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:

\[c = w(70 - d)\]

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.

In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 8,000. Coordinates are (days of free time, consumption). A straight, downward sloping line connects points (0, 6,300) and (70, 0). There are two parallel, downward-sloping, convex curves. One is tangent to the straight line at point A (34, 3,240). The other lies below this one at all points.
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Days of work 0 10 20 30 40 50 60 70
Free days 70 60 50 40 30 20 10 0
Consumption ($) 0 900 1,800 2,700 3,600 4,500 5,400 6,300

Figure 3.10 Your preferred choice of free time and consumption.

The budget constraint and the feasible set: In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 8,000. Coordinates are (days of free time, consumption). A straight, downward sloping line connects points (0, 6,300) and (70, 0).
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Days of work 0 10 20 30 40 50 60 70
Free days 70 60 50 40 30 20 10 0
Consumption ($) 0 900 1,800 2,700 3,600 4,500 5,400 6,300

The budget constraint and the feasible set

The budget constraint shows the maximum amount of consumption you can have for each number of free days. It is your feasible frontier, and the area below it is the feasible set. The equation of the budget constraint is c = w(70 − d). The wage is w = 90, so the budget constraint is c = 90(70 − d).

The marginal rate of transformation: In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 8,000. Coordinates are (days of free time, consumption). A straight, downward sloping line connects points (0, 6,300) and (70, 0).
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Days of work 0 10 20 30 40 50 60 70
Free days 70 60 50 40 30 20 10 0
Consumption ($) 0 900 1,800 2,700 3,600 4,500 5,400 6,300

The marginal rate of transformation

The wage is $90, so the slope of the budget constraint is –90. $90 is your MRT (the rate at which you can transform free days into consumption), and it is also the opportunity cost of a free day.

Your ideal plan: In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 8,000. Coordinates are (days of free time, consumption). A straight, downward sloping line connects points (0, 6,300) and (70, 0). There are two parallel, downward-sloping, convex curves. One is tangent to the straight line at point A (34, 3,240). The other lies below this one at all points.
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Days of work 0 10 20 30 40 50 60 70
Free days 70 60 50 40 30 20 10 0
Consumption ($) 0 900 1,800 2,700 3,600 4,500 5,400 6,300

Your ideal plan

With these indifference curves, your ideal plan for the break would be at point A, with 34 free days, and earnings of $3,240. At this point, your MRS is equal to the absolute value of the slope of the budget constraint, which is the wage ($90).

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:

\[c = 90(70 - d) + \text{1,000}\]
In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 7,000. Coordinates are (days of free time, consumption). A straight, downward sloping line connects points (0, 6,300) and (70, 0). Point A (34, 3,240) lies on this line. Another straight line connects points (0, 7,336) and (81.51, 0) and lies above the previous line at all points. Point B (39.4, 3,790) lies on the higher straight line. There are three parallel, downward-sloping, convex curves. From bottom to top, they are labelled IC1, IC2, and IC3. IC3 is tangent to the highest straight line at point B. IC2 is tangent to the lowest straight line at point A. IC1 lies below IC2 at all points.
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Figure 3.11 The effect of additional income on your choice of free time and consumption.

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 IC3 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.

In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 8,000. Coordinates are (days of free time, consumption). A straight, downward sloping line connects points (0, 6,300) and (70, 0). Point A (34, 3,240) lies on this line. Another straight line connects points (0, 7,336) and (81.51, 0) and lies above the previous line at all points. Point B (34, 4,240) lies on the higher straight line. There are two parallel, downward-sloping, convex curves. The higher curve is tangent to the highest straight line at point B. The lower curve is tangent to the lowest straight line at point A.
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Figure 3.12 A student with different preferences: their MRS doesn’t change when consumption rises.

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.

In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 8,000. Coordinates are (days of free time, consumption). A straight, downward sloping line connects points (0, 6,300) and (70, 0). Another straight line connects points (0, 9,100) and (70, 0). Point D (30, 5,200) lies on this line. There are two parallel, downward-sloping, convex curves. One is tangent to the upper straight line at point D. The other is tangent to the lower straight line at point A (34, 3,240).
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Figure 3.13a The effect of a wage rise on your choice of free time and consumption.

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 IC2. The higher wage enables you to reach point D on IC4. Figure 3.13b shows how we can decompose the change from A to D into two parts that correspond to these two effects.

