Queen's University Economics DepartmentQueen's University Economics Department
Economics 320A - Macroeconomic Theory II
Fall 1998
Assignment 4
Instructor: Marc-André Letendre
Deadline: December 3, 1998 (in class)
Question 1 (70 points)
Consider the growth model where
member h of generation t has the utility function
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uht = ln(cht(t))+bln(cht(t+1)) b = 0.5, |
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and receives lifetime labor endowment
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[Dht(t), Dht(t+1)] = [9, 3]. |
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The competitive firms are
represented by the aggregate production function
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Y(t) = g(t) K(t)0.4 L(t)0.6, |
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where g(t) = (1.05)g(t-1) and g(0) = 1.
The number of (identical) people born in period t is N(t) = 100 and is
constant over time.
(a) 10 points
Show that the production function with neutral technical change
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Y(t) = 1.05t K(t)0.4 L(t)0.6, |
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is equivalent to the following production function
with labor-augmenting technical change
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Y(t) = K(t)0.4 (1.0847t L(t))0.6. |
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(b) 10 points
What is the effective-labor force in period t, denoted EL(t)?
(c) 10 points
Derive the equation describing the evolution of capital along an equilibrium
path.
(d) 10 points
Calculate the growth rate of capital, output and aggregate consumption
in a steady-state equilibrium.
(e) 10 points
Suppose that the initial capital stock is K(1) = 5.
Calculate the capital-effective labor ratio in period 1 and
in steady state.
(f) 20 points
Use a phase diagram to explain the evolution of capital during the transition
from period 1 to the steady state and rank the growth rates of L, EL and
K during that period.
Question 2 (30 points)
(a) 15 points
Consider an economy with the same structure as in question 1 but with different
parameter values.
The equation describing the evolution of capital along an equilibrium
path is
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K(t+1) = 3.3 K(t)0.4 1.03t. |
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Generate and graph
the equilibrium paths for capital of a poor country which has K(1) = 1 and
for a rich country which has K(1) = 25. Restrict your attention to the first 20 periods.
What does your graph tell you regarding
the growth rates of capital in each country during the transition?
(b) 15 points
What are the predictions of the model regarding cross-country difference
in growth rate and level of capital in the long run.
Are they realistic?
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