A proprietor of a company finds that it takes x units of labor and y units of ca

Question

A proprietor of a company finds that it takes x units of labor and y units of capital to produce f(x,y)=100x3/4y1/4 units of their product. If a unit of labor costs $300, a unit of capital costs $500, and $200,000 is budgeted for production, a) determine how may units should be expended on labor and how many units should be expended on capital in order to maximize production. b) give a definition of the marginal productivity of money. c) compute the marginal productivity for this particular case.

Explanation / Answer

a)

It is given that total budgeted cost of the production is $200,000.

Therefore, 300x+500y=200000

Hence, y=400-3x/5

Now, for maximum value of the production function, f(x,y)=100x3/4y1/4, its derivative with respect to x has will be 0.

Therefore, d/dx(100x3/4y1/4)=0

Or, d/dx{100x3/4(400-3x/5)1/4}=0

Or, 100[(3/4)x-1/4(400-3x/5)1/4+x3/4(1/4)(400-3x/5)-3/4(-3/5)=0

Or, 400-3x/5=x/5

Or, x=500.

Now, substituting x=500 in the equation, 300x+500y=200000, we get y=100.

Hence, 500 unis should be expended on labour and 100 units should be expended on capital.

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