A firm uses two inputs, X1 and X2, to produce a good that is described by the production function: Q = f (X1, X2) = X1/2X1/4 12 The firm sells its output at $80 per unit: P = $ 80. The cost of input 1, X1, is $4. The cost of input 2. X2, is $2. (a) Solve for the profit - maximizing input mix, output, and profit. (b) Check the second-order conditions to verify that the profit is at a maximum
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\beta < 1. 16. A firm uses two inputs, X1 and X2, to produce a good that is described by the production function: Q = f (X1, X2) = X1/2X1/4 12 The firm sells its output at $80 per unit: P = $ 80. The cost of input 1, X1, is $4. The cost of input 2. X2, is $2. (a) Solve for the profit - maximizing input mix, output, and profit. (b) Check the second-order conditions to verify that the profit is at a maximum
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- A firm can manufacture a product according to the production function Q=3K1/2L1/2, and capital is fixed at 4. a) When the firm hires 16 units of labor. The average product of labor is......... b) when the firm hires 16 units of labor, the marginal product of labor is....... C) if the firm can sell its output at a price of $ 10 per unit and can hire labor at $ 10 per unit , it should hire?............units of labor maximize the profits.Juan Valdez owns a coffee farm in Colombia. His production function is: f(x1,x2)=(x1−1)^0.25 x2^0.5 Assume the price of input 1 is r and the price of input 2 is w. (a) Write down an expression for the technical rate of substitution. (b) Find Juan's demand for inputs conditional on the quantity y of coffee Juan wants to produce. (c) Find Juan's cost function. (d) What is the supply function of Juan's firm?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,
- 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=1A firm uses capital and labor to produce output according to the pproduction function q = 4pKL, for which MPL = 2pK=L and MPK = 2 L=K. a. If the wage w=$4 and the rental rate of capital r=$1,what is the least expensive way to produce 16 units of output? b. What is the minimum cost of producing 16 units? c. Show that for any level of output q, the minimum cost of producingq is $q.A firm requires labor and capital to produce a given output. Labor costs $8 per hour and capital costs $12 per hour. At the current level of output, the marginal product of labor is 48 and the marginal product of capital is 60. To minimize production costs, in which way should the firm allocate its resources?
- Let 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.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…Catalina Films produces video shorts using digital editing equipment (K) and editors (L). The firm has the production function Q(K, L)=KxL, where Q is the hours of edited footage. The wage is $25, and the rental rate of capital is $50. The firm wants to produce 3,000 units of output at the lowest possible cost.a) Find the marginal product of each input.b) Determine whether the production function exhibits diminishing marginal product to each input.c) Find the marginal rate of technical substitution(MRTSLK)d) How does MRTSLK change as more L, is used holding output constant.e) Find the least costly combination of labor and capital to produce 3000 units
- A firm produces basketballs with two variable inputs - labour (L) and plastic (K) - and has the following production function: f (L , K) = 6L1/3K1/6 If the price of output is 4, the price of labour is $2 and the price of plastic is $3, calculate the a) profit-maximizing level of each input b) the total production and c) the profit.Suppose that the Travnikar Corporation manufactures widgets. Analysts that work for the company have found a function C(x) which reports the cost to produce x widgets and a function R(x) which reports the total revenue from selling x widgets. Currently, the company is producing 500 widgets, but the analysts find that R'(500)=85 and C'(500)=79. In order to maximize profits, should the analysts recommend increasing or decreasing production?The total cost (in dollars) to produce qq units of a good is given by the function:C(q)=7.6q+44000C(q)=7.6q+44000A. What is the total cost to produce q=4800q=4800 units?Total cost = $ B. What is the cost of the 48014801st item?Cost of the 48014801st item = $