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

## 3.4 The feasible set

- opportunity cost
- What you lose when you choose one action rather than the next best alternative. Example: ‘I decided to go on vacation rather than take a summer job. The job was boring and badly paid, so the opportunity cost of going on vacation was low.’

We know from Karim’s preferences that he would like both his consumption spending and his free time to be as high as possible. But not all combinations of consumption and free time are available to him—his choice is limited to what he could achieve by working for a wage of €30. So he faces a dilemma: the more free time he takes, the less consumption he can have. In other words, free time has an **opportunity cost**: to get more free time, Karim has to forgo the opportunity of higher consumption.

### Building block

For an introduction to opportunity cost, read Section 2.2.

The total amount of consumption he can have depends on the number of hours of free time he takes. Remember that if he works for \(h\) hours at a wage \(w\), his income is \(y = wh\). So if he takes \(t\) hours of free time, he will work for \((24 − t)\) hour per day, and his maximum level of consumption, \(c\), is then given by:

\[c = w(24 - t)\]- budget constraint
- An equation that represents all combinations of goods and services one could acquire that would exactly exhaust one’s budgetary resources.

We will call this his **budget constraint**, because it shows what he can afford to buy.

Figure 3.6 shows the two goods that Karim cares about: hours of free time (\(t\)) on the horizontal axis, and consumption (\(c\)) on the vertical axis. In the table in Figure 3.6, we have calculated the free time corresponding to a given level of working hours varying between 0 and 16 hours per day, and his maximum consumption, when his wage is \(w\) = €30.

Hours of work | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 |

Free time (h) | 24 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 |

Consumption (€) | 0 | 60 | 120 | 180 | 240 | 300 | 360 | 420 | 480 |

When we plot the points shown in the table, we get a downward-sloping straight line: this is the graph of the budget constraint. The equation of the budget constraint is:

\[c = 30(24 - t)\]The budget constraint in Figure 3.6 is the mirror image of the relationship between income and hours of work in Figure 3.3. It is a straight line, with a constant slope corresponding to the wage, but it slopes downward, which means that the slope is negative. At any point on the line, if free time increases by one hour, the maximum possible consumption *decreases* by €30—so the slope is –30.

Figure 3.6 shows which combinations of consumption and free time are feasible choices for Karim, and which are not. None of the points that lie above the budget constraint are feasible. For example, point C represents 12 hours of free time and consumption of €450. This is not feasible because if he takes 12 hours of free time his earnings will be only €360.

All the points that lie on or below the budget constraint, in the shaded area, are feasible. For example, Karim could choose point D, with 18 hours of free time and €70 of consumption. From what we know about his preferences, we would not expect him to do so, because he would then consume less than he earned—and remember that he only cares about his consumption and free time so has no other use for his earnings. By choosing a combination below the budget constraint, Karim would be giving up something that is freely available to him. He could obtain more consumption without sacrificing any free time, or have more time without reducing consumption. But point D is nevertheless a feasible choice for him.

- feasible set
- All of the combinations of goods or outcomes that a decision-maker could choose, given the economic, physical, or other constraints that they face.
*See also: feasible frontier.* - feasible frontier
- The curve or line made of points that defines the maximum feasible quantity of one good for a given quantity of the other.
*See also: feasible set.* - marginal rate of transformation (MRT)
- The quantity of a good that must be sacrificed to acquire one additional unit of another good. At any point, it is the absolute value of the slope of the feasible frontier.
*See also: marginal rate of substitution.*

The shaded area represents Karim’s **feasible set**. A set is merely a collection of things—and in this case, the set contains all the feasible combinations of free time and consumption. In principle, he could choose any point in this set if he wished to do so. The budget constraint is the upper boundary of the feasible set, which is also called the **feasible frontier**.

The slope of the feasible frontier—corresponding to the wage—determines the size of the trade-off Karim faces: how much consumption must be given up in return for an extra hour of free time. The opportunity cost of one hour of free time is equal to the wage: it is the €30 of consumption that he could have had by working instead.

Another way to describe the trade-off is to say that the feasible frontier shows the **marginal rate of transformation (MRT)**: the rate at which Karim can transform free time into consumption. When Karim’s wage is €30 per hour, the rate at which he can transform free time into consumption is €30 per hour. If the wage were to increase to €35, the budget constraint would become steeper, with a slope of –35, and his marginal rate of transformation would be €35 per hour.

We have now identified two trade-offs:

*The marginal rate of substitution (MRS)*: As explained in the previous section, the MRS measures the trade-off that Karim is willing to make between consumption and free time.*The marginal rate of transformation (MRT)*: In contrast, this measures the trade-off that Karim is constrained to make by the feasible frontier.

The next section will demonstrate how the choice Karim makes between his consumption and his free time strikes a balance between these two trade-offs.

**Question 3.6** Choose the correct answer(s)

Make a copy of Figure 3.6, which shows Karim’s budget constraint when his wage is €30. Add another line to show his budget constraint if his wage increases to €40. Suppose he chooses 16 hours of free time and consumption of €240. Mark this as point E on the diagram.

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

- When the wage rises, point E is still feasible, because it is still inside the feasible frontier. Karim will earn €320 a day so can afford to spend €240 on consumption.
- With 12 hours of free time, consumption of €400 is not feasible when the wage is €30.
- The budget constraint pivots outwards around (24, 0). 24 hours of free time and no consumption is always feasible in principle.
- All the points in the original feasible set, plus all the points between the two budget constraints, are feasible at the higher wage.

**Exercise 3.3** A student budget problem

Zoë, a student planning her budget for her first term at university in London, has found out from a survey on a student website that in 2021, UK students spent an average of £88 per month on socializing and eating out: £47 on ‘going out’ and £41 on ‘takeaways and eating out’. With these numbers to guide her, she decides to set aside £240 for social activities and entertainment during the term. She estimates that a night out socializing with friends will cost £16 on average, but she also wants to be able to go to the cinema regularly, and cinema tickets cost £10 each.

- If Zoë used the whole budget for nights out, how many could she have?
- If she spent the whole budget on cinema tickets, how many could she buy?
- Complete the table below to show how many
*nights out*Zoë could have if she decided to buy the number of*cinema tickets*shown. (Not all of the answers are whole numbers.) - Use the information in the table to draw Zoe’s budget constraint, with cinema tickets on the horizontal axis and nights out on the vertical axis. What is the marginal rate of transformation?

Cinema tickets, c |
0 | 4 | 8 | 12 | 16 | 20 | 24 |

Nights out, n |