Julio receives utility from consuming food (F) and clothing (C) as given by the utility function U(F,C) = FC. In addition, the price of food is \(2 per unit, the price of clothing is \)10 per unit, and Julio's weekly income is $50.

a. What is Julio's marginal rate of substitution of food for clothing when utility is maximized? Explain.

b. Suppose instead that Julio is consuming a bundle with more food and less clothing than his utility-maximizing bundle. Would his marginal rate of substitution of food for clothing be greater than or less than your answer in part a? Explain.

For the given utility function, Julio will maximize his utility when:

$$\begin{gathered} \frac{M{U}_C}{P_P}=\frac{M{U}_F}{P_F} \\ \frac{F}{10}=\frac{C}{2} \\ F=5C \end{gathered}$$

The consumption bundle can be computed from the budget line as:

2F + 10C = 50

(2x5C)+10C=50

C=2.5

then F=12.5

Putting the relation in the utility function:

$U(C,F)=FC=12.5\times 2.5=31.25$

The marginal rate of substitution is:

$$\begin{gathered} {\mathrm{MRS}}_{\mathrm{F},\mathrm{C}}=\frac{{\mathrm{MU}}_{\mathrm{F}}}{{\mathrm{MU}}_{\mathrm{C}}}=\frac{{\mathrm{P}}_{\mathrm{F}}}{{\mathrm{P}}_{\mathrm{C}}} \\ =\frac{\mathrm{C}}{\mathrm{F}}=\frac{{\mathrm{P}}_{\mathrm{F}}}{{\mathrm{P}}_{\mathrm{C}}} \\ =\frac{2.5}{12.5}=\frac{1}{5} \end{gathered}$$

Julio's marginal rate of substitution is 0.2.

The marginal rate of substitution of two goods is the ratio of the prices of the goods.

The utility-maximizing consumption bundle is (12.5, 2.5). If the consumption for food increases from 12.5 and consumption for cloth decreases from 2.5, the marginal substitution rate remains the same.

The MRS does not change because prices of per unit food and cloth remain the same after the consumption bundle changes.