1. Given the following cost and production functions, find where production is optimized. C = 25x₁ + 5x2 Q = 5x₁²5x75 2. A firm wants to minimize the cost of producing 100 units of output given that the cost of labor (w) is $25 and the cost of capital (r) is $5. Output is produced according to the Cobb-Douglas production function: Q = 5K-5L.5.
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- 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…Suppose a firm with a production function given by Q = 30K0.5L0.5 produces 1,800 units of output. The firm pays a wage of $50 per unit and pays a rental rate of capital of $120 per unit. MPL = 15K0.5/L0.5 ; MPK = 15L0.5/K0.5 . How many units of labor and capital should the firm employ minimize the cost of producing 1,800 units of output, and what would be the total cost?Assume that a firm with 1600 units of capital and a cobb-douglas production function Q = L^1/2K^1/2 pays labor $10 per unit. How much labor would it hire if it could sell its output at $50 per unit?
- Suppose that production for good X is characterized by the following production function, Q = K0.5L0.5, where K is the fixed input in the short run. If the per-unit rental rate of capital, r, is $25 and the per-unit wage, w, is $15, then the average total cost of using 81 units of capital and 9 units of labor is: $5. $75. $80. There is insufficient information to determine the average total costs.Let the production function be Q = 4* K1/4 L1/4 assume that both factors are variable. (a) Derive the contingent demand functions for K and L (b) Substitute the contingent demand functions in the total cost that you minimized in part a) to obtain the total cost function. (c) Find the amount of K and L necessary to produce Q=8 when v=16 and w=1 (with minimal possible cost). (d) Find the average and marginal cost functions.Suppose that production for good X is characterized by the following production function, Q = K0.5L0.5, where K is the fixed input in the short run. If the per-unit rental rate of capital, r, is $15 and the per-unit wage, w, is $25, then the average fixed cost of using 9 units of capital and 81 units of labor is $75. incalculable since there is insufficient information to determine the average fixed costs. $80. $5.
- Consider a production function Q = (K1/2) + (L1/2) + (M1/2) where L, K, M represent labor, capital, and material. Input prices for L, K, M are 1, 1, and 1. The firm wants to produce 12 units of output. a) What is the total cost if K = 4? b) What is the total cost if K = 4 and L = 9?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.Suppose a firm with a production function given by Q = K0.4L0.6 produces 100 units of output. The firm pays a wage of $20 per units and pays a rental rate of capital of $40 per unit. (Note: MPL = 0.6K0.4L-0.4 and MPK = 0.4K-0.6L0.6 ) What is the minimum cost of producing 100 units of output?
- A firm produces output using labor, L, and capital, K, measured in man-hours and machine-hours, respectively. The firm’s production function f(L, K) = √ √ LK, with associated marginal product functions MPL = √K and MPK = 2 L√ √L . The price of labor is $10 per man-hour and the price of capital is $40 2 K per machine hour. The firm’s production target is 100 units of output per day. (a) Does this firm’s cost minimization problem have an interior or a corner solution? Justify your answer. (b) How many man-hours and how many machine-hours per day are used in the firm’s cost minimizing input combination? (c) What is the minimum cost for this firm of producing 100 units of output per day? *Please fully explain out math processA firm’s technology (recipe for combining factors of production) has an impact on its costs. Given a firm has a technology Q(L,K) = L0.8K0.2 (Where L represents labour and K represents Capital), which of the following is/are necessarily true? A firm’s technology (recipe for combining factors of production) has an impact on its costs. Given a firm has a technology Q(L,K) = L0.8K0.2 (Where L represents labour and K represents Capital), which of the following is/are necessarily true? Costs are exponentially increasing (by an exponent of 0.6) The elasticity of substitution between the factors is σ=1 The firm will spend equal amounts on the Labour and Capital factors Returns to scale are decreasing Optimum amounts of inputs for a given output is found by the interaction of cost functions and technologyEconomics Suppose that production for good X is characterized by the following production function, Q = 4K0.5L0.5, where K is the fixed input in the short run. If the per-unit rental rate of capital, r, is $12 and the per-unit wage, w, is $20, then the average total cost of using 25 units of capital and 49 units of labor is Multiple Choice $6.25. $9.14.Correct $10.07. incalculable since there is insufficient information to determine the average total costs.