Chapter 6, Problem 3CP

(a)

To determine

Determine the marginal rate of substitution (MRS).

Explanation of Solution

The utility function is given as follows:

$\text{U}\left(\text{X,Y}\right)=\left(\text{XY}\right)+\left(\text{Y}\right)$        (1)

Since the marginal utility of X $\left({\text{MU}}_{\text{X}}\right)$ is $\text{Y}$, and the marginal utility of Y $\left({\text{MU}}_{\text{Y}}\right)$ is $\left(\text{X}+1\right)$, MRS between X and Y can be represented as follows:

$\begin{array}{c}{\text{MRS}}_{\text{XY}}=\frac{{\text{MU}}_{\text{X}}}{{\text{MU}}_{\text{Y}}}\\ {\text{MRS}}_{{}_{\text{XY}}}=\frac{\text{Y}}{\left(\text{X}+1\right)}\end{array}$

Thus, E’s ${\text{MRS}}_{{}_{\text{XY}}}$ is $\frac{\text{Y}}{\left(\text{X}+1\right)}$.

This is clear from the above expression that when X is increased, and the level of utility is unchanged, Y is ready to give up less Y in order to accommodate more X. That means, Person E’s utility function satisfies the declining MRS property. Hence, Person E’s preferences do satisfy the declining MRS property.

(b)

To determine

Determine the best choice of Person E, when price of X is $4 and price of Y is $1.

Explanation of Solution

Since the income is $20, the price of X is $4, and the price of Y is $1, the budget equation can be represented as follows:

$\text{4X}+\text{Y}=20$        (2)

Let determine the best choice of Person E between X and Y using the tangency condition (equalize the marginal utility with the price ratio) as follows:

$\begin{array}{c}\frac{{\text{MU}}_{\text{X}}}{{\text{MU}}_{\text{Y}}}=\frac{{\text{P}}_{\text{X}}}{{\text{P}}_{\text{Y}}}\\ \frac{\text{Y}}{\left(\text{X}+1\right)}=\frac{4}{1}\end{array}$

Or

$\begin{array}{c}\frac{\text{Y}}{\left(\text{X}+1\right)}=\frac{4}{1}\\ \text{Y}=4\text{X}+4\end{array}$

Substitute the value of Y into Equation 2:

 $\begin{array}{c}\text{4X}+\text{Y}=20\\ \text{4X}+4\text{X}+4=20\\ \text{8X}=20-4\\ \text{8X}=16\\ \text{X}=\frac{16}{8}\\ \text{X}=2\end{array}$

Thus, X is 2.

Substitute the value of X into the functional form of Y.

$\begin{array}{c}\text{Y}=4\text{X}+4\\ \text{Y}=\left(4\times 2\right)+4\\ \text{Y}=12\end{array}$

Thus, Y is 2.

Therefore, the best choice for X is 2 and Y is 12, when the price of X is $4, and the price of Y is $1.

Substitute the respective values in to Equation 1 (utility function):

$\begin{array}{c}\text{U}\left(\text{X,Y}\right)=\left(\text{XY}\right)+\left(\text{Y}\right)\\ =\left(2\times 12\right)+12\\ =36\end{array}$

Therefore, in this case, the total utility is 36.

Person E’s (own) price elasticity of demand:

The price elasticity of demand can be calculated using the following formula:

${\text{E}}_{\text{D}}=\frac{\Delta \text{Y}}{\Delta {\text{P}}_{\text{Y}}}\times \frac{{\text{P}}_{\text{Y}}}{\text{Y}}$        (3)

Let assume, the price of good X is ${\text{P}}_{\text{X}}$, and the price of good Y is ${\text{P}}_{\text{Y}}$, and the income is $M. Therefore, the budget equation can be represented as follows:

${\text{P}}_{\text{X}}\text{X}+{\text{P}}_{\text{Y}}\text{Y}=\text{M}$        (4)

Use the tangency condition with the indifference curve and the budget line as follows:

$\begin{array}{c}\frac{\text{Y}}{\left(\text{X}+1\right)}=\frac{{\text{P}}_{\text{X}}}{{\text{P}}_{\text{Y}}}\\ {\text{P}}_{\text{X}}\text{X}+{\text{P}}_{\text{X}}={\text{P}}_{\text{Y}}\text{Y}\\ {\text{P}}_{\text{X}}\text{X}={\text{P}}_{\text{Y}}\text{Y}-{\text{P}}_{\text{X}}\end{array}$

Substitute the value of ${\text{P}}_{\text{X}}\text{X}$ into Equation 4:

$\begin{array}{c}{\text{P}}_{\text{X}}\text{X}+{\text{P}}_{\text{Y}}\text{Y}=\text{M}\\ {\text{P}}_{\text{Y}}\text{Y}-{\text{P}}_{\text{X}}+{\text{P}}_{\text{Y}}\text{Y}=\text{M}\\ 2{\text{P}}_{\text{Y}}\text{Y}=\text{M}+{\text{P}}_{\text{X}}\\ \text{Y}=\frac{\text{M}+{\text{P}}_{\text{X}}}{2{\text{P}}_{\text{Y}}}\end{array}$

Now use Equation 3 to get the own price elasticity of demand as follows:

