3. Answer all parts (a)-(e) of this question. Consider the following production function: Q = 3KL, with marginal produc MPL = K L3 and MPK Define the unit prices of L and K as w and r, r L3 KS Finally, assume that the firm takes the price of its product, p, as given and maximiz Imagine capital is fixed at Ko = 1 in the short run. What is the sh function of the firm? Explain briefly. (a) (b) What is the long-run cost function of the firm? What is the long-run supply function of the firm?
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part C D E
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- A firm uses labour, L and capital K, to produce a single product, X. capital is fixed but labour is variable. The firm’s production function is:X=-0.2L3 + 18L2 + 1620L.Where X is the number of units of the product per week, and L is the number of persons employed.A. A t what weekly output is marginal cost equal to average variable cost?B. if the price of the product is $0.20 per unit, what is the maximum weekly wage that the firm would pay rather than close down?Suppose, the demand and supply curve in a US manufacturing firm are provided as follows: ES = 20 + 2w ED = 70 − 3w where E is the level of employment and w is the hourly wage. Let’s assume this firm shows the representative wage of the manufacturing industry. Suppose the price of each unit of capital used in this industry is $25. The price of output is constant at $50 per unit. The production function is f(E,K) = E½K ½ , so that the marginal product of labor is MPE = (½)(K/E) ½ If the current capital stock is fixed at 1,600 units, how much labor should the industry employ in the short run? How much profit will the industry earn?Given: Q = -0.0250 L3 + 1.5 L2 a) Derive the AP and MP equations (4 pts) b) What is the level of Q where AP and MP of labor is maximum? 2. The chief economist for Lanux Corporation, a large appliance manufacturer, estimated the firm’s short-run cost function for vacuum cleaners using an average variable cost function of the form AVC = a + bQ + cQ2 where AVC = dollars per television and Q = number of vacuum cleaners produced each month. Total fixed cost each month is $180,000. The following results were obtained: DEPENDENT VARIABLE: AVC R-SQUARE F-RATIO P-VALUE ON F OBSERVATIONS: 19 0.7360 39.428 0.0001 PARAMETER STANDARD VARIABLE ESTIMATE ERROR T-RATIO P-VALUE INTERCEPT 191.93 54.65 3.512…
- Consider the production functions of three different Firms utilizing inputs labor (L) and capital (K) in producing goods X, Y, and Z given below. The three firms face the same fixed price for labor and capital at 5 per unit and 10 per unit, respectively. X = KL2 – L3; Y = 10K1.5L0.5; Z = K0.5L0.5 a. Show and discuss which Firm(s) exhibit(s) diminishing marginal productivity of capital b. Solve for the returns to scale of each Firm and interpret your results c. Solve for Firm Y’s MRTSLK and interpret your computed value. Comment on what happens to the MRTSLK if Firm Y decides to hire more labor d. Mathematically show at what stage of production Firm X will experience the law of diminishing marginal productivity/ returns if it chooses to employ 30 units of capital. Graphically illustrate your results2.1 Consider N firms each with the constant-returns-to-scale production function Y = F (K, AL), or (using the intensive form) Y = ALf (k). Assume f'(•) > 0, f"(•) < 0. Assume that all firms can hire labor at wage wA and rent capital at cost r, and that all firms have the same value of A.(a) Consider the problem of a firm trying to produce Y units of output at minimum cost. Show that the cost-minimizing level of k is uniquely defined and is independent of Y, and that all firms therefore choose the same value of k.(b) Show that the total output of the N cost-minimizing firms equals the output that a single firm with the same production function has if it uses all the labor and capital used by the N firms. 2.2 The elasticity of substitution with constant-relative-risk-aversion utility. Consider an individual who lives for two periods and whose utility is given by equation (2.43). Let P1 and P2 denote the prices of consumption in the two periods, and let W denote the value of the…The Long-run production function is given by; Y = 180 L0.8 K1.8Where, Y = Output (mt/day), L = Labour (hours/mt) K = Capita (Rs/mt)a) Calculate Marginal Product of Labour (MPL) and Marginal Product of Capital (MPK), if L=12 and K=20 .b) Derive the equation for Isoquent and graphically show it by assuming L= 10, 15, 20 25 and 30. c) Determine factor intensity and returns to scale of this production function. d) Prove that the elasticity of labour is 0.8 and elasticity of capital is 1.8 .
- The Long-run production function is given by; Y = 180 L0.8 K1.8Where, Y = Output (mt/day), L = Labour (hours/mt) K = Capita (Rs/mt)a) Calculate Marginal Product of Labour (MPL) and Marginal Product of Capital (MPK), if L=12 and K=20 b) Derive the equation for Isoquent and graphically show it by assuming L= 10, 15, 20 25 and 30. c) Determine factor intensity and returns to scale of this production function. d) Prove that the elasticity of labour is 0.8 and elasticity of capital is 1.8 a) According to the Wall Street Journal, Mitsubishi Motors recently announced a major restructuring plan in an attempt to reverse declining global sales. Suppose that as part of the restructuring planMitsubishi conducts an analysis of how labour and capital are used in its production process. Prior to restructuring Mitsubishi’s marginal rate of technical substitution is 0.15 ( in absolute value). Tohire workers. Suppose that Mitsubishi must pay the competitive hourly wage of US$ 15. In the study of…The Long-run production function is given by; Y = 180 L0.8 K1.8Where, Y = Output (mt/day), L = Labour (hours/mt) K = Capita (Rs/mt)a) Calculate Marginal Product of Labour (MPL) and Marginal Product of Capital (MPK), if L=12 and K=20 b) Derive the equation for Isoquent and graphically show it by assuming L= 10, 15, 20 25 and 30. c) Determine factor intensity and returns to scale of this production function. d) Prove that the elasticity of labour is 0.8 and elasticity of capital is 1.8 a)According to the Wall Street Journal, Mitsubishi Motors recently announced a major restructuring plan in an attempt to reverse declining global sales. Suppose that as part of the restructuring plan Mitsubishi conducts an analysis of how labour and capital are used in its production process. Prior to restructuring Mitsubishi’s marginal rate of technical substitution is 0.15 ( in absolute value). To hire workers. Suppose that Mitsubishi must pay the competitive hourly wage of US$ 15. In the study of…Production Function. Consider the Cobb-Douglas production function discussed in class:F(K, L) = AK1/3 L2/3. Suppose that parameters are initially A = 1, K = 150, and L = 10. D) Suppose that the quantity of labor L doubles. Calculate Y, w, r, Y/L, and K/L. Com-ment on how and why these numbers changed relative to (c) and why they did so.
- A purely competitive firm has a single variable input < (labor), with the wage rate. W0 per period. Its fixed inputs cost the firm a total of F dollars per period. The price of the product is P0. (a) write the production function, revenue function, cost function, and profit function of the firm. (b) what is the first-order condition for profit maximization ? Give this condition an economic interpretation. (c) What economic circumstances would ensure taht profit is maximized rather tahtn minimized?Suppose there are two primary factors of production, labour (L) and capital (K), and there are nosecondary factors of production. The production function for this economy is simply Y = cKL,where c is a constant and α= 0.5. Which of the following is true? 1.This economy exhibits constant marginal product of labour 2.This economy exhibits decreasing returns to scale 3.This economy exhibits constant marginal product of capital 4.This economy exhibits diminishing marginal product of labourConsider 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…