In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 7,000. Coordinates are (days of free time, consumption). There are three straight lines. Two are parallel to each other. The lower of the two connects points (0, 6,300) and (70, 0). The higher of the two connects points (0, 7,697), C (41.5, 3959) and (85.49, 0). The third straight line is steeper than the other two and connects points (0, 9,100) and (70, 0). Point D (30, 5,200) lies on this line. There are two parallel, downward-sloping, convex curves. One is tangent to the steeper straight line at point D and the higher of the two parallel lines at point C. The other is tangent to the lower straight line at point A (34, 3,240). The horizontal distance between points A and C is the income effect. The horizontal distance between C and D is the substitution effect. The horizontal distance between points A and D is the overall effect.
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Figure 3.13b The effect of a wage rise on your choice of free time and consumption.

A rise in wages: In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 8,000. Coordinates are (days of free time, consumption). A straight, downward sloping line connects points (0, 6,300) and (70, 0). Another straight line connects points (0, 9,100) and (70, 0). A downward-sloping, convex curve is tangent to straight line at point A (34, 3,240).
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A rise in wages

When the wage is $90, your best choice of free days and consumption is at point A. The steeper line shows your new budget constraint when the wage rises to $130 per day. Your feasible set has expanded.

Now you can reach a higher indifference curve: In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 7,000. Coordinates are (days of free time, consumption). A straight, downward sloping line connects points (0, 6,300) and (70, 0). Another straight line connects points (0, 9,100) and (70, 0). Point D (30, 5,200) lies on this line. There are two parallel, downward-sloping, convex curves. One is tangent to the upper straight line at point D (30, 5,200). The other is tangent to the lower straight line at point A (34, 3,240).
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Now you can reach a higher indifference curve

Point D on IC4 gives you the highest utility. At point D, your MRS is equal to the new wage, $130. You have only 30 free days, but your consumption has risen to $5,200.

If there was no change in opportunity cost of free time: In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 7,000. Coordinates are (days of free time, consumption). There are three straight lines. Two are parallel to each other. The lower of the two connects points (0, 6,300) and (70, 0). The higher of the two connects points (0, 7,697), C (41.5, 3959) and (85.49, 0). The third straight line is steeper than the other two and connects points (0, 9,100) and (70, 0). Point D (30, 5,200) lies on this line. There are two parallel, downward-sloping, convex curves. One is tangent to the steeper straight line at point D and the higher of the two parallel lines at point C. The other is tangent to the lower straight line at point A (34, 3,240).
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If there was no change in opportunity cost of free time

The dotted line shows what would happen if you had enough income to reach IC4 without a change in the opportunity cost of free time. You would choose C, with more free time.

The income effect: In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 7,000. Coordinates are (days of free time, consumption). There are three straight lines. Two are parallel to each other. The lower of the two connects points (0, 6,300) and (70, 0). The higher of the two connects points (0, 7,697), C (41.5, 3959) and (85.49, 0). The third straight line is steeper than the other two and connects points (0, 9,100) and (70, 0). Point D (30, 5,200) lies on this line. There are two parallel, downward-sloping, convex curves. One is tangent to the steeper straight line at point D and the higher of the two parallel lines at point C. The other is tangent to the lower straight line at point A (34, 3,240). The horizontal distance between points A and C is the income effect.
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The income effect

The shift from A to C is called the income effect of the wage rise; on its own it would cause you to take more free time.

The substitution effect: In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 7,000. Coordinates are (days of free time, consumption). There are three straight lines. Two are parallel to each other. The lower of the two connects points (0, 6,300) and (70, 0). The higher of the two connects points (0, 7,697), C (41.5, 3959) and (85.49, 0). The third straight line is steeper than the other two and connects points (0, 9,100) and (70, 0). Point D (30, 5,200) lies on this line. There are two parallel, downward-sloping, convex curves. One is tangent to the steeper straight line at point D and the higher of the two parallel lines at point C. The other is tangent to the lower straight line at point A (34, 3,240). The horizontal distance between points A and C is the income effect. The horizontal distance between C and D is the substitution effect.
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The substitution effect

The rise in the opportunity cost of free time makes the budget constraint steeper. This causes you to choose D rather than C, with less free time. This is called the substitution effect of the wage rise.

