Unit 5 The rules of the game: Who gets what and why

5.4 Setting up a model: Technology and preferences

We will now develop a model that allows us to study differing institutions and how they affect who does what, and who gets what, in an economic interaction. The key idea is that institutions (the rules of the game) affect the choices open to people (their feasible sets) and the power that members of some groups can exercise over others. As summarized in the title of this unit—The rules of the game: Who gets what and why—different rules of the game distribute income among members of society in different ways. The institutions affect the efficiency and fairness of the resulting allocations of the game.

We model an interaction between a farmer, Angela, who produces grain, and Bruno, who owns the land that Angela farms. The amount of grain produced depends on how many hours Angela works each day. We can think of the grain they each obtain as their income from the interaction; it is Angela’s only source of income, so if she receives too little grain, she will starve.

Depending on the rules of the game, and her alternative options, how much power Bruno has over Angela differs. His power ranges from being able to threaten her physically and coerce her to work long hours, while receiving little of the grain she produces, to collecting a payment from her for using his land when she has the power simply to say no and walk away. His power is diminished when the rules change and they can negotiate a mutually agreeable bargain.

The nature and extent of Bruno’s and Angela’s power determines how many hours Angela works and how the grain she produces is divided between the two. Different rules of the game result in different pay-offs for each player. This is another case where we use a two-person game to represent how entire groups of people interact in society—landowners and renters, for example.

preferences
A description of the relative values a person places on each possible outcome of a choice or decision they have to make.
technology
The description of a process that uses a set of materials and other inputs, including the work of people and machines, to produce an output.
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.
production function
A production function is a graphical or mathematical description of the relationship between the quantities of the inputs to a production process and the amount of output produced.
indifference curve
A curve that joins together all the combinations of goods that provide a given level of utility to the individual.

While the institutions differ, the preferences of Bruno and Angela, and the technology that Angela uses to produce grain, are the same in each case:

  • Angela wants: the best-for-her feasible combination of grain and free time, according to her preferences (and her resulting indifference curves).
  • Bruno wants: as much grain as possible (he is not doing any work).
  • The feasible set of hours of Angela’s work and the total amount of grain to be divided among the two, as given by the farming technology (the production function).

Angela’s and Bruno’s preferences

We assume that (unlike the experimental participants in Unit 4, and others) our two actors are entirely self-interested: their preferences concern only what they get for themselves.

Angela values both grain (her income, which she consumes) and free time. We can model her preferences in the same way as for Karim in Unit 3, by drawing indifference curves.

Each point in Figure 5.3a shows a combination of grain (measured in bushels) and free time, and the indifference curves join together combinations that she values equally. For example, Angela is indifferent between having 16 hours of free time and consuming 33 bushels of grain, and having only 10 hours of free time but consuming 56 bushels of grain: both of these combinations are on the indifference curve IC3. But if we move from any point on IC3 to another point above and to the right, she prefers that combination because it gives her more of both goods. Higher indifference curves, like IC4 and IC5, correspond to higher levels of utility.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. There are five parallel, downward-sloping, convex indifference curves, labelled IC1, IC2, IC3, IC4 and IC5 from the lower-most to the upper-most. Utility increases further away from the origin.
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Figure 5.3a Angela’s indifference curves for free time and grain.

Mathematically, the slope of the indifference curves is a negative number. But the MRS is a positive quantity representing the size of the trade-off. We often say ‘the MRS is equal to the slope’, when strictly speaking we mean the absolute value of the slope.

marginal rate of substitution (MRS)
The trade-off that a person is willing to make between two goods. At any point, the MRS is the absolute value of the slope of the indifference curve. See also: marginal rate of transformation.
marginal utility
The additional utility resulting from a one-unit increase in the amount of a good.

The slope of the indifference curve at any point corresponds to the marginal rate of substitution (MRS) between grain and free time. It represents the trade-off Angela is willing to make between the two goods. The steeper the indifference curve, the more she values free time relative to grain.

The more free time she has (further to the right), the flatter the curves are—as she values free time less. So she would be unwilling to give up much grain for an extra hour of free time. In other words, when she already has plenty of free time, its marginal utility—the additional utility she would get from an extra hour—is low compared with the marginal utility of grain. Her marginal utility from free time diminishes as the amount of free time increases.

