91Ó°ÊÓ

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:

$U(C,F)=FC=12.5×2.5=31.25$

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.