Question

A consumer has $\mathbf{\backslash (}\mathbf{300}$to spend on goods$\mathbf{X}$and$\mathbf{Y}$.The market prices of these two goods are${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash )}\mathbf{15}\mathbf{\text{\&}}{\mathbf{P}}_{\mathbf{y}}\mathbf{=}\mathbf{\backslash (}\mathbf{5}$.

a.What is the market rate of substitution between goods$\mathbf{X}$and$\mathbf{Y}$?

b.Illustrate the consumer’s opportunity set in a carefully labelled diagram.

c.Show how the consumer’s opportunity set changes if income increases

by$\mathbf{\backslash )}\mathbf{300}$. How does the$\mathbf{\$}\mathbf{300}$increase in income alter the market rate of

substitution between goods$\mathbf{X}$and$\mathbf{Y}$?

Short answer

1. The Market Rate of Substitution between Good$\mathrm{X}$ and$\mathrm{Y}$ Good is $3$units.
2. Mapped the graph illustrating the consumer’s opportunity set.
3. The market rate substitution will not change even if the consumer's income will change.

Step-by-step solution

Find the market rate of substitution

a.

To determine the market rate of substitution between Good$\mathrm{X}$ and Good $\mathrm{Y}$, use the following equation:

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

Here,

$P_x$is the price ofGood$X$

$P_y$is the price ofGood$Y$

$M$is the Market rate of Substitution.

We have,

$\begin{array}{l}{\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}15\\ {\mathrm{P}}_{\mathrm{y}}=\mathrm{\$}5\end{array}$

By substituting the following we will get,

$\begin{array}{c}\mathrm{M}=\frac{{\mathrm{P}}_{\mathrm{x}}}{{\mathrm{P}}_{\mathrm{y}}}\\ \Rightarrow \frac{\mathrm{\$}15}{\mathrm{\$}5}\\ \Rightarrow 3\end{array}$

Therefore, the Market Rate of Substitution betweenGood$\mathbf{X}$ andGood$\mathbf{Y}$is$\mathbf{3}$ units.

Draw the illustration of consumer’s opportunity set:

b.

The given is the illustration of the consumer's opportunity set. Here, the shaded area is the one that the consumer will afford to purchase from the combination of Good $X$ andGood $Y$in the diagram, it is the budget set.

The budget line is the upper boundary of the budget set.

The budget line separates the affordable goods that the consumer can purchase from the unaffordable bundle of goods.

The unaffordable goods are, mainly any point outside the shaded area.

Find the Vertical and horizontal incept of opportunity set:

c.

To find: Vertical incept of opportunity set

Given: Income increases to $\mathrm{\$}300$and the price ofGood$\mathrm{Y}$is$\mathrm{\$}5$

Use the following equation:

$Y=\frac{M}{P_y}$

Here,

$Y$is the vertical incept of opportunity set.

$M$is the consumer’s income

$P_y$is the Price ofGood$Y$

By substituting the following we will get,

$\begin{array}{l}\mathrm{Y}=\frac{\mathrm{M}}{{\mathrm{P}}_{\mathrm{y}}}\\ \mathrm{Y}=\frac{8300}{5}\\ \mathrm{Y}=60\end{array}$

Therefore, the vertical incept of opportunity set is 60.

To find: The Horizontal incept of Opportunity Set.

Given that the income increases to $\mathrm{\$}300$ and the price ofGood$X$ is$\mathrm{\$}15$

Use the following equation:

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

Here,

$X$is the vertical incept of opportunity set.

$M$is the consumer’s income

$P_x$is the Price ofGood$X$

By substituting the following we will get,

$\begin{array}{c}\mathrm{X}=\frac{\mathrm{M}}{{\mathrm{P}}_{\mathrm{x}}}\\ =\frac{300}{15}\\ =20\end{array}$

Therefore, the horizontal incept of opportunity set is 60.

Hence, we can say that, the market rate substitution will not change even if the consumer's income will change because it is independent to whatever happens to the income and will only change if the price of commodity changes.