M.A. 1005-01/02 Macroeconomic Theory FINAL EXAM

发布时间：2022-12-14

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Fall 2020

M.A. 1005-01/02

Macroeconomic Theory

FINAL EXAM

1. (40 points) Consider an economy with the following structure. All markets are perfectly competitive, prices are fully flexible (no price rigidities), and technology (total factor productivity) is constant. The representative household’s preferences are given by

with β ≡ 1/(1+ρ) and utility function u(ct )= lnct .

Households obtain income from labor and from the assets they hold.

The production function has the form yt = F(kt , lt ) = kt(α) l with 0<α<1.

Firms hire labor and rent capital each period. Labor supply is inelastic, so lt = 1 at all t. Capital accumulation depends on investment It (net of depreciation): kt+1 −(1−δ)kt = It .

There is also a government that purchases in each period a constant quantity of goods g and finances those purchases with lump-sum taxes T, with a balanced budget each period. Hence the government budget constraint is Tt = gt with gt = g , i.e. T = g at all t; the household’s budget constraint is at+# = 业t lt + (1 + rt )at 一 ct 一 Tt ; the resource constraint for the economy is ct + It + gt = yt.

a) - Write down the equilibrium conditions and combine them into the two dynamic equations for consumption and capital.

- Obtain the equations for stationary consumption and stationary capital, and write the analytical solution for the steady state k* and c* .

- Draw the phase diagram, indicating the direction of movements (arrows) in the different regions, the steady state that represents the long-run equilibrium, and (approximately) the saddle path leading to it.

b) Consider the following sequence of events (starting from the steady state): at time t the government credibly announces a law that will decrease permanently the level of future government purchases to g$ 想 g , starting from period t+2 (i.e. agents expect now gt+j = g$ forj=2, 3, …).

- What will be the long-run effects of the fall in g? Show it in the phase diagram, justify it analytically and explain briefly.

- Would consumption change at time t? Explain very briefly how (qualitatively) and why households react. - Qualitatively, what will the transitional dynamics for consumption and capital (and hence output) look

like? Show it in the phase diagram, as well as in a graph with the approximate time path of k, y, c and g.

Explain briefly.

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2. (40 points) Consider a model similar to the one in the first exercise but with the following modification: the production function is yt = kt(α) (At lt )1−α . Firms take efficiency At as given when deciding how much capital and labor to use in production; but as a result of learning-by-doing and knowledge spillovers, At is actually a function of the capital stock used: in particular, we assume that in the aggregate At = bkt , where b is a positive parameter. Note that this is like an externality, meaning firms do not take into account the contribution that their capital is making to aggregate productivity: they make their decisions taking At as

given; then in the aggregate general equilibrium we have the additional condition At = bkt . Labor supply is inelastic, so lt = 1 at all t.

a) Taking into account all the conditions, show what determines the growth rate of consumption, the rate of return and the growth rate of capital, the level of aggregate output.

b) Show that there is a balanced growth path where consumption, capital and output grow at the same constant rate (assuming that the parameter values make it positive).

c) Suppose the government intervenes to subsidize investment (financing the program with lump-sum taxes), such that the households’ budget constraint becomes

!t+# = #t $t + (1 + (t )!t − +t + /!t − ,t

where s is the subsidy (per unit invested) and ,t are taxes. Agents take ,t as given in their decisions; but then in the aggregate, to balance the government budget, we have ,t = /!t .

- Show how this policy will affect the growth rate of the economy.

- Could this policy be justified in terms of efficiency (i.e. improving social welfare)? Explain very briefly why or why not.

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3. (20 points) Consider a New Keynesian model of the economy with nominal rigidities (where the aggregate price level adjusts slowly) and without capital (so y = c). We have seen that, taking log-linear approximations, the equilibrium dynamics can be represented by the following two conditions (respectively from the supply-side and the demand-side)

πt = βEt (πt+1 ) +κt

t = Et (t+1 ) − [it − Et (πt+1 ) − rtn ]

together with a monetary policy rule that determines the path of nominal interest rates it

it = ρ+φππt +φy t +νt (assume φπ > 1, φy ≥ 0 )

where π is inflation, ỹ is the output gap, i.e. the deviation of log output from the log of ‘natural’ output (the equilibrium output if prices were flexible), rn is the natural (flexible prices) real interest rate, and νt is a stochastic component capturing monetary shocks. We assume parameters such that the equilibrium is unique (no indeterminacy). In steady state both π = 0 and ỹ = 0. We also assume a production function Yt = At Lt , where Yt is actual output, At is productivity (constant in steady state), Lt is aggregate hours worked. Use these conditions and the logic of the underlying model (that is, the behavior of firms in setting prices, and of households in choosing their consumption) to answer qualitatively the following question. a) Suppose at time t the economy receives a negative productivity shock (i.e. At lower than its steady state – the shock is temporary but with some persistence). Explain briefly what would happen (and why) to current and expected inflation, to the output gap, to the actual output level and to aggregate hours worked.

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