The sum of the income and substitution effects: In this diagram, the horizontal axis shows days of free time, and ranges from 0 to 70. The vertical axis shows consumption in dollars, and ranges from 0 to 7,000. Coordinates are (days of free time, consumption). There are three straight lines. Two are parallel to each other. The lower of the two connects points (0, 6,300) and (70, 0). The higher of the two connects points (0, 7,697), C (41.5, 3959) and (85.49, 0). The third straight line is steeper than the other two and connects points (0, 9,100) and (70, 0). Point D (30, 5,200) lies on this line. There are two parallel, downward-sloping, convex curves. One is tangent to the steeper straight line at point D and the higher of the two parallel lines at point C. The other is tangent to the lower straight line at point A (34, 3,240). The horizontal distance between points A and C is the income effect. The horizontal distance between C and D is the substitution effect. The horizontal distance between points A and D is the overall effect.
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The sum of the income and substitution effects

The overall effect of the wage rise depends on the sum of the income and substitution effects. In this case, the negative substitution effect is bigger, so with the higher wage you take less free time.

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.

In this diagram, the horizontal axis shows hours of free time, and ranges from 8 to 24. The vertical axis shows consumption in dollars, and ranges from 0 to 250. A straight line connects points (0, 240) and (24, 0).
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Figure 3.14 The feasible set for a worker whose wage is $15.

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

  • The slope of the budget constraint is the negative of the wage rate (–15).
  • A gift of $60 would make the budget constraint steeper, with the intercept on the vertical axis increasing to $300.
  • A decrease in the wage rate would make the feasible set smaller and reduce the worker’s utility.
  • A decrease in the wage rate would reduce the opportunity cost of free time, so the worker would choose to work fewer hours.
  • 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.

In this diagram, the horizontal axis shows the number of cinema tickets, and ranges between 0 and 24. The vertical axis shows the number of nights out, and ranges between 0 and 16. Coordinates are (number of cinema tickets, number of nights out). A straight line connects points (0, 15) and (16, 0). Point C (10, 6) lies on this line. A downward sloping, convex curve labelled IC2 is tangent to this straight line at point C. Another straight line connects points (0, 15) and (24, 0). Point A (13, 7) lies on this line. A downward sloping, convex curve labelled IC1 is tangent to this straight line at point A.
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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.

  1. How has Zoë’s marginal rate of transformation changed? Has the opportunity cost of a cinema ticket increased or decreased?
  2. How has Zoë’s spending power changed: can she afford more, or less, socializing and entertainment?
  3. 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?
  4. To decompose the effect of the price change into an income and substitution effect, copy the diagram and draw a straight line tangent to IC2, with the flatter slope of the original budget constraint. Mark the tangency point B.
  5. 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?
  6. 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?
  7. 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?

Extension 3.7 Mathematics of income and substitution effects

In the main part of this section we solved the summer break constrained choice problem diagrammatically, and decomposed the effect of a wage change into income and substitution effects. Here, we apply the calculus method described in Extension 3.5 to solve this problem, and then determine the effects of changes in wage or income mathematically. Lastly, we explain a numerical method for decomposing the wage change into income and substitution effects.

In the constrained choice problem in the main part of this section, you decide how many of the 70 days of your summer break to spend working. You choose your consumption, \(c\), and your days of free time, \(t\), to achieve the highest possible utility within your feasible set.

Here we use \(t\) rather than \(d\) for days for free time to avoid confusion with the differentiation symbol.

Your wage rate is \(w\), and you may also have some unearned income, \(I\).

The summer break constrained choice problem

Choose \(t\) and \(c\) to maximize \(u(t, c)\) subject to the constraint \(c=w(70 - t) + I\).

We will represent your preferences using the utility function:

\[u(t, c) = t(c+600)\]

Even in a mathematical analysis, it is helpful to draw a diagram to keep track of the economic interpretation. Figure E3.4 illustrates the decomposition into income and substitution effects with this utility function, when the wage rises from $96 to $150. The utility function is different from the one we used to draw Figure 3.13b, so the shape of the indifference curves, and the exact numbers, are slightly different, but it is a very similar problem.

In this diagram, the horizontal axis shows day of free time, ranging from 0 to 70, and the vertical axis shows consumption spending in dollars, ranging from 0 to 8000. Coordinates are (days of free time, consumption spending). The original budget constraint is a downward-sloping line connecting the points (0, 6720) and (70, 0). The new budget constraint is a downward-sloping line connecting the points (0, 10500) and (70, 0). A dotted downward-sloping line is parallel to the original budget constraint and connects the points (0, 8280) and (70, 1560). There are two downward-sloping convex curves that do not intersect. The lower curve is tangent to the original budget constraint at A (38, 3072). The upper curve is tangent to the new budget constraint at D (37, 4950) and tangent to the dotted line at C (46, 3864). The horizontal distance from points A to C is the income effect. The horizontal distance from points C to D is the income effect.
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Figure E3.4 A wage increase from $96 to $150 reduces your choice of free time from 38 days (point A) to 37 days (point D).

