M1-TSE. Macro I. 2010-2011. Homework 3 Toulouse School of Economics Ernesto Pasten (
[email protected] ) Frank Portier (
[email protected] )
Macroeconomics I
Homework 3 October 20, 2010
1. Household choose consumption Ct and labor Nts to maximize "∞ # X E0 β t U (Ct , Nt ) t=0
subject to At+1 = Rt At + Wt Nt − Ct + Πt − Tt where Πt denotes dividends (profits) obtained from firms which the households own and Tt denotes lump-sum taxes that are used to finance government spending: Tt = Gt . Firms in this economy choose labor demand Ntd to maximize profits Πt = Yt − Wt Nt Yt = θt Ntα where θt is the current level of technology. There is no capital in this model, so all output is either consumed by households or used for government spending: Yt = Ct + Gt . (a) What are the optimality conditions for the households choice of Nt and Ct ?. Explain what these conditions mean in words. What is the optimality condition for the firm’s choice of Nt ? (b) Suppose U (C, N ) = log (C − v (N )) where v(N ) =
N 1+φ . 1+φ
Write down the equilibrium conditions for this economy. Describe the effect of an increase in government spending Gt on wages, labor, consumption and output in this economy. Explain these findings. (c) Now assume that U (C, N ) = log (C) − v (N ) . 1
M1-TSE. Macro I. 2010-2011. Homework 3
How does this change your answer to part b? Explain. (d) Now suppose that the government taxes labor in order to pay for government spending. In particular: Gt = Tt = τ t Wt Nt . Assume again that U (C, N ) = log (C) − v (N ) . What is the effect of an increase in government spending on wages, labor, consumption and output in this economy. How does it compare to the case where taxes are lump-sum as in part c? 2. Consider the following general equilibrium model. Households choose labor supply Nt , bond holdings Bt and money balances Mt to solve #) " (∞ 1+φ X N M t+s − t+s max Et β s log(Ct+s ) + γ log P 1+φ t+s s=0 subject to the budget constraint in nominal terms: Bt+1 + Mt = (1 + it ) Bt + Wt Nt + Mt−1 + Πt − Pt Ct + TtM where Bt are nominal bonds, it is the nominal interest rate, Wt is the nominal wage, and Pt Ct are nominal expenditures on consumption. Here Pt is the price index. The term Πt denotes nominal profits (dividends) that the households receive from firms. There are many firms in this economy.Individual firm i chooses its price Pt (i) one period in advance to maximize expected profits Et−1 Πt (i) = Et−1 {Pt (i) Yt (i) − Wt Nt (i)} subject to the downward sloping demand curve: −ε Pt (i) Yt (i) = Yt Pt where Yt is total demand and Yt (i) is demand for firm i and ε > 1. The production function is: Yt (i) = θt Nt (i) . Assume that each firm in the economy chooses its price Pt (i) to satisfy: Pt (i) = Et−1 {µM Ct } where µ = ε/ (ε − 1) and the nominal marginal cost of production is M Ct = Wt /θt (a) Households choose their bond holdings Bt , their labor supply Nt and their real balances Mt /Pt to satisfy: 1 1 = β (1 + it+1 ) Et Pt C t Pt+1 Ct+1 2
M1-TSE. Macro I. 2010-2011. Homework 3 Wt 1 Pt C t = βEt
Ntφ = 1 P t Ct
1 Pt+1 Ct+1
+
γ . Mt
Interpret each of these conditions. (b) Assume that the growth rate of technology satisfies: log θt − log θt−1 = vtθ where vtθ is a mean zero iid shock to the growth rate of technology. The monetary authority sets the growth rate of money to satisfy: log Mt − log Mt−1 = vtM + σvtθ where σ denotes how the monetary authority allows the money growth rate to respond to changes in technology. In the equilibrium, we will have Nt (i) Pt (i) Yt (i) Ct
= = = =
Nt , Pt , Yt , Yt .
Show that in equilibrium these conditions imply: Mt = ΦPt Yt for some constant Φ. (c) Show that in equilibrium the quantity of labor satisfies implies that labor supply satisfies N 1+φ =
1 Wt /θt . µ Et−1 (Wt /θt )
Now take expectations at time t − 1 to show that Et−1 Nt = N ∗ for some constant level of labor N ∗ which we interpet as the natural rate of employment. What is N ∗ ? How does it vary with the markup µ? What is the intution here? (d) How does output growth and inflation today and tommorrow respond to a money shock vtM and a technology shock vtθ ? Explain. (e) Suppose the goal of the monetary authority is to stabilize employment at its natural rate N ∗ . How should it choose σ? Explain.
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