Question 1(4×10pts)
Describe and graph how each of the following affect the k̇ = 0 and ċ= 0 locus. How does c and k
react immediately after change and what is the new steady state? Note: k and c will denote per
intensive unit of labor (same as k̂ and ĉ in the lecture slides).
a) Permanent rise in productivity growth rate g.
b) Rise in the preference for today’s consumption θ.
c) Proportional downward shift of production function f(k).
d) Decrease in the depreciation rate of capital δ.
Question 2(5pts+10pts+3×5pts+10pts+5pts+5pts+10pts)
Consider an economy with infinitely lived representative households which provide labor
services in exchange for wages, receive interest income on assets, purchase goods for
consumption and save by accumulating additional assets. We will modify here the standard
Ramsey model by assuming that government purchases affect utility from private consumption
and that government purchases and private consumption are perfect substitutes. Thus, the
representative household maximizes its lifetime welfare:
subject to its flow budget constraint and the No-Ponzi-Game condition, where n is the rate of
population growth, θ > 0, ρ > 0 and ρ > n. Assume further that the government purchases per
capita are gt = Gt/Lt, which are financed by a constant tax on consumption 1 > τc > 0, and the
government budget is balanced. The productive sector of the economy has competitive firms
which produce goods, pay wages for labor input and make rental payments for capital inputs.
The firms have neoclassical production function, expressed in per capita terms yt = Akα
t where 0
< α < 1 and capital depreciates at the rate δ > 0.
a) Specify the household’s dynamic optimization problem.
b) Derive the first order conditions of the household’s optimization problem.
c) Obtain the Euler equation.
d) Write down government’s budget constraint, the government spending per capita, and ġ.
e) Rewrite the Euler equation in terms of c, r, θ, and ρ. How does the tax affect the
consumption choice?
f) Write down and solve the problem of a profit-maximizing representative firm. Using the
results above specify the competitive market equilibrium.
g) Derive the conditions for the steady-state level of capital and consumption per capita and
draw the phase diagram.
h) Find the value of k* for α =0.5, A=4, δ=0.4, and ρ=0.6.
i) Assume that the economy is initially at a steady state with k* and c* > 0. What are the
effects of a temporary increase in government purchases on the paths of consumption,
capital and interest rate (draw their behavior over time).
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