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Suppose we're given the following cobb-Douglas production function: P(L,K) = 30L0R20 Where L is the units of labor, K is units of capital, and P of comma is total units that can be produced with this labor/capital combination Suppose each unit of labor costs $900, and each unit of capital costs $1,800
Suppose we're given the following cobb-Douglas production function: P(L,K) = 30L0R20 Where L is the units of labor, K is units of capital, and P of comma is total units that can be produced with this labor/capital combination Suppose each unit of labor costs $900, and each unit of capital costs $1,800. Also suppose a total of 1,080,000 is available to be invested in capital and labor combined. 1. How many units of labor and capital should be purchased to maximize production subject to the budgetary constraint? 2. What is the maximum number of units of production under the given budgetary conditions?
Expert Solution
Production function;
P = 30L0.8 K0.2
Cost of labor, w = $900
Cost of capital, r = $1800
Total cost available, C = 1080000
1. Marginal rate of technical substitution;
MRTS = MPL / MPK
= d/dL (30L0.8 K0.2) / d/dK (30L0.8 K0.2)
= 24K0.2/L0.2 / 6L0.8/K0.8
MRTS = 4K/L
Profit is maximised when;
MRTS = w/r
4K/L = 900/1800
4K/L = 1/2
8K = L
Putting this in cost function;
C = wL + rK
1080000 = 900(8K) + 1800K
1080000 = 7200K + 1800K
1080000 = 9000K
K = 120
L = 960
2. The maximum number of units which can be produced is;
P = 30L0.8 K0.2
= 30 (960)0.8 (120)0.2
= 30 * 243.12 * 2.6
P = 18963.36
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