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Solutions for HW 3, Econ 902, FALL 2014
1. Dadkhah: Ch. 8: 1 - See attached pages.
2. Dadkhah: Ch. 9: 4, 5, 6 - See attached pages.
3. Dadkhah: Ch. 10: 1, 2, 3 - See attached pages.
4. Chiang: Ch. 12: 1 - See attached pages.
5. Consider the following simple representation of the tradeoff between taxes and income. The total
output (income) of an economy, Y , can be described as coming from the level of productivity, A, which
represents the effectiveness with which capital (machinery, factories, etc.) are used, the amount of
capital, K, and the level of government expenditure, G.
Y = AK α G1−α
(1)
(a) Graph this function of Y against K assuming that 0 < α < 1. Why should it make sense that α
is greater than zero but less than one? Take the first and second derivatives with respect to K
treating everything else as a constant and graph your results. Provide economic interpretations
of each.
SOLUTION: The graph of the function should look like this:
The assumption that α is greater than zero but less than one, 0 < α < 1, implies a positive
marginal product of capital but with diminishing returns. These implications are intuitive. More
machinery and factories should be able to produce more output, but the additional gains from
more capital diminishes as the size of the capital stock grows since labor is not changing. The
derivatives are as follows:
dY
dK
d2 Y
dK 2
= αAK α−1 G1−α > 0
=
(α − 1) αAK α−2 G1−α < 0
The graphs of these two derivatives are as follows:
Note that the second derivative is negative, but increases in value as K gets larger. I left the
numbers on the y-axis in the graph. The numbers are not important and depend entirely on
the numbers entered into an Excel spread sheet for A, G, α, and K. However, it is the negative
value for the second derivative which is important. It says that the slope of the first derivative is
negative, meaning there are diminishing returns to capital. In addition, as K gets larger the rate
at which the M P K is falling is getting smaller in absolute value. That is, the slope of the MPK
function is getting flatter.
(b) Draw another graph for Y against G. Interpret this diagram. How does G affect Y? Does the
effect increase or decrease? Why might that make sense? Take the first and second derivatives
with respect to G treating everything else as a constant and graph your results. Provide economic
interpretations of each.
SOLUTION: These graphs are identical in the shape of the curves to those in part a.
Government has a positive effect on output, but at a diminishing (decreasing) rate. There are a
number of ways to justify these assumptions. One way, we discussed in class. The government
provides goods and services that make firms and workers more productive such as a legal system
that protects property rights and respects contract law, police protection, national defense, education, and health care. All of these lead to increases in the amount the economy can produce
for a given amount of physical capital. However, the gains from increased government expenditure diminish as adding more and more police, for example, makes people feel more secure, but
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the additional benefit of the one-millionth cop in my neighborhood does not add much whereas
putting the first one or two officers there makes a tremendous difference.
dY
dG
d2 Y
dG2
=
(1 − α)AK α G−α > 0
= −α (1 − α) AK α G−α−1 < 0
The graphs of these two derivatives are as follows:
The intution here is the same as above. Government expenditures have a positive marginal
product, thus a positive first derivative, but the marginal product diminishes as G gets larger,
thus a negative second derivative.
(c) Now consider how government expenditures are actually paid for, i.e. taxes. Thus, disposable
income (income net of taxes) is:
Ydisposable = AK α G1−α − T
(2)
if the government finances everything through taxes. Rewrite the expression in terms of just
taxes and capital as determining disposable income. To do this, assume that the government
does not have a government budget surplus or deficit, such that G = T . Substititute T for G in
the above expression and take the derivative of disposable income with respect to T . Your firstorder condition should have two components corresponding to a marginal benefit and a marginal
cost. Provide economic interpretations of these. Now graph disposable income against the level
of taxation and interpret what this graph represents.
SOLUTION: Substituing for G using T we have:
Ydisposable = AK α T 1−α − T
(3)
Taking the first derivative with respect to taxes, T , we get:
dYdisposable
= (1 − α) AK α T −α − 1
dT
(4)
Interpreting this expression was probably challenging for most people. I will explain this in some
detail and hopefully find some time in class to go over it. Look at the first derivate again. The
first component, (1 − α) AK α T −α , represents the marginal benefit of increased taxation because
it shows how output rises as the government spends (taxes) more. The second component is just
−1 which represents how disposable income falls with increased government spending (taxation).
For every unit of government expenditure, disposable income falls by one unit. This can be easier
to see in a graph. The graph below shows these two components separately as follows:
The curved line is the marginal product of government expenditure, just like the corresponding
graph in part b. The straight line at one shows the cost in terms of income taken away by the
government to fund expenditures. On the left portion of the graph, before the two intersect, the
marginal benefits of higher government expenditure exceed the cost. That means that disposable
income can be increased by raising taxes in that portion of the graph. The reason is that higher
government expenditure will lead to output increases, Y , greater than the additional loss in taxes.
