NAME: ________________________ PSU Email:[email protected]

Homework #4 Intermediate Microeconomics ECON 302, Section 001

Due: Wednesday, November 4, 2015 at 8AM 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

LKY = . 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:

5$2$ 21 == PandP .

Output at plant 1 is defined by: 2111 25.0100 LLY −= and 11 5.0100 LMPL −=

Output at plant 2 is defined by 2222 375.075 LLY −= and 22 75.075 LMPL −= 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 2P 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 2P . 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 2P from Part B.

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

22

3

32

3 2

48

LKMPL KLMPK

rwLKY

=

=

===

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:

LKMRTS = __________ 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:

qqMC qqVC

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:

XMU YMU

XYXU

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.