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

## 3.5 Decision-making and scarcity

We have described Karim’s preferences for different combinations of consumption and free time, and worked out which combinations are feasible for him. The third step in modelling decision-making about hours of work is to bring the decision-maker’s preferences and feasible set together, to determine the combination of consumption and free time that will be chosen. Figure 3.7a brings together Karim’s feasible frontier (Figure 3.6) and indifference curves (Figure 3.4). Recall that the indifference curves indicate what Karim prefers, and their slopes show the trade-offs that he is willing to make; the feasible frontier is the constraint on his choice, and its slope shows the trade-off he is constrained to make.

Figure 3.7a shows four indifference curves, labelled IC_{1} to IC_{4}. IC_{4} represents the highest level of utility because it is the furthest away from the origin. No combination of consumption and free time on IC_{4} is feasible, however, because the whole indifference curve lies outside the feasible set. Suppose that Karim considers choosing a combination somewhere in the feasible set, on IC_{1}. The steps in Figure 3.7a show how he can increase his utility by moving to points on higher indifference curves until he reaches a feasible choice that maximizes his utility.

Karim maximizes his utility at point E, where his indifference curve is tangent to the feasible frontier (they touch but do not cross). The model predicts that Karim will:

- choose to spend 7 hours each day working, and 17 hours on other activities
- have €210 to spend on consumption.

In Figure 3.7a, Karim reaches the highest attainable indifference curve at E: the point on the feasible frontier where the indifference curve and the feasible frontier have the same slope. Now, remember that the slopes represent the two trade-offs facing Karim:

*The slope of the indifference curve represents the MRS*: It is the trade-off he is willing to make between free time and consumption.*The slope of the frontier represents the MRT*: It is the trade-off that he is constrained to make between free time and consumption because it is not possible to go beyond the feasible frontier.

Karim achieves the highest possible utility where the two trade-offs just balance (E). His best combination of consumption and free time is at the point where the marginal rate of transformation is equal to the marginal rate of substitution.

Figure 3.7b shows the MRS (slope of indifference curve) and MRT (slope of feasible frontier) at the points shown in Figure 3.7a. At B and D, the amount of consumption Karim is willing to trade for an hour of free time (MRS) is greater than the amount he would have to trade (MRT), so he prefers to increase his free time. At A, the MRT is greater than the MRS, so he prefers to decrease his free time. And, as expected, at E the MRS and MRT are equal.

B | D | E | A | |
---|---|---|---|---|

Free time | 9.5 | 12 | 17 | 21.9 |

Consumption | 435 | 360 | 210 | 63 |

MRT | 30 | 30 | 30 | 30 |

MRS | 223 | 105 | 30 | 2 |

- constrained choice problem
- A problem in which a decision-maker chooses the values of one or more variables to achieve an objective (such as maximizing profit, or utility) subject to a constraint that determines the feasible set (such as the demand curve, or budget constraint).

We have modelled Karim’s decision on working hours as what we call a **constrained choice problem**: a decision-maker (Karim) pursues an objective (utility maximization, in this case) subject to a constraint (his feasible frontier).

In our example, both free time and consumption are scarce for Karim because:

*Free time and consumption are goods*: Karim values both of them.*Each has an opportunity cost*: More of one good means less of the other.

In constrained choice problems, the solution is the choice that best satisfies the individual’s objectives. If we assume that utility maximization is Karim’s goal, then the best combination of consumption and free time is a point on the feasible frontier at which:

\[\text{MRS} = \text{MRT}\]The table in Figure 3.8 summarizes Karim’s trade-offs.