In Figure E3.4, you begin with a wage of $96 per day, and an unearned income of zero. You choose the combination of consumption and free time that maximizes your utility, which is at point A. If the wage increases to $150, your choice moves to point D. In the diagram, the overall effect of the wage increase on free time is a small reduction, and this is decomposed into a positive income effect and a negative substitution effect.

We now prove these results mathematically. We will:

  • solve the constrained choice problem for this utility function
  • examine how changes in \(w\) and \(I\) affect the solution
  • calculate the effect on free time in the case when \(I=0\) and the wage rises from $96 to $150
  • and then decompose this change into an income effect and a substitution effect.

Solving the constrained choice problem

Remember that your best choice is a point on the budget constraint where MRS = MRT. The MRS is the ratio of the marginal utilities:

\[\text{MRS} = \left| \frac{\partial u}{\partial t} \left/ \frac{\partial u}{\partial c} \right.\right| = \frac{c+600}{t}\]

and the MRT is the wage, \(w\) (the absolute value of the slope of the feasible frontier). Hence your chosen point \((t^*,\ c^*)\) satisfies the pair of simultaneous equations:

\[\begin{align} \frac{c+600}{t}&=w \\ c&=w(70-t)+I \end{align}\]

Using the second equation to substitute for \(c\) in the first equation and solving for \(t\) gives:

\[t^*=35+\frac{I+600}{2w}\]

and substituting this back into the first equation determines \(c^*\):

\[c^*=wt^*-600=35w+\frac{I-600}{2}\]

For the calculations below, it is helpful to note that the utility you get at your best choice is \(u=(c^*+600)t^*\). Since (from the first of the simultaneous equations) \(c^*+600=wt^*\), we can write your utility in terms of \(t^*\) only:

\[\begin{align} u=w{t^*}^2 \end{align}\]

How the solution changes if w or I changes

The equations above show that the solution \((t^*,\ c^*)\)—that is, your utility-maximizing choice of \(c\) and \(t\)—are functions of the wage, \(w\), and unearned income, \(I\). To work out what would happen if \(w\) changed, we can differentiate the functions with respect to \(w\), holding \(I\) constant—and similarly for changes in \(I\).

An increase in w:

Considering first the effect on free time:

\[\begin{align*} t^*(w, I)&= 35 + \frac{I+600}{2w} \\ \Rightarrow \frac{\partial t^*}{\partial w}&= - \frac{I+600}{2w^2} \end{align*}\]

The sign of the derivative tells us the direction of the effect on \(t^*\) of a small (infinitesimal) increase in \(w\). In this case it is negative—for all values of \(w\) and \(I>0\). We can deduce that with this utility function, any wage increase will reduce the choice of free time.

What about consumption?

\[\begin{align} c^{*}(w, I) &= 35w + \frac{I - 600}{2} \\ \Rightarrow \frac{\partial c^{*}}{\partial w} &= 35 \end{align}\]

This partial derivative is positive for all \(w\) and \(I\), so a rise in the wage always increases chosen consumption.

An increase in I:

Using the same method for \(I\):

\[\begin{align} t^{*}(w, I) &= 35 + \frac{I + 600}{2w} \Rightarrow \frac{\partial t}{\partial I} = \frac{1}{2w} > 0 \\ c^{*}(w, I) &= 35w + \frac{I - 600}{2} \Rightarrow \frac{\partial c}{\partial I} = \frac{1}{2} \end{align}\]

Both partial derivatives are positive for all \(I\) and \(w>0\)—so any increase in unearned income will result in more free time and consumption.

How general are these results?

The expressions for the partial derivatives are specific to the particular utility function we have used. But for most plausible utility functions, increases in income raise both consumption and free time, and increases in the wage raise consumption. However, the effect of the wage on free time depends on the balance between the income and substitution effects.

How free time changes when I = 0 and the wage rises from $96 to $150

Initially \(w=96\) and \(I=0\). Substituting these numbers into the equations for the solution above, we find the best choice in this situation, corresponding to point A in Figure E3.4:

\[t_A=35+600/192 = 38.125 \text{ and } c_A=35\times 96-300=3,060\]

After the wage rise, \(w=150\) and \(I=0\). The best choice is point D:

\[t_D=35+600/300 = 37 \text{ and } c_D=35\times 150-300=4,950\]

The overall effect of the wage rise on free time is a decrease from 38.125 days to 37 days.

