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Summer 2016 ECN 303
Problem Set #1
Due at the beginning of class on Monday, May 23. Give complete answers and show your
work. The assignment will be graded on a credit/no credit basis. In order to receive credit for
the assignment, you must demonstrate a good faith effort on each of the questions. I will post
correct answers so that you may self-assess in preparing for Exam I.
1.
a. Why must an isoquant be downward sloping when both labor and capital have
positive marginal products? Explain and illustrate graphically.
If the marginal product of labor is positive, then when we increase the level
of labor (say from π³π to π³π in the graph below) holding everything else constant this will increase total
output (πΈ must be > πΈπ at point π© in the graph). To keep the level of output at the original level, we need to stay on the same isoquant. To do so, since the
marginal product of capital is positive we would then need to reduce the
amount of capital being used to π²π. So, to keep output constant, when the level of one input increases the level of the other input must decrease. This
negative relationship between the inputs implies that the isoquant will have a
negative slope, i.e., be downward sloping, and that the inputs are
substitutable.
2
b. Why do isoquants not intersect? Explain and illustrate graphically.
Suppose we draw isoquants for two levels of output 1Q and 2Q with 2 1Q Q .
In addition, suppose that these isoquants crossed at some point A as in the
following diagram.
Labor
Capital
A B
C
Q1
Q2
Point A implies that a given input combination can produce two different
outputs. We define the production function, however, as showing the
maximum amount of output that a given input combination can produce.
Furthermore, because B is on Q2 and C is on Q1, input combination B
produces more output than input combination C. This is not possible if the
inputs have positive marginal products, however, since point C contains
more of both inputs and therefore should achieve a higher level of output.
Hence intersecting isoquants are neither consistent with the basic definition
of the production function nor with the assumption that the inputs possess
positive marginal products.
3
2. Consider the production data given in Table 5.2 of the textbook. Quantities of labor are
given in the top row, quantities of capital are given in the left-most column, and output
levels are recorded in the body of the table.
a. Do labor and capital both have positive marginal products? Explain and provide
numerical support for your answer.
Labor and capital both have positive marginal products in the table. In
order to show this pick any column and any row in the table and calculate
the difference between one entry and the next to get the change in output due
to a 1 unit change in the input (i.e., the marginal product). In the table
below, I hold capital equal to 3 and calculate laborβs marginal product for
that amount of capital. Likewise I hold labor equal to 4 and calculate
capitalβs marginal product for that amount of labor. Both input marginal
products are clearly positive.
K = 3 L = 4
L Q π΄π·π³
K Q π΄π·π²
1 17 --
1 20 --
2 24 7
2 28 8
3 30 6
3 35 7
4 35 5
4 40 5
5 39 4
5 45 5
6 42 3
6 49 4
4
b. Do the inputs exhibit diminishing marginal returns? Explain and provide
numerical support for your answer.
Yes, both inputs exhibit diminishing marginal returns. In order to show this,
hold one input constant and examine what happens to the marginal product
of the other input as it is increased. In the table above, with capital fixed at 3
units, laborβs marginal product decreases from 7 to 3 as labor is increased.
Likewise, with labor fixed at 4 units, capitalβs marginal product decreases
from 8 to 4 as capital usage is increased.
c. In a well-labelled graph, draw the isoquant corresponding to an output level of
π = 35.
Note that all four input combinations labelled in my graph can be found in
the production function table given in the problem set-up.
d. Is the isoquant drawn in part c convex to the origin of the graph? Why or why
not? Why are isoquants typically convex? Explain.
Yes, the isoquant in the graph of part c is convex to the origin of the graph.
Isoquants for real world production processes will be convex as long as the
inputs in question are subject to the law of diminishing marginal returns.
The law of diminishing marginal returns states that the marginal product of
an input decreases as more of that input is employed, holding other input
usage levels constant. By definition, the absolute value of the slope of the
isoquant evaluated at a particular input combination is π΄πΉπ»πΊ =π΄π·π³
π΄π·π².
Movement along the isoquant implies input substitution. Movement down
5
the isoquant for example means labor is being substituted for capital. By
virtue of diminishing marginal returns, as more labor is combined with less
capital, π΄π·π³ necessarily decreases and π΄π·π² necessarily increases. Because
its numerator decreases and its denominator increases, the π΄πΉπ»πΊ must
decrease. Hence, the slope of the isoquant decreases as labor is substituted
for capital, which implies that the isoquant is convex to the origin.
