# 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 IC1 to IC4. IC4 represents the highest level of utility because it is the furthest away from the origin. No combination of consumption and free time on IC4 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 IC1. 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.

Figure 3.7a How many hours does Karim decide to work?

Which point will Karim choose?

The diagram brings together Karim’s indifference curves and his feasible frontier.

Feasible combinations

On the indifference curve IC1, all combinations between A and B are feasible because they lie in the feasible set. Suppose Karim chooses one of these combinations.

Could do better

All combinations in the area between IC1 and the feasible frontier are feasible, and give higher utility than combinations on IC1. For example, a movement to C would increase Karim’s utility.

Could do better

Moving from IC1 to point C on IC2 increases Karim’s utility. Switching from B to D would raise his utility by an equivalent amount.

But again, Karim can raise his utility by moving into the area between IC2 and the frontier. He can continue to find feasible combinations on higher indifference curves, until he reaches E.

At E, he has 17 hours of free time per day and €210 for consumption. Karim maximizes his utility: he is on the highest indifference curve obtainable, given the feasible frontier.

MRS = MRT

At E, the indifference curve is tangent to the feasible frontier. The marginal rate of substitution (corresponding to the slope of the indifference curve) is equal to the marginal rate of transformation (corresponding to the slope of the frontier).

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

Figure 3.7b How many hours does Karim decide to work?

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.

1. 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)?
2. 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.)
3. 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.

1. Suppose that the only things Alexei cares about are his final grade and his free time. Are these two goods scarce for Alexei? Explain.
2. Draw a graph showing how Alexei’s grade depends on his hours of study (with study hours on the horizontal axis).
3. 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.
4. 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?
5. 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?
6. 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).

• Karim will choose a point where the marginal rate of substitution equals the marginal rate of transformation.
• C is below the feasible frontier but D is on the feasible frontier. Therefore, Karim may choose point D as his best choice.
• The other workers who have downward-sloping indifference curves, whatever their slope, would choose point E.
• At E, Karim has the highest ratio of consumption per hour of free time per day.
• 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.

### Extension 3.5 Solving the constrained choice problem for consumption and free time

Constrained choice problems arise in many economic models. Although diagrams can help us understand them, as they do in the main part of this section, solving them mathematically gives us additional insight, and also helps to ensure that we are not misled by the particular way we draw a diagram. In this extension, we describe two methods for solving them using calculus. One uses the analysis and economic insight from Extensions 3.3 and 3.4; the other is a mathematical substitution method that can sometimes be used for constrained choice problems.

Karim wants as much consumption as possible while sacrificing the least possible amount of free time. This is shown diagrammatically in Figure 3.7a (reproduced here as Figure E3.3): he maximizes his utility by choosing point E where an indifference curve is tangential to the feasible frontier. At point E, his marginal rate of substitution (MRS) is equal to his marginal rate of transformation (MRT). In this extension, we show how to formulate Karim’s decision mathematically as a constrained choice problem, and solve it to find his best combination of consumption and free time.

Figure E3.3 How many hours does Karim decide to work?

