Chapter 5, Problem 5.1P

a)

To determine

To find: The utility function in term of 0.75 liter containers and 2 liter containers.

Explanation of Solution

If both goods are substitute goods, the utility function will be in the following form:

$U=\alpha x+\beta y$

Therfore, from the question, the value of Alpha will be 0.75 and the value of Beta will be 2.

Thus, the utility function is:

U = 0.75x + 2y

b)

To determine

To find: Person E’s demand function.

Explanation of Solution

$\begin{array}{l}MUx=\frac{dU}{dx}\\ MUx=\frac{d\left(0.75x+2y\right)}{dx}=0.75\\ MUy=\frac{d\left(0.75x+2y\right)}{dy}=2\\ \end{array}$

In the case of substitute goods, consumer will spend all income on x if ratio of prices Px/Py is less than the marginal utility’s ratio. If it will not be less than the ratio of marginal utility, it will buy zero quantity.

$\begin{array}{l}Quantityofx=\frac{I}{Px}\\ Ratioofprices,\frac{Px}{Py}\le \frac{0.75}{2}\\ Px\le \frac{3}{8}Py\end{array}$

Thus, demand function is:

$x=\{\begin{array}{l}\frac{I}{Px}whenPx\le \frac{3}{8}Py\\ 0whenPx>\frac{3}{8}Py\end{array}$

c)

To determine

To find: Demand curve for good x.

Explanation of Solution

Demand curve is represented by red line.

d)

To determine

To find: Reason to the shift in demand curve for x.

Explanation of Solution

In the diagram, it can be seen that demand curve have two potion that is touched with x axis and other is overlapped on vertical axis. So, the portion that is touched with x axis will shift the curve rightwards.

e)

To determine

To find: The compensated demand curve.

Explanation of Solution

The compensated demand curve occurs in the case of substitution effect and not in the case of income effect. Moreover, in the following case, the substitution effect exists so the shape of demand curve will be same as the demand curve of part “d”.