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 𝑳𝟏 to 𝑳𝟐 in the graph below) holding everything else constant this will increase total

output (𝑸 must be > 𝑸𝟏 at point 𝑩 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 𝑲𝟐. 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.

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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.

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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 labor’s marginal product for

that amount of capital. Likewise I hold labor equal to 4 and calculate

capital’s marginal product for that amount of labor. Both input marginal

products are clearly positive.

K = 3 L = 4

L Q 𝑴𝑷𝑳

K Q 𝑴𝑷𝑲

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

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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, labor’s marginal product decreases from 7 to 3 as labor is increased.

Likewise, with labor fixed at 4 units, capital’s 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

𝑄 = 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 𝑴𝑹𝑻𝑺 =𝑴𝑷𝑳

𝑴𝑷𝑲.

Movement along the isoquant implies input substitution. Movement down

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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, 𝑴𝑷𝑳 necessarily decreases and 𝑴𝑷𝑲 necessarily increases. Because

its numerator decreases and its denominator increases, the 𝑴𝑹𝑻𝑺 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 𝐿 = 10 and 𝐾 = 15, how much output is produced? 𝑸 = 𝟓𝟎𝟎(𝟏𝟎).𝟔(𝟏𝟓).𝟖 = 𝟏𝟕, 𝟑𝟕𝟏. 𝟕𝟑

b. If 𝐿 = 10 and 𝐾 = 15, what is 𝑀𝑃𝐿? What is 𝑀𝑃𝐾?

In a Cobb-Douglas production function, 𝑴𝑷𝑳 = 𝜶 × 𝑨𝑷𝑳 and 𝑴𝑷𝑲 = 𝜷 ×𝑨𝑷𝑲. Given L= 𝟏𝟎, 𝑲 = 𝟏𝟓, and 𝑸 = 𝟏𝟕, 𝟑𝟕𝟏. 𝟕𝟑, labor's marginal product is 𝑴𝑷𝑳 =

. 𝟔 × (𝟏𝟕,𝟑𝟕𝟏.𝟕𝟑

𝟏𝟎) = 𝟏, 𝟎𝟒𝟐. 𝟑 and capital's marginal product is 𝑴𝑷𝑲 =. 𝟖 ×

(𝟏𝟕,𝟑𝟕𝟏.𝟕𝟑

𝟏𝟓) = 𝟗𝟐𝟔. 𝟒𝟗 .

c. Given 𝐿 = 10 and 𝐾 = 15, does the law of diminishing marginal returns hold for labor? Why or why not?

We know from b that 𝑴𝑷𝑳 = 𝟏, 𝟎𝟒𝟐. 𝟑 at {𝑳 = 𝟏𝟎 & 𝑲 = 𝟏𝟓}. So what happens to labor’s marginal product as we increase labor and hold capital

fixed? Labor’s marginal product at {𝑳 = 𝟏𝟏 & 𝑲 = 𝟏𝟓}, for example, is

𝑴𝑷𝑳 =. 𝟔 × (𝟏𝟖,𝟑𝟗𝟒.𝟏

𝟏𝟏) = 𝟏, 𝟎𝟎𝟑. 𝟑 which is a bit less than 𝑴𝑷𝑳 at {𝑳 =

𝟏𝟎 & 𝑲 = 𝟏𝟓}, 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: 𝑸 = 𝑨𝑳𝜶𝑲𝜷. Using calculus, the marginal

product of labor is 𝑴𝑷𝑳 =𝝏𝑸

𝝏𝑳= 𝜶𝑨𝑳𝜶−𝟏𝑲𝜷. The derivative of 𝑴𝑷𝑳 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 labor’s marginal product: 𝝏𝑴𝑷𝑳

𝝏𝑳= (𝜶 − 𝟏)𝜶𝑨𝑳𝜶−𝟐𝑲𝜷

This derivative must be less than 0 if 𝟎 < 𝜶 < 𝟏. 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

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between 0 and 1. Labor’s exponent in this problem is 0.6, which is less than 1

and more than 0 and therefore implies that labor’s marginal product

decreases as labor is increased while capital is held constant.]

d. Given 𝐿 = 10 and 𝐾 = 15, what is the marginal rate of technical substitution?

