Question

A consumer must divide $\mathbf{\backslash (}\mathbf{600}$between the consumption of product$\mathbf{X}$and product$\mathbf{Y}$. The relevant market prices are${\mathbf{P}}_{\mathbf{x}}\mathbf{=}\mathbf{\backslash )}\mathbf{10}\mathbf{\text{\&}}{\mathbf{P}}_{\mathbf{y}}\mathbf{=}\mathbf{\backslash (}\mathbf{40}$

a.Write the equation for the consumer’s budget line.

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

c.Show how the consumer’s opportunity set changes when the price of good$\mathbf{X}$increases to$\mathbf{\backslash )}\mathbf{20}$. How does this change alter the market rate of substitution between goods$\mathbf{X}$and$\mathbf{Y}$?

Short answer

1. The equation for the consumer's budget line is$10\mathrm{X}+40\mathrm{Y}=600$
2. Graphed the consumer’s opportunity set.
3. The consumers will be more interested in purchasing Good$\mathrm{Y}$ than Good $X$.

Step-by-step solution

Find the equation for the consumer’s budget line:

a.

We use the following budget equation:

${\mathrm{P}}_{\mathrm{x}}\mathrm{X}+{\mathrm{P}}_{\mathrm{y}}\mathrm{Y}=\mathrm{M}$

Here,

$P_x$is the price ofGood$X$

$P_y$is the price ofGood$Y$

$M$is the Consumer’s Income.

We have,

$\begin{array}{l}{\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}10\\ {\mathrm{P}}_{\mathrm{y}}=\mathrm{\$}40\\ \mathrm{M}=\mathrm{\$}600\end{array}$

By substituting the following we will get,

$\begin{array}{l}{P}_{x}X+P_{y}Y=M\\ 10X+40Y=600\end{array}$

Therefore, the equation for the consumer's budget line is $\mathbf{10}\mathit{X}\mathbf{+}\mathbf{40}\mathit{Y}\mathbf{=}\mathbf{600}$

Draw the illustration of consumer’s opportunity set:

b.

The upper boundary line that is shown in the diagram is the budget line, it represents the combination of Good $X$and Good$Y$ .

While the slope of the curve in the diagram, is the Marginal Rate of Substitution. The readiness of a consumer to substitute one good from another with the same purpose of consumption and with equal satisfaction is defined as Marginal Rate of Substitution.

Using the following equation, we can determine the Marginal Rate of Substitution:

$\mathrm{MRS}=\frac{-{\mathrm{P}}_{\mathrm{x}}}{{\mathrm{P}}_{\mathrm{y}}}$

The relevant market prices are given as,

$\begin{array}{l}{\mathrm{P}}_{\mathrm{x}}=\mathrm{\$}10\\ {\mathrm{P}}_{\mathrm{y}}=\mathrm{\$}40\end{array}$

By substituting the following we will get,

$\begin{array}{l}\mathrm{MRS}=\frac{-{\mathrm{P}}_{\mathrm{x}}}{{\mathrm{P}}_{\mathrm{y}}}\\ \mathrm{MRS}=\frac{-{\mathrm{P}}_{\mathrm{x}}}{{\mathrm{P}}_{\mathrm{y}}}=\frac{10}{40}\\ \mathrm{MRS}=-0.25\end{array}$

Therefore, the marginal rate of substitution is $(-0.25)$.

Find the Marginal rate of substitution with the increased price:

c.

Since, the price of Good $X$increased to $\mathrm{\$}20$, the new equation for the consumer's budget line will be:

$\text{20}\mathrm{X}+40\mathrm{Y}=600$

Therefore, we can calculate as:

$\begin{array}{l}\mathrm{MRS}=\frac{-{\mathrm{P}}_{\mathrm{x}}}{{\mathrm{P}}_{\mathrm{y}}}\\ \mathrm{MRS}=\frac{-{\mathrm{P}}_{\mathrm{x}}}{{\mathrm{P}}_{\mathrm{y}}}=\frac{20}{40}\\ \mathrm{MRS}=-0.5\end{array}$

Hence, we can conclude that the consumers will be more willing to purchase Good $\mathbf{Y}$ than Good $\mathrm{X}$.