3. Consider the following Cobb-Douglas production function: 8.06.0500 KLQ
a. If πΏ = 10 and πΎ = 15, how much output is produced? πΈ = πππ(ππ).π(ππ).π = ππ, πππ. ππ
b. If πΏ = 10 and πΎ = 15, what is πππΏ? What is πππΎ?
In a Cobb-Douglas production function, π΄π·π³ = πΆ Γ π¨π·π³ and π΄π·π² = π· Γπ¨π·π². Given L= ππ, π² = ππ, and πΈ = ππ, πππ. ππ, labor's marginal product is π΄π·π³ =
. π Γ (ππ,πππ.ππ
ππ) = π, πππ. π and capital's marginal product is π΄π·π² =. π Γ
(ππ,πππ.ππ
ππ) = πππ. ππ .
c. Given πΏ = 10 and πΎ = 15, does the law of diminishing marginal returns hold for labor? Why or why not?
We know from b that π΄π·π³ = π, πππ. π at {π³ = ππ & π² = ππ}. So what happens to laborβs marginal product as we increase labor and hold capital
fixed? Laborβs marginal product at {π³ = ππ & π² = ππ}, for example, is
π΄π·π³ =. π Γ (ππ,πππ.π
ππ) = π, πππ. π which is a bit less than π΄π·π³ at {π³ =
ππ & π² = ππ}, so the marginal product of labor is decreasing as labor is increased and capital is held fixed.
[Note we can generalize this result as follows. Consider a general Cobb-
Douglas production function: πΈ = π¨π³πΆπ²π·. Using calculus, the marginal
product of labor is π΄π·π³ =ππΈ
ππ³= πΆπ¨π³πΆβππ²π·. The derivative of π΄π·π³ with
respect to labor usage tells us how the marginal product of labor changes as
the amount of labor is increased, holding capital constant. If this derivative
is negative, then, by implication, diminishing marginal returns prevails. So,
taking the derivative of laborβs marginal product: ππ΄π·π³
ππ³= (πΆ β π)πΆπ¨π³πΆβππ²π·
This derivative must be less than 0 if π < πΆ < π. So it turns out that, given a Cobb-Douglas production function, diminishing marginal returns always
holds for an input as long as its exponent in the production function is
6
between 0 and 1. Laborβs exponent in this problem is 0.6, which is less than 1
and more than 0 and therefore implies that laborβs marginal product
decreases as labor is increased while capital is held constant.]
d. Given πΏ = 10 and πΎ = 15, what is the marginal rate of technical substitution?
We showed in class that, given a Cobb-Douglas production function,
π΄πΉπ»πΊ = (πΆ
π·) Γ (
π²
π³). Therefore, with 10 units of labor and 15 units of capital
the π΄πΉπ»πΊ = (.π
.π) Γ (
ππ
ππ) = π. πππ. An alternative approach to answering
this question would be to note that regardless of the type of production
function, the π΄πΉπ»πΊ =π΄π·π³
π΄π·π² by definition. Taking the values determined in
part b, π΄πΉπ»πΊ =ππππ.π
πππ.ππ= π. πππ.
e. Consider an alternative input combination where πΏ = 20 and πΎ = 8.919055. Is this input combination on the same isoquant as input combination {πΏ = 10, πΎ =15}? Why or why not?
πΈ = πππ(ππ).π(π. ππππππ).π = ππ, πππ. ππ. Both the input combination in a and the input combination in e produce the same amount of output, hence
both input combinations are on the same isoquant.
f. What type of returns to scale does this production function exhibit? Why?
Returns to scale are easily evaluated in a Cobb-Douglas production function.
As we showed in class, all we need to do is add the input exponents together
and compare with 1. If the sum exceeds 1, there are increasing returns to
scale. If the sum equals 1, there are constant returns to scale. If the sum is
less than 1, there are decreasing returns to scale. In this problem, πΆ + π· =. π+. π = π. π > π. Therefore, production exhibits increasing returns to scale in this problem.
g. Double the input amounts used in part a and calculate the resulting percentage
change in output. What is the returns to scale elasticity here?
First note that doubling the input amounts from part a implies πΈ =πππ(ππ).π(ππ).π = ππ, πππ. π. The returns to scale elasticity is defined as
πΉπ»πΊπ =%βπΈ
%βπ°πππππ. %βπΈ = (
πππππ.πβπππππ.ππ
πππππ.ππ) Γ πππ = πππ. π. Since we
doubled the inputs, the percent change in the inputs is 100%. Hence the
returns to scale elasticity is πΉπ»πΊπ =πππ.π%
πππ%= π. πππ, which confirms the
assertion in part f that the production function exhibits increasing returns to
scale.
