Chapter 5, Problem 5.8P

To determine

(a)

To derive: the demand curve for x and to verify that demand for x is independent of the level of income at an interior optimum.

Explanation of Solution

The quasi-linear utility function is of form:

  $\begin{array}{l}U(x,y)=\sqrt{x}+y\\ Also,M{U}_x=\frac{1}{2\sqrt{x}}\\ M{U}_y=1\end{array}$

Prices are given to be P_x and P_y.

Now,

  $\begin{array}{l}\frac{{\text{MU}}_{\text{x}}}{{\text{MU}}_{\text{y}}}\text{=}\frac{{\text{P}}_{\text{x}}}{{\text{P}}_{\text{y}}}\\ \frac{\raisebox{1ex}{$\text{1}$}\!\left/ \!\raisebox{-1ex}{$\text{2}\sqrt{\text{x}}$}\right.}{\text{1}}\text{=}\frac{{\text{P}}_{\text{x}}}{{\text{P}}_{\text{y}}}\\ {\text{P}}_{\text{y}}\text{=}\text{2}{\text{P}}_{\text{x}}\sqrt{\text{x}}\\ \sqrt{\text{x}}\text{=}\frac{{\text{P}}_{\text{y}}}{\text{2}{\text{P}}_{\text{x}}}\\ \\ \text{Squaring}\text{both}\text{sides,}\\ x=\frac{1}{4}{\left(\frac{P_y}{P_x}\right)}^2\end{array}$

It can be seen from the demand curve of x, that the demand for x is independent of level of income of the consumer.

To determine

(b)

To derive: the demand curve for y and analyze whether y is a normal good.

Explanation of Solution

The equation of the budget line can be written as:

  $\begin{array}{l}{\text{P}}_{\text{x}}\text{.x}\text{+}{\text{P}}_{\text{y}}\text{.y}\text{=}\text{I}\\ \text{Putting}\text{the}\text{value}\text{of}\text{x,}\\ {\text{P}}_{\text{x}}\text{.}\frac{\text{1}}{\text{4}}{\left(\frac{{\text{P}}_{\text{y}}}{{\text{P}}_{\text{x}}}\right)}^{\text{2}}\text{+}{\text{P}}_{\text{y}}\text{.y}\text{=}\text{I}\\ \text{y}\text{=}\frac{\text{1}}{{\text{P}}_{\text{y}}}\left[\text{I}\text{-}\frac{{\left({\text{P}}_{\text{y}}\right)}^{\text{2}}}{{\text{4P}}_{\text{x}}}\right]\\ \text{y}\text{=}\frac{\text{I}}{{\text{P}}_{\text{y}}}\text{-}\frac{{\text{P}}_{\text{y}}}{{\text{4P}}_{\text{x}}}\end{array}$

For analyzing whether y is a normal good, differentiating y with respect to income,

  $\begin{array}{l}\frac{\text{dy}}{\text{dI}}\text{=}\frac{\text{1}}{{\text{P}}_{\text{y}}}\\ \frac{\text{dy}}{\text{dI}}\text{>}\text{0}\text{.}\text{It}\text{shows}\text{that}\text{y}\text{is}\text{a}\text{normal}\text{good}\text{.}\end{array}$

Now, differentiating y with respect to P_x,

  $\begin{array}{l}\frac{\text{dy}}{{\text{dP}}_{\text{x}}}\text{=}\frac{{\text{P}}_{\text{y}}}{\text{4}{\text{P}}_{\text{x}}{}^{\text{2}}}\\ \frac{\text{dy}}{{\text{dP}}_{\text{x}}}\text{>}\text{0}\text{.}\text{Hence,}\text{when}{\text{P}}_{\text{x}}\text{rises,}\text{the}\text{demand}\text{for}\text{y}\text{also}\text{rises}\text{.}\end{array}$