A firm has the following production function: q=KL The firm will produce 64 units of output...

A firm has the following production function: q=KL The firm will produce 64 units of output and faces prices for labor and capital of $4 and $1 respectively. What is the optimal quantity of labor and capital the firm should employ in order to minimize the cost of producing 64 units of output? What is the minimum cost of producing 64 units of output? Show the firms optimal production decision on a graph.

Solutions

Expert Solution

Production function : q = KL

q = 64 units .

w = 4$

r = 1$

Firm's cost minimization condition :

Tangency point between isocost and isoquant : MRTS = MP_L / MP_K = w / r

So , MP_L = = K

MP_K = = L

MP_L / MP_K = w / r

or , K/L = 4/1

or , K = 4L

q = KL = 4L²

( q= 64 ) L = 4 units . K = 16 units

Minimum cost or total cost = wL + rK = 16 + 16 = 32 $

Rahul Sunny answered 1 year ago

Next > < Previous

Related Solutions

A firm has a production function Q = KL, where Q is the quantity of output,...

A firm has a production function Q = KL, where Q is the quantity of output, K is the amount of capital and L is the amount of labor. MPL=K and MPK=L . a) Suppose that capital is fixed at K=10 in short run. In this case, the marginal product of labor is MPL=10. Does this production function exhibit diminishing marginal returns to labor? b) Suppose that in the short run, K is fixed at 10. The interest rate is...

27) A firm has a production function Q = KL, where Q is the quantity of...

27) A firm has a production function Q = KL, where Q is the quantity of output, K is the amount of capital and L is the amount of labor. MPL=K and MPK=L. a) Suppose that capital is fixed at K=10 in short run. In this case, the marginal product of labor is MPL=10. Does this production function exhibit diminishing marginal returns to labor? b) Suppose that in the short run, K is fixed at 10. The interest rate is...

The production function of a firm is given as Q = 50√KL. Here Q is the...

The production function of a firm is given as Q = 50√KL. Here Q is the output produced, K is the capital input and L is the labor input. Take the partial derivative of the long-term cost function according to the wage, interpret the function you find. Do the same for the rent cost of the capital (take derivative according to r). Interpret the function you find.

A firm has a production function of Q = KL + L, where MPL = K...

A firm has a production function of Q = KL + L, where MPL = K + 1 and MPK = L. The wage rate (W) is $100 per worker and the rental (R) is $100 per unit of capital. a. In the short run, capital (K) is fixed at 4 and the firm produces 100 units of output. What is the firm's total cost? b. In the long run, what is the total cost of producing 100 units of...

Short Run Cost Curves: Consider a firm with the following production function: q=(KL)/20 a. For a...

Short Run Cost Curves: Consider a firm with the following production function: q=(KL)/20 a. For a short-run situation in which K=10, wage = 4 and cost of capital = 1, derive expressions for short run total cost and short run average cost for this production function. b. Plot the short-run total cost curve and label it “TC1”. Now suppose that the cost of capital goes up to 2. (Continue to assume we’re in the short run and K can not...

Suppose a firm has the production function, Q = 12KL. It has a contract to produce...

Suppose a firm has the production function, Q = 12KL. It has a contract to produce 144 units of output per day. One way to do is to use 3 units of capital (K) and 4 units of labor (L). Suppose a unit of labor costs $2 (wage = w) while a unit of capital costs $3 (r). What is the slope of the isocost line? A) 2/3 B) 1/4 C) 1/3 D) 3/2 At this combination of inputs, is...

Suppose a firm has the production function, Q = 12KL. It has a contract to produce...

Suppose a firm has the production function, Q = 12KL. It has a contract to produce 144 units of output per day. One way to do is to use 3 units of capital (K) and 4 units of labor (L). Suppose a unit of labor costs $2 (wage = w) while a unit of capital costs $3 (r). What is the Marginal Rate of Technical Substitution (MRTS) at this specific input ratio, i.e., K = 3 and L = 4?...

Suppose a firm has the production function, Q = 12KL. It has a contract to produce...

Suppose a firm has the production function, Q = 12KL. It has a contract to produce 144 units of output per day. One way to do is to use 3 units of capital (K) and 4 units of labor (L). Suppose a unit of labor costs $2 (wage = w) while a unit of capital costs $3 (r). What will be the marginal product of capital (MPK) when K = 3 units and L = 4 units? A) MPK =...

A firm has production function q = 100 L + KL− L^2 − K^2 The price...

A firm has production function q = 100 L + KL− L^2 − K^2 The price of the good is $1. The wage is $10, and the price of capital is $30. Assume that the firm is a price - taker in a perfectly competitive market. a. What will the firm’s profit maximizing choice of capital and labor be? b. Suppose that the firm’s capital is fixed in the short-run and wage rises to $20. What is the firm’s new...

25) A U.S. firm currently produces 200 units of output according to the production function q...

25) A U.S. firm currently produces 200 units of output according to the production function q = L0.5K0.5 and faces input prices equal to wU.S. = rU.S = $11. Should the U.S. firm move their company abroad where they will face input prices equal to wabroad = $6.50 and rabroad = $15.00? A) Yes, because the total costs will fall from $3,859 to $2,810. B) No, because the total costs will increase from $2,810 to $3,859. C) No, because the...

Subjects