The trade-off | Where to find it on the diagram | |
---|---|---|

MRS | The amount of consumption, in euros, that Karim is willing to trade for an hour of free time | The slope of the indifference curve |

MRT | The amount of consumption Karim would gain (or lose) by giving up (or taking) another hour of free time | The slope of the budget constraint (the feasible frontier) which is equal in absolute value to the wage |

We sometimes say that ‘the MRS is equal to the slope of the indifference curve’, and ‘the MRT is equal to the slope of the feasible frontier’. But the MRS and MRT are both positive numbers, and the slopes are negative. Strictly speaking, we mean that the MRS and MRT are equal to the slopes in *absolute value*.

**Exercise 3.4** Zoë’s constrained choice problem

Remember the choice facing Zoë, from Exercise 3.3, who has a budget of £240 for cinema tickets (costing £10 each) and nights out socializing with friends (£16 each on average). We can think of her problem as similar to Karim’s: both cinema tickets and nights out are goods for her; she would like to have as many of both as possible, but her choice is limited by her budget constraint.

- Using the diagram of the feasible set from Exercise 3.3, sketch indifference curves to show her preferences for the two goods, assuming they have a similar shape to Karim’s (downward-sloping and becoming flatter as free time increases). Explain why it is reasonable to assume that Zoë’s indifference curves also have this shape (Hint: think about how the MRS changes as you move along an indifference curve.) Can you think of any reason why they might have a different shape (for example, straight lines, or curves that become steeper as the good on the horizontal axis increases)?
- For the indifference curves you have drawn, mark her most-preferred combination of cinema tickets and nights out on the diagram. (This is only an illustrative sketch, but make sure that all the indifference curves have plausible shapes and do not cross.)
- What is Zoë’s marginal rate of substitution at this point? How do you know?

**Exercise 3.5** Alexei’s constrained choice problem

Consider the situation of Alexei, a student, who knows that his final grade from the course will depend on the average number of hours he studies in a day. If he does not study, his grade will be zero. For each hour that he studies, his grade will increase by 8 percentage points, up to a maximum of 12 hours per day. After that, more study will not raise his grade any further.

- Suppose that the only things Alexei cares about are his final grade and his free time. Are these two goods scarce for Alexei? Explain.
- Draw a graph showing how Alexei’s grade depends on his hours of study (with study hours on the horizontal axis).
- In a diagram with hours of free time on the horizontal axis and grade on the vertical axis, sketch indifference curves to represent Alexei’s preferences, assuming as before that they are downward-sloping and become flatter as free time increases.
- Add Alexei’s feasible frontier and feasible set to the diagram. (Hint: it should be the mirror image of the graph in Question 2 of this exercise.) What is his marginal rate of transformation between free time and grade points?
- Mark Alexei’s preferred choice on your diagram. (Your answer will depend on how you have drawn the indifference curves). How many hours per day does he choose to study?
- Draw another set of indifference curves (using a separate diagram) to show that Alexei’s preferences could lead him to choose exactly 12 hours of free time per day, and that in this special case, his MRS could be less than his MRT. Would he ever decide to have fewer than 12 hours of free time?

**Question 3.7** Choose the correct answer(s)

Figure 3.7a shows Karim’s feasible frontier and his indifference curves for consumption and hours of free time per day. Suppose that other workers with similar qualifications have the same feasible frontier, but that their indifference curves may have different slopes, depending on their preferences.

Using this diagram, read the following statements and choose the correct option(s).

- If Karim were at a point on the feasible frontier where MRS ≠ MRT, then he would be willing to give up more of one good than would actually be necessary to get some of the other. Therefore, he will choose to do so until he reaches a point where MRS = MRT.
- Along the feasible frontier, Karim would be on a higher indifference curve at E than at D. Therefore point D is not his best choice.
- Consumers with flatter indifference curves (more willing to sacrifice more hours of free time for the amount of consumption) have a lower marginal rate of substitution. Therefore, they will choose combinations to the left of E (such as D) where their indifference curves are tangent to the feasible frontier.
- The points along the feasible frontier to the left of E have higher consumption relative to hours of free time, but Karim prefers point E, where the marginal rate of substitution equals the marginal rate of transformation.