Ag Bus 313
Problem Set 4
200 Points
Due: 5:00 pm on 3/2/16
Dr. Hurley
Directions: Please answer all items on this homework. You must show all your work. Unless
otherwise stated, please simplify your answer.
1. Suppose you sold a product y at an output price of $30. Suppose that the production of y
uses an input x that costs $240 per unit. Answer the following questions based on the
following production function: y = f(x) = 38x-x2. Please show how you found your answer.
(45 Points)
a. Set-up the constrained profit maximization problem. What is the Lagrangean and first
order condition for this problem?
b. Solve the first order conditions to find the optimal input level, optimal output level, and
profit.
c. Verify that your optimal input in part b is truly the optimal input by using MVP=MIC.
d. Verify that your optimal input in part b is truly the optimal input by using MPP = w/p.
e. Verify that your optimal output in part b is truly the optimal output by using MR=MC.
f. What stage is your optimal solution in?
g. Set-up the unconstrained profit maximization problem with respect to input and find
where the optimal input occurs.
h. Set-up the unconstrained profit maximization problem with respect to output and find
where the optimal output occurs.
i. How much profit would you have lost if you produced at the maximum revenue point?
2. Suppose you sold a product y at an output price of $5. Suppose that the production of y uses
two inputs x1 and x2. The cost of input 1 is $120 while the cost of input 2 is $80 per unit.
Answer the following questions based on the following production function: y = f(x1,x2) =
42x1-3x12 + 32x2-4x22. Please show how you found your answer. (25 Points)
a. Set-up the constrained profit maximization problem.
b. What is the Lagrangean and first order condition for this problem?
c. Solve the first order conditions to find the optimal input level, optimal output level,
and profit.
d. Solve for the optimal inputs in part a by using MPPx1 = w1/p and MPPx2 = w2/p; this
should verify your answer in part c is correct.
e. How much profit would you have lost if you produced at the maximum revenue
point?
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Last revised: 12/29/15
3. Suppose you sold a product y at an output price of $8. Suppose that the production of y uses
two inputs x1 and x2. The cost of input 1 is $4 while the cost of input 2 is $256 per unit.
Answer the following questions based on the following production function: y = f(x1,x2) =
40x12/5x21/5. Please show how you found your answer. (25 Points)
a. Set-up the constrained profit maximization problem.
b. What is the Lagrangean and first order condition for this problem?
c. Solve the first order conditions to find the optimal input level, optimal output level,
and profit.
d. Solve for the optimal inputs in part a by using MPPx1 = w1/p and MPPx2 = w2/p; this
should verify your answer in part c is correct.
e. What are the returns to scale of this production function?
4. Suppose you have 20,000 acres of land to allocate to corn and soybeans. The production
function for corn is Y1 = 20x11/4, where Y1 is the amount of bushels of corn and x1 is the
amount of land used for corn. You also know that the production function for soybeans is Y2
= 40x21/4, where Y2 is the amount of bushels of soybeans and x2 is the amount of land used
for soybeans. Assume that the price of corn is 320 and the price of soybeans is 160. Also
assume the input cost is 1.6. (45 Points)
a. Set-up the maximization problem.
b. Set-up the Lagrangean and give the first order conditions.
c. Please find the production possibility frontier (PPF) using soybeans as the dependent
variable and corn as the independent variable?
d. What is the marginal rate of product transformation (MRPT) for the PPF?
e. What is the optimal output given the price ratio?
f. Solve for the optimal input use by using the MVP’s.
g. How much revenue and profit are being brought in at the optimal outputs and input
allocation?
h. Please sketch the solution on a graph. (Note, all the numbers on the graph may not
come out to be integer numbers.)
i. What is the MRPT at Y1 = 200? Please explain what this means in an economic
sense.
5. Suppose you have the following game: (15 Points)
Farmer A
Small
Large
Harvest
(41, 42)
(36, 26)
Farmer B
Don’t Harvest
(25, 35)
(29, 31)
a. Identify the key components of this game.
b. Are there any dominant or dominated strategies?
c. What are the Nash Equilibriums? Please explain. (Hint: There are two.)
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Last revised: 12/29/15
6. Suppose you have the following game: (10 Points)
Processor A
Small
Medium
Large
Raise Price
(80,89)
(56,95)
(53,83)
Processor B
Keep Prices Constant Lower Prices
(2,16)
(42,17)
(52,69)
(17,28)
(66,73)
(7,10)
a. Are there any dominant or dominated strategies?
b. What are the Nash Equilibriums? Please explain.
7. Suppose you have two large meat processors trying to decide what size processing plant they
should build. Assume that they are making their choice on plant size simultaneously. Each
has the same strategy set—build a small plant, build a medium size plant, or build a large
plant. Suppose you have the following strategy sets that map to the following payoffs: (20
Points)
—If both firms choose small, they equally split $240 million
—If both firms choose medium, they equally split $200 million
—If both firms choose large, the equally split $120 million
—If one firm chooses small and the other chooses medium, the firm that chooses small gets
$80 million and the firm that chooses medium gets $130 million
—If one firm chooses small and the other chooses large, the firm that chooses small gets $70
million and the firm that chooses large gets $110 million
—If one firm chooses medium and the other chooses large, the firm that chooses medium
gets $65 million and the firm that chooses large gets $75 million
a.
b.
c.
d.
Please put this game in Matrix form.
Are there any dominant or dominated strategies? Please explain.
What is the Nash Equilibrium? Please explain
Does this game have a Prisoner’s Dilemma? Please explain.
8. Please answer the following questions given the game below: (15 Points)
a. What are the four components of the game tree below?
b. What is the Subgame Perfect Nash Equilibrium? Please explain how you found
your answer.
c. Would you get the same outcome if this was a simultaneous move game?
Please explain. (Hint: you could solve this game using the matrix form of a
game.)
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Last revised: 12/29/15
Raise
Producer B
Same
Lower
Exit
Raise
Small
Same
Lower
Medium
Exit
Producer A
Raise
44, 4
66, 43
24, 37
82, 83
15, 72
80, 45
16, 90
45, 96
21, 67
Large
Same
Lower
Gigantic
Exit
Raise
Same
Lower
Exit
10, 93
14, 75
72, 20
4, 18
40, 16
11, 45
68, 58
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Last revised: 5/16/13