Question

A consumer is in equilibrium at point $\mathbf{A}$in the accompanying figure. The price

of good$\mathbf{X}$is$\mathbf{\$}\mathbf{5}$.

a.What is the price of good$\mathbf{Y}$?

b.What is the consumer’s income?

c.At point$\mathbf{A}$, how many units of good$\mathbf{X}$does the consumer purchase?

d.Suppose the budget line changes so that the consumer achieves a new equilibrium at point$\mathbf{B}$. What change in the economic environment led to this new equilibrium? Is the consumer better off or worse off as a result of the price change?

Short answer

1. The price of Good $\mathrm{Y}$is${\mathrm{P}}_{\mathrm{y}}=\mathrm{\$}5$ .
2. The consumer’s income is$\mathrm{M}=\mathrm{\$}100$
3. At point $\mathrm{A}$, the consumer purchase $10$units of good$\mathrm{X}$
4. We see the reduction in the quantity of good $\mathrm{X}$is smaller than the increase in the quantity of good $\mathrm{Y}$. So, it can be concluded that the consumer is better off as more is always preferred to less.

Step-by-step solution

Find the price of Good Y

a.

The maximum quantity of a commodity or good $\mathrm{X}$that is affordable with the given income $\mathrm{M}$ is represented by following equation:

$\mathrm{X}=\frac{\mathrm{M}}{{\mathrm{P}}_{\mathrm{x}}}$

Here,

$X=20\text{units}$that is the maximum affordable quantity of good$X$

$M$is the consumer’s income

$P_x=\$5$that is the Price ofGood$X$

By substituting the following we will get,

$\begin{array}{l}\mathrm{X}=\frac{\mathrm{M}}{{\mathrm{P}}_{\mathrm{X}}}\\ 20=\frac{\mathrm{M}}{\mathrm{\$}5}\\ \mathrm{M}=\mathrm{\$}100\end{array}$

Also,the maximum affordable quantity of good Y is 20 units..

Hence, the price of good$Y$ is

$\begin{array}{c}Y=\frac{M}{P_Y}\\ 20=\frac{\$100}{P_Y}\\ P_Y=\$5\end{array}$

Find the consumer’s income

b.

The maximum quantity of a commodity or good $X$that is affordable with the given income$M$ is represented byfollowing equation:

$X=\frac{M}{P_x}$

Here,

$X=20\text{units}$that is the maximum affordable quantity of good$X$

$M$is the consumer’s income

$P_x=\$5$that is the Price ofGood$X$

By substituting the following we will get,

$\begin{array}{l}\mathrm{X}=\frac{\mathrm{M}}{{\mathrm{P}}_{\mathrm{X}}}\\ 20=\frac{\mathrm{M}}{\mathrm{\$}5}\\ \mathrm{M}=\mathrm{\$}100\end{array}$

Hence, the consumer’s income is $\mathrm{M}=\mathrm{\$}100$ .

Find the consumer’s income 

c.

At the initial equilibrium point $\mathrm{A}$,the consumer consumes$10$ units of good$Y$ at $5$each. Hence, the money income spent in consuming good $Y$is$\mathrm{\$}50$ .

The remaining$\mathrm{\$}50$are used in consuming good $X$which is priced at $\mathrm{\$}5$per unit.

Therefore, the consumer consumes $10$units of good $\mathrm{X}(\mathrm{\$}50/\mathrm{\$}5)$.

Find if it is the consumer better off or worse off as a result of the price change

d.

At point $B$, the consumer is able to consume a more than twice the units of good $Y$.

Here, the maximum affordable quantity of good $X$is still $20$units. and, the maximum affordable quantity of good$Y$ has increased to $40$units.

This implies that the price of good$Y$has fallen which has enabled the consumer to consume more units of relatively inexpensive good $Y$.

To analyze if the consumer is better off or worse off, first compute the reduction in the quantity consumed of good$X$ .

Since the consumer can consumes$40$ units of good $Y$as the maximum affordable quantity with unchanged income of$\mathrm{\$}100$ , the price of good$\mathrm{Y}$ has fallen to:

$\begin{array}{l}\mathrm{Y}=\frac{\mathrm{M}}{{\mathrm{P}}_{\mathrm{Y}}}\\ 40=\frac{\mathrm{\$}100}{{\mathrm{P}}_{\mathrm{Y}}}\\ {\mathrm{P}}_{\mathrm{Y}}=\mathrm{\$}2.5\end{array}$

This implies that consumer spent$\mathrm{\$}62.50$ on the purchase of good$Y$ ( $25$units at$\mathrm{\$}2.50$ per unit), remaining$\mathrm{\$}37.50$ are used in consuming good$X$ which is still priced at $\mathrm{\$}5$per unit.

Therefore, the consumer consumes$7.5$ units of good$\mathrm{X}(\mathrm{\$}37.50/\mathrm{\$}5)$

Since the reduction in the quantity of good $X$is smaller than the increase in the quantity of good$Y$, it can be concluded that the consumer is better off as more is always preferred to less.