Economics 501
Problem Set 1 Suggested Solutions
Fall 2006
Prof. Daniel
Problem Set 1 Suggested Solutions
1. Abel & Bernanke Ch. 3, Numerical Problem #1 (p. 106)
a. To find the growth of total factor productivity, you must first calculate the value
of A in the production function. This is given by A = Y/(K.3N.7). The growth rate of
A can then be calculated as [(At+1 – At)/At] × 100%. The result is:
`
% increase in
A
A
1960 11.981
—
1970 14.152
18.1%
1980 14.746
4.2%
1990 16.335
10.8%
2000 18.779
15.0%
b. Calculate the marginal product of labor by seeing what happens to output when
you add 1.0 to N; call this Y2, and the original level of output Y1. (A more precise
method is to take the derivative of output with respect to N; dY/dN = 0.7A(K/N)0.3.
The result is the same after rounding.)
1960
1970
1980
1990
2000
Y1
2377
3578
4901
6708
9191
Y2
2402
3610
4935
6747
9239
MPN
25
32
34
39
48
2. Abel & Bernanke Ch. 3, Numerical Problem #6 (p. 106)
a. Since w = 4.5 K0.5 N–0.5, N–0.5 = 4.5 K0.5/w, so N = 20.25 K/w2. When K = 25, N =
506.25/w2. If t = 0, then NS = 100w2. Setting labor demand equal to labor supply
gives 506.25/w2 = 100w2, so w4 = 5.0625, or w = 1.5. Then NS = 100 (1.5)2 = 225.
(Check: N = 506.25/1.52 = 225.]) Y = 45N0.5 = 45(225)0.5 = 675. The total after-tax
wage income of workers is (1 – t) w NS = 1.5 × 225 = 337.5.
b. If t = 0.6, then NS = 100 [(1 – 0.6) w]2 = 16w2. The marginal product of labor is
MPN = 22.5/N0.5, so N = 100 [(1 – 0.6) × 22.5/N0.5]2, so N2 = 8100, so N = 90.
Then Y = 45N0.5 = 45(90)0.5 = 426.91. Then w = 22.5/900.5 = 2.37. The total aftertax wage income of workers is (1 – t)w NS = 0.4 × 2.37 × 90 = 85.38. Note that
there’s a big decline in output and income, although the wage is higher.
c. A minimum wage of 2 is binding if the tax rate is zero. Then N = 506.25/22 =
126.6, NS = 100 × 22 = 400. Unemployment is 273.4. Income of workers is wN =
Economics 501
Problem Set 1 Suggested Solutions
Fall 2006
Prof. Daniel
2 × 126.6 = 253.2, which is lower than without a minimum wage, because
employment has declined so much.
1/ 3
2/3
3. Suppose that the production function for the U.S. is Yt = At K t N t . Assume
that real GDP is $9,000 billion, the capital stock is $8,000 billion, and that there
are 150 million workers.
a. What is the implied level of total factor productivity? How would you
interpret total factor productivity?
9 × 1012 = A(8 × 1012 )1 / 3 (150 × 10 6 ) 2 / 3
9 × 1012
⇒ A = 1/ 3
8 150 2 / 31012 / 31012 / 3
9 × 10 4
= 1/ 3
8 150 2 / 3
A = 1,594.
TFP is essentially a residual that we interpret as capturing technology, human
capital, management, et cetera.
b. What are the marginal products of capital and labor? (Hint: be careful with
units.)
MPK = α AK α −1 N 1−α
1
= 1,594(8 × 1012 ) −2 / 3 (150 ×106 ) 2 / 3
3
⎛ 8 × 1012 ⎞
1
= 1,594 ⎜
6 ⎟
3
⎝ 150 × 10 ⎠
−2 / 3
MPK = 0.375.
MPN = (1 − α ) AK α N −α
= (1 − 1 / 3)1,594(8 × 1012 )1 / 3 (150 × 10 6 ) −1 / 3
⎛ 8 × 1012
2
= 1,594⎜⎜
6
3
⎝ 150 × 10
⎞
⎟⎟
⎠
1/ 3
⎛ 8 × 1012
2
= 1,594⎜⎜
6
3
⎝ 150 × 10
⎞
⎟⎟
⎠
1/ 3
MPN = $40,000.
c. How would real GDP change if:
Economics 501
Problem Set 1 Suggested Solutions
•
•
•
Fall 2006
Prof. Daniel
Labor increased by 5 percent while all other factors remained
unchanged.
∆Y = ∆N × MPN = (0.05 × 150,000,000) × $40,000 = $300,000,000,000 . This
is about 3.3% of GDP.
Capital increased by 5 percent while all other factors remained
unchanged. ∆Y = ∆K × MPK = (0.05 × 8 × 1012 ) × 0.375 = $150,000,000,000 .
This is about 1.6% of GDP.
TFP increased by 5 percent while all other factors remained
unchanged.
∆Y = ∆AF ( K , N ) = (0.05 × 1,594) × (8 × 1012 ) (150 × 10 6 ) = $450,000,000,000.
