**Unit 3** Doing the best you can: Scarcity, wellbeing, and working hours

## 3.7 Income and substitution effects on hours of work and free time

When we use the model of constrained choice to analyse the effects of a wage rise (as for Karim in Section 3.6) we find that living standards (represented by utility in the model) always rise. Hours of work may either rise or fall, depending on which of two opposing effects dominates. In this section, we analyse these two effects in more detail, using another example to disentangle them.

Imagine that you are planning how to spend the ten-week summer break before your next year at college. You have the opportunity to work in a local shop, where you would be paid $90 per day. But you also want to have time to meet friends, take a holiday, and study for next year’s courses. How many days should you work during the break?

We will suppose that, like Karim, you care about two goods: the consumption that you can enjoy from your earnings, and the number of days of free time you will have. You are making a plan for the whole summer break, so we will focus on the total amounts of consumption and free time, rather than daily and weekly averages.

You have a total of 70 days in which you can either work or take free time. If \(w\) is the *daily* wage, and you take \(d\) days of free time during the break, then you work for \((70 − d)\) days, and your maximum level of consumption, \(c\), is given by the budget constraint:

Figure 3.10 shows your budget constraint when the daily wage is $90, and your feasible set.

The slope of the budget constraint corresponds to the wage: for each additional free day, total consumption must decrease by $90. The area under the budget constraint is your feasible set. Your problem is very similar to Karim’s problem: the marginal rate at which you can transform days of free time into consumption, which is also the opportunity cost of a free day, is constant and is equal to your wage: it is $90 for every day you work.

What will you choose? That depends on your preferences, and they will depend, in turn, on your situation. For example, if you are able to live rent-free with your family, income will be less important to you than to a student who has to pay for accommodation, and you will place a relatively high value on free time. If part of your earnings will be needed to pay for consumption during the next semester at college, you may be less willing to substitute free time for consumption. Your preferred choice of free time and consumption will be the combination on the feasible frontier that is on the highest possible indifference curve. In Figure 3.10, we have drawn indifference curves of the typical shape. Work through the steps to find your most-preferred choice.

If your indifference curves have the same shape as the ones in Figure 3.10, then you would choose point A, with 34 free days during the break. You plan to spend 36 days at work in the shop, with total earnings of $3,240.

Like Karim, you are balancing two trade-offs: your MRS, the rate at which you are willing to swap days of free time for additional consumption; and your MRT, the rate at which you are able to transform free days into consumption—which is equal to the daily wage. Your utility-maximizing combination of consumption and free time is the point on the budget constraint where:

\[\text{MRS} = \text{MRT} = w\]While considering this decision, you receive an email. A mysterious benefactor would like to give you $1,000 to spend as you like (all you have to do is provide your banking details). You realize at once that this will affect your plan. The new situation is shown in Figure 3.11: for each level of free time, your total income (earnings plus the mystery gift) is $1,000 higher than before. So the budget constraint is shifted upwards by $1,000—the feasible set has expanded. Your budget constraint is now:

\[c = 90(70 - d) + \text{1,000}\]The extra income of $1,000 does not change your opportunity cost of time: each hour of free time still reduces your consumption by $90 (the wage). Your new ideal choice is at B, with 39 days of free time. B is the point on IC_{3} where the MRS is equal to $90. With the indifference curves shown in this diagram, your response to the extra income is not simply to spend $1,000 more; you increase consumption by less than $1,000, and you take some extra free time. A student with different preferences might not choose to increase their free time: Figure 3.12 shows a case in which the MRS at each value of free time is the same on both indifference curves. This student chooses to keep their free time the same, and consume $1,000 more.

- income effect
- The effect that an increase in income has on an individual’s demand for a good (the amount that the person chooses to buy) because it expands the feasible set of purchases. When the price of a good changes, this has an income effect because it expands or shrinks the feasible set, and it also has a substitution effect.
*See also: substitution effect.*

Figures 3.11 and 3.12 show examples of the **income effect**: the effect of additional income on the choice of free time. Your income effect, shown in Figure 3.11, is positive—extra income raises the level of free time you will choose. For the student in Figure 3.12, the income effect is zero. We assume that for most goods the income effect will be either positive or zero, but not negative: if your income increased, you would not choose to have less of something that you valued.

- substitution effect
- When the price of a good changes, the substitution effect is the change in the consumption of the good that occurs because of the change in the good’s relative price. The price change also has an income effect, because it expands or shrinks the feasible set.
*See also: income effect.*

The $1,000 gift is unearned income, so it *only* has an income effect. Since the MRT has not changed, there is no increased incentive to work. So there is no **substitution effect**: the opportunity cost of a free day is still $90, and you have no reason to substitute consumption for free time.

You suddenly realize that it might not be wise to give the mysterious stranger your bank account details (perhaps it is a hoax). With regret you return to the original plan, and decide to work for 34 days during the break. But suddenly your fortunes improve. You hear of a vacancy at the superstore, where you would be paid $130 per day. Now your budget constraint is:

\[c = 130(70 - d)\]Figure 3.13a shows how the budget constraint changes when the wage rises from $90 to $130 per day. For each day of free time you give up, your consumption can now rise by $130 rather than $90, so the budget constraint becomes steeper. It pivots around the point (70, 0)—whatever the wage, your consumption will be zero if you don’t work. Your feasible set has expanded. And now you achieve the highest possible utility at point D, with only 30 free days, but consumption of $5,200. You send off your application to the superstore.

