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Suppose that a certain factory output is given by the Cobb-Douglas production function Q(K, L) = 60K¹/312/3 units, where K is the level of capital and L the size of the labor force need to maximize the factory's output.

Suppose that a certain factory output is given by the Cobb-Douglas production function Q(K, L) = 60K¹/312/3 units, where K is the level of capital and L the size of the labor force need to maximize the factory's output.

Microeconomic Theory
Microeconomic Theory
12th Edition
ISBN:9781337517942
Author:NICHOLSON
Publisher:NICHOLSON
Chapter9: Production Functions
Section: Chapter Questions
Problem 9.2P
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Given the solution for a to c. Show clear working for questions d to f

Suppose that a certain factory output is given by the Cobb-Douglas production function Q(K,L) = 60K¹/312/3 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. (d) Write down the first-order conditions. (e) Determine how many units should be expended on labour and how many units should be expended on capital in order to maximize production. (f) What is the maximum production level? (g) Use the bordered Hessian to prove that the level of production is indeed maximized.
Transcribed Image Text:Suppose that a certain factory output is given by the Cobb-Douglas production function Q(K,L) = 60K¹/312/3 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. (d) Write down the first-order conditions. (e) Determine how many units should be expended on labour and how many units should be expended on capital in order to maximize production. (f) What is the maximum production level? (g) Use the bordered Hessian to prove that the level of production is indeed maximized.
4:08 AO Expert Solution bartleby.com/questions-and- • Q is the output level • K is the level of capital input • L is the level of labor input Step 1 The Cobb-Douglas production function is a widely used function in economics to model the relationship between the inputs used in production and the resulting output. It is given by the following functional form: Q=A* K^a* L^B where: • A is a productivity factor • a and ẞ are parameters that measure the share of output produced by each input ²01² -40K(1/3) L(-4/3) Ə²Q/ƏKƏL = 40 / 3k K(-2/3) L(-1/3) = The parameters a and ß are typically assumed to be positive and less than one, which implies that the production function H = exhibits decreasing returns to scale. In other words, as the inputs are increased proportionately, the resulting increase in output is less than proportionate. Step 2 (a) To determine whether the Cobb-Douglas production function is concave, convex, strictly concave, strictly convex, or neither, we need to compute the second-order partial derivatives of the function. Q(K,L) = 60K(1/3)* L(2/3) Taking the first-order partial derivatives, we get: ?Q / ?к = 20K(-2/3)(2/3) aQ/L = 40K(1/3) (-1/3) Taking the second-order partial derivatives, we get: a²Q/ Ək² = -40K K(-5/3) (2/3) a²Q/ ak² a² QƏKƏL ²/ KA²² det(H) = (a²Q/ƏK²) (A²Q/ ƏL²)-(0²Q / ƏKƏL)² ŵ LTE+ ↓↑ To determine the concavity or convexity of the function, we can compute the determinant of the Hessian matrix: (b) The budget constraint is: $ 200K + $ 100L = $ 200,000 all 30% (c) The Lagrange function is: L(K, L, X) = 60K(1/3) (2/3) + >($ 200K + + :D ||| det(H) = -40K (-5/3) T (2/3), Substituting the second-order partial derivatives, we get: *(-40K(¹/3) L(-4/3) - (40/3K-2/3) L(-1/3) 2 det(H) = 3200/9K(-8/3) (-10/3) Since det(H) is always positive, the Cobb-Douglas production function is strictly convex. $ 100L WAS THIS HELPFUL? $ 200,000) SAVE
Transcribed Image Text:4:08 AO Expert Solution bartleby.com/questions-and- • Q is the output level • K is the level of capital input • L is the level of labor input Step 1 The Cobb-Douglas production function is a widely used function in economics to model the relationship between the inputs used in production and the resulting output. It is given by the following functional form: Q=A* K^a* L^B where: • A is a productivity factor • a and ẞ are parameters that measure the share of output produced by each input ²01² -40K(1/3) L(-4/3) Ə²Q/ƏKƏL = 40 / 3k K(-2/3) L(-1/3) = The parameters a and ß are typically assumed to be positive and less than one, which implies that the production function H = exhibits decreasing returns to scale. In other words, as the inputs are increased proportionately, the resulting increase in output is less than proportionate. Step 2 (a) To determine whether the Cobb-Douglas production function is concave, convex, strictly concave, strictly convex, or neither, we need to compute the second-order partial derivatives of the function. Q(K,L) = 60K(1/3)* L(2/3) Taking the first-order partial derivatives, we get: ?Q / ?к = 20K(-2/3)(2/3) aQ/L = 40K(1/3) (-1/3) Taking the second-order partial derivatives, we get: a²Q/ Ək² = -40K K(-5/3) (2/3) a²Q/ ak² a² QƏKƏL ²/ KA²² det(H) = (a²Q/ƏK²) (A²Q/ ƏL²)-(0²Q / ƏKƏL)² ŵ LTE+ ↓↑ To determine the concavity or convexity of the function, we can compute the determinant of the Hessian matrix: (b) The budget constraint is: $ 200K + $ 100L = $ 200,000 all 30% (c) The Lagrange function is: L(K, L, X) = 60K(1/3) (2/3) + >($ 200K + + :D ||| det(H) = -40K (-5/3) T (2/3), Substituting the second-order partial derivatives, we get: *(-40K(¹/3) L(-4/3) - (40/3K-2/3) L(-1/3) 2 det(H) = 3200/9K(-8/3) (-10/3) Since det(H) is always positive, the Cobb-Douglas production function is strictly convex. $ 100L WAS THIS HELPFUL? $ 200,000) SAVE
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