both U and v are strictly increasing and strictly concave. Firms produce using aconstant returns to scale production function F (k, N), so the aggregate feasibilitycondition is now:F (kt, Nt) = ct+ kt+1− (1 − δ)kt.Consider the social planner’s problem in this environment.(a) Write down the Lagrangian and find the optimality conditions for the choicesof ct, kt+1and Ntat any date t.(b) Taking ratios of your optimality conditions, find the household Euler equa-tion relating the marginal utilities of consumption at dates t and t + 1 to themarginal product of capital. Also find the equation relating the marginal rateof substitution between consumption and leisure to the marginal product oflabor.(c) Find the equations determining the steady state, and characterize the steadystate as sharply as you can. How does having elastic labor supply (v 6= 0)affect the steady state levels of consumption and capital?(d) Now consider the special case (like in class) with δ = 1 and F (k, N) = AkαN1−αand U (c) + v(1 − N) = log c + γ log(1 − N). Show that the optimal solutionis to save a constant fraction s of output, ct= (1 − s)F (kt, Nt) and work aconstant number of hours Nt=¯N. Find expressions for s and¯N.(e) Continuing with the example from the previous part, suppose that there is anincrease in productivity A. How will that affect the optimal levels of consump-tion, labor, and capital, both on impact of the change and over


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