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

$$L=50x^{\frac{3}{5}}y^{\frac{2}{5}}-\lambda(200x+100y-30000)$$

$$\frac{\varDelta L}{\varDelta x}=30(\frac{y}{x})^{\frac{2}{5}}-200\lambda$$

$$\frac{\varDelta L}{\varDelta y}=20(\frac{x}{y})^{\frac{3}{5}}-100\lambda$$

$$200x+100y=30000$$

$$x=109$$

$$y=82$$