Karim’s utility function (the one shown in Figure E3.3) is

$u(t,c)=(t-6)^2(c-45) \text{ for } t>6, c>45$

and the equation of the feasible frontier (his budget constraint) is $$c= w(24-t)$$, where $$w$$ is the wage (equal to 30 in Figure E3.3). ### Karim’s constrained choice problem Choose $$t$$ and $$c$$ to maximize $$u(t, c)$$ subject to the constraint, $$c=w(24\ –\ t)$$. In constrained choice problems, the function we want to maximize (or in some problems minimize) is called the ‘objective function’. Karim’s objective is to maximize his utility. Sometimes in this sort of problem, the constraint is written as an inequality: $$c \leq w(24\ –\ t)$$, which can be interpreted as saying that his choice must lie in the feasible set. But because his utility depends positively on $$t$$ and $$c$$, we know that he will want to choose a point on the frontier. So we can write the constraint as an equation, which makes the problem easier to solve mathematically. ### Two methods for solving Karim’s constrained choice problem One way to solve Karim’s problem is to use the constraint to substitute for $$c$$ in terms of $$t$$ in the utility function, so that utility is expressed as a function of the single variable $$t$$: $u = (t-6)^2(w(24-t)-45)$ Then we can maximize this expression with respect to $$t$$ by equating its derivative to zero. Using the product rule for differentiation: \begin{align} \frac{du}{dt} = 2(t - 6)(w (24 - t) - 45) - w (t - 6)^{2} &= 0 \\ \Rightarrow 2(w(24 - t) - 45) &= w(t - 6) \end{align} Rearranging to solve this equation for $$t$$, we obtain: $t=18-\frac{30}{w}$ In the case when $$w=30$$, this gives ­us $$t=17$$, and we can substitute back into the budget constraint to determine the corresponding level of consumption: $$c = 30(24-17) = 210$$. To check that we have found a maximum point (rather than a minimum or a point of inflection) we should consider the sign of the second derivative $$d^2u/dt^2$$ at this point. Differentiating the expression above for $$du/dt$$, you can check that $$d^2u/dt^2=90(23-2t)$$, which is negative when $$t=17$$so this is indeed a maximum. In summary, to maximize his utility subject to the budget constraint, Karim should choose to take 17 hours of free time, and work for 7 hours, obtaining consumption of210. This is point E in Figure E3.3.

An alternative method is to apply what we know about the solution from drawing indifference curves and the budget constraint. That is, the best combination of $$c$$ and $$t$$ for Karim is at the tangency point of the budget constraint and an indifference curve. So it satisfies two conditions:

• It is on the budget constraint.
• It is at a point where $$\text{MRS}=\text{MRT}$$.

We know from Extension 3.3 that the MRS is equal to the ratio of the marginal utilities, $$\frac{\partial u}{\partial t}/\frac{\partial u}{\partial c}$$:

$\frac{\partial u}{\partial t} = 2(t-6)(c-45) \ \text{and} \ \frac{\partial u}{\partial c} = (t-6)^2 \Rightarrow \text{MRT} = \frac{2(c-45)}{(t-6)}$

and from Extension 3.4 that the MRT is $$w$$. So the two conditions satisfied by Karin’s best combination are:

$c = w(24-t)$ $\frac{2(c-45)}{(t-6)} = w$

This gives us a pair of simultaneous equations for $$t$$ and $$c$$. The easiest way to solve them is to use the first equation to substitute for $$c$$ in the second. You can check that this gives us the same equation for $$t$$ that we obtained using the first method.

### The first-order condition

You will encounter many other examples of constrained choice problems in economic models. Typically, we find the solution at a point on the constraint that satisfies a condition known as the ‘first-order condition’.

In Karim’s problem the solution is on the budget constraint, and satisfies the first-order condition MRS = MRT. It is described as ‘first-order’ because it involves first derivatives. Using the substitution method, we obtained the first-order condition by setting the first derivative of the objective function to zero.

Note that when we used the substitution method, we also checked the ‘second-order condition’: we checked that we had found a maximum point using the second derivative of the objective function. We didn’t check the second-order condition for the other method, because there we were using our economic understanding of the problem (helped by a diagram) to deduce that the tangency point must be the maximum.

### Exercise E3.4 Solving the constrained problem using two methods

Karim’s friend has the utility function $$u(t,c)=t^{3}c$$ and faces an hourly wage of \$25.

1. Assuming that Karim’s friend wants to maximize utility, use the substitution method to find how many hours of free time ($$t$$) he would choose, and the corresponding amount he can consume ($$c$$).
2. Verify that you obtain the same answer using the MRS = MRT method.
3. Using the general expressions $$u(t,c)=t^a c^b$$ and $$c=w(24-t)$$, show that the substitution method and the MRS = MRT methods obtain the same result.

Read more: Sections 8.1 to 8.3 of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.