7
4. Suppose that ππΆ0 = $2000, π€ = $90, and π = $60.
a. Graph the isocost line corresponding to a total cost of $2,000. Be sure to label the axes
and points of interest such as intercepts in the graph appropriately.
b. The input price ratio, π€/π, is a possible rate of input substitution. True or false? Explain.
True. The input price ratio tells us the amount of capital that can replace 1 unit of
labor with no change in total cost. Likewise the input price ratio tells us how much
less capital must be used in order to afford employment of one more unit of labor
with no change in total cost. In short, the input price ratio is the input substitution
rate at which total cost is constant. In the example, 1.5 units of capital may be
substituted for 1 unit of labor, and vice versa, with no change in total cost.
c. If the price of labor and the price of capital both increase by 10%, how does that
affect the isocost line corresponding to total cost of $2,000? Explain and illustrate
graphically.
If both input prices increase by 10%, then this causes the $2,000 isocost line to shift
in toward the origin. The slope of the line remains the same because $ππ
$ππ=
$ππ
$ππ=
8
π. π, thus the input substitution rate at which total cost remains the constant has not
changed. This should seem reasonable because, while both input prices increased,
the relative expense of the two inputs remained unchanged. The inward shift of the
new isocost line reflects that less of both inputs can be purchased with two thousand
dollars when both inputs become more expensive.
d. If only the price of labor increases by 10%, how does that affect the isocost line
for ππΆ0 = $2000?
If only the price of labor increases by 10%, this reduces the horizontal intercept of the
isocost line to 20.2 and increases the slope of the isocost line to πβ²
π=
$ππ
$ππ= π. ππ. These
changes cause the isocost line to rotate down from its original position as in the graph
below.
9
In this scenario, the higher price of labor means that less labor and capital can be
afforded with $2000 and that the rate at which capital can be substituted for a unit
of labor with no change in total cost has increased to 1.65.
5. A firm operates with a technology that is characterized by a standard set of negatively-
sloped, convex isoquants. At the current level of production, laborβs marginal product is
20 and capitalβs marginal product is 10. A unit of labor costs π€ = $15 per hour while a
unit of capital costs π = $12 per hour. Is the firm producing its current level of output at
minimum cost? If yes, explain why. If no, show why not and indicate whether the firm
should be using (i) more capital and less labor, or (ii) less capital and more labor.
Long run cost minimization requires that inputs be employed such that
π΄πΉπ»πΊ =π
π
The input price ratio in the problem is
π
π=
$ππ
$ππ= π. ππ
In order to obtain the MRTS at the current input combination, recall the economic
definition of the MRTS, i.e.,
π΄πΉπ»πΊ =π΄π·π³π΄π·π²
We are given the information that at the current production level π΄π·π³ = ππ and π΄π·π² =
ππ. Hence
π΄πΉπ»πΊ =π΄π·π³π΄π·π²
=ππ
ππ= π
10
Bringing together the MRTS and the input price ratio for comparison,
π = π΄πΉπ»πΊ >π
π= π. ππ
This tells us that at the current input combination labor is twice as productive as capital yet
only 25% more expensive than capital. Per class discussion, the firm should substitute
labor for capital at the rate of the π΄πΉπ»πΊ. For example, substitution of 1 unit of labor for 2
units of capital, by definition of the π΄πΉπ»πΊ, maintains production at the current level but
reduces cost by $9 (the extra unit of labor costs $15 but 2 fewer units of capital saves $24
(= π Γ βπ² = $ππ Γ (βπ)) in capital expense for a net cost reduction of $9). The firm
should substitute labor for capital at the rate of the MRTS as long as
π΄πΉπ»πΊ >π
π
As more labor and less capital is employed, the MRTS decreases (due to diminishing
marginal returns). Once the firm achieves the input mix at which
π΄πΉπ»πΊ = π. ππ
the firm will have minimized its cost of production.
6. The XYZ Corporation uses two homogeneous inputsβlabor and capitalβin the
production of its product. A unit of labor costs XYZ $80 per day and a unit of capital
costs them $60 per day. Presently, XYZ wishes to produce 200 units of output per day.
