FInd the profit maximizing levels of K and L as functions of r,w, and p. b) Suppose that r = w= $1 and p =$4 . What is the profit maximizing level of output,y?
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Consider two firms that produce a single output good,y, using two inputs :Capital, K , and labor, L, the prices of each unit of capital and labor are r and w,respectively. The output good y sells for $p per unit.
Firm A's production function is y = fa(K,L) = K1/4L1/4. The profit function is equals to : K1/4L1/4 - rK -wL.
a) FInd the profit maximizing levels of K and L as functions of r,w, and p.
b) Suppose that r = w= $1 and p =$4 . What is the profit maximizing level of output,y?
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- Consider two firms that produce a single output good,y, using two inputs :Capital, K , and labor, L, the prices of each unit of capital and labor are r and w,respectively. The output good y sells for $p per unit. Firm A's production function is y = fa(K,L) = K1/4L1/4. The profit function is equals to : K1/4L1/4 - rK -wL. a) FInd the profit maximizing levels of K and L as functions of r,w, and p. b) Suppose that r = w= $1 and p =$4 . What is the profit maximizing level of output,y?A competitive firm’s production function is given by y= f(x1,x2)=x1ax2b If a=b=0.5 , the price of factor 1 is 12, and the price of factor 2 is 3, find the cost minimizing input combinations and the total cost of producing 40 units of output. Redo part (1), this time by first deriving the firm’s conditional factor demand functions and the cost function.Problem #1 Consider two firms that produce a single output good, y, using two inputs: Capital, K, and labor, L. the prices of each unit of capital and labor are r and w, The output good y sells for $p per unit. Firm A’s production function is y = fA(K,L) = K1/4 L1/4. The profit function is thus: ðA(K,L) = K1/4 L1/4 – rK -wL Find the profit maximizing levels of K and L as functions of r, w, and p. Suppose that r = w = $1 and p = $4. What is the profit maximizing level of output, y ?
- Consider the production functions of three different Firms utilizing inputs labor (L) and capital (K) in producing goods X, Y, and Z given below. The three firms face the same fixed price for labor and capital at 5 per unit and 10 per unit, respectively. X = KL2 – L3; Y = 10K1.5L0.5; Z = K0.5L0.5 a. Derive the short-run cost function of Firm Z if 25 units of capital are employed. Suppose that good Z is sold at a perfectly competitive price of 10 per unit, calculate Firm Z’s profit and discuss if the Firm Z should continue to operate.b. Derive the long-run cost function of Firm YA competitive firm’s production function is given by y= f(x1,x2)= 4x11/2 + 10x21/2 a) The price of factor 1 is 1, the price of factor 2 is 1, and the price of output is 2. Find the profit-maximizing quantities of x1 and x2? What is the profit-maximizing quantity of output? b) Redo part (a), this time by first deriving the firm’s factor demand functions and the supply function, and then substituting the prices in these functions.Consider a price-taking firm whose production function is given by q = 3 L1/5 K1/9 where L and K denote respectively the amount of labour and capital the firm uses to produce q units of output. Suppose the price of labour is w = 16, the price of capital is 24 and the price of the firm's output is p=225 . Find the firm's cost function. Then enter below the value of the firm's marginal cost at the point where q = 100.
- The Director of ABC Enterprise hires labour (L) and rents capital equipment (K) in a competitive market to produce mango juice. At the moment, the wage rate of labour is GH¢2 per hour and capital is rented at GH¢5 per hour. Also, the unit price of mango juice is GH¢0.75 and total cost of production is GH¢1,000. Suppose the firm’s production function (Q) follows a Cobb-Douglas specification given as: 0.5 0.5 ?=14? ? +10 Determine the optimal input usage and the maximum profit that ABC Enterprise would obtain at the optimal input levels.A firm employs labor and capital by paying $40 per unit of labor employed and $200 per hour to rent a unit of capital. The production function is given by: Q=60L-2L^2+180K-3K^2, where Q is total output. Determine the firm's optimal combination of capital (K) and labor (L)?Suppose that the market for apples is perfectly competitive. Production of apples requires two inputs:workers (L) and land (K). The production function for apples is:F(K,L)=3K^(1/3)* L^(2/3) A) Suppose that W=4, R=16, and in the short-run, K=27. Find the firm’s short-run cost function SRC(q)and short-run marginal cost function SRMC(q).B) What is the optimal production of apples if P=4? What are consumer surplus and profit?C) Find the firm’s short-run supply of apples Q(P) when W=4, R=16 and K=27.D) Find the firm’s long-run cost function LRC(q) and long-run marginal cost function LRMC(q) when W=4and R=16. How do these compare to the firm’s short-run cost function and marginal cost function?E) Find the firm’s long-run supply of apples q(P) when W=4 and R=16.
- Suppose a price taking firm’s production function is given by q= √L, where is labor. When the wage rate is 1.5, and the output price changes from 2 to 4, the welfare gain of the firm (gain in the producer surplus) is?Suppose the long-run production function for a competitive firm is f(x1,x2)= min {x1,2x2}. The cost per unit of the first input is w1 and the cost of the second input is w2. .a. Find the cheapest input bundle, i.e. amount of labor and capital, that yields the given output level of y. .b. Draw the conditional input demand functions for labor and capital in the x1-y and x2- y spaces. .c. Write down the formula and draw the graph of the firm’s total cost function as a function of y, using the conditional input demand functions. What is the relationship between the returns to production scale and the behavior of the total costs? .d. Write down the formula and draw the graph of the average cost function, as a function of y. .e. Write down the formula and draw the graph of the marginal cost function, as a function of y.Suppose the production function for a competitive firm is y=f(x1,x2)= x1 1/4 x2 1/4 . The cost per unit of the first input is w1 and the cost of the second input is w2. A: What are the returns to scale of this production function? B: Find the cheapest input bundle, x1 and x2, that yields the given output level of y. C: Write down the formula of the firm’s total costs as a function of y. D: Are the average costs increasing, constant or decreasing in y? Are the marginal costs increasing, constant or decreasing in y?