Question

Suppose that the production function is given by y = 2x^0.5. The price of x is $3 and the price of y is $4. Derive the corresponding VMP and AVP functions. What is MFC? Solve for the profit-maximizing level for input use x.

Answer 1

Let $P_x=\$3, P_y=\$4.$

Then, the total value of the product (TVP) can be written as follows:

$$TVP=P_y\cdot y.$$

The VMP can be found as follows:

$$VMP=\dfrac{dTVP}{dx}=P_y\dfrac{d}{dx}(2x^{0.5})=P_y\cdot\dfrac{1}{\sqrt{x}}.$$

The total factor cost (TFC) can be written as follows:

$$TFC=P_x\cdot x.$$

The profit function can be written as:

$$\Pi=TVP-TFC=P_y\cdot y-P_x\cdot x.$$

By the definition, the marginal factor cost (MFC) equals:

$$MFC=\dfrac{dTFC}{dx}=\dfrac{d}{dx}(P_x\cdot x)=P_x=\$3.$$

Average value product (AVP) can be written as:

$$AVP=P_y\cdot2x^{0.5}\cdot\dfrac{1}{x}.$$

The profit-maximizing level:

$$\dfrac{d}{dx}\Pi=0,$$ $$P_y\dfrac{dy}{dx}-P_x=0,$$ $$P_y\dfrac{d}{dx}(2x^{0.5})-P_x=0,$$ $$P_y\cdot\dfrac{1}{\sqrt{x}}-P_x=0,$$ $$x=(\dfrac{P_y}{P_x})^2=(\dfrac{\$4}{\$3})^2=1.77$$