Question

Farmer Billy has a plot of land on Long island. On farmer billy's land, he grows corn and soy beans for harvest and feeds his diary cattle the corn so they produce milk. FARMER billy enters the market with the soy beans and milk for sale. Farmer billy also consumes some of the soy beans and milk he produces. His preferences for soy beans and milk may be described by Cobb-Douglas, where soy beans assume a fixed 40% of his consumption expenditures and he only consumes soy beans and milk. Farmer billy can sell one cup of soy beans for $3 and one quart of milk for $2. Farmer billy enteres the market with 400 cups of soy beans and 200 quarts of milk..Which of the following are farmer billy's real endownments?

Answer 1

Solution:

Let soybeans be a commodity x, and milk a commodity y.

Then the Cobb-Douglas function will have the form:

$$U(x,y)=x^{0.4}y^{0.6}$$

$$p_x=3, p_y=2$$

The farmer will maximize utility from these two benefits under the following condition

$$\frac{MU_x}{p_x}=\frac{MU_y}{p_y}$$

$$MU_x=\frac{0.4y^{0.6}}{x^{0.6}}$$

$$MU_y=\frac{0.6x^{0.4}}{y^{0.4}}$$

$$\frac{0.4y^{0.6}}{3x^{0.6}}=\frac{0.6x^{0.4}}{2y^{0.4}}$$

$$y=2.25x$$

Consequently, a quart of milk brings him 2.25 times more benefits than a cup of soy.

We will find the income of a farmer entering the market with these two products.

$$TR= \displaystyle\sum_{i=1}^n p_iQ_i$$

$$TR=3\times300+2\times200=900+400=1300$$