Your automobile assembly plant has a Cobb-Douglas production function given by q

Question

Your automobile assembly plant has a Cobb-Douglas production function given by

q = x0.4y0.6

where q is the number of automobiles it produces per year, x is the number of employees, and y is the daily operating budget (in dollars). Annual operating costs amount to an average of $25,000 per employee plus the operating budget of $365y. Assume that you wish to produce 4,000 automobiles per year at a minimum cost. How many employees should you hire? HINT [See Example 5.] (Round your answer to the nearest employee.)

x = __________ employees

This is all the information that was given in the question.

Explanation / Answer

Ans) These are the important equations:

(1) q = 0.4x0.6y
(2) r = 25000x + 365y (r is the annual operating costs)

Since q is given (q=4000), we can express y in terms of x:

4000 = 0.4x0.6y
y = 4000/[(0.4x)(0.6)]
y = 50000/3x ----> we substitute this to equation 2

r = 25000x + 365(50000/3x)
r = 25000x + 6083333.33x^-1
r should be minimized. In calculus, we differentiate this and equate to zero. Then we can determine x, which is the number of employees.

dr/dx = 25000 - 6083333.33x^-2 = 0
6083333.33/x^2 = 25000
25000x^2 = 6083333.33
x^2 = 243.33
x = 15.59
x=16

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