If the vertical distance between the curves seems to get wider as you move from left to right, this is an optical illusion. You may want to convince yourself by measuring it.

We make an assumption about Angela’s preferences: her indifference curves are vertical shifts of each other. This means, firstly, that the vertical distance between two curves is the same whatever her amount of free time. The arrows in Figure 5.3b show that the distance between IC3 and IC4 is the same whether she has 12 or 18 hours of free time.

Preferences where the slope of all indifference curves is the same for a given horizontal axis value are called quasi-linear. The extension of this section shows you the mathematical form and properties of quasi-linear utility functions.

Secondly, for each level of free time, the slope is the same on every indifference curve. The tangents to the indifference curves where the amount of free time is 18 hours are all parallel to each other. In other words, Angela’s MRS depends on the amount of free time she has, but does not change if she receives more or less grain.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. There are fivedownward-sloping, convex indifference curves that do not cross. Point Z lies on the second highest indifference curve at 12 hours of free time. The vertical distance between the second and third highest indifference curves at 12 hours of free time is 17 bushels, and it’s the same as the vertical distance between the same curves at 18 hours of free time.
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Figure 5.3b Angela’s MRS depends on her amount of free time but not on the amount of grain.

We adopt this assumption to simplify the analysis. In particular, it allows us to measure in bushels of grain how much Angela’s utility differs between one allocation and another. For example, we can say that she prefers point Z to point Y by the equivalent of 17 bushels of grain.

Figure 5.3c shows Bruno’s preferences using the same axes. How long Angela spends producing grain does not matter to him—he doesn’t care how much free time she has. He is interested only in the amount of grain that he, as the landowner, receives—the more the better. So his indifference curves are horizontal.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. There are five parallel, horizontal lines. Utility increases further upwards from the horizontal axis.
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Figure 5.3c Bruno’s preferences for grain and Angela’s free time.

Angela’s technology

We set aside until Unit 6 the important fact that the amount of work done depends not only on hours of work but also on how hard and carefully the person works.

The feasible combinations of grain, and free time for Angela, are determined by the farm’s technology for producing grain. Figure 5.4 shows the production function, which tells us how the amount of grain produced (the output) depends on how much work Angela does, (the input, measured in hours per day). It is similar to the grain production function in Section 1.6; the main difference is that in Section 1.6, the input is the total number of farmers working the land, whereas here it is the number of hours worked per day on one farm by a single farmer.

average product
The average product of an input is total output divided by the total amount of the input. For example, the average product of a worker (also known as labour productivity) is total output divided by the number of workers employed to produce it.
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.

If Angela works for five hours a day, she produces 37 bushels of grain (point T in the figure). Her average product of labour is 37/5 = 7.4 bushels. The average product corresponds to the slope of the ray from the origin to point T. Her production function (again like the one in Section 1.6) has a concave shape: the average product of an hour’s work diminishes as the number of hours increases. As before, this happens because the amount of land available is fixed: working twice as many hours on the same amount of land would not double its output.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). An upward-sloping, convex curve starts from the origin, passes through point T (5, 37), and is labelled production function.
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Working hours 0 1 2 3 4 5 6 8 12 16 24
Grain 0 17 24 29 34 37 41 46 54 60 64

Figure 5.4 Angela’s production function.

Using the information in Figure 5.4, we can work out which combinations of grain production and free time for Angela are feasible. As in Unit 3, what we call free time is all of the time that is not spent working to produce grain—it includes time for eating, sleeping, and everything else that we don’t count as farm work, as well as her leisure time. We know, for example, that by working for five hours, Angela could produce 37 bushels of grain. So 19 hours of free time and 37 bushels of grain are feasible. In the table in Figure 5.5, we have worked out the amounts of free time corresponding to each quantity of grain produced. Then we have plotted each combination of grain and free time to obtain the feasible frontier.

The feasible frontier in Figure 5.5 is the mirror image of the production function in Figure 5.4, with free time instead of hours of work on the horizontal axis. It shows how much grain can be produced and consumed for each possible amount of free time.

