Answer to Question #206734 in Microeconomics for Master P

Question

Sarah and Jane are two representative individuals living in an economy that produces two goods, X and Y. Sarahʹs and Janeʹs utility functions are given as

Sarah:

Jane:

Jane marginal utilities

Sarah marginal utilities

Answer 1

The complete question is

a.

Marginal rate of substitution is the rate at which a good is substituted for another. Marginal rate of substitution is the slope of the indifference curves. Marginal rate of substitution is calculated by the ratio of marginal utility of two goods. The following shows the calculations:

$MRS_J=\frac{MU_X}{MU_Y}$

$=\frac{50X^{-0.5}Y^{0.5}}{50X^{0.5}Y^{-0.5}}$

$=\frac {Y}{X}$

$MRS_S=\frac{MU_X}{MU_Y}$

$=\frac{50X^{-0.6}Y^{0.6}}{50X^{0.6}Y^{-0.6}}$

$=\frac {Y}{X}$

b.

The equilibrium condition for maximizing utility happens when the slope of the indifference curve is equal to the slope of the budget line. The slope of the indifference curve is the ratio of the marginal utilities of the two goods. The slope of the budget line is the ratio of the price of the two goods. When marginal rate of substitution is equal to the ratio of the price of goods, the consumer is said to maximize its utility. 

Person J:

$MRS_J=\frac{MU_X}{MU_Y}=\frac{P_X}{P_Y}$

$\frac{50X^{-0.5}Y^{0.5}}{50X^{0.5}Y^{-0.5}}=\frac{10}{20}$

$\frac{Y}{X}=\frac{1}{2}$

$X=2Y$

Putting this equation in the budget line equation for person J, the following is obtained:

$XP_X+YP_Y=M\\X\times10+Y\times20=600\\10\times2Y\times 20Y=600$

$Y=\frac{600}{40}\\Y=15$

Putting the value of Y in the budget line equation, the following is obtained:

$X=2\times Y\\=2\times 15\\=30$

Person S:

The following shows the equilibrium condition for person S:

$MRS_S=\frac{MU_X}{MU_Y}=\frac{P_X}{P_Y}$

$\frac{50X^{-0.6}Y^{0.6}}{50X^{0.6}Y^{-0.6}}=\frac{10}{20}$

$\frac{Y}{X}=\frac{1}{2}$

$X=2Y$

Putting the relation between X and Y in the budget line equation for person S, the following is obtained:

$XP_X+YP_Y=M\\X\times10+Y\times20=700\\10\times2Y\times 20Y=700$

$Y=\frac{700}{40}\\Y=17.5$

Now, putting the value of Y in the relationship between X and Y, the following is obtained:

$X=2\times Y\\=2\times 17.5\\=35$

c.

In an economy, there are two parts, consumption and production. The total consumption should be less than or equal to the total production in the economy to avoid the import of goods. 

The calculated values in part b maximize the utility for both person J and person S. The ratio of marginal utilities are also same. Hence, the values satisfy the conditions for equilibrium of exchange.