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.

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:

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

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Figure 3.6 The budget constraint and the feasible set.

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:

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.

Extension 3.4 The marginal rate of transformation

This extension uses calculus to measure the MRT as the slope of the feasible frontier. The example of the feasible set and the MRT in the main part of this section is very simple; we explain here how to generalize these concepts to the case of a worker whose feasible frontier is not straight. A case like this arises in Unit 5.

We defined Karim’s marginal rate of transformation as the rate at which he can transform free time into consumption. Diagrammatically, it is the absolute value of the slope of the feasible frontier.

Karim’s feasible frontier is his budget constraint; all combinations of free time \(t\), and consumption \(c\), on or below the frontier are feasible choices for him. Calculating his MRT is easy in this case, because his budget constraint is straight line: it has a constant slope:

\[c= w(24-t) \Rightarrow \frac{dc}{dt}=-w\]

So Karim’s MRT is the wage, \(w\). It is constant: that is to say, it is the same for every point on the feasible frontier. However, feasible sets arise in many economic models, and the feasible frontier is not always a straight line. We give an example here and there is another in Unit 5. As in Extension 3.3, we will use the calculus approach to marginal quantities to measure the MRT.

Consider the case of Marina, who makes her living writing short stories that she sells to magazines. Like Karim, she values consumption \(c\), and free time \(t\), and her consumption depends on the hours she works. However, unlike Karim, she doesn’t receive a fixed wage for an hour’s work. Her productivity, and hence her income per hour, is greatest for the first hour or two she works each morning, and declines the more hours she spends writing each day.

Suppose that, if she works for \(h\) hours, her daily income is given by a function \(f(h)\), which increases from 0 to $400 as \(h\) rises from 0 to 16 hours. We could describe it as her production function: it tells us how much income she produces, given her working hours. The properties of \(f\) are:

\[f(0)=0;\ f(16)=400;\ f'(h)>0; \text{ and } f''(h)<0\]

The condition that the second derivative \(f''(h)\) is negative captures the property that she gets gradually less productive as hours increase: income continues to rise, but at a decreasing rate.

Marina’s consumption is \(c=f(h)\). We can write it in terms of free time rather than working hours by substituting \(h=24-t\). This gives us the equation of the feasible frontier—that is, the maximum consumption \(c\) she can have if she takes \(t\) hours of free time:

\[c=f(24-t)\]

Figure E3.2 shows the shape of the feasible frontier when \(f\) has these properties.

The actual function we used to draw it is \(f(h)=400(1-(1-h/16)^{a})^{1/a}\), where the parameter \(a\) is equal to 1.6.

The main difference from Karim’s feasible frontier in Figure 3.6 is that the slope changes as we move along it. At point B, where Marina has 22 hours of free time and works for 2 hours per day, the tangent to the curve is steep, and it becomes flatter as her daily hours of work rise.

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Figure E3.2 Marina’s feasible frontier.

To calculate the MRT we simply differentiate the equation of the feasible frontier with respect to \(t\):

\[c=f(24-t) \Rightarrow \frac{dc}{dt} = f'(24-t) \frac{d}{dt}(24-t)\]

using the composite function rule (also called the chain rule). Hence:

\[\begin{align} \frac{dc}{dt} &= -f^{\prime}(24 - t) \\ \text{MRT} &= f^{\prime}(24 - t) \end{align}\]

As free time \(t\) increases, the marginal rate of transformation rises:

\[\frac{d\text{MRT}}{dt} = \frac{d}{dt}f^{\prime}(24-t) = -f^{\prime\prime}(24-t) > 0\]

(using the chain rule again).

Remember that Karim’s MRT—the rate at which he can transform free time into consumption—is his wage. We could interpret Marina’s MRT similarly: it is the rate at which she can transform free time into income, so it is effectively her wage, but this ‘wage’ declines from minute to minute as working time rises. With the function we used to draw Figure E3.2, her effective hourly wage (MRT) at point B where \(t=22\) is $42.81; but at A, where she works much longer hours and has only 16 hours of free time, it has fallen to $19.16 per hour.

Exercise E3.3 Calculating and interpreting the marginal rate of transformation

Suppose Marina’s feasible frontier has the equation \(f(t) = 100\ln(25\ –\ t)\).

1. Sketch her feasible frontier, with consumption per day ($) on the vertical axis and hours of free time per day on the horizontal axis for \(t\) = 8,…, 24.
2. Using calculus, derive an expression for Marina’s MRT. Use your sketch and calculus to explain how the MRT changes with Marina’s hours of free time.
3. Calculate Marina’s MRT at 16 and 22 hours of free time and compare it with the MRTs of the feasible frontier in Figure E3.2. Discuss some factors that might affect an individual’s MRT.

Read more: Section 7.2 (for the composite function rule) of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed, 2023). Manchester: Manchester University Press.