Like the MRS, the MRT is a positive number although the slope is negative. More accurately, the MRT equals the absolute value of the slope.

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.
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.’

The slope of the feasible frontier is the marginal rate of transformation (MRT) of free time into grain. At each point on the frontier, it tells us the trade-off Angela faces: how much grain Angela would have to give up to get one more unit of free time. The table in Figure 5.5 shows that if Angela were to reduce her free time from 19 hours (at point T) to 18 hours, her production of grain would increase from 37 to 41 bushels. So the marginal rate of transformation at point T is four bushels per hour. Equivalently the opportunity cost of an hour of free time at point A is four bushels of grain.

Karim’s feasible frontier in Unit 3 is a straight line, so for him the MRT is the same at every point on the frontier. For Angela, the MRT changes: the more free time she takes, the greater is the MRT—when she already has a lot of free time the opportunity cost of taking another hour is higher: how much grain she has to give up.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64), T (19, 37) and (24, 0) and is labelled feasible frontier.
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Free time 24 23 22 21 20 19 18 16 12 8 0
Grain 0 17 24 29 34 37 41 46 54 60 64

Figure 5.5 Angela’s feasible frontier.

Question 5.1 Choose the correct answer(s)

Read the following statements about Figure 5.3b and Figure 5.4 and choose the correct option(s).

  • Angela’s indifference curves are flatter when she has more free time because she has diminishing marginal utility of grain.
  • For a particular amount of free time, the slope of Angela’s indifference curves are the same.
  • We can derive Angela’s feasible frontier from her production function.
  • The slope of the feasible frontier is the MRS.
  • The indifference curves are flatter for combinations to the right, but the reason is diminishing marginal utility of free time, not of grain. When Angela has more free time, she values free time less relative to grain, compared to when she has very little free time.
  • Angela’s indifference curves have this particular property, so her MRS depends on her amount of free time but not on the amount of grain.
  • The production function shows how much grain Angela can produce for each hour she works. Since there are 24 hours in a day, we can use this information to derive the feasible frontier, which shows the possible combinations of leisure time and grain which are available to Angela.
  • The slope of the feasible frontier is the MRT. The MRS is the slope of Angela’s indifference curves.

Extension 5.4 The properties of concave production functions and quasi-linear preferences

We analyse the mathematical properties of Angela’s technology and preferences, using calculus. Before working through this extension you should read the introduction to production functions in Extension 2.4. You will also need to be familiar with the mathematical analysis of preferences in Extension 3.3 (essential) and the marginal rate of transformation in Extension 3.4.

Angela’s technology and preferences have properties that can plausibly be assumed to hold in many economic models. Her production function has a concave shape, like the agricultural production function in Unit 1: it slopes upward, but gets gradually flatter as working hours increase. Her preferences, like those of Karim in Unit 3, are convex (as discussed in Extension 3.3): in other words, her indifference curves slope downward, getting gradually flatter as we move to the right. In addition, her preferences have a special property that is useful in some models: they are quasi-linear.

In this extension, we describe these properties mathematically. First, it is helpful to clarify what we mean by ‘concave’ and ‘convex’.

Concave and convex functions

concave, concave function
A function, \(f(x)\), is said to be concave if its second derivative is negative for all values of x.
convex, convex function
A function, \(f(x)\), is said to be convex if its second derivative is positive for all values of x.

A function, \(f(x)\), is said to be concave if its second derivative is negative for all values of \(x\); that is, if \(f''(x)\leq0\). If \(f''(x)<0\) for all values of \(x\), the function is strictly concave.

If a function is concave, its slope \(f'(x)\) decreases as \(x\) increases.

Conversely, a function, \(f(x)\), is said to be convex if its second derivative is positive for all values of \(x\); that is, if \(f''(x)\geq0\). If \(f''(x)>0\) for all \(x\), it is strictly convex.

If a function is convex, its slope \(f'(x)\) increases as \(x\) increases.

Figure E5.1 compares the graphs of four possible functions, \(y=f(x)\). The top two are both increasing functions (their slopes are positive), as we would generally expect production functions to be. The left-hand one is concave—like all the examples of production functions we use in Units 1 to 5. Its slope decreases as \(x\) increases