4. A two-output-two-input firm has a transformation function of the type:91 + 92 = z4z4.where q and q are outputs and z and 2z are inputs.(a) Is this production function consistent with non-increasing returns to scale? Explain why or why not.(b) Fix z = l and z, = 1. Draw the transformation function between g and q2.(c) Fix q = = 1. Draw the transformation function (isoquant) for inputs z, and z2.(d) Find profit maximising z1. z2.91.92 at prices equal to: Pi = 1, p2 = 2 and w =maximisation problem, think about your picture in (b). With a transformation frontier of this form and such prices, does the firmproduce both outputs? If not, rephrase the maximisation problem accordingly.1, wz = 2. Hint: before you set up the profit

4. A two-output-two-input firm has a transformation function of the type: 91 +92 = 21/42/4. where q, and q are outputs and z₁ and 2₂ are inputs. (a) Is this production function consistent with non-increasing returns to scale? Explain why or why not. (b) Fix 2₁ = 1 and 2₂ = 1. Draw the transformation function between ₁ and 2. (c) Fix q₁ q2 = 1. Draw the transformation function (isoquant) for inputs z₁ and 2₂. (d) Find profit maximising 21.22.91-92 at prices equal to: P₁ = 1, P2= 2 and ₁= 1, 2 = 2. Hint: before you set up the profit maximisation problem, think about your picture in (b). With a transformation frontier of this form and such prices, does the firm produce both outputs? If not, rephrase the maximisation problem accordingly.

4. A two-output-two-input firm has a transformation function of the type: 91 +92 = 21/42/4. where q, and q are outputs and z₁ and 2₂ are inputs. (a) Is this production function consistent with non-increasing returns to scale? Explain why or why not. (b) Fix 2₁ = 1 and 2₂ = 1. Draw the transformation function between ₁ and 2. (c) Fix q₁ q2 = 1. Draw the transformation function (isoquant) for inputs z₁ and 2₂. (d) Find profit maximising 21.22.91-92 at prices equal to: P₁ = 1, P2= 2 and ₁= 1, 2 = 2. Hint: before you set up the profit maximisation problem, think about your picture in (b). With a transformation frontier of this form and such prices, does the firm produce both outputs? If not, rephrase the maximisation problem accordingly.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter9: Production Functions

Section: Chapter Questions

Problem 9.7P

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1. Which of the following production functions do not exhibit constant returns to scale? A. F(K, L) AK L¹-a B. F(K, L) AK + BL C. F(K,L)=L D. F(K,L) = K(1-e-(L/K)) E. None of the above
4) Does these production functions exhibit decreasing, constant, or in- creasing returns to scale? Prove your answer. a) q= b) KL (K+L) q=100√ML+M+L
(5) Consider the Cobb-Douglas production function f(¤1,T2) = Arfx3. (a) For what values of a and b does this production function exhibit increasing returns to scale? What about decreasing returns? Constant returns? (b) Find the marginal product of x1 and r2.
Answer two part. I need clear answer for "Are the law of returan capable"?
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.
(a) Suppose we have the following production function: Q=K²/³L2/³ Does the technology have constant returns to scale (CRS)? If not, what? Show your work and explain what it means.< (b) What is the firm's cost minimi on problem in the long-run? Explain and illustrate in a figure using isocost and isoquants how termine optimal inputs K and L. ← (c) Now, consider the short-run where K is fixed at 8 (so that K²/³ = 4). Let r = $20 and w = $10. Show then discuss how short-run average costs change as Qrises. You can use math or a table/figure to do this. (d) Suppose P = $15. Show that optimal Q = 64 in the short-run. You can use math or a spreadsheet. Show details. < (e) Now suppose P = $5. Should the firm shut-down and set Q=0? Explain. <
Which of the following production functions represent increasing returns to scale? 1. f(K, L) = 2K + 3L II. f(K, L) =[K¹/² + [¹/2]³ III. f(K, L) = min{4K, L} IV. f(K, L) = 5KL-K²/2 - 1²/2 (1) only (II) only (II) and (IV) All of them (1) and (III)
Given the following production function: q = 10KL. Assume that w = 25, r = 75 and C = 1200. (a) Mathematically find the minimum cost combination of capital and labour to produce a given level of output.b) Does the production function in part (a) show increasing returns to scale, decreasing returns to scale, or constant returns to scale? Explain. (c) Using isoquants and isocosts, graphically illustrate the effect of an increase in the wage rate, assuming the firm is producing at the same level of output.
Under what conditions do the following production functions exhibit decreasing, constant, or increasing returns to scale? a) q=L+K b) q=LaKb c) q= L+ LaKb + K
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Suppose a firm uses capital K and labor L to produce output Q with the following production function: 1 Q=4KL + = 1² 3 (a) Find the marginal product of capital and the marginal product of labor. (b) Find the average production of capital and the average production of labor. (c) Does this production function have increasing, constant, or decreasing returns to scale? Explain. (d) Find the marginal rate of technical substitution (RTS). (e) Use your result in (d) to find the elasticity of substitution. (f) Find the contingent demand functions for capital and labor if w is the price of labor input and v is the price of capital input.