Compare the outcomes in Figure 3.11 and 3.13a. With an increase in unearned income, you want to work fewer days, while the wage increase in Figure 3.13a makes you decide to increase your days of work. Why does this happen? Because the substitution effect of the wage increase is bigger than the income effect:

*The feasible set expands, raising your potential utility*: For each level of free time, you can have more consumption. For a given level of free time, your MRS is higher at a point with higher income: you are now more willing to sacrifice consumption for extra free time. This is the income effect shown in Figure 3.11—you respond to additional income by taking more free time as well as increasing consumption.*The budget constraint is steeper*: The opportunity cost of free time is now higher. In other words, the marginal rate at which you can transform time into income (the MRT) has increased. And that means you have an incentive to work more—to decrease your free time. This is the substitution effect, and in Figure 3.13a it outweighs the income effect.

We can measure the income and substitution effects more precisely, using the diagram. Before the wage rise, you are at A on IC_{2}. The higher wage enables you to reach point D on IC_{4}. Figure 3.13b shows how we can decompose the change from A to D into two parts that correspond to these two effects.

### Income and substitution effects

We can now describe the income and substitution effects more precisely. A wage rise:

- raises your income for each level of free time, increasing the level of utility you can achieve
- increases the opportunity cost of free time.

So it has two effects on your choice of free time:

*The income effect*(because the budget constraint shifts outwards): the effect that the additional income would have if there were no change in the opportunity cost*The substitution effect*(because the slope of the budget constraint, the MRT, rises): the effect of the change in the opportunity cost, given the new level of utility.

Figure 3.13b shows that with indifference curves of this typical shape, a substitution effect will always be negative: with a higher opportunity cost of free time, you choose a point on the indifference curve with a higher MRS, which is a point with less free time (and more consumption). The overall effect of a wage rise depends on the sum of the income and substitution effects. In Figure 3.13b, the negative substitution effect is bigger than the positive income effect, so free time falls.

Whether the substitution effect is big enough to outweigh the income effect depends on how easy it is to substitute between working time and consumption. A student deciding how to spend the summer break may have quite a lot of flexibility in how they use their time and hence their willingness to substitute between consumption and free time. This will be reflected in the shape of their indifference curves. Even a small change in the wage available might then have a substantial effect on their choice of free time.

But for a person with many domestic responsibilities, giving up free time may be more difficult. The additional incentive to work from a small change in wages might then have little effect on their decision: the substitution effect will be smaller, and the income effect is more likely to dominate.

**Question 3.9** Choose the correct answer(s)

Figure 3.14 depicts the feasible daily consumption and free time for a worker whose hourly wage is $15.

Read the following statements and choose the correct option(s).

- For every additional hour of free time, the worker has $15 less to spend on consumption, so the slope of the budget constraint is −15.
- A gift would shift the budget constraint outward in a parallel manner, as the consumer could consume $60 more at any given level of free time.
- A decrease in the wage would pivot the budget constraint inwards around the point (24,0). The feasible set would be smaller and the worker would not be able to reach the levels of utility that were possible when the wage was $15.
- With a decrease in the wage, the opportunity cost of free time would be lower, but without seeing the indifference curves we cannot tell how hours of work would change. They could rise or fall, depending on the relative size of the income and substitution effects.

**Exercise 3.7** Zoë’s problem: The price of a cinema ticket increases

In Exercise 3.4, you determined the best choice of social activities and entertainment for Zoë, given her budget of £240 and prices £10 and £16 for cinema tickets and nights out, respectively. The figure below shows her budget constraint and a possible set of indifference curves; with these preferences her best choice is at point A, with 13 cinema tickets and (approximately) seven nights out.

Suppose she discovers that cinema tickets in London are likely to be more expensive than elsewhere in the UK: the price is £15, rather than £10. The price increase pivots her budget constraint around the point (0, 15): she can still afford the point (0, 15), but now, if she spends all of her budget on cinema tickets, she will only be able to buy 16 tickets.

- How has Zoë’s marginal rate of transformation changed? Has the opportunity cost of a cinema ticket increased or decreased?
- How has Zoë’s spending power changed: can she afford more, or less, socializing and entertainment?
- Point C is now her utility-maximizing choice. What has happened to her utility, and to the number of cinema tickets she chooses to buy?
- To decompose the effect of the price change into an income and substitution effect, copy the diagram and draw a straight line tangent to IC
_{2}, with the flatter slope of the original budget constraint. Mark the tangency point B. - The shift from A to B is the income effect: the price increase lowers Zoë’s spending power, almost as if she had less income to spend. How does it affect the number of tickets Zoë buys?
- The shift from B to C is the substitution effect: the effect of the change in the opportunity cost of cinema tickets, given the new level of utility. How does it affect the number of tickets she buys?
- Can you explain why the substitution effect works in the same way for Zoë and Karim, but the income effect works in the opposite way?