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Suppose a Cobb-Douglas Production function is given by the following: where L is units of labor, K is units of capital, and P(L, K) is total units that can be produced with this labor/capital combination. Suppose each unit of labor costs $700 and each unit of capital costs $1,400. Further suppose a total of $175,000 is available to be invested in labor and capital (combined). A) How many units of labor and capital should be "purchased" to maximize production subject to your budgetary constraint? Units of labor, L = P(L, K) = 20L0.6 K0.4 Units of capital, K = B) What is the maximum number of units of production under the given budgetary conditions? (Round your answer to the nearest whole unit.) Max production= = units

Suppose a Cobb-Douglas Production function is given by the following: where L is units of labor, K is units of capital, and P(L, K) is total units that can be produced with this labor/capital combination. Suppose each unit of labor costs $700 and each unit of capital costs $1,400. Further suppose a total of $175,000 is available to be invested in labor and capital (combined). A) How many units of labor and capital should be "purchased" to maximize production subject to your budgetary constraint? Units of labor, L = P(L, K) = 20L0.6 K0.4 Units of capital, K = B) What is the maximum number of units of production under the given budgetary conditions? (Round your answer to the nearest whole unit.) Max production= = units

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Cobb-Douglas Production Function Example** Suppose a Cobb-Douglas Production function is given by the following: \[ P(L, K) = 20L^{0.6}K^{0.4} \] where \( L \) is units of labor, \( K \) is units of capital, and \( P(L, K) \) is the total units that can be produced with this labor/capital combination. Suppose each unit of labor costs $700 and each unit of capital costs $1,400. Further suppose a total of $175,000 is available to be invested in labor and capital (combined). **A) How many units of labor and capital should be "purchased" to maximize production subject to your budgetary constraint?** - Units of labor, \( L \) = [_________] - Units of capital, \( K \) = [_________] **B) What is the maximum number of units of production under the given budgetary conditions? (Round your answer to the nearest whole unit.)** - Max production = [_________] units
Transcribed Image Text:**Cobb-Douglas Production Function Example** Suppose a Cobb-Douglas Production function is given by the following: \[ P(L, K) = 20L^{0.6}K^{0.4} \] where \( L \) is units of labor, \( K \) is units of capital, and \( P(L, K) \) is the total units that can be produced with this labor/capital combination. Suppose each unit of labor costs $700 and each unit of capital costs $1,400. Further suppose a total of $175,000 is available to be invested in labor and capital (combined). **A) How many units of labor and capital should be "purchased" to maximize production subject to your budgetary constraint?** - Units of labor, \( L \) = [_________] - Units of capital, \( K \) = [_________] **B) What is the maximum number of units of production under the given budgetary conditions? (Round your answer to the nearest whole unit.)** - Max production = [_________] units
Expert Solution
Step 1: Analysis and Introduction

Given Information:

Cobb Douglas Production Function P open parentheses L comma K close parentheses equals 20 L to the power of 0.6 end exponent K to the power of 0.4 end exponent.

Here, L is the unit of Labor and K is the unit of capital.

Each unit of labor costs $700 and each unit of capital costs $1400.

Total cost is $175,000.

To find:

The unit of labor and the unit of capital which gives the maximum production.

Solution Process:

Step 1: Use Lagrange multiplier, form the function L(x,y,λ) and partially differentiate on each variable.

Step 2: Equate each derivative to zero and find the critical point of L(x,y,λ).

Step 3: Find the function value at the critical point and classify them into maximum and minimum.

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