Question

a firm produces a product with labor and capital as inputs. The production function is described...

a firm produces a product with labor and capital as inputs. The production function is described by Q=LK. the marginal products associated with this production function are MPL=K and MPK=L. let w=1 and r=1 be the prices of labor and capital, respectively

a) find the equation for the firms long-run total cost curve curve as a function of quantity Q

b) solve the firms short-run cost-minimization problem when capital is fixed at a quantity of 5 units (ie.,K=5). derive the equation for the firms short-run total cost curve as a function of quantity Q and graph it together with the long run total cost curve.

c) how do the graphs of the long-run and short-run total cost curves change when w=1 and r=4?

d) how do the graphs of the long run and short run total cost curves change when w=4, and r=1?

Homework Answers

Answer #1

Answer:-

a) Cost-minimizing quantities of inputs are equal to L= √Q√(r/w) and K= √Q/ √(r/w). Hence, in the long-run the total cost of producing Qunits of output is equal to TC(Q) = 2√(Qrw). For w= 1 and r= 1 we have TC(Q) = 2√Q.

b) When capital is fixed at a quantity of 5 units (i.e., K= 5) we have Q= K*L= 5 L. Hence, in the short-run the total cost of producing Qunits of output is equal to STC(Q) = 5 + Q/5.

d)Answer:-

When w= 4 and r= 1 we have TC(Q) = 4√Qand STC(Q) = 5+ 4Q/5

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
A firm produces a product with labor and capital. Its production function is described by Q...
A firm produces a product with labor and capital. Its production function is described by Q = min(L, K). Let w and r be the prices of labor and capital, respectively. a) Find the equation for the firm’s long-run total cost curve as a function of quantity Q and input prices, w and r. b) Find the solution to the firm’s short-run cost minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5). Derive...
A firm produces output according to the production function. Q=sqrt(L*K) The associated marginal products are MPL...
A firm produces output according to the production function. Q=sqrt(L*K) The associated marginal products are MPL = .5*sqrt(K/L) and MPK = .5*sqrt(L/K) (a) Does this production function have increasing, decreasing, or constant marginal returns to labor? (b) Does this production function have increasing, decreasing or constant returns to scale? (c) Find the firm's short-run total cost function when K=16. The price of labor is w and the price of capital is r. (d) Find the firm's long-run total cost function...
Consider a firm that used only two inputs, capital (K) and labor (L), to produce output....
Consider a firm that used only two inputs, capital (K) and labor (L), to produce output. The production function is given by: Q = 60L^(2/3)K^(1/3) . a.Find the returns to scale of this production function. b. Derive the Marginal Rate of Technical Substitutions (MRTS) between capital and labor. Does the law of diminishing MRTS hold? Why? Derive the equation for a sample isoquant (Q=120) and draw the isoquant. Be sure to label as many points as you can. c. Compute...
A hat manufacturing firm has the following production function with capital and labor being the inputs:...
A hat manufacturing firm has the following production function with capital and labor being the inputs: Q = min(5L,3K) (it has a fixed-proportions production function). If w is the cost of a unit of labor and r is the cost of a unit of capital, derive the firm’s optimal inputs, long-run total cost curve, average cost curve, and marginal cost curve in terms of the input prices and Q. b) A firm has the linear production function Q = 2L...
(2) Consider the production function f(L, K) = 2K √ L. The marginal products of labor...
(2) Consider the production function f(L, K) = 2K √ L. The marginal products of labor and capital for this function are given by MPL = K √ L , MPK = 2√ L. Prices of inputs are w = 1 per hour of labor and r = 4 per machine hour. For the following questions suppose that the firm currently uses K = 2 machine hours, and that this can’t be changed in the short–run. (e) What is the...
For each of the following production functions (a and b) find the following equations (i-iii) in...
For each of the following production functions (a and b) find the following equations (i-iii) in terms of Q0 ,w and r. i) MRTS L,K ii) Long-run capital and labor demand curve. iii) Long-run total cost curve. iv) Short -run capital and labor demand curve if the firm is stuck with K = 9. v) Short- run total cost if the firm is stuck with K =9 . (a ) Q = 5L^(1/2) * K ^(1/2) (b) Q = LK...
A firm’s production function is Q! = min(4L ,5K ). The price of labor is w...
A firm’s production function is Q! = min(4L ,5K ). The price of labor is w and the price of capital is r. a) Derive the demand function of labor and capital respectively. How does the demand of capital change with the price of capital? b) Derive the long-run total cost function. Write down the equation of the long-run expansion path. c) Suppose capital is fixed at K = 8 in the short run. Derive the short-run total cost function....
III. A production process uses two inputs, labor and capital which can be written as: Q...
III. A production process uses two inputs, labor and capital which can be written as: Q = 5LK, MPL = 5K and MPK = 5L 'w' = $150 and 'r' = $1000. Find the least cost combination of L and K when output Q = 1000 tons per day. What is the total cost of producing 1000 tons per day using this least cost combination?
Suppose Cool T-Shirts Co produces T-shirts and employs labor (L) and capital (K) in production. Suppose...
Suppose Cool T-Shirts Co produces T-shirts and employs labor (L) and capital (K) in production. Suppose production function for Cool T-Shirts Co is Q=K*L, and Cool T-Shirts Co wants to produce Q=625. Suppose marginal product of labor (MPL) and marginal product of capital (MPK) are as follows: MPL=K and MPK=L. Suppose Cool T-Shirts Co pays workers $10 per hour (w=$10) and interest rate on capital is $250 (r=250). What is the cost-minimizing input combination if Cool T-Shirts Co wants to...
The production function for a firm is Q = −0.6L 3 + 18L 2K + 10L...
The production function for a firm is Q = −0.6L 3 + 18L 2K + 10L where Q is the amount of output, L is the number of labor hours per week, and K is the amount of capital. (a)Use Excel to calculate the total short run output Q(L) for L = 0, 1, 2...20, given that capital is fixed in the short run at K = 1. (b) Use Excel to calculate the total long run output Q(L) for...