Download Spring, 2006

ECON 204
INTERMEDIATE MACROECONOMIC THEORY
Spring, 2006
ASSIGNMENT 1
Instructions: The assignment is due by 3 pm on Feb 1st in the assignment box for this
course located in the hall beside room BEC 360. Late assignments will not be marked. If
you cannot turn in an assignment due to extenuating circumstances, you must contact the
instructor as soon as possible and present documentation. Assignments will be marked on
the quality of the response to the questions, clarity of exposition, and quality of
presentation. If you cannot pick up the returned assignment in class, please see our TA
Yang Lu at her office hour:
Question 1: (Chapter 3: National Income)
(10 marks)
Assume that in a closed economy, equilibrium GDP (Y) is 5,000. Consumption is given
by C=500+0.6(Y-T). Taxes (T) are equal to 600. Government spending (G) is equal to
1,000. Investment is determined by the equation I=2,160-100r, where r is the real interest
rate.
Answer:
Question 2: (Chapter 4: Economic Growth I)
(10 marks)
Suppose that a country is initially operating at a steady-state level of capital per worker
and zero technology progress. If a birth-control campaign is successful in significantly
reducing the country’s population growth, what will happen to the growth rates of capital
per worker, output per worker (living standards), and total output on the path to the new
steady state? At the new steady state, how will each of these three growth rates compare
with their initial steady-state values? (You can try to draw some graphs, like what we did
in class, to demonstrate the dynamics of these growth rates. Here we concern growth
rates not levels.)
Answer:
To TA: I don’t expect that students can draw the above diagrams. If they can describe the
answer by their own words, it is enough.
Question 4: (Chapter 5: Economic Growth II)
(15 marks)
Textbook page 128: question2.
Answer:
To solve this problem, it is useful to establish what we know about the U.S. economy: A
Cobb-Douglas production function has the form y  k  , where  is capital’s share of
income. The question tells us that   0.3 , so we know that the production function is
y  k 0.3 . The depreciation rate   0.04 . The capital-output ratio K/Y=2.5. Because
k / y  [ K /( L * E )] /[Y /( L * E )]  K / Y , we also know that k / y =2.5. (That is, the
capital-output ratio is the same in terms of effective workers as it is in levels.)
a. Begin with the steady-state condition, sy  (  n  g )k . Rewriting this equation leads
to a formula for saving in the steady state: s  (  n  g )( k / y ). Plugging in the values
established above: s  (0.04  0.03) * 2.5  0.175 . The initial saving rate is 17.5 percent.
b. We know from Chapter 3 that with a Cobb-Douglas production function, capital’s
share of income   MPK ( K / Y ). Rewriting, we have: MPK   /( K / Y ) . Plugging in
the values established above, we find: MPK  0.3 / 2.5  0.12.
c. We know that at the Golden Rule steady state: MPK  n  g   . Plugging in the
values established above: MPK  (0.03  0.04)  0.07. At the Golden Rule steady state,
the marginal product of capital is 7 percent, whereas it is 12 percent in the initial steady
state. Hence, from the initial steady state we need to increase k to achieve the Golden
Rule steady state.
d. We know from Chapter 3 that for a Cobb-Douglas production function,
MPK   (Y / K ) . Solving this for the capital-output ratio, we find:
K / Y  0.3 / 0.07  4.29 . In the Golden Rule steady state, the capital-output ratio equals
4.29, compared to the current capital-output ratio of 2.5.
e. We know from part (a) that in the steady state s  (  n  g )( k / y ) , where k / y is the
steady-state capital-output ratio. In the introduction to this answer, we showed that
k / y  K / Y , and in part (d) we found that the Golden Rule K / Y  4.29. Plugging in
this value and those established above: s  (0.04  0.03)( 4.29)  0.3. To reach the Golden
Rule steady state, the saving rate must rise from 17.5 to 30 percent.
Question 4: (Chapter 5: Economic Growth II)
(10 marks)
Suppose the economy stays at a steady state with labor growth n and technological
progress g. The production function is Y=F(K,LE). Prove:
a) The capital-output ratio (K/Y) is constant;
Answer:
We know that the income share of capital =  =MPK*K/Y. In the steady state, we know
from part a) that the capital-output ratio K/Y is constant. We also know that MPK is a
function of k, which is constant at a steady state; therefore the MPK itself must be
constant in the steady state. Thus, capital’s share of income is constant. Because the
production function in the Solow model is constant return to scale, labor’s share of
income =1-  =1-[the income share of capital]. So, if the income share of capital is
constant, then labor’s share is constant as well.
c) The capital income and labor income grow at n+g.
We know that at a steady state, total income grows at n+g – the rate of population growth
plus the rate of technological change. In part (b) we showed that labor’s and capital’s
share of income is constant, If the shares are constant, and total income grows at n+g,
then labor income and capital income must also grow at the rate n+g.