# Baltimore City Community College Problem Set Production Function Worksheet Consider the production function ?? = ??(??,??) = ???? ???? ?? , where A, a, b are positi

Baltimore City Community College Problem Set Production Function Worksheet Consider the production function ?? = ??(??,??) = ????

????

??

, where A, a, b are positive constants.

a. Compute the marginal products of labor and capital, the average product of labor, and the

marginal rate of technical substitution of labor for capital

b. For what values of A, a, and b are there diminishing marginal returns to labor L? What about

diminishing marginal returns to capital?

c. For what values of A, a, and b does the production process have diminishing marginal rate of

technical substitution?

d. Is this function homogeneous? Explain.

e. For what values of A, a, and b are there constant/decreasing/increasing returns to scale?

2) Suppose a firm has the following production function: ?? = 2??? + ??

a. Compute the marginal products of labor and capital, the average product of labor, and the

marginal rate of technical substitution of labor for capital.

b. Are there diminishing returns to labor? What about to capital?

c. Is there diminishing marginal rate of technical substitution (MRTS)?

d. Draw a couple of the functions isoquants: for instance ??1 = 2 and ??2 = 4

e. Calculate MRTS on these two isoquants at K=1/4 and K=1. Mark these 4 points in the graph.

f. What type of returns to scale does this production function have?

3) For each of the 9 production functions listed below, compute marginal products and check if they are

diminishing. Is the cross-partial ??

2??

???????? positive, zero, or negative? (i.e. how does ?????? vary with K, and

?????? with ???)

Compute |MRTS| and check if there is diminishing MRTS.

Finally, pick the isoquant map that you think best describes the function (out of the 8 options A-H).

[Hint: intersections with the axes will probably be relevant, as will the variation in MRTS. Note

that the isoquants are not labeled, and you should not make any assumptions about their values

(other than the fact that they increase as we move away from the origin). At least one graph

corresponds to multiple production functions, others may not correspond to any.]

a) ?? = 12?? + ??

2

b) ?? = ????? [Hint: if you are trying to decide between 2 graphs, you could compare the slope of

isoquants along a 45-degree line from the origin (i.e. where L=K) and see which appears closer

to the MRTS value you found.]

c) ?? = ln ?? +

1

2

ln??

d) ?? = ????? + ?? [Hint: remember to be flexible: sometimes you can determine how a function

g(x,y) varies with x simply by noticing whether x appears in the denominator or in the

numerator. Other times youll want to check whether ????

???? is positive or negative.]

e) ?? = ??? + ???

f) ?? = (??? + ???)

2

g) ?? = 2?? + ??

h) ?? = ?? + 2??

i) ?? = ??? + 2? Econ 100B, Summer I, 2020, Tocoian

PROBLEM SET 1

Due by Monday at 11am

1) Consider the production function ?? = ??(??, ??) = ?????? ?? ?? , where A, a, b are positive constants.

a. Compute the marginal products of labor and capital, the average product of labor, and the

marginal rate of technical substitution of labor for capital

b. For what values of A, a, and b are there diminishing marginal returns to labor L? What about

diminishing marginal returns to capital?

c. For what values of A, a, and b does the production process have diminishing marginal rate of

technical substitution?

d. Is this function homogeneous? Explain.

e. For what values of A, a, and b are there constant/decreasing/increasing returns to scale?

2) Suppose a firm has the following production function: ?? = 2??? + ??

a. Compute the marginal products of labor and capital, the average product of labor, and the

marginal rate of technical substitution of labor for capital.

b. Are there diminishing returns to labor? What about to capital?

c. Is there diminishing marginal rate of technical substitution (MRTS)?

d. Draw a couple of the functions isoquants: for instance ??1 = 2 and ??2 = 4

e. Calculate MRTS on these two isoquants at K=1/4 and K=1. Mark these 4 points in the graph.

f. What type of returns to scale does this production function have?

3) For each of the 9 production functions listed below, compute marginal products and check if they are

??2 ??

diminishing. Is the cross-partial

positive, zero, or negative? (i.e. how does ?????? vary with K, and

????????

?????? with ???)

Compute |MRTS| and check if there is diminishing MRTS.

Finally, pick the isoquant map that you think best describes the function (out of the 8 options A-H).

[Hint: intersections with the axes will probably be relevant, as will the variation in MRTS. Note

that the isoquants are not labeled, and you should not make any assumptions about their values

(other than the fact that they increase as we move away from the origin). At least one graph

corresponds to multiple production functions, others may not correspond to any.]

a) ?? = 12?? + ?? 2

b) ?? = ????? [Hint: if you are trying to decide between 2 graphs, you could compare the slope of

isoquants along a 45-degree line from the origin (i.e. where L=K) and see which appears closer

to the MRTS value you found.]

1

c) ?? = ln ?? + 2 ln ??

d) ?? = ????? + ?? [Hint: remember to be flexible: sometimes you can determine how a function

g(x,y) varies with x simply by noticing whether x appears in the denominator or in the

????

numerator. Other times youll want to check whether ???? is positive or negative.]

e) ?? = ??? + ???

f)

g)

h)

i)

??

??

??

??

2

= (??? + ???)

= 2?? + ??

= ?? + 2??

= ??? + 2??

1

Econ 100B, Summer I, 2020, Tocoian

A

B

C

D

E

F

2

Econ 100B, Summer I, 2020, Tocoian

G

H

4) Returns to scale and homogeneity

For each of the following production functions, determine what type of returns to scale it has, and

whether it is homogeneous (and, if so, of what degree).

[You can use the unspecified parameter s, or a numerical value such as s=2 if you prefer.]

a) ?? = ??(??, ??) = 2?? + ??

b) ?? = ??(??, ??) = ??? + 2??

c) ?? = ??(??, ??) = min(2??, 20??)

2

d) ?? = ??(??, ??) = (min(2??, 20??))

e) ?? = ??(??1 , ??2 , ??) = min(??1 + ??2 , 20??)

f) ?? = ??(??, ??) = ??3 + ????

2

g) ?? = ??(??, ??) = (??? + ???)

h) ?? = ??(??, ??) = ????? + ??

3

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