HW4

Suppose that a firm has a production function f(L,K)= min{L,K}. From the following combination of labor and capital (L,K), which one belongs to the same iso-quant as (3,90)? (15,15) (30,3) (1,4) (90,4) (3,1)

(30,3) f(3,90)=min{3,90}=3; the only combination (L,K) that produces 3 from the options is (30,3)

Average Fixed Cost AF(q)

F/q

Formula for Marginal Cost MC(q)

MC(q)=C'(q)

The marginal rate of technical substitution of L for K at (L,K) is equal to?

The negative of the slope of the tangent to the iso-quant through (L,K) at (L,K)

A call center has a production function: f(L,K)=20L+80K. The maximal amount of calls that the call center may receive given that L=2 and K=3 is?

280 f(2,3)=20(2)+80(3)=280.

Consider a firm that has production function f(L,K)= 3L2/3K1/3. What is the expression for the marginal rate of technical substitution MRTS(LK) at (L,K)?

2K/L

Consider a firm that has production function f(L,K)=5L1/3K2/3. What is the expression for this firm's Marginal Product of labor?

5K^2/3/3L^2/3.

Consider a firm with production function f(L,K)=2L+6K. Assume that capital is fixed at K=6. Assume also that the price of capital r=10 and the price of labor w=2. Then, the average cost of producing q units is?

AC(q)=24/q+1

Consider a firm with production function f(L,K)=3L^(1/3)K^(2/3). Assume that capital is fixed at K=1. Assume also that the price of capital r=5 and the price of labor w=3. Then, the average cost of producing q units is?

AC(q)=5/q+(q^2)/9.

Formula for Average Cost AC(q)

AC(q)=C(q)/q

Consider a firm with production function f(L,K)=3L^(1/3)K^(2/3). Assume that capital is fixed at K=1. Assume also that the price of capital r=5 and the price of labor w=3. Then, the average fixed cost of producing q units is?

AF(q)=5/q If K=1, then F=rK=5. Thus, AF(q)=F/q=5/q

Consider a firm with production function f(L,K)=3L^(1/3)K^(2/3). Assume that capital is fixed at K=1. Assume also that the price of capital r=5 and the price of labor w=3. Then, the average variable cost of producing q units is?

AVC(q)=(q^2)/9

Consider a firm with production function f(L,K)=3L+8K. Assume that capital is fixed at K=12. Assume also that the price of capital r=10 and the price of labor w=3. Then, the average variable cost of producing q units is?

AVC(q)=1-(96/q)

Consider a firm with production function f(L,K)=2L+6K. Assume that capital is fixed at K=6. Assume also that the price of capital r=10 and the price of labor w=2. Then, the cost of producing q units is?

C(q)=24+q

Consider a firm with production function f(L,K)=3L^(1/3)K^(2/3). Assume that capital is fixed at K=1. Assume also that the price of capital r=5 and the price of labor w=3. Then, the cost of producing q units is?

C(q)=5+((q^3)/9).

Formula for Cost (Total Cost) C(q)

C(q)=F+VC(q)

Consider a firm with production function f(L,K)=3L+8K. Assume that capital is fixed at K=12. Assume also that the price of capital r=10 and the price of labor w=3. Then, the average fixed cost of producing q units is?

F=120/q. If K=12, then F=rK=120. Then AF(q)=F/q=120/q.

Consider a firm with production function f(L,K)=3L^(1/3)K^(2/3). Assume that capital is fixed at K=1. Assume also that the price of capital r=5 and the price of labor w=3. Then, the fixed cost of producing q units is?

F=5. If K=1, then F=rK=5.

Consider a firm with production function f(L,K)= 2L+6K. Assume that capital is fixed at K=6. Assume also that the price of capital r=10 and the price of labor w=2. Then, the fixed cost of producing q units is?

F=60.

The following production function represents an industry in which there is free entry: f(L,K)=100L^(1/2)K^(1/3).

False. The function is a Cobb-Douglas function with alpha=1/2 and beta=1/3. Since alpha+beta=5/6<1 then it satisfies decreasing returns to scale. Free entry would require constant returns to scale.

