X2^(1/4). The prices of the inputs wland w2. 1.Find the firm's demand for inputs (as a functi of output and the input prices) | 2. Find the cost function of the firm.
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- Suppose that the production function is given by y = 2x0.5 The price of x is $3 and the price of y is $4. Derive the corresponding VMP and AVP functions. What is MFC? Solve for the profit-maximizing level for input use x.A perfectly competitive rm produces output q with capital K and labor L according to the production function: q = f(K, L) = 4K 1 4L 1 4 The price of labor is w = 4, and the price of capital is r = 4. The rm has xed costs equal to 8. The current market price is 8. (a) Prove that this production function has decreasing returns to scale. (b) Find the optimal (cost-minimizing) ratio of capital to labor inputs K L for any level of output (use scale expansion path). (c) For any q, nd the cost minimizing inputs as functions of output: i.e., derive functions L(q) and K(q).(a) Find the conditional input demand function for inputs 1 and 2, as well as the cost function for this firm. (b) What is the firm's marginal cost when it is producing y units of output?
- Only typed answer A firm has the production function Q = K^0.8L^0.2. The wage is W and the rental rate on capital is R. Derive the long-run demand functions for capital and labor. Using Lagrangian is optional.Consider a production function in such a form: Q= f (K,L)= 3KL Calculate the elasticity of substitution (Ϭ) for this firm.Consider a production function of two inputs, labor and capital given by Q=LK. Let w =5 and r = 1, where w is the price per unit of labor and r is the price per unit of capital. a) Suppose that the firm is required to produce Q units of output. Show how the cost-minimizing quantity of labor depends on the quantity Q? b) If the firm is operating in the short run, with K fixed at 10 units, calculate the short-run total cost of producing 100 units of output?
- 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)?Suppose the long-run production function for a competitive firm is f(x1,x2)= min {3x1,2x2}. 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: 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? C: Write down the formulas and draw the graphs of the average cost and marginal cost functions, as functions of y.Consider the following shortrun production function: Q =100L - L2 where Q is the output level and L is labour input. If the price of output in the market is K.sh 50 and labour costs K.sh 1200 per hour, how many hours would the firm use to maximize profits. What is the profit maximizing level of output?
- 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 = $25.d) Determine this firm’s minimum cost of producing 120 units.e) Now suppose that the firm’s production goal is left as the variable Q. Come up with the firm’s costfunction C(Q). Show your work.Suppose the production function for high quality brandy is given by : Q = √KLWhere q is the output of brandy per week and L is labor hours per week ., in the short run K is fixed at 100, so the short run production function is Q = 10√La. If the capital rents for 10$ and the wage are 5$per hour ., write the short run total cost function. b. How much will the firm produce at a price of 20$ per bottles of brandy ?c. How many labor hours will be hired per week?A firm faces a production function given byq = √klwhere q is the output, k is the firm’s amount of capital equipment and ? is the amount of labour-time employed.(a) In the short run, the amount of capital equipment is fixed at k = 100. The rental rate for k is v = $1, and the wage rate for l is w = $2.Calculate the firm’s short-run average cost (SAC) and short-run marginal cost (SMC) functions. Graph the SAC and the SMC curves for the firm. (b) Where does the SMC curve intersect the SAC curve? Explain why the SMC curve will always intersect the SAC curve at its lowest point.(c) Calculate the long-run total cost of production. For w = $2, v = $1, graph the long-run total cost curve. Show that this is an envelope for the short-run curves computed in part (a).
