You are a manager for Tesla and recently hired an economist to work with engineering and operations experts to estimate the production function for lithium battery production, and the production function is Q = 5 K 3/4 L 1/4. Tesla has already spent $12,000 on the 10 units of capital it owns and cannot acquire more capital in the short run; if workers are paid $ 1,000 per day and batteries can be sold for $10,000 each:
a) What is the profit maximizing level of the output and labor usage? b) What is the maximum profit?
We have a production function Q = 5 K^(3/4) L^(1/4)
Here K = 10 and is fixed. Hence Q = 5*10^(3/4) L^(1/4) or Q = 28.117L^(1/4)
This gives L* = (Q/28.117)^4
Cost function is C = wL + rK
C = 1000*(Q/28.117)^4 + 12000
C = 0.0016Q^4 + 12000
Profit = revenue - cost
= 10000Q - 0.0016Q^4 - 12000
Profit is maximum when its derivative is 0
10000 - 0.0016*4Q^3 = 0
This gives Q = 116 units and L = 290 units
Profit = $858298
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