Efficiency experts at the company have determined that the cost minimizing input
combination for the company at this production level is L = 8 and K = 12.
a. Provide a graph depicting the 200 unit isoquant and the isocost line that the firm
is on at the cost minimizing input combination. Explicitly identify the cost
minimizing combination of labor and capital in the graph. Calculate the total cost
of production using the current input combination and label the actual numerical
values of the isocost line intercepts in your graph.
First, letβs work out the total cost implied at this input combination. π»πͺπ = ππ³π +
ππ²π = ($ππ Γ π) + ($ππ Γ ππ) = $π, πππ. Next, determine the intercepts of the
isocost line. Accordingly, π»πͺπ
π=
$ππππ
$ππ= ππ. ππ gives the vertical intercept and
π»πͺπ
π=
$ππππ
$ππ= ππ gives the horizontal intercept. The appropriate graph is given
below.
11
b. What does the marginal rate of technical substitution equal at input combination
{πΏ = 8 and πΎ = 12}? Explain.
In this problem, we donβt have enough information to calculate the π΄πΉπ»πΊ directly,
but we do know that if input combination π¨~{π³ = π, π² = ππ} does minimize cost,
then it must be the case that π΄πΉπ»πΊ =π
π at input combination π¨. Given that
π = $ππ and π = $ππ,π
π=
$ππ
$ππ= π. ππ. Therefore, the π΄πΉπ»πΊ must equal 1.33 at
the input combination {π³ = π, π² = ππ}.
c. Suppose that workers at the company demand a 35% wage increase (raising daily,
per unit labor cost to $108). What is the total cost of producing 200 units of
output using {πΏ = 8 and πΎ = 12} now? Show graphically what happens to the isocost line going through {πΏ = 8 and πΎ = 12} after the change in the price of labor (be sure to provide the numerical values of the isocost lineβs new
intercepts).
Given the wage increase to πβ² = $πππ, the initial input combination now costs out
at π»πͺπ = πβ²π³π + ππ²π = ($πππ Γ π) + ($ππ Γ ππ) = $π, πππ. By implication, the
intercept values of the iso-cost line containing input combination π¨ change to π»πͺπ
π=
$ππππ
$ππ= ππ. π and
π»πͺπ
π=
$ππππ
$πππ= ππ. ππ. The dotted line in the graph shows
the isocost line passing through input combination A at input prices πβ² =
$πππ and π = $ππ.
12
d. Illustrate graphically the substitution effect that the change in the price of labor
has on the usage of labor and capital. What will ultimately happen to the total
cost of production once optimal substitution has taken place?
The increase in the price of labor breaks the equality between the MRTS and the
input price ratio that prevailed in parts a and b. Specifically, π. ππ = π΄πΉπ»πΊπ¨ <πβ²
π=
$πππ
$ππ= π. π. This says that laborβs productivity is 33% greater than capitalβs
productivity at the margin but laborβs expense is 80% greater than that of capital.
With the inequality running in this direction, the firm should substitute capital for
labor at the rate of the MRTS. So, for example, if a unit of labor is replaced with
1.33 units of capital, production is unchanged at Q = 200, but total cost is reduced
by -$28.20 (i.e., βπ»πͺ = πβ²βπ³ + πβπ² = ($πππ Γ (βπ)) + ($ππ Γ (π. ππ)) =
β$πππ + $ππ. ππ = βππ. ππ). As the firm substitutes capital for labor in this
fashion, π΄π·π³ increases and π΄π·π² decreases due to diminishing marginal returns all
of which causes the MRTS to increase towards 1.8 in the process. The firm should
continue to substitute capital for labor until the MRTS is driven back into equality
with the input price ratio, as at point B in the graph below. Once point B is reached,
the total cost of producing Q = 200 can be reduced no further. Using the vertical
intercepts in the graph below we can deduce that $ππππ = π»πͺπ < π»πͺπ < π»πͺπ =
$ππππ. This tells us that some of the increased cost transmitted via the wage
increase can be avoided by the firm via substitution away from labor and toward
capital. Nevertheless, the original total cost value of $1360 can no longer be
achieved, even with the optimal input adjustment. This implication should seem
intuitively reasonable. That is, if the firm had been minimizing total cost in the first
13
place at the lower wage, then when the wage increases with no offsetting change in
the price of capital, total cost must necessarily increase by some degree. Input
adjustment/response to the price change allows the firm to avoid some but not all of
the cost hit of the wage increase.
The graph reveals the substitution effect for labor to be βπ³ = π³π β π < π and the
substitution effect for capital to be βπ² = π²π β ππ > π.