Formula for Fixed Cost F

K*r

Consider a firm with production function f(L,K)=3L^(1/3)K^(2/3). Assume that capital is fixed at K=1. Then, the amount of labor necessary to produce q units is?

L(q,1)=(q^3)/27 If q=3L1/3K2/3, and K=1, then L=q3/27.

Consider a firm with production function f(L,K)=3L+8K. Assume that capital is fixed at K=12. Then, the amount of labor necessary to produce q units is?

L(q,12)= (q/3)-32 If q=3L+8K and K=12, then L=(q/3)-32

Consider a firm with production function f(L,K)=3L^(1/3)K^(2/3). Assume that capital is fixed at K=1. Assume also that the price of capital r=5 and the price of labor w=3. Then, the marginal cost of producing q units is?

MC(q)=(q^2)/3.

Consider a firm with production function f(L,K)=3L+8K. Assume that capital is fixed at K=12. Assume also that the price of capital r=10 and the price of labor w=3. Then, the marginal cost of producing q units is?

MC(q)=1

Consider a firm that has production function f(L,K)= 3L2/3K1/3. What is the value of the Marginal Product of capital when L=1 and K=8?

MPK(L,K)= 1/4.

Consider a firm that has production function f(L,K)=6L1/3K2/3. What is the expression for this firm's Marginal Product of capital?

MPK(L,K)= 10L1/3/3K1/3.

Consider a firm that has production function f(L,K)=5L1/3K2/3. What is the value of the Marginal Product of labor when L=1 and K=8?

MPL(L,K)= 20/3.

The MRTSLK(L,K) for a certain firm is constant and equal to 2. Then, if the firm substitutes 2 units of labor for two unit of capital?

Production increases MRTSLK(L,K) is the amount of capital that the firm can substitute with one unit of labor so the production remains constant. That means that in this particular problem if the firm substitutes one unit of labor for two of capital the production remains constant. Thus, substituting two units of labor for two units of capital increases production.

Consider a firm that has production function f(L,K)=5L^1/3K^2/3. Does this production function satisfy the law of decreasing marginal returns of capital?

True. Recall that MPK(L,K)=df/dK(L,K)=(10/3)L1/3K-1/3=10L^1/3/3K^1/3. So for a fixed L, if K increases MPK(L,K) decreases. Actually, marginal returns of capital are always decreasing for this function

The following production function satisfies increasing returns to scale: f(L,K)=100LK.

True. f(2L,2K)=100(2L)(2K)=4f(L,K). Then, f(2L,2K)>2f(L,K). Alternatively, the function is a Cobb-Douglas function with alpha=1 and beta=1. Since alpha+beta=2>1 then it satisfies increasing returns to scale.

The following production function satisfies constant returns to scale: f(L,K)=3L^αK^(1-α).

True. f(2L,2K)=3(2L) ^α (2K) ^(1-α) = 2 ^α2^(1-α) f(L,K)=2f(L,K). Then, f(2L,2K)=2f(L,K). The function is a Cobb-Douglas function with alpha=1 and beta=1-alpha. Since alpha+beta=1 then it satisfies constant returns to scale.

Average Variable Cost AVC(q)

VC(q)/q

Consider a firm with production function f(L,K)= 4L^(2/3)K^(1/3). Assume that capital is fixed at K=1. Assume also that the price of capital r=4 and the price of labor w=2. Then, the variable cost of producing q units is?

VC(q)= (q^3/2)/4

Consider a firm with production function f(L,K)=2L+6K. Assume that capital is fixed at K=6. Assume also that the price of capital r=10 and the price of labor w=2. Then, the variable cost of producing q units is?

VC(q)= q-36

A firm's production function:

associates with each combination of inputs (L,K) the maximal amount of output that the firm is able to produce with (L,K).

A call center has a production function: f(L,K)=80L+200K. If capital is fixed at K=4, what is the expression for the maximal production as a function of labor?

f(L,4)=80L+800 if K=4, then f(L,4)=80L+800.

APL(L,K) is the slope of the line that interpolates the production function and the origin; it is increasing for small L.

true

Formula for Variable Cost VC(q)

w*L