# Weighted Marginal Utility

Fiona now faces an income constraint.

### According to the following information on the marginal utility and price of ice cream and chocolate, which one do you think she should choose?

The marginal utility of a chocolate is 72 and the price is R3.

The marginal utility of ice cream is 100 and the price of an ice cream is R5.

Before a decision can be made, we need know what the marginal utility per rand is that she obtains from the chocolate and ice cream. In other words, what is the "bang for her buck" (value for the money she spends).

The marginal utility per rand, also called the weighted marginal utility, is calculated as follows:

$$\text{Weighted Marginal Utility} = {\text{marginal utility} \over \text{price}}$$

In symbols it can be written as:

$$MU_p = {MU \over P}$$

In the case of Fiona, the weighted marginal utility of the chocolate and ice cream are as follows:

chocolate:

$$\text{} = {72 \over 3}$$

$$\text{} = {\text{24 marginal utilities per rand}}$$

ice cream:

$$\text{} = {100 \over 5}$$

$$\text{} = {\text{20 marginal utilities per rand}}$$

### Based on the above information about the weighted marginal utilities help Fiona decide.

If she can only buy one of them, which one should she buy?

Since her marginal utility per rand is higher for chocolate (24) than for ice cream (20), she should choose chocolate as it gives her more utilities per rand spend.

Fiona's weighted marginal utility for units of chocolate and ice cream is indicated in the following table, given a price of R3 for a chocolate and R5 for an ice cream:

### Weighted marginal utility

 Chocolate (P = 3) Ice cream (P = 5) Units MU MU/P MU MU/P 1 72 24 120 24 2 66 22 115 23 3 54 18 100 20 4 42 14 85 17 5 36 12 70 14 6 27 9 65 13 7 24 8 60 12 8 18 6 45 9 9 9 3 35 7

The first column shows the number of units. The second column indicates the marginal utility she derives from eating units of chocolate. The third column provides her weighted marginal utility, given a price of R3 for a chocolate. The fourth column indicates her marginal utility for units of ice cream and in the fifth column her weighted marginal utility for an ice cream, given a price of R5 for an ice cream.

Note how her marginal utility for chocolate and ice cream decreases as she consumes more units. This is because of the law of diminishing marginal utility. Also note how her marginal utility per rand (weighted marginal utility) for ice cream and chocolate decreases as she consumes more.

### Help Fiona decide how to spend her income to reach maximum satisfaction.

Assuming that Fiona has R50, how should she spend her income in order to maximise her satisfaction?

Let's see how Fiona should this decision:

• Her first purchase can be either a chocolate or an ice cream. Why? Because both chocolate and ice cream have a weighted marginal utility of 24.
• Let's assume she buys the chocolate first. Her second purchase should then be an ice cream since it has a higher weighted marginal utility (24) than a second chocolate (22).
• Her third purchase should be a second ice cream since it has a higher weighted marginal (23) utility than a second chocolate (22).
• Her fourth purchase should be a second chocolate (22) and a third ice cream (20).

She should then continue to purchase chocolate and ice cream based on the marginal utilities per rand until her income is spend.

The process that Fiona follows can be written as the following rule:

The utility-maximising choice between goods occurs where the weighted marginal utility (marginal utility per rand) is the same for both goods.

In symbols it is written as

$${MU_a \over P_a} = {MU_b \over P_b}$$

In the case of Fiona who has R50 to spend, this occurs where the weighted marginal utility for ice cream and chocolate is 12. She therefore maximises her utility by buying five units of chocolate and seven units of ice cream as indicated in the following table.

### Weighted marginal utility

Chocolate (P = 3) Ice cream (P = 5)
Units MU MU/P MU MU/P
1 72 24 120 24
2 66 22 115 23
3 54 18 100 20
4 42 14 85 17
5 36 12 70 14
6 27 9 65 13
7 24 8 60 12
8 18 6 45 9
9 9 3 35 7

### A consumer has R30 available and wants to purchase cups of tea and chocolates. The price of tea is R5 and the price chocolate is R10. How many quantities can the consumer buy if she spends her entire income on chocolates?

Correct.

The price of chocolate is R10 per unit, the quantity of chocolate that can be bought is three units (R30 ÷ R10).

Think again.

The price of chocolate is R10 per unit, the quantity of chocolate that can be bought is three units (R30 ÷ R10).

Think again.

The price of chocolate is R10 per unit, the quantity of chocolate that can be bought is three units (R30 ÷ R10).

### The table below is based on Glenda's consumption of coffee and tea. Complete the missing values of weighted marginal utilities.

Utils Coffee (P = 3) Tea (P = 4)
MU MU÷P MU MU÷P
1 30 10 48
2 24 40 10
3 18 6 32 8
4 12 4 16
5 6 12 3
Utils Coffee (P = 3) Tea (P = 4)
MU MU÷P MU MU÷P
1 30 10 48 12
2 24 8 40 10
3 18 6 32 8
4 12 4 16 4
5 6 2 12 3

### Given the price of goods and services, which of the following indicates consumer equilibrium?

1. The consumer spends his or her income in such a way that he or she attains the highest possible total utility.
2. The consumer spends his or her income in such a way that the weighted marginal utility (marginal utility per rand) is the same for the goods.
3. The consumer spends his or her income in such a way that the marginal utilities are the same for the goods.

a and b

It is indeed the case that if the consumer has spent his or her income in such a way that the highest possible total utility is reached, consumer equilibrium exists.
Consumer equilibrium is also the point where the weighted marginal product is the same for the different goods.

### The table below indicates Glenda's consumption of coffee and tea. Use the table to answer the following questions:

 Utils Coffee (P = 3) Tea (P = 4) MU MU÷P MU MU÷P 1 30 10 48 2 24 40 10 3 18 6 32 8 4 12 4 16 5 6 12 3

If Glenda has already consumed two cups of coffee and three cups of tea and she must now decide whether to buy another cup of tea or another cup of coffee, what advice would you give her to ensure that she maximises her satisfaction?

Think again.

By purchasing the third cup of coffee, she gains six utils per rand (weighted marginal utility), whereas if she purchases a fourth cup of tea, she gains only four utils per rand.

Correct.

By purchasing the third cup of coffee, she gains six utils per rand (weighted marginal utility), whereas if she purchases a fourth cup of tea, she gains only four utils per rand.

### If Glenda has R28 to spend on tea and coffee, at which combination of coffee and tea would she be in equilibrium if he spends her total income of R28?

Correct.

The condition for consumer equilibrium is that weighted marginal utilities must be equal. Glenda will be at equilibrium at four units of coffee and four units of tea. This is because the weighted marginal utilities are the same for coffee and tea, and then she has spent her total available income of R28.

Although the weighted marginal utilities are the same at two units of coffee and three units of tea Glenda has not spent her total available income of R28.

Think again.

The condition for consumer equilibrium is that weighted marginal utilities must be equal. Glenda will be at equilibrium at four units of coffee and four units of tea. This is because the weighted marginal utilities are the same for coffee and tea, and then she has spent her total available income of R28.

Although the weighted marginal utilities are the same at two units of coffee and three units of tea Glenda has not spent her total available income of R28.

Think again.

The condition for consumer equilibrium is that weighted marginal utilities must be equal. Glenda will be at equilibrium at four units of coffee and four units of tea. This is because the weighted marginal utilities are the same for coffee and tea, and then she has spent her total available income of R28.

Although the weighted marginal utilities are the same at two units of coffee and three units of tea Glenda has not spent her total available income of R28.