03 Jun Question Problem #1 Consider the following production function Q = [L(2 ? L)]2 for L ? [0, 2]. i) Find the average product of labor
Question
Problem #1
Consider the following production function
Q = [L(2 ? L)]2
for L ? [0, 2].
i) Find the average product of labor
ii) Find the marginal product of labor
iii) For what value of L is the marginal product of labor the same as the average product?
Problem #2
Which of the following production functions exhibits constant returns to scale?
i) Q = F (K, L) = 2 + K? 3L
+
ii) Q = F (K, L) = K + KL
1
iii) Q = F (x1 , x2 , …, xN ) = (x? + x? + … + x? ) ?
1
2
N
iv) Q = F (K, L) = K 2 L2
Problem #3
Consider the following cost function
C(Q) = 200 ? 20Q + Q2 .
i) Find the average costs
ii) Find the marginal costs
iii) Plot both the marginal and the average costs. Be precise in the neighborhood of the lowest value of the average costs
Problem #4
Let the production function be given by
1
2
Q = F (K, L) = AK 3 L 3 .
Assume that the rental price of capital is r and the prevailing wage is w.
1
i) Find the optimal amounts of labor and capital hired by a cost-minimizing
rm that needs to produce Q0 units of output.
ii) How does the optimal ratio of K change with A? Would you interpret L increases in A as capital saving technological progress?
iii) Find the total costs of production as a function of r, w, and Q0 .
iv) Imagine that A doubles and at the same time the factor prices r and w
double what is the impact on the total cost of production of Q0 units of output?
Problem #5
Consider the following production function
1
1
Q = AK 2 L 2 .
¯
i) Imagine that capital is xed at K. Assume that factor prices are r and w.
What is the optimal amount of labor hired by a cost minimizing rm interested
in producing Q0 units of output?
ii) How many units of capital would this rm hire if it was free to select its
capital optimally?
¯
iii) For what level of output, Q0 , is the original value of K optimal?
Problem #6
Consider the following production function
1
Q = F (K, L) = (K ? + L? ) ? .
Assume that the price of output is equal to p.
i) Assume that the rm employs K units of capital and L units of labor.
Assume that the factor rental prices are r and w respectively. What is the level
of prot earned?
ii) Find the marginal product of labor
iii) Find the marginal product of capital
iv) State the conditions that describe the optimal amounts of K and L
v) What is the maximum amount of prot earned?
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