Chapter 10.A, Problem 6P

To determine

Concept introduction:

Budget Line: It is defined as the combination of all the goods that a consumer can purchase, exhausting all his income. The formula for the budget line is:

$M=X{P}_x+Y{P}_y$

Here,

• $X$ is the quantity of good X.

• $Y$ is the quantity of good Y.

• $M$ is the total income.

• $P_x$ is the price of good X.

• $P_y$ is the price of good Y.

Slope of the Budget line: It is defined as the ration of price, in the above case the slope is $\frac{-P_x}{P_y}$

Marginal Rate of Substitution: It is defined as the quantity of goods sacrificed for the additional unit of another good. The formula for it is:

$MR{S}_{xy}=\frac{M{U}_x}{M{U}_y}$

Here,

• $MR{S}_{xy}$ is the marginal rate of substitution of $x$ and $y$

• $M{U}_x$ is the marginal utility of $x$

• $M{U}_y$ is the marginal utility of $y$

Optimality Rule: According to the indifference curve approach, the consumer achieves its optimum bundle at appoint where

$\begin{array}{l}M=X{P}_x+Y{P}_y\\ MR{S}_{xy}=\frac{P_x}{P_y}\end{array}$

Explanation of Solution

a. Mrs. Kory’s optimum bundle for chocolates and albums.

It is given that the total income is $50. Cost of the album and chocolate is $10 and $2 respectively. The marginal rate of substitution of the album in place of chocolate is 1.

By optimality rule,

$\begin{array}{l}M=A{P}_a+C{P}_c\\ \$50=4\times \$10+5\times \$2\\ \$50=\$50\end{array}$

$MR{S}_{ac}=\frac{P_a}{P_c}$ ...................(1)

Here,

• $MR{S}_{ac}$ is the marginal rate of substitution of $a$ and $c$

• $P_a$ is the price of album.

• $P_c$ is the price of chocolate.

• $A$ is the quantity of album.

• $C$ is the quantity of chocolate.

Substitute $10 for $P_a$ $2 for $P_c$ and 1 for $MR{S}_{ac}$ in equation (1)

$\begin{array}{l}1\ne \frac{\$10}{\$2}\\ 1\ne 5\end{array}$

• So, the bundle (4, 5) is on the budget line, but is not the optimal solution as it does not satisfy the optimality condition.

• In the above case, Mrs. Kory values the album more than that of chocolate so, $MR{S}_{ac}$ is less than the price ratio.

• Hence, she should buy more chocolates and less albums such that the optimal condition is fulfilled.

Conclusion:

Bundle (4, 5) is not an optimal bundle, Mrs. Kory must buy more cups of hot chocolate and less albums.

b. Mrs. Kory’s optimum bundle for chocolates and albums

It is given that the total income is $50 and cost of the album and chocolate is $10 and $2 respectively.

By optimality rule,

$\begin{array}{l}M=A{P}_a+C{P}_c\\ \$50=2\times \$10+15\times \$2\\ \$50=\$50\end{array}$

$MR{S}_{ac}=\frac{P_a}{P_c}$ ............(2)

$\begin{array}{l}MR{S}_{ac}=\frac{M{U}_a}{M{U}_c}\\ MR{S}_{ac}=\frac{25}{5}\\ MR{S}_{ac}=5\end{array}$

Here,

• $M{U}_a$ is the marginal utility of album.

• $M{U}_c$ is the marginal utility of chocolate.

Substitute $10 for $M{U}_a$ $2 for $M{U}_c$ and 5 for $MR{S}_{ac}$ in equation (2):

$\begin{array}{l}5=\frac{\$10}{\$2}\\ 5=5\end{array}$

• So, the bundle (2, 15) is on the budget line and is an optimal solution, as it satisfies the optimality condition.

• In the above case, Mrs. Kory is exchanging five chocolates for one more album. And the price of the album is five times that of the chocolate, so the optimality exists.

Conclusion:

Bundle (4, 5) is not an optimal bundle, Mrs. Kory must buy more cups of hot chocolate and less albums.

c. Mrs. Kory’s optimum bundle for chocolates and albums

It is given that the total income is $50 and cost of the album and chocolate is $10 and $2 respectively. The marginal rate of substitution of the album in place of the chocolate is 5

By optimality rule,

$\begin{array}{l}M\ne A{P}_a+C{P}_c\\ \$50\ne 1\times \$10+10\times \$2\\ \$50\ne \$30\end{array}$

Substitute $10 for $P_a$ $2 for $P_c$ and 5 for $MR{S}_{ac}$ in equation (1)

$\begin{array}{l}5=\frac{\$10}{\$2}\\ 5=5\end{array}$

So, the bundle (1, 10) is not on the budget line, no need to check the second condition

Conclusion:

Bundle (1, 10) is not an optimal bundle, as it is not on the budget line.