Show that the quantity of labor(L) and capital(K) that a firm demand decreases with a factor’s own factor price (w for labor and r for capital) and increases with the output price (P) when the production function is a Cobb-Douglas of the form ? = ?????

Show that the quantity of labor(L) and capital(K) that a firm demand decreases with a factor’s own factor price (w for labor and r for capital) and increases with the output price (P) when the production function is a Cobb-Douglas of the form ? = ?????

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter10: Cost Functions

Section: Chapter Questions

Problem 10.7P

See similar textbooks

Similar questions

1) A firm uses two inputs K and L of capital and Labor respectively, to produce a single output Q according to the specific type of production function: Q= F (K, L) =k°L". a,b >0 The prices of capital and labor are rand w, respectively. (1) Find the cost minimizing input of capital (K) (1i) Find the cost minimizing input of Labor (L) Note: These are also called the demand functions of K and L. (li) Write down the cost function of the firm.
The total cost function of a firm is given by: Where q is total output, W and V are unit prices of labour (L) and capital (K) respectively a. Find the conditional input demands for labour (L) and capital (K) b. From your result in (a), derive the underlying production function for q Consider a firm that produces cars with the following production function: q = min(ak, bL) a. Find the Marginal Rate of Technical Substitution of labour for capital for this firm b. If per unit prices of capital (K) and labour (L) are r and w respectively, find the input demand functions for K and L; as well as the long run cost function c. Present your solutions graphically. (Thus, draw the Isoquant and Isocost of this firm)
A firm uses labor (L) and capital (K) to produce rocking chairs (Q) with the following production function Q=LK. The wage (w) is $10 and the rate of capital (r) is $20. The target number of rocking chairs to produce is 800. It is the short run and the amount of K is fixed at 5. What the optimal values for L* and K* in the short run? Enter the number for the the optimal amount of L in the short run? Enter the number for the the "optimal" amount of K in the short run?
Suppose the long-run production function for a competitive firm is f(x1,x2)= min {x1,2x2}. 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. Draw the conditional input demand functions for labor and capital in the x1-y and x2- y spaces. .c. 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? .d. Write down the formula and draw the graph of the average cost function, as a function of y. .e. Write down the formula and draw the graph of the marginal cost function, as a function of y.
1/2 Consider a firm with the production function ƒ(x₁, x₂) = x¹/²x₂. The price of the two inputs is ₁ = 2 and w₂ = 1. If x₁ = x₂ = 16, the marginal product of input 1 is When ₁ is increasing and 2 stays the same, the marginal product of input 1 is Constant Decreasing Increasing This production function has Constant returns to scale Decreasing returns to scale O Increasing returns to scale None of the other answers is correct Does the production function have a diminishing technical rate of substitution? No Yes
Given the production function: Where CES stands for Constant Elasticity of Substitution. K is capital and L is labor. The price per unit produced is p, interest (unit price K) is r and wages (unit price L) are w. a) Find the profit function π(K,L) and simplify b) Find the first order conditions to maximize the profit c) Find the value function for L and K, i.e the functions L* = L*(p,r,w) and K* = K*(p,r,w) and simplify. What is the meaning of the value functions d) Find the best production quantity and the maximum profit, i.e Q*(P,r,w) and π*(P,r,w) simplify results
Consider a firm that produces widgets according to the following Cobb-Douglas production function: Q = A * L^α * K^β where: Q is the quantity of output, L is the quantity of labor, K is the quantity of capital, A is a scale parameter (total factor productivity), α and β are the output elasticities of labor and capital respectively. Given that A = 1, α = 0.6, β = 0.4, L = 16 and K = 9, a) Calculate the quantity of output Q. b) If the firm increases the quantity of labor (L) to 20 while keeping the quantity of capital (K) constant, what will be the new quantity of output?
Suppose the production function is Cobb-Douglas and f(x1;x2)=x11/2x23/2 Write an expression for the marginal product of x1. Does marginal product of x1 increase for small increases in x1, holding x2 fixed? Explain Does an increase in the amount of x2 lead to decrease the marginal product of x1? Explain What is the the technical rate of substitution between x2 and x1? What is the type of returns to scale of this production function? (Increasing, decreasing, constant)
The production function for a product is given by q = 10 K1/2L1/2 where K is capital, and L is labor and q is output. a) Find the marginal product of labor and the marginal product of capital. b) Find the marginal rate of technical substitution between labor and capital. c) Denote the wage of labor by w and the rental of capital by What is the cost minimization condition for a firm? Show it diagrammatically. d) Now suppose w =30 and r = 120. What is the minimum cost of producing q=1000. (You must show your work by clearly writing the equations that you use to derive the cost minimizing levels of L and K.) e) Now suppose that the firm is in the short run and cannot vary the amount of capital. That is, it must use the same amount of capital as in part d. However, the firm wants to produce 1200 units of output. How much labor should it use to minimize its cost and what is the minimum cost of producing q =1200?
Consider a production function of three inputs, labor, capital, and materials, given by Q = LKM. The marginal products associated with this production function are as follows: MPL = KM, MPK = LM, and MPM = LK. Let w = 5, r = 1, and m = 2, where m is the price per unit of materials.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. Show how the cost- minimizing quantity of capital depends on the quantity Q. Show how the cost - minimizing quantity of materials depends on the quantity Q. b) Find the equation of the firms long-run total cost curve.c) Find the equation of the firms long-run average cost curve.d) Suppose that the firm is required to produce Q units of output, but that its capital is fixed at a quantity of 50 units (ie., K 50). Show how the cost- minimizing quantity of labor depends on the quantity Q. Show how the cost- minimizing quantity of materials depends on the quantity Q. e)…
Suppose that a certain factory output is given by the Cobb-Douglas production function Q(K, L) = 60K ¹/³ [2/3 units, where K is the level of capital and L the size of the labor force need to maximize the factory's output. (a) Determine whether the Cobb-Douglas production function is concave, convex, strictly concave, strictly convex or neither. If a unit of labour costs $100, unit of capital $200, and $200,000 is budgeted for production (b) Formulate the problem as a constrained optimization problem. (c) Write down the Lagrange function.
Consider 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…