# ECON 302, Section 001 – Assume that you have the following production schedule where labor is the only variable input

Question

NAME: ________________________ PSU Email:_________________@psu.edu

Homework #4

Intermediate Microeconomics ECON 302, Section 001

1. Assume that you have the following production schedule where labor is the only variable input. Fill in

the missing values for the MPL and APL for workers 1 – 4.

A. (4 Points) Complete the table below:

Number of Workers Units Produced Marginal Product Average Product

0 0 ----- ----

1 20

2 35

3 45

4 50

5 45

B. (2 Points) At which worker does diminishing marginal returns begin?

C. (2 Points) Define the concept of free disposal of inputs. Does it apply anywhere in this problem?

2. (2 Points) You have been doing research for a firm and have discovered that the firm’s average

product of capital is decreasing. If capital is the only variable input, what does this imply about the firms

marginal product of capital?

3. (2 Points) Use the space below to draw a production function where the average product of labor is

increasing up to the 10th worker and decreasing afterwards.

Output

# of workers (L)

4. Assume that the production function for a firm is defined as: 5

2

3

1

Y = K L .

A. (2 Points) If capital is fixed in the short run at 27 units (K = 27), what is the short-run production

function?

Y = __________________

B. (2 Points) Does this function exhibit increasing, decreasing, or constant returns to scale in the long

run? To receive full credit, you must show some work.

CIRCLE ONE: INCREASING DECREASING CONSTANT

C. (3 Points) Assume that production is occurring in the long run. Use the space below to draw the

isoquants representing output levels of 10 and 20 units respectively. Include at least three points on

each isoquant and fully label the graph.

5. Acme Corporation must decide how to allocate labor across two product lines. Each product line has

its own factory and the firm has 100 workers. Assume the price of each good is given by:

$2 $5 P1 = and P2 = .

Output at plant 1 is defined by: 2

1 1 25 1 Y = 100L ? 0. L and 1 1 MPL = 100 ? 0.5L

Output at plant 2 is defined by 2

2 2 375 2 Y = 75L ? 0. L and 2 2 MPL = 75 ? 0.75L

A. (4 Points) What is the efficient distribution of labor between the two factories in the short run given

the current prices of output.

Labor in Factory 1 = __________

Labor in Factory 2 = __________

B. Assume P2 falls to $3. How does the distribution of labor change within the firm?

Labor in Factory 1: __________

Labor in Factory 2: __________

C. (3 Points) In the space provided, sketch a graph of the marginal revenue products associated with

this firm. Be sure to label the efficient labor distribution before and after the change in P2 . Use the left

side of the graph for Good 1 and the right side for Good 2.

D. (2 Points) Return to the given information in Part A. Now assume that the firm replaces several

machines used to produce Good 1, making the workers associated with that product more productive.

Compare and contrast the effects of this change in technology with the effect of the change in P2 from

Part B.

6. Assume that you have the following production function and input prices in the long run:

2 2

3

2 3

3

2

8 4

MPL K L

MPK KL

Y K L w r

=

=

= = =

A. (2 Points) Derive a general formula for the Marginal Rate of Technical Substitution of Labor which

will be in terms of capital and labor:

MRTS LK = __________

B. (1 Point) If the firm uses 10 units of capital and 8 workers, how much will be produced?

Y = __________

C. (1 Point) Continue to assume that the firm uses an input bundle of (8, 10). How much does this

bundle cost?

COST = __________

D. (2 Points) What condition (or rule) must be met for a particular bundle to represent production at

lowest cost? State your answer as an equation specific to this problem.

E. (1 Point) Does the current input bundle (8, 10) represent production at lowest cost? Circle one.

YES NO

F. (2 Points) If the firm keeps costs constant, what input bundle represents production at lowest cost?

K? = _____

L= _____

G. (1 Point) Does output increase or decrease at this new bundle? Circle one.

INCREASE DECREASE

H. (2 Points) Rather than keep costs constant, suppose the firm decides to keep output constant and

change cost. What input bundle represents production at least cost while holding output constant at

the level seen at the beginning of this question?

K = _____

L= _____

I. (2 Points) How much does the firm save by using this new input bundle?

SAVINGS = __________

J. (4 Points) In the space below, graph the initial bundle and its location on an isoquant and isocost line.

Add the points from Parts F and H to the graph with their corresponding isoquants and isocost lines. Be

sure to fully label the graph and all of the y-intercepts for your isocost lines.

K. (2 Points) Using your answer from Part H, derive a cost function in terms of quantity.

C(Q) = _________________

7. Assume a firm has the following costs in a perfectly competitive industry:

MC q q

VC q q

FC

( ) 8

( ) 4

100

2

=

=

=

A. (2 Points) What is the breakeven quantity for this firm?

q = __________

B. (2 Points) Calculate the level of profit this firm makes when P* = $25.

Profit = __________

C. (4 Points) In the space below, draw the average cost, average variable cost, and marginal cost curves.

Show the breakeven quantity, breakeven price, and current market price as points on the graph.

8. Assume a consumer has the following preferences:

MU X

MU Y

U X XY

Y

X

=

= +

= +

20

20

Her income is $10 and she faces prices for X and Y equal to $1 each.

A. (2 Points) Solve for her optimal consumption bundle.

X = __________

Y= ___________

B. (2 Points) = What is her utility level at the optimal bundle?

U = __________

C. (2 Points) Assume the price of Y falls to $0.50. What is her consumption bundle after the price

change?

X = __________

Y = __________

D. (2 Points) Now assume the price of Y falls to $0.25. What is her consumption bundle after the price

change?

X = __________

Y = __________

E. (2 Points) Decompose the total effect of the price change into the substitution and income effects.

Substitution Effect = __________

Income Effect = __________

F. What is the compensating variation associated with this change in price?

CV = __________

G. (2 Points) Solve for the demand of X.

X = __________

9. (1 Point) Please initial each of the following statements. Failure to complete this will result in a one

point deduction.

____ The assignment is open book/open note. Students may discuss this assignment with other

students in the class, but must turn in legible, handwritten, and original answers. The posting of

questions from this assignment to an online forum will be considered cheating. Answers closely

matching replies to such a post will be scrutinized. Students caught cheating will receive a zero on the

assignment and may receive additional punishment under Faculty Senate Rule 49-20.

____ This assignment must be turned in at the beginning of class on Wednesday, November 4th. It will

be considered late once the instructor starts lecturing. Late assignments will not be accepted and

students will receive a score of zero on that assignment.

____ All work must be stapled, in the proper order, in the upper-left corner of the paper prior to being

submitted. Six points will be deducted if the instructor staples or restaples the work for you.

____ Your name and nine-digit Penn State ID number must appear at the top of the first page.

____ Be sure to fully answer the entire question. Some questions have multiple parts which ask you to

explain your reasoning. Providing the correct answer without an explanation (when asked for) will not

earn students full credit.

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