Second Midterm Exam
The production function for the Cosmos firm can be written as: Q = 500L + 90L2 – .
- Graph the total product function.
Q Q = 500L + 90L2
The production function for the Fringle firm can be written as: Q = 5K1/2L1/2
- Graph the isoquant for Q = 350.
Q = 350
- Assume the price per unit of K = $500 and the price per unit of L = $1200. Calculate the least cost way of making Q = 350. How much K and L will you use? What are your costs?
Min. C=1200L + 500K s.t Q=5K1/2L1/2
ℓ=1200L+ 500K+l[70- K1/2L1/2]]
Foc w.r.t = K = 500-0.5lK-1/2L1/2=0
F.o.c w.r.t = L= 1200- lK1/2L-1/2=0
F.o.c w.r.t = l = 70- K1/2L1/2=0
- Now assume that the price per unit of L drops to $900 and the price per unit of K stays the same. Calculate the least cost way of making Q = 350. How much K and L will you use? What is the cost?
Given that 70- K1/2L1/2=0
L=19.4 =20 units
- Now assume that you have been given $100,000 to make as much output as possible. The price per unit of K is $500 and the price per unit of L is $1000. What is the maximum amount of Q you can make? How much K and L will you use?
In this case we maximize Q subject to costs
Foc r w.r.t = K= 2.5K^-.5L^.5-500 =0
Foc r w.r.t = L = 2.5 K^.5L^-.5-1000=0
Foc r w.r.t = l = 100000-500K-1000L=0
- Now assume that the price per unit of L falls to $600 and the price per unit of K stays the same. How much Q can you now make for $100,000? How much K and L will you use?
Max Q=5K1/2L1/2 s.t
First order conditions
L=83.33 = 84 units
- Now assume that you have been given K = 6400 free of charge and each unit of L costs $2000 and each unit of Q can be sold for $100. How much K and L will you use to maximize profits? What are your maximum profits?
- What is the marginal product of L at the profit maximizing amount of L?
Sara has the following production function: Q = 10K1/2L1/2. She faces the following demand function: Q = 11,000 – 5P. The price per unit of K is $2,000 and the price per unit of L is $3,000.
- Calculate the amount of K and L that Felicia will use to maximize profits.
Min C, st. Q
ℓ = 2000K + 3000L + ?(Q – 10PK1/2L1/2)
Foc w.r.t K = 2000 – 5?PK-1/2L1/2 =0———————————- (1)
Foc w.r.t L = 3000 – 5?PK1/2L-1/2 =0————————————-(2)
Foc w.r.t ? = Q – 10PK1/2L1/2
Therefore, Devide eq (i) with (ii), you get:
2/3 = L/K, hence L = 2/3K, replace L in eq (i)
2000 = 5P(2/3)
P = 600
- Calculate the profit maximizing amount of output and the profit maximizing price.
- Calculate the maximum profits of the firm.
- Calculate the marginal revenue at the profit maximizing amount of Q.
- Calculate the marginal product of labor at the K and L in 15.
- At what price and quantity would total revenue be a maximum? What is the maximum revenue?