This is about 5% of GDP. This is an important feature of TFP. An increase in
TFP increases output one for one.
Labor and capital both increased by 5% while TFP remained
unchanged.
∆Y = ∆N × MPN + ∆K × MPK = $300,000,000,000 + $150,000,000,000 = $450,000,000,000.
This again is about 5% of GDP. Another important feature of the Cobb
Douglas production function is that it exhibits constant returns to scale. An
equal percent increase in both inputs increases output by the same amount.
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•
2/3
4. The profit maximization problem with a tax on output becomes:
a. Max ∏ = (1-t)AKαN1-α – w N – ucK. Take the partial derivative of profit with
respect to N and set this equal to zero. (1-t)(1-α)AKαN.-α – w = 0. Labor demand is
given by treating the N as ND (that is, labor demand) and solving the equation for
ND =K( (1-t)(1-α)A/w)1/α). Now labor demand becomes the after-tax marginal
product of labor.
b. Labor demand with the tax is lower.
c. The graph of labor demand with the tax shows a lower demand curve.
5. Consider a representative consumer whose preferences are represented by the
utility function U (c, l ) , where c is consumption and l is leisure. The consumer
derives income from wages w and dividend income a. Suppose that the
government imposes a proportional income tax on the representative consumer’s
wage income. That is, the consumer’s wage income is w(1 − t )( h − l ) , where t is
the tax rate, and h is the total amount of time available. What effect does the
income tax have on consumption and labor supply? (A graphical argument is
sufficient.) Explain your result in terms of income and substitution effects.
The graph for this problem is shown below. An increase in the tax rate reduces the
after-tax wage, which induces an inward swing in the budget line. The income effect
of an increase in the tax rate makes the individual want to reduce both leisure and
consumption. The substitution effect works in the opposite direction: the opportunity
cost of leisure falls, so the individual would like to consume more leisure. In the
Economics 501
Problem Set 1 Suggested Solutions
Fall 2006
Prof. Daniel
graph below, the substitution effect dominates the income effect, and labor supply
falls.
c
a
l
6. Suppose that a consumer can earn a higher wage rate for working “overtime.”
That is, for the first q hours the consumer works, she receives a real wage of w1,
and for hours worked beyond q, she receives w2, where w2 > w1. Suppose that the
consumer pays no taxes and receives only wage income, and she is free to choose
hours of work.
a. Draw the consumer’s budget constraint, and show her optimal choice of
consumption and leisure.
The graph below depicts the optimal choice of consumption and leisure. The
indifference curve touches the steep portion of the budget constraint, indicating
that the individual chooses to work overtime. Note that it is also possible to draw
the indifference curve such that the individual chooses not to work overtime. In
this case, the indifference curve would touch the flatter portion of the budget
constraint.
Economics 501
Problem Set 1 Suggested Solutions
Fall 2006
Prof. Daniel
c
c*
a-T
l*
l-q
l
b. Show that the consumer would never work q hours, or anything very close to
q hours. Explain the intuition behind this.
The kink in the budget line means that it is impossible for the indifference curve
to meet at exactly l – q (at l – q the individual works q hours). It would be like
trying to make the surface of a ball touch the corner of a box.
c. Determine what happens if the overtime wage rate w2 increases. Explain you
results in terms of income and substitution effects. You will need to consider
the case of a worker who initially works overtime, and a worker who initially
does not work overtime.
The first graph below shows the case of a worker who initially works overtime.
The effect of an increase in the overtime wage is just the standard analysis of a
wage increase. The slope of the budget constraint rises, and the individual works
more if the substitution effect of the wage increase dominates the income effect.
In the second graph, the individual initially does not work overtime. In this case,
the increase in the overtime wage induces the individual to switch to radically
increase her labor supply.
Economics 501
Problem Set 1 Suggested Solutions
Fall 2006
Prof. Daniel
c
a-T
l-q
l
Individual initially works overtime
c
a-T
l-q
Individual initially does not work overtime
l
Economics 501
Problem Set 1 Suggested Solutions
Fall 2006
Prof. Daniel
6. Consumption is the largest component. Over time government expenditure has
fallen as a share, while investment and consumption have risen. Real net exports have
fallen both as a fraction of GDP and as a level.
Consumption/GDP
Government Expenditures/GDP
Jan-03
Jan-99
Jan-95
Jan-91
Jan-87
Jan-83
Jan-79
Jan-75
Jan-71
Jan-67
Jan-63
Jan-59
Jan-55
Jan-51
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
Jan-47
Components of GDP/GDP
Investment/GDP
Net Exports/GDP
Real Consumption
Real Government Expenditures
Real Investment
Real Net Exports
Jan-03
Jan-99
Jan-95
Jan-91
Jan-87
Jan-83
Jan-79
Jan-75
Jan-71
Jan-67
Jan-63
Jan-59
Jan-55
Jan-51
9000.0
8000.0
7000.0
6000.0
5000.0
4000.0
3000.0
2000.0
1000.0
0.0
-1000.0
-2000.0
Jan-47
Real Components of GDP