1. For each of the following production functions, determine if the tech- nology

Question

1. For each of the following production functions, determine if the tech-

nology exhibits increasing, decreasing, or constant returns to scale.

a. f(L;K) = L + K

b. f(L;K) = pL + pK

c. f(L:K) = LK + L + K

d. f(L;K) = pKL + L + K

2. Draw isoquant maps for the following technologies.

i) f(L;K) = LK

ii) g(L;K) = L + 2K

iii) h(L;K) = min(2L;K)

3. Frisbees are produced according to the production function q = 2K+L

where q =output of frisbees per hour, K =capital input per hour, L =labor

input per hour.

a) If K = 10, how much L is needed to produce 100 frisbees per hour?

b) If K = 25, how much L is needed to produce 100 frisbees per hour?

c) Graph the q = 100 isoquant. Indicate the points on that isoquant

de.ned in part a and part b. What is the RTS along this isoquant? Explain

why the RTS is the same at every point on the isoquant.

d) Graph the q = 50 and q = 200 isoquants for this production function

also. Describe the shape of the entire isoquant map.

e) Suppose technical progress resulted in the production function for

frisbees becoming q = 3K + 1:5L. Answer part a through part d for this

new production function and discuss how it compares to the previous case.

4. Consider the production function f(L;K) = L + K.

a. Suppose K is .xed at 2. Find algebraic expressions for the total

product of labor function TP(L), the average product of labor AP(L), and

the marginal product of labor MP(L).

b. Graph the functions in part a.

5. A .rm uses capital and labor to produce output according to the

production function q = 4pKL, for which MPL = 2

p

K=L and MPK =

2

p

L=K.

a. If the wage w = $4 and the rental rate of capital r = $1, what is the

least expensive way to produce 16 units of output?

b. What is the minimum cost of producing 16 units?

c. Show that for any level of output q, the minimum cost of producing

q is $q.

d. Explain how a 10% wage tax would a

Explanation / Answer

For each of the following production functions, determine if the technology exhibits increasing, decreasing, or constant returns to scale.??a. f(L,K) = L + K??b. f(L,K) = ?L + ?K??c. f(L,K) = LK + L + K??d. f(L,K) = (?KL) + L + K

Solution:

We substitute L = 1 and K = 1

f(L, K) = 1 + 1 = 2

Now, substitute L = 2 and K = 2

f(L, K) = 2 + 2 = 4

Since output is doubled, by doubling labor and capital, therefore it is constant returns to scale.

We substitute L = 1 and K = 1

f(L, K) = ?1 + ?1 = 1 + 1 = 2

Now, substitute L = 2 and K = 2

f(L, K) = ?2 +? 2 = 1.41 + 1.41 = 2.82

Since output is less than doubled, by doubling labor and capital, therefore it is decreasing returns to scale.

We substitute L = 1 and K = 1

f(L, K) = 1*1 + 1 + 1 = 3

Now, substitute L = 2 and K = 2

f(L, K) = 2*2 + 2 + 2 = 8

Since output is greater than doubled, by doubling labor and capital, therefore it is increasing returns to scale.

We substitute L = 1 and K = 1

f(L, K) = ?1*1 + 1 + 1 = 3

Now, substitute L = 2 and K = 2

f(L, K) = ?2*2 + 2 + 2 = 6

Since output is doubled, by doubling labor and capital, therefore it is constant returns to scale.

??Draw isoquant maps for the following technologies.??i) f(L;K) = LK??ii) g(L;K) = L + 2K??iii) h(L;K) = min(2L;K)?

Solution:

i)

ii)

iii)

?Frisbees are produced according to the production function q = 2K+L
where q =output of frisbees per hour, K =capital input per hour, L =labor
input per hour.

a) If K = 10, how much L is needed to produce 100 frisbees per hour?

b) If K = 25, how much L is needed to produce 100 frisbees per hour?

c) Graph the q = 100 isoquant. Indicate the points on that isoquant
de

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