$\begin{array}{c}{\text{E}}_{\text{D}}=\frac{\Delta \text{Y}}{\Delta {\text{P}}_{\text{Y}}}\times \frac{{\text{P}}_{\text{Y}}}{\text{Y}}\\ {\text{E}}_{\text{D}}=\frac{\text{M}+{\text{P}}_{\text{X}}}{2{\text{P}}_{{}_{\text{Y}}}^2}\times \frac{{\text{P}}_{\text{Y}}}{\text{Y}}\\ {\text{E}}_{\text{D}}=\frac{\text{M}+{\text{P}}_{\text{X}}}{2{\text{P}}_{\text{Y}}}\times \frac{1}{\text{Y}}\\ {\text{E}}_{\text{D}}=\text{Y}\left(\frac{1}{\text{Y}}\right)\\ {\text{E}}_{\text{D}}=1\end{array}$

Thus, Person E’s own price elasticity of demand is 1 for any ${\text{P}}_{\text{Y}}$> 0. As the expenditure is maximized when the own price elasticity of demand is 1, Person E’s expenditure on Y is the same for any ${\text{P}}_{\text{Y}}$> 0. That is, Person E’s expenditure on good Y is the largest at all positive prices of Y.

(c)

To determine

Determine the new consumption bundle, when the price of Y rises to $4.

Explanation of Solution

Since the income is 20, the price of X is $4, and the price of Y is $4, the new budget equation can be represented as follows:

$\text{4X}+4\text{Y}=20$        (5)

Let determine the new best choice of Person E between X and Y using the tangency condition (equalize the marginal utility with the price ratio) as follows:

$\begin{array}{c}\frac{{\text{MU}}_{\text{X}}}{{\text{MU}}_{\text{Y}}}=\frac{{\text{P}}_{\text{X}}}{{\text{P}}_{\text{Y}}}\\ \frac{\text{Y}}{\left(\text{X}+1\right)}=\frac{4}{4}\end{array}$

Or

$\begin{array}{c}\frac{\text{Y}}{\left(\text{X}+1\right)}=\frac{4}{4}\\ \frac{\text{Y}}{\left(\text{X}+1\right)}=1\\ \text{Y}=\text{X}+1\end{array}$

Substitute the value of Y into Equation 5:

 $\begin{array}{c}\text{4X}+4\text{Y}=20\\ \text{4X}+4\left(\text{X}+1\right)=20\\ \text{4X}+\text{4X}+4=20\\ \text{8X}=20-4\\ \text{8X}=16\\ \text{X}=\frac{16}{8}\\ \text{X}=2\end{array}$

Thus, X is 2.

Substitute the value of X into the functional form of Y.

$\begin{array}{c}\text{Y}=\text{X}+1\\ \text{Y}=2+1\\ \text{Y}=3\end{array}$

Thus, Y is 3.

Therefore, the best new choice for X is 2 and Y is 3, when the price of Y rises to $4.

Income effect and substitution effect:

Substitute the respective values in to Equation 1 (utility function):

$\begin{array}{c}\text{U}\left(\text{X,Y}\right)=\left(\text{XY}\right)+\left(\text{Y}\right)\\ =\left(2\times 3\right)+3\\ =9\end{array}$

Therefore, in this case, the total utility is 9.

In this case, Person E consumes 12 units of Y when price is $1, and 3 units of Y at the price increases to $4. That means, Person E’s change in uncompensated compensation of Y when the price raise from $1 to $4 is $9\left(12-3\right)$.

Now, to determine the substitution effect and the income effect, it is important to find that how much Person E will consumed, if Person E was constrained to be on the original indifference curve ( when Utility is 36), at the new prices as follows:

Use the tangency condition (equalize the marginal utility with the price ratio) as follows:

$\begin{array}{c}\frac{\text{Y}}{\left(\text{X}+1\right)}=\frac{4}{4}\\ \frac{\text{Y}}{\left(\text{X}+1\right)}=1\\ \text{Y}=\text{X}+1\end{array}$

Substitute the respective value of Y into Equation 1, when the utility is 36:

$\begin{array}{c}\text{U}\left(\text{X,Y}\right)=\left(\text{XY}\right)+\left(\text{Y}\right)\\ 36=\left[\text{X}\left(\text{X}+1\right)\right]+\left(\text{X}+1\right)\\ 36={\text{X}}^2+2\text{X}+1\\ \text{X}=5\end{array}$

Thus, X is 5. Substitute the value of X into the functional form of Y.

$\begin{array}{c}\text{Y}=\text{X}+1\\ \text{Y}=5+1\\ \text{Y}=6\end{array}$

Thus, Y is 6.

Therefore, the best new choice for X is 2 and Y is 6, when the price of Y rises to $4 (at the original indifference curve)

Substitute the respective values in to Equation 1 (utility function):

$\begin{array}{c}\text{U}\left(\text{X,Y}\right)=\left(\text{XY}\right)+\left(\text{Y}\right)\\ =\left(5\times 6\right)+6\\ =36\end{array}$

Therefore, in this case, the total utility is 36.

Substitute the respective values into budget equation:

$\begin{array}{c}\text{4X}+4\text{Y}=\text{M}\\ \left(\text{4}\times \text{5}\right)+\left(4\times 6\right)=\$44\end{array}$

Thus, Person E needs $44 to purchase the 5 units of X and 6 units of Y at new prices.

Therefore, the substitution effect on Y is the change in consumption of Y in these two situations. That is, the substitution effect on Y is $6\left(12-6\right)$ .

Let subtract the substitution effect from the uncompensated total effect to get the income effect. Hence, the income effect is $3\left(9-6\right)$.

The compensating variation is the difference between the initial income and the additional income to take the old utility level at the new prices. Hence, the compensating variation is $\$24\left(\$44-\$20\right)$