To the right of the intersection, the increase in output from higher government expenditures gets
smaller such that the marginal increase in output does not make up for the loss in disposable
income through taxes. (Do not be concerned about the actual numbers on the graph, it is the
shapes of the curves and interpretations that matter.) Combining these two curves and simply
graphing (1 − α) AK α T −α − 1 as a whole shows us this:
The graph looks the same as the marginal benefit portion in the previous graph but shifted down
after substracting off the marginal cost (1) at all levels of G. Notice that where it falls to zero
corresponds to the intersection in the previous graph. That intersection shows exactly where the
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marginal benefit of increased taxation equals the marginal cost. The following graph shows the
function describing disposable income: Ydisposable = AK α T 1−α − T
Notice how the graph rises then falls. The rising portion on the left corresponds to where the
marginal benefits in the previous graph exceed the marginal cost in terms of taxes. It shows
that disposable income increases when government spending is initially low, reaches a peak (the
maximum level of disposable income), and then declines as the loss in taxes outweighs the gains
from production. Notice that the slope is zero at the maximum point. With that information
we can solve for the level of taxes that yields the highest level of disposable income (This was
not required on the homework). Since the first derivative is the slope of this function, we simply
need to find the level of taxes that occurs when the slope is exactly zero. So, setting the first
derivative equal to zero we have:
dYdisposable
= (1 − α) AK α T −α − 1 = 0
dT
(5)
Solving:
(1 − α) AK α T −α
(1 − α) AK
α
=
1
= Tα
Tα
=
(1 − α) AK α
T∗
=
[(1 − α) AK α ]
1/α
Thus, we have the level of taxes, T ∗ (where * indicates the optimal level), that generates the
highest disposable income. Using the numbers that I used for the graphs above where α = 4/5,
K = 10, and A = 1, implies that
T∗
=
T∗
=
T∗
=
1/α
[(1 − α) AK α ]
h
i1/(4/5)
(1 − 4/5) (1)101/5
h
i5/4
(1/5) 104/5
= 1.337
That generates a level of disposable income of 5.350 and a total income of 6.687 or precisely 20%
of income goes to taxes.
6. (a) Josh’s intertemporal two-period budget constaint equates the present value of lifetime consumption to the present value of lifetime income:
C1 +
C2
Y2
= Y1 +
1+r
1+r
(b) The lagrangian is:
C2
Y2
− C1 −
L = ln C1 + ln C2 + λ Y1 +
1+r
1+r
The first order conditions:
L1 ⇒
L2 ⇒
Lλ ⇒
C1 +
1
C1
1
C2
C2
1+r
=
=
=
λ
λ/ (1 + r)
Y2
Y1 + 1+r
Solving the first order conditions yield:
C1
C2
B
Y2
= 12 Y1 + 1+r
=
= (1 + r) C1
=
= Y1 − C1
=
3
55, 000
60, 500
−45, 000
(c) The new results would be:
C1
C2
B
=
=
=
57, 500
60, 375
−47, 500
With a decrease in the interest rate, Josh will consume more in period 1 and less in period 2.
Furthermore, savings will decrease (Josh will borrow more). Since Josh is a net borrower, the
decrease in the interest lessens how much he must repay in the second period, and thus gain more
wealth through the income effect. This effect will try to increase consumption in both periods
and decrease savings. However, borrowing is now relatively cheaper. This is the substitution
effect. The cheaper cost of borrowing will increase his incentive to borrow more which will
decrease savings, increase period 1 consumption and decrease period 2 consumption. The net
result, considering the income and subsitution effect for a net borrower is period 1 consumption
will unambigiously rise, savings will unambigiously fall, and the change in period 2 consumption
depends or the relatively magnitudes of the income and substitution effects. In our case, the
substitution effect dominates and period 2 consumption falls.
(d) We saw from part (a) that if there were no borrowing constraint, then the optimal level of
borrowing would be 45,000. Since this exceeds the maximum borrowing limit, the borrowing
constraint will bind. Josh will borrow the max and consume his period 1 income. Thus, C1 will
be 40,000 and B equals 30,000. Finally, C2 then becomes Y2 − (1 + r) B = 77, 000. See the graph
for more details.
(e) A drop in the period 2 income will decrease both current and future consumption through a pure
income effect. Furthermore, we expect savings to rise, or in our case, borrowing to fall. If we
reoptimize, we find
Y2
= 30, 000
C1 = 21 Y1 + 1+r
C2 = (1 + r) C1
= 33, 000
B = Y1 − C1
= −20, 000
Since Josh would not optimally borrow in excess of 30,000, the borrowing constraint does not
bind. Again, see the graph for more details.
7. (a) Optimization problem using a Lagrangian,
maxA,B,C
ln A + ln B + ln C + λ (24 − A − B − C)
FOC:
1
A
1
B
1
C
=λ
=λ
=λ
24 − A − B − C = 0
The first order conditions imply that the marginal utility of each component, A, B, C, will equal
the shadow value of wealth. The final FOC is the time constraint.
(b) Using the first order conditions, we have a symmetric equilibrium, where A = B = C = 8.
(c) We can think of the coefficients in the time constraint as their respective marginal cost. Thus, if
the HA increases, Sue will devote less time on micro and leisure, and more on macro.
(d)
maxA,B,C HA ln A + HB ln B + ln C + λ (24 − A − B − C)
FOC:
HA
A =λ
HB
B =λ
1
C =λ
24 − A − B − C = 0
Solving for A, B and C yields:
A∗
B∗
C∗
=
=
=
4
HA
(1+HA +HB ) 24
HB
(1+HA +HB ) 24
1
(1+HA +HB ) 24
Using values of HA = 50 and HB = 150, we have: A∗ = 5.97; B ∗ = 17.9; and C ∗ = 0.13.
(e) We have the following partial derivatives:
w.r.tHA
A∗
B∗
C∗
1+HB
24
(1+HA +HB )2
HB
− (1+H +H )2 24
A
B
− (1+H 1+H )2 24
A
B
w.r.tHB
A
− (1+HH+H
24
2
A
B)
1+HA
2 24
(1+HA +HB )
− (1+H 1+H )2 24
A
B
The partial derivates indicate that as the amount of homework for a given subject rises, it will
increase the time spent on that individual suject (dA/dHA > 0 and dB/dHB > 0), but decrease
the time spent on the remaining subject and leisure (∂A/∂HB < 0 and ∂B/∂HA < 0).
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