We showed in class that, given a Cobb-Douglas production function,

𝑴𝑹𝑻𝑺 = (𝜶

𝜷) × (

𝑲

𝑳). Therefore, with 10 units of labor and 15 units of capital

the 𝑴𝑹𝑻𝑺 = (.𝟔

.𝟖) × (

𝟏𝟓

𝟏𝟎) = 𝟏. 𝟏𝟐𝟓. An alternative approach to answering

this question would be to note that regardless of the type of production

function, the 𝑴𝑹𝑻𝑺 =𝑴𝑷𝑳

𝑴𝑷𝑲 by definition. Taking the values determined in

part b, 𝑴𝑹𝑻𝑺 =𝟏𝟎𝟒𝟐.𝟑

𝟗𝟐𝟔.𝟒𝟗= 𝟏. 𝟏𝟐𝟓.

e. Consider an alternative input combination where 𝐿 = 20 and 𝐾 = 8.919055. Is this input combination on the same isoquant as input combination {𝐿 = 10, 𝐾 =15}? Why or why not?

𝑸 = 𝟓𝟎𝟎(𝟐𝟎).𝟔(𝟖. 𝟗𝟏𝟗𝟎𝟓𝟓).𝟖 = 𝟏𝟕, 𝟑𝟕𝟏. 𝟕𝟑. 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, 𝜶 + 𝜷 =. 𝟔+. 𝟖 = 𝟏. 𝟒 > 𝟏. 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 𝑸 =𝟓𝟎𝟎(𝟐𝟎).𝟔(𝟑𝟎).𝟖 = 𝟒𝟓, 𝟖𝟒𝟒. 𝟑. The returns to scale elasticity is defined as

𝑹𝑻𝑺𝝐 =%∆𝑸

%∆𝑰𝒏𝒑𝒖𝒕𝒔. %∆𝑸 = (

𝟒𝟓𝟖𝟒𝟒.𝟑−𝟏𝟕𝟑𝟕𝟏.𝟕𝟑

𝟏𝟕𝟑𝟕𝟏.𝟕𝟑) × 𝟏𝟎𝟎 = 𝟏𝟔𝟑. 𝟗. Since we

doubled the inputs, the percent change in the inputs is 100%. Hence the

returns to scale elasticity is 𝑹𝑻𝑺𝝐 =𝟏𝟔𝟑.𝟗%

𝟏𝟎𝟎%= 𝟏. 𝟔𝟑𝟗, which confirms the

assertion in part f that the production function exhibits increasing returns to

scale.

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4. Suppose that 𝑇𝐶0 = $2000, 𝑤 = $90, and 𝑟 = $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, 𝑤/𝑟, 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 $𝟗𝟗

$𝟔𝟔=

$𝟗𝟎

$𝟔𝟎=

8

𝟏. 𝟓, 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 𝑇𝐶0 = $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 𝒘′

𝒓=

$𝟗𝟗

$𝟔𝟎= 𝟏. 𝟔𝟓. These

changes cause the isocost line to rotate down from its original position as in the graph

below.

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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, labor’s marginal product is

20 and capital’s marginal product is 10. A unit of labor costs 𝑤 = $15 per hour while a

unit of capital costs 𝑟 = $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

𝑴𝑹𝑻𝑺 =𝒘

𝒓

The input price ratio in the problem is

𝒘

𝒓=

$𝟏𝟓

$𝟏𝟐= 𝟏. 𝟐𝟓

In order to obtain the MRTS at the current input combination, recall the economic

definition of the MRTS, i.e.,

𝑴𝑹𝑻𝑺 =𝑴𝑷𝑳𝑴𝑷𝑲

We are given the information that at the current production level 𝑴𝑷𝑳 = 𝟐𝟎 and 𝑴𝑷𝑲 =

𝟏𝟎. Hence

𝑴𝑹𝑻𝑺 =𝑴𝑷𝑳𝑴𝑷𝑲

=𝟐𝟎

𝟏𝟎= 𝟐

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Bringing together the MRTS and the input price ratio for comparison,

𝟐 = 𝑴𝑹𝑻𝑺 >𝒘

𝒓= 𝟏. 𝟐𝟓

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 𝑴𝑹𝑻𝑺. For example, substitution of 1 unit of labor for 2

units of capital, by definition of the 𝑴𝑹𝑻𝑺, 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

(= 𝒓 × ∆𝑲 = $𝟏𝟐 × (−𝟐)) 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

𝑴𝑹𝑻𝑺 >𝒘

𝒓

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

𝑴𝑹𝑻𝑺 = 𝟏. 𝟐𝟓

the firm will have minimized its cost of production.

6. The XYZ Corporation uses two homogeneous inputs—labor and capital—in 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, let’s work out the total cost implied at this input combination. 𝑻𝑪𝟎 = 𝒘𝑳𝟎 +

𝒓𝑲𝟎 = ($𝟖𝟎 × 𝟖) + ($𝟔𝟎 × 𝟏𝟐) = $𝟏, 𝟑𝟔𝟎. Next, determine the intercepts of the

isocost line. Accordingly, 𝑻𝑪𝟎

𝒓=

$𝟏𝟑𝟔𝟎

$𝟔𝟎= 𝟐𝟐. 𝟔𝟕 gives the vertical intercept and

𝑻𝑪𝟎

𝒘=

$𝟏𝟑𝟔𝟎

$𝟖𝟎= 𝟏𝟕 gives the horizontal intercept. The appropriate graph is given

below.

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b. What does the marginal rate of technical substitution equal at input combination

{𝐿 = 8 and 𝐾 = 12}? Explain.

In this problem, we don’t have enough information to calculate the 𝑴𝑹𝑻𝑺 directly,

but we do know that if input combination 𝑨~{𝑳 = 𝟖, 𝑲 = 𝟏𝟐} does minimize cost,

then it must be the case that 𝑴𝑹𝑻𝑺 =𝒘

𝒓 at input combination 𝑨. Given that

𝒘 = $𝟖𝟎 and 𝒓 = $𝟔𝟎,𝒘

𝒓=

$𝟖𝟎

$𝟔𝟎= 𝟏. 𝟑𝟑. Therefore, the 𝑴𝑹𝑻𝑺 must equal 1.33 at

the input combination {𝑳 = 𝟖, 𝑲 = 𝟏𝟐}.

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 {𝐿 = 8 and 𝐾 = 12} now? Show graphically what happens to the isocost line going through {𝐿 = 8 and 𝐾 = 12} after the change in the price of labor (be sure to provide the numerical values of the isocost line’s new

intercepts).

Given the wage increase to 𝒘′ = $𝟏𝟎𝟖, the initial input combination now costs out

at 𝑻𝑪𝟏 = 𝒘′𝑳𝟎 + 𝒓𝑲𝟎 = ($𝟏𝟎𝟖 × 𝟖) + ($𝟔𝟎 × 𝟏𝟐) = $𝟏, 𝟓𝟖𝟒. By implication, the

intercept values of the iso-cost line containing input combination 𝑨 change to 𝑻𝑪𝟏

𝒓=

$𝟏𝟓𝟖𝟒

$𝟔𝟎= 𝟐𝟔. 𝟒 and

𝑻𝑪𝟏

𝒘=

$𝟏𝟓𝟖𝟒

$𝟏𝟎𝟖= 𝟏𝟒. 𝟔𝟕. The dotted line in the graph shows

the isocost line passing through input combination A at input prices 𝒘′ =

$𝟏𝟎𝟖 and 𝒓 = $𝟔𝟎.

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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, 𝟏. 𝟑𝟑 = 𝑴𝑹𝑻𝑺𝑨 <𝒘′

𝒓=

$𝟏𝟎𝟖

$𝟔𝟎= 𝟏. 𝟖. This says that labor’s productivity is 33% greater than capital’s

productivity at the margin but labor’s 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., ∆𝑻𝑪 = 𝒘′∆𝑳 + 𝒓∆𝑲 = ($𝟏𝟎𝟖 × (−𝟏)) + ($𝟔𝟎 × (𝟏. 𝟑𝟑)) =

−$𝟏𝟎𝟖 + $𝟕𝟗. 𝟖𝟎 = −𝟐𝟖. 𝟐𝟎). As the firm substitutes capital for labor in this

fashion, 𝑴𝑷𝑳 increases and 𝑴𝑷𝑲 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 $𝟏𝟑𝟔𝟎 = 𝑻𝑪𝟎 < 𝑻𝑪𝟐 < 𝑻𝑪𝟏 =

$𝟏𝟓𝟖𝟒. 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

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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 ∆𝑳 = 𝑳𝟏 − 𝟖 < 𝟎 and the

substitution effect for capital to be ∆𝑲 = 𝑲𝟏 − 𝟏𝟐 > 𝟎.