Consider a firm which utilizes the neo-classical production function given by the equation f(K, L) = AK L1 where K is the capital stock and L is the labor supply employed. Assuming this firm rents its factors of production from perfectly competitive factor markets, the rental prices of capital and labor are denoted by and. Derive this firm's cost functioc(w, ry)n for a generic output level y by solving its cost minimization problem.
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- Suppose the hourly wage is $10 and the price of each unit of capital is $25. The price of output is constant at $50 per unit. The production function isf(E, K) = E 1/2K 1/2so that the marginal product of labor isMPE = (1/2)(K/E) 1/2If the current capital stock is fixed at 1,600 units, how much labor should the firm employ in the short run? How much profit will the firm earn?Consider a Cobb-Douglas production function:f(l, k) = Alα k1−α,where A is the total factor of productivity (a constant greater than 1), 0 < α < 1, lrepresentslabor, and k represents capital. The following sub-questions will guide you through showing thatthe elasticity of substitution is constant.a) Find the marginal product of labor. Verify that this production function exhibits diminishingmarginal productivity of labor. b) Find the marginal product of capital. Verify that this production function exhibits diminishingmarginal productivity of capital. c) Find the marginal rate of technical substitution. Write your answer as MRT S = . . . d) In part (C), you should’ve found the MRTS as a function of the input ratio, kl. Take theabsolute value of both sides and solve for the input ratio, so that the expression gives theinput ratio as a function of MRTS (i.e. kl = . . .). Take the log of both sides, then take thederivative with respect to the log of MRTS. Is the elasticity of…dien There are two factors in a production function y = x 113 x₂¹1³. The market price of each unit ofy is p=3, and the factor prices are w₁=1 and W₂=2 for x₁ and X₂ respectively. variable (a). Calculate your cost function as a function of y if X, and x₂ are both barible factors. (b). Now derive the functions of average and marginal cost and plot them against quantity (C). Solve for your optional output of y. Calculate the ratio of two factors (X₁/X₂) (d). In the short run the fixed factor is set at X₂=1. What is the new optimal output level now?
- 1) A firm uses two inputs K and L of capital and Labor respectively, to produce a single output Q according to the specific type of production function: Q= F (K, L) =k°L". a,b >0 The prices of capital and labor are rand w, respectively. (1) Find the cost minimizing input of capital (K) (1i) Find the cost minimizing input of Labor (L) Note: These are also called the demand functions of K and L. (li) Write down the cost function of the firm.Consider the following short-run production function (where L – labour, Q - output): Q = 10L - 0.5L^2Suppose that output can be sold for $10 per unit. Also assume that the firm can obtain as much of the variable input as it needs at $20 per unit.Determine the marginal value of product with respect to labour.Determine the marginal cost of labour (factor cost function).Determine the optimal value of L, given that the objective is to maximize profits.Goleta Brewing Company hires only two types of labor, managers and brewing assistants (denoted M and B, respectively). GBC has the following Cobb-Douglas production function F(M,B) = M.5 B.5 and wants to produce 10 barrels of pale ale this week. If the wage of managers is $50 per hour and the wage of brewing assistants is $10 per hour, how many managers and brewing assistants should the firm hire (round to nearest whole number)? How does your answer change when the wage of managers decreases to $30 per hour and the wage of brewing assistants remains constant. Is this result consistent with your intuition?
- 100% 1 / 4 - Consider a delivery company whose output is "number of packages deliv- ered" from two inputs, trucks and drivers. Assuming that only one driver can drive one truck, draw (and correctly label) a production isoquant for the delivery company. 2. Consider a Cobb-Douglas production function, f(K, L) = Kª L* ,where K is capital, L is labour, and (a, 6) are parameters. Describe in words how to choose capital and labour to maximize the production of output. 3. Consider a canola producer with a two-input production function: VSN q = f(S, N) = S+ N where S is seed and N is nitrogen. Calculate the marginal product of ni- trogen. Does the marginal product of nitrogen depend on the amount of seed used? Explain. 4. Assume that all you knew about a two-input production function, f(K, L), was the following: a2 f(K, L) > 0. Is that enough to know that profit maximization is or is not possible? How know? Explain. do you 5. When a farm faces a downward sloping demand curve, its production…Consider a firm that uses labour and capital as inputs for production accord- ing to some production technology y = f(K, L). Let c(y, w, r) be the cost of producing y units of output if the wage rate is w and the cost of capital is r. Let L* and K* be the optimal capital and labour demand for producing y units. Prove that dc(y, w, r) dc(y, w, r) _ K. L* and = K*. dw drSuppose a price-taking profit-maximising firm produces output in the short run using only labour as a variable factor. Output (Q) and labour input (L) are related by the equation: Q Where & a positive constant, the price of output is given and denoted by p, and the wage is given and denoted by w. The firm seeks to maximise profits, given by pQ – wL. Show that the first order condition for a profit maximum may be written as: イー apL' (4)文 and hence show that in this case the firm's demand for labour is given by Further, by examining the second-order conditions explain carefully the requirement that
- Consider a firm that produces output (y) using only two inputs (A and B). The firm has a perfect substitutes production technology and a production function of the formf(A, B) = 5A + 8B. %3D Suppose the firm is currently using 200 units of input B. What is the marginal product of inputA when A 100? 500 What is the marginal product whenA 150? 750 Give your answers to two decimal places.Consider the following production function: x = f0.K) = A e°k where xis the output e is the labour input, kis the capital input, and A, bare positive constants. (a) Set up the cost minimization problem and solve for the first order conditions using the Lagrange Method. Let wbe the wage rate and rthe rental rate of capital. (b) Using your answer in (a), find how much labour and capital would the firm use to produce x outputs, given wage rate wand rental rate of capital ? what are these functions called? (e) Using your answer in (b), find the minimum cost it takes to produce xoutputs, What is this function called?Tom Petersen is considering renovating bar stools at Dreamland. The productionfunction for new bar stools is given byq= 0.5L0.5where q is the number of stools produced during the renovation week and l represents thea number of worker hours employed during the period. The firm is a price taker for both bar stools(which sell for P) and workers (which can be hired at a wage rate of w per hour).a. What is the cost function for this firm C(w,q)?b. What is the supply function for bar stools q(P,w)?c. What is the profit function for this firm (P,w)?d. What is the firm’s demand function for labor function L(P,w)?