Answer to Question #214467 in Math for hunka

Question

A cobb douglas production function for a new company is given by F(x,y)=50x^3/5y^2/5. Where x represents the units of labor and y represents the units of capital. Suppose the units of labor and the capital cost is $200 and $100 each respectively and the budget constraint is $30,000. Find the maximum production level for this manufacturer.

Answer 1

The production level for this manufacturer is at maximum, if MPx/MPy = x/y.

$MPx = F'(x) = 30(y/x)^{2/5},$

$MPy = F'(y) = 20(x/y)^{3/5},$

$\frac{30(y/x)^{2/5}}{20(x/y)^{3/5}} = \frac{200} {100} ,$

1.5y/x = 2,

x = 0.75y,

If the budget constraint is $30,000, then:

200×(0.75y) + 100y = 30,000,

250y = 30,000,

y = 120 units,

x = 0.75×120 = 90 units.

$F(x, y) = 50×90^{3/5}×120^{2/5} = 5,048.8.$