Chapter 6, Problem 16QP

(a)

To determine

Complete the table.

Explanation of Solution

Marginal utility is the additional unit of satisfaction derived from the consumption of one more additional unit of goods and services. It can be calculated by substituting each value in the following equation:

$\text{Marginal Utility}={\text{Total Utility}}_{\text{Present}}-{\text{Total Utility}}_{\text{Previous}}$ (1)

For calculating marginal utility from total utility, substitute the values in Equation (1) with the help of Table 1.

$\begin{array}{c}\text{Marginal Utility}=\text{140}-\text{0}\\ =140\end{array}$

Thus, the marginal utility for the first additional unit of the first-run movie is 140. Likewise, each marginal utility can be calculated by substituting each value in Equation (1). It is shown in Table 1.

Table 1

First-Run Movies			Bottles of Wine
Quantity	Total Utility	Marginal Utility	Quantity	Total Utility	Marginal Utility
0	0	0	0	0	0
1	140	140	1	180	180
2	260	120	2	340	160
3	360	100	3	460	120
4	440	80	4	510	50
5	500	60	5	540	30

Economics Concept Introduction

Marginal utility: Marginal utility refers to the additional unit of satisfaction derived from the consumption of one more additional unit of goods and services.

(b)

To determine

Utility maximizing combination of the first-run movies and wine.

Explanation of Solution

The consumer spends $50 for entertainment and wine per month, where the price for both movie and a bottle of wine is $10. The utility is the maximization of a consumer, which is at the point where the last dollar spent on both the commodities provides the same level of marginal utility of dollar. This can be calculated by dividing each marginal utility with price of the commodity. Marginal utility per dollar is shown in Table 2.

Table 2

First-Run Movies			Bottles of Wine
Quantity	Marginal Utility	Marginal Utility per Dollar (P=$10)	Quantity	Marginal Utility	Marginal Utility per Dollar (P=$10)
0	0	0	0	0	0
1	140	14	1	180	18
2	120	12	2	160	16
3	100	10	3	120	12
4	80	8	4	50	5
5	60	6	5	30	3

Thus, the utility maximizing combination of the consumer is 2 movies and 3 bottles of wine, where he gets the same level of marginal utility per dollar.

Economics Concept Introduction

Utility maximization: The utility is maximized at the point where the last dollar spent on both the commodities provides the same level of marginal utility of the dollar.

Marginal utility: Marginal utility refers to the additional unit of satisfaction derived from the consumption of one more additional unit of goods and services.

(c)

To determine

Equilibrium allocation between movie and wine when the price of wine decreases.

Explanation of Solution

When the price of wine decreases to $5 per bottle, the consumer will watch 3 movies and consume 4 bottles of wine at $50. This is because the consumer will not purchase goods when marginal utility per dollar is lower than its price.

Thus, consumer’s equilibrium allocation between movie and wine is 3 and 4, respectively.

Economics Concept Introduction

Marginal utility: Marginal utility refers to the additional unit of satisfaction derived from the consumption of one more additional unit of goods and services.

(d)

To determine

Calculate the elasticity of demand for wine.

Explanation of Solution

When the price of wine decreases as $5, the consumer will buy 4 bottles of wine instead of 3. The elasticity of demand for wine using the midpoint method can be calculated using the following formula:

${\text{Elasticity}}_{\text{Midpoint formula}}=\frac{\frac{\left({\text{Quantity}}_{\text{new}}-{\text{Quantity}}_{\text{old}}\right)}{\left(\frac{{\text{Quantity}}_{\text{new}}+{\text{Quantity}}_{\text{old}}}{2}\right)}}{\frac{\left({\text{Price}}_{\text{new}}-{\text{Price}}_{\text{old}}\right)}{\left(\frac{{\text{Price}}_{\text{new}}+{\text{Price}}_{\text{old}}}{2}\right)}}$ (1)

For calculating the elasticity of demand for wine, substitute the respective values in Equation (1).

$\begin{array}{c}\text{Elasticity of wine}=\frac{\frac{\left(4-\text{3}\right)}{\left(\frac{\text{4}+\text{3}}{2}\right)}}{\frac{\left(\text{10}-\text{5}\right)}{\left(\frac{\text{10}+\text{5}}{2}\right)}}\\ =\frac{\frac{1}{\frac{7}{2}}}{\frac{5}{\frac{15}{2}}}\\ =\frac{\frac{1}{3.5}}{\frac{5}{7.5}}\\ =\frac{0.285}{0.667}\\ =0.427\end{array}$

Thus, the elasticity of demand for wine is 0.427.

Economics Concept Introduction

Elasticity of demand: Elasticity of demand refers to the responsiveness or the change in quantity demanded due to the change in price.