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- 8. If a firm has a production function of the form: Q = L0.5K2, what can we conclude? (a) The firm has a quasi-concave production function and will experience constant marginal returns to labour and increasing marginal returns to capital. (b) The firm has a quasi-concave production function and will experience decreasing marginal returns to labour and increasing marginal returns to capital. (c) The firm has a convex production function and will experience increasing marginal returns to labour and increasing marginal returns to capital. (d) The firm has a convex production function and will experience constant marginal returns to labour and increasing marginal returns to capital. (e) The firm has a linear production function and will experience decreasing marginal returns to labour and increasing marginal returns to capital.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.1. Consider the production functions for good X and good Y: X = K? +L ; Y= 20VKL A. Show and discuss which production function(s) exhibit(s) diminishing marginal productivity of labor and capital. B. Show and discuss which production function(s) exhibit(s) diminishing MRTSLK. C. Suppose capital and labor are doubled, solve for the returns to scale of each production function, and interpret your results.
- 6. Consider the following Cobb - Douglas utility function: U = χαγβην *Note, it should be assumed that a, B.y > 0 Show that this production function can exhibit increasing returns to scale globally while maintaining diminishing returns for each individual input.1. Let y = f(x1, x2)=x11/2+ X1X2 be a firm's production function, where x20, x220. - a. Write down the firm's production possibility set, and its input requirement set.“ b. Is this production function concave, quasi-concave? c. Is this production function homogenous, homothetic? + d. Find its returns to scale when x1=1, and x2=1.eSuppose 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).
- 1) In connection with the Law of the Minimum and the Law of Diminishing Returns, describe how the production function model can be a helpful tool in managing an agricultural production system.Exercise A.5 A company operates with production function q-K¹/2 [1/2 a) What type of returns to scale does this production function present? As production varies, how would you expect long-term average costs to evolve? b) Assume that the firm's capital equipment is fixed and equal to K = 100 units. Find the firm's short-run production function and the marginal productivity of labour and graph these functions. As production varies, how would you expect marginal costs to evolve in the short term? c) If the prices of capital and labour are r = 8€ and w = 2€ respectively, obtain the expression of the short-run total cost function of the firm. b) Obtain the expressions of the functions of short-term average total costs, average variable costs and marginal costs of this company. Find the production level for which the average total cost is minimized and the corresponding average total cost level. d) Indicate the value of the average total cost, the average variable cost and the marginal cost…(a) What is the long-run cost function if the production function is q = L + K? (b) Consider a general production function q = f(L, K) - Set up the Lagrangian that minimizes short-run cost. Show that the Lagrangian multiplier represents the marginal cost of production. - Set up the Lagrangian that minimizes long-run cost. Show that the Lagrangian multiplier represents the marginal cost of production.
- 1. Suppose firms use two inputs, capital, K, and labour L to produce output. Their production functions are written as a function of K and L: F(K, L). For each of the following production functions, determine i) whether the production function exhibits increasing, decreasing, or constant returns to scale, and ii) if both inputs double, by what factor (how many times) output will increase. (a) F(K,L) = 5K² +3L² (b) F(K,L) = 8K0.4 0.3 (c) F (K, L) = K² (2K³ – L³)P MC АТC b D Q5 Q1 Q2Q3 Q4 MR Q7. Provide your own examples for the questions below. a. An example for economies of scale using production function. Explain why. Explanation: b. An example for diseconomies of scale using production function. Explain why. Explanation: c. A real example of a firm that enjoys economies of scale and explain why this happened. Explanation:2. Which of the following production functions with inputs K, L, and H exhibits Constant Returns to Scale (CRS)? (Assume Ā > 0 is a constant) a) Y = AK/2L/2 b) Y = ĀK/2L/3H/4 c) Y = ĀK/2 – L/H d) Y = ĀKL/2H\/2