Given a firm which uses 2 inputs, X1 and X2; to produce a good that is described by the production function: Q = f (X1 X2) = X11/2 X21/4 The firm sells its ouput at N$80 per units. Given cost of input 1, X1; is N$4; and the cost of input 2, X2; is N$2 Solve for the profit maximizing input mix, output, and profit.
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Given a firm which uses 2 inputs, X1 and X2; to produce a good that is described by
the production function: Q = f (X1 X2) = X11/2 X21/4
The firm sells its ouput at N$80 per units. Given cost of input 1, X1; is N$4; and
the cost of input 2, X2; is N$2
Solve for the profit maximizing input mix, output, and profit.
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- A firm’s production function is - y = f(X1, X2)= X11/2 + X1X2 , Where X1≥0, X2≥0 1. Write down the firm’s production possibility set, and its input requirement set. 2. Is this production function concave, quasi-concave? 3. Is this production function homogenous? 4. Find its returns to scale when X1=1, and X2=1Q.No.3. Consider the production function: (3) Y = 0.75X + 0.0042X2 – 0.000023X3 (a) At what level of X, the output will be maximum? (b) If input price is 0.15$ and output price is 4$ then at what level of X, profit will be maximum?A firm has the production function F(L, K) = L^1/2 + K^1/2The price of labor is $10 and the price of capital is $15. The firm has a production goal of Q = 100 units ofoutput.a) Neatly specify this firm’s cost minimization problem, using the particulars associated with this problem.b) Give two equations that an interior solution satisfies, tailoring your equations to the particulars of thisproblem.c) Solve the two equations for the firm’s optimal choice. Show your work.
- Suppose the utility function of a person consuming two commodities X and Y with income Birr 600 is given by U =2xy. If the per unit price of X is Birr 20 and per unit price of Y is Birr 40. a) Calculate the utility maximizing level of consumption of X1 and X2. b) Find the MRSX, Y at the optimum.If the production function of a firm is given by Q=,and the input prices are r = Birr 8 per unit and w = Birr 2 per unit,Suppose that a firm’s production technology is described by theproduction function f(x1, x2) = (x1)^2x2, where x1 denotes the quantity ofinput 1 and x2 denotes the quantity of input 2. Let the price of input 1 be$1 and the price of input 2 be $4.a. Derive the conditional input demand functions for bothinputs.b. Derive the firm’s cost functionLet y = f(x1, x2)=x11/2 + x1x2 be a firm’s production function, where x1≥0, x2≥0. Write down the firm’s production possibility set, and its input requirement set. Is this production function concave, quasi-concave? Is this production function homogenous? Find its returns to scale when x1=1, and x2=1.
- Suppose that Marie produces milk q using her own labor l and cattle k using the production functionq = f(k, l) = k2/3ℓ1/3Although Marie does not need to pay anyone to use either input, the opportunity costs of labor and cattle are w = 1 and v = 16, respectively, and P is the price of milk. a) Suppose that Marie’s stock of cattle is fixed at k0 = 8. Set up her short run cost minimization problem and find her labor demand ℓ(q) and cost function SC(q). b) Find Marie’s short run marginal cost SMC(q) and average cost SAC(q) functions and find the quantity at which short run average cost is minimized. c) Set up Marie’s short run profit maximization problem and find her short run supply curve q(P).A firm produces a good using two inputs, capital (K) and labour (L). For every unit of output being produced, the ratio of capital to labour must be , where is a positive parameter. Meanwhile, the total cost of production must equal $1,000. If the rental rate of capital is $20 per unit of capital, and the hourly wage is $10 per unit of labour, how much capital does the firm use for it production? a) k=100a/(2/a+1) b) k=100/a(2a+1) c) k=100/(2/a+1) d) k=100/(2a+1)A firm is jointly owned by Juan and Roda. The firm’s production function requires two inputs: effort by Juan, denoted by x, and effort by Roda, denoted by y. Effort is only observable by the person who exerts it. The cost to Juan of a unit of his effort is c j = 2 and the cost to Roda for a unit of her effort is cr = 2. The price received for the goods is p = 2. The production of the firm is given by Q = 10(ln(x + 1) + ln(y + 1)). Assume that both Juan and Roda are risk-neutral rational agents.a) What are the socially optimal amounts of effort x* and y*? What is the total surplus in that case? (Hint: Solve the problem of a social planner that cares equally for Juan and Roda.)b) Suppose that Juan and Roda have a contract that specifies that Juan pays a fixed amount w = 15 to Roda and that Juan gets to keep and sell all the output. What is the total surplus now? How much of that surplus goes to Juan? To Roda?c) Now suppose that the contract between Juan and Roda specifies that the total…
- The manager of Don Teeta Company Limited hires labour (L) and rents capital equipment (K) ina very competitive market. Currently, the wage rate of labour is GH¢2 per hour and capital isrented at GH¢5 per hour, the unit price of the product is GH¢0.75 and total cost of production isGH¢1,000. Suppose the firm’s production function (Q) is as follows:? = 14?0.5?0.5 + 10Determine the optimal input usage and the maximum profit.Question A Suppose the short-run production function is q = 1L0.5. If the marginal cost of producing the 10th unit is $8, what is the wage per unit of labor? Question B A consumer has the utility function U(q1,q2) = q10.5 + q2Assume p2 = 1 and Y = 100. What is the equivalent variation of a price increase for good 1 from 1 to 4?Suppose that a firm has production function F(L, K) = L2/3 K1/3 for producing widgets, thewage rate for labor is w = $400, and the rental rate of capital is r = $25.d) Determine this firm’s minimum cost of producing 120 units.e) Now suppose that the firm’s production goal is left as the variable Q. Come up with the firm’s costfunction C(Q). Show your work.