Chapter 7, Problem 4P

Subpart (a):

To determine

Utility maximizing output.

Explanation of Solution

According to the maximization of the utility method, the marginal utility received from the last dollar spent on each commodity should be equal. Thus, it compares the utility of the dollar. The utility per dollar spent can be calculated by dividing the marginal utility of the unit by its price.

The utility maximizing combination of goods can be calculated at the point where the last dollar spent on each commodity yields the same level of utility. Thus, the marginal utility per dollar spent on each commodity can be calculated using the following formula as follows:

$\text{Marginal utility per dollar spent}=\frac{\text{Marginal utility from unit}}{\text{Price of unit}}$ (1)

To calculate the marginal utility (MU) per dollar for the first unit of good A, substitute the respective values in equation (1).

Table -1 shows the value of marginal utility per dollar for the different goods and their prices that are obtained by using equation (1).

Table -1

	MU per dollar unit 1	MU per dollar unit 2	MU per dollar unit 3	MU per dollar unit 4	MU per dollar unit 5	MU per dollar unit 6	MU per dollar unit 7	MU per dollar unit 8
Good A	4.00	3.00	2.50	2.00	1.50	1.00	0.83	0.67
Good B	4.00	2.50	2.00	1.50	1.17	0.83	0.33	0.17
Good C	3.75	3.00	2.00	1.75	1.25	1.00	0.88	0.75
Good D	1.5.	1.25	1.00	0.75	0.54	0.29	0.17	0.08
savings	5.00	4.00	3.00	2.00	1.00	0.50	0.25	0.13

Since, the marginal utility of a dollar is 5 for 1 unit of saving, he will go for it. Then, he will go for the consumption of Good A, B and another unit of savings which gives 4 utils of marginal utility per dollar. Then, the consumption of the first unit of good C will take place which gives 3.75 utils of marginal utility per dollar. Then, he will go for the 2^nd unit of Good A and C along with the third unit of saving which provides 3 utils of marginal utility per dollar. Then, he will consume the 3^rd unit of Good A and 2^nd unit of Good B for marginal utility per dollar worth 2.50 utils. Finally, the 4^th unit of Good A and savings along with 3^rd unit of Good B and C which gives 2 utils of marginal utility per dollar is consumed. As a result, his total budget would exhaust and cease his consumption, where his utility would be at the maximum.

The utility maximizing output combination is the combination where the marginal utility per dollar spent on each commodity is equal. Analyzing the table, we can see that 4 units of good A and D, 3 units of Good B and C and 4 units of savings will provide an individual with the marginal utility per dollar worth 2 utils. Since, no unit of good D provides marginal utility per dollar worth 2 utils, no unit of good D will be consumed.

Economics Concept Introduction

Concept introduction:

Utility: It can be explained as the benefit or the satisfaction which is derived from the consumption of a good or service by the consumer.

Marginal utility: It is the extra satisfaction that a consumer derives from the consumption of an additional unit of the specific good or service.

Utility maximizing combination: The utility maximizing combination is such that the last rupee spent on each commodity should yield the same level of an additional utility or the marginal utility spent on each commodity should be equal.

Utility per dollar spent: It is the utility of one dollar spent on the unit. It can be calculated by dividing the marginal utility with the initial price.

Subpart (b):

To determine

Utility maximizing output.

Explanation of Solution

According to the marginal utility per dollar, the consumer consumes the goods and services until when it gives the equal marginal utility per dollar which are worth 2 utils. It is at 4 units of savings, the marginal utility per dollar equals to 2 units and thus, the consumer will choose to save $4.

Economics Concept Introduction

Concept introduction:

Utility: It can be explained as the benefit or the satisfaction which is derived from the consumption of a good or service by the consumer.

Marginal utility: It is the extra satisfaction that a consumer derives from the consumption of an additional unit of the specific good or service.

Utility maximizing combination: The utility maximizing combination is such that the last rupee spent on each commodity should yield the same level of an additional utility or the marginal utility spent on each commodity should be equal.

Utility per dollar spent: It is the utility of one dollar spent on the unit. It can be calculated by dividing the marginal utility with the initial price.

Subpart (c):

To determine

Utility maximizing output.

Explanation of Solution

The utility maximizing combination is 4Units of Good A, 3 units of Good B, 3 units of Good C, 0 unit of Good D and 4 Unit of savings. With this combination, the total budget of the consumer gets exhausted. This can be verified by substituting the values as follows:

$\begin{array}{c}\text{Total budget}=\left(\begin{array}{l}{\text{Units}}_{\text{Good A}}+{\text{Units}}_{\text{Good B}}+{\text{Units}}_{\text{Good C}}\\ +{\text{Units}}_{\text{Good D}}+{\text{Units}}_{\text{Savings}}\end{array}\right)\\ =\left(4\times 18\right)+\left(3\times 6\right)+\left(3\times 4\right)+\left(0\times 24\right)+\left(4\times 1\right)\\ =72+18+12+0+4\\ =106\end{array}$

Since the total budget is $106 which is equal to the cost of the utility maximizing combination, the total budget gets exhausted with this combination of goods and savings.

Economics Concept Introduction

Concept introduction:

Utility: It can be explained as the benefit or the satisfaction which is derived from the consumption of a good or service by the consumer.

Marginal utility: It is the extra satisfaction that a consumer derives from the consumption of an additional unit of the specific good or service.

Utility maximizing combination: The utility maximizing combination is such that the last rupee spent on each commodity should yield the same level of an additional utility or the marginal utility spent on each commodity should be equal.

Utility per dollar spent: It is the utility of one dollar spent on the unit. It can be calculated by dividing the marginal utility with the initial price.