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- Economics A firm has the production function q = K0.25L0.25 , where K is capital, L is labor, and q is output. The price of capital is v = 4 and the price of labor is w = 4. [You may need this result: if Yα = Z. Then Y = Z1/α]. (i) Derive the firm’s short-run cost function if capital is fixed at K = 16. (ii) Derive the firm’s long-run cost function. (iii) Someone claims that the firm’s1. A firm producing hockey sticks has a production function given by q = 2Vkl The price of labor is "w", the price of capital is "v". If the capital used for producing hockey sticks is fixed at "k1" in the short run. The short run cost wq? + vk, 4k1 function is SC = a. Given q, w, and v, how should the capital stock "k1" be chosen to minimize total cost? b. Use your results from part (a) to calculate the long-run total cost of hockey stick production.Suppose the representative firm of the economy has a production function of the form F(K, N) = AKO.5N 0.5 The marginal product of labor is then given by MP N = 0.5AK0.5N –0.5 . The current capital 3. stock is K = 40. (a) Holding fixed capital at 40, draw a graph of output as a function of labor. What are some important features of this graph? (b) If A = 4 what is the Labor Demand Curve, ND(w), as a function of the real wage w? (c) Suppose labor is supplied inelastically with NS(w)= 10. What is the equilibrium wage w, employment level N, and full employment output Y ? (d) Suppose that productivity unexpectedly increases to A = 6. What is the new equilibrium wage w, employment level N, and full employment output Y ? (e) Depict this change graphically. Denote the original labor supply and demand curves by NS and ND respectively. Denote the new labor demand curve after the productivity shock by ND' . (Does not need to be to scale).
- Question 1. Suppose a small firm employs Capital (K) and labour (L) at its production of output. The production function of the firm is: q = -56³ + 10L²K + 100LK0.5. Assume that K-25 in the short run. a. What is its short-run production function? b. What are the firm's marginal product of labour and average product of labour in the short run? c. In the short run, when marginal output is maximized, what is the value of L? d. Calculate the short-run elasticity of output with respect to labour when L=10 and K-25 (Hint: the firm's elasticity of output with respect to labour in the short run is a function of marginal product of labour and average product of labour)5. Selpats produces pens with the production function f(k, l) = 200vkl. The price of capital is wk = 40 and the price of labor is wL = 10. (i) Determine the optimal input bundle for producing y pens in the long run and find the long-run cost function. (ii) Determine the optimal input bundle for producing y = 10,000 pens in the long run. (iii) Now suppose that capital is fixed at k = 25 in the short run. Find the short-run total cost function and the short-run marginal cost function. Explain the relationship between the short-run and long-run marginal costs when y = 10,000. (iv) Suppose that the price of output is p. Which value of y maximizes short-run profit? р.Question 8 Consider the following short-run production function, where labor (L) is the variable input and capital (K) is fixed: Q (L) = 6L² – 0.4L After which value of labor do marginal returns diminish? 10
- F (L, K) = L0.2K0.7， The wage rate (price per unit of labour) is w = 2 and the capital rental rate (price per unit of capital) is r = 7. Does this production function exhibit increasing, decreasing or constant returns to scale? Explain. What is the marginal productivity of labour and the marginal productivity of capital for (L,K) = (1,1)? Would a firm (which minimises costs) use this combination of labour and capital? Explain. If your answer is yes, then what would be the quantity of production for which the company would use this combination?A firm engaged in the manufacture of RTWS faces the short-run production function Q = 250L - 5L², where L is the number of units of labor and Q is the number of RTWs produced annually. d.) How many RTWS can be produced by the firm in a year if there are 10 units of labor? e.) Compute the marginal product of the 40th unit of labor. f.) How many RTWs can be produced by the firm in a year if there are 40 units of labor? g.) Sketch the graph of the production function.8. Consider a representative firm that faces a constant returns to scale production function Y = zF(K, Nd), where Y is output of consumption goods, z is TFP, K is physical capital, and Nª is labour input. The amount of capital is assumed to be given and fixed. The production function exhibits a positive marginal product of labour, as well as diminishing returns to labour. The firm seeks to maximize profits. (a) Explain what the profit maximizing condition for the firm is and show the profit maximizing labor demand on the graph with total revenue and total variable cost curves. On another graph show the firm's demand for labour curve and the profit maximizing labor input. (b) Suppose that the government now imposes an output tax Ty < 1 per unit of con- sumption good produced. What is the effect of this tax on the firm's demand for labor? Explain.
- 2. Suppose a firm faces the input prices of labor and capital of w = $15 and r = $2, respectively. The firm's production function is q = 4K°.5L°5. The firm currently has 40o units of capital, the amount of which cannot be easily changed. The firm has flexibility in adjusting the amount of labor used in the production process. Find the short-run average cost of producing 400 units of output. Round up your answer to the nearest cent.2. A firm producing hockey sticks has a production function given by q = 2Vkl The price of labor is "w", the price of capital is "v". For any given level of output "q": a. Calculate the firm's long-run total, average and marginal cost function. b. Please show the cost function is homogeneous of degree 1 in input prices. c. Please show the cost function is concave in v. Suppose now that capital used for producing hockey sticks is fixed at "k1" in the short run. d. Calculate the firm's short-run total costs as a function of q, w, v, and k1.Consider the production function: F(L, K) = L0:2K0.7. %3D The wage rate (price per unit of labour) is w = 2 and the capital rental rate (price per unit of capital) is r = 7. (a) Does this production function exhibit increasing, decreasing or constant returns to scale? Explain. What is the marginal productivity of labour and the marginal productivity of capital for (L, K) = (1,1)? Would a firm (which minimises costs) use this combination of labour and capital? Explain. If your answer is yes, then what would be the quantity of production for which the company would use this combination? r (b) Compute the quantity of labour and capital that this firm would use to produce y = 2 at the minimum cost. How much would this cost be? What is the average cost and the marginal cost for that production level? Hint: for this part, you can use directly (without providing the derivation) any results derived in the lecture or tutorials. (c) Derive the equation of the isoquant for y = 2 (with K in the…