Decomposing the overall effect of the wage rise on free time

The overall effect of the wage rise is that free time decreases:

\[t_D-t_A=37-38.125=-1.125\]

We do the decomposition in four steps:

1. What is your utility after the wage rise?

After the wage rise, you are at point D where \(t_D= 37\). Your utility is:

\[u_D=150t_D^2=205,350\]

In Figure E3.4, this is the level of utility on the indifference curve through D.

2. What change in income would have the same effect on utility as the wage rise?

Suppose that the wage had remained at \(w=96\), but unearned income had increased from 0 to \(J\). Your best choice of free time would be:

\[t=35+\frac{J+600}{192} \text { with utility } u=96t^2\]

If this gives you the same utility as the wage rise, you will be at point C in the diagram with:

\[t_C=35+\frac{J+600}{192} \text { with utility } u_C=96t_C^2=u_D=150t_D^2.\]

To find the value of \(J\), we solve the equation \(96t_C^2=150t_D^2\) using the expressions for \(t_C\) and \(t_D\) above. First note that (since the numbers have been carefully chosen) we can simplify the equation:

\[\begin{gather*} 96t_C^2=150t_D^2 \Rightarrow 16t_C^2=25t_D^2 \Rightarrow (4t_C)^2 = (5t_D)^2 \\ \text {and hence:} \ 4t_C = 5t_D \end{gather*}\]

(The other algebraic possibility, \(4t_C = -5t_D\), makes no economic sense.) Substituting for \(t_C\) and \(t_D\) and solving for \(J\):

\[\begin{align*} 4 \left (35+\frac{J+600}{192}\right)&= 5\times 37 \\ \Rightarrow J&= 1,560 \end{align*}\]

So increasing unearned income from 0 to $1,560, while keeping the wage at $96, would have the same effect on utility as the wage rise from $96 to $150.

3. Find the income effect

The income effect of the wage rise is the change in free time that would occur if you achieved the same level of utility through an increase in unearned income—that is, if you received additional income of \(J\) = $1,560. It corresponds to the change in free time between points A and C:

\[\begin{gather*} t_A=38.125 \text{ and } t_C = 35 + \frac{600+J}{192} =46.25 \\ \Rightarrow \text{ income effect }= t_C-t_A = 8.125 \end{gather*}\]

So if you only considered the income effect of a rise in the wage from $96 to $150, you would take 8.125 additional days of free time.

4. Find the substitution effect

However, your choice is also affected by the increase in the opportunity cost of free time—which causes a substitution effect. Free time costs more, so you substitute consumption for free time.

Since the overall effect of the wage rise is the sum of the income and substitution effects, and we know both the overall effect and the income effect, we can easily deduce the substitution effect:

     
Overall effect \(t_D-t_A\) = 37 − 38.125 = −1.125
Income effect \(t_C -t_A\) = 46.25 − 38.125 = +8.125
Substitution effect \(t_D -t_C\) = −1.125 − 8.125 = −9.25

The substitution effect is the effect on the choice of free time of changing the wage, while keeping utility constant.

The income and substitution effects on free time act in opposite directions

In our example, the income effect is positive (a wage rise increases free time) while the substitution effect is negative. This is true for most plausible utility functions. Whether the overall effect on free time is positive or negative depends on which of the two component effects dominates.

We showed above that with the utility function \(u(t, c) = t(c+600)\), an increase in the wage always reduces free time: \(\frac{\partial t^*}{\partial w}=-\frac{I+600}{2w^2}<0\). But with other utility functions we get different results. For example, if \(u(t, c) = t(c-200)\):

\[\begin{equation*} \frac{\partial t^*}{\partial w} = -\frac{I -200}{2w^2} \begin{cases} < 0 & \text{if} \ I > 200 \\ = 0 & \text{if} \ I = 200 \\ > 0 & \text{if} \ I < 200 \end{cases} \end{equation*}\]

In this case, the positive income effect dominates when unearned income \(I\) is low, and the negative substitution effect dominates when \(I\) is high.

Exercise E3.5 Income and substitution effects

Suppose your friend has the utility function, \(u(t,c) = tc\), and has the budget constraint, \(c = w(24-t) + I\).

  1. If the wage is $16 and \(I\) = $160, how many hours of free time would your friend choose? What would the corresponding level of consumption be?
  2. Suppose the wage rises to $25, while income remains at $160. How many hours of free time would your friend choose now, and what would the corresponding level of consumption be?
  3. Find the size of the income and substitution effect (measured in hours of free time) corresponding to this wage rise.

Read more: Sections 14.1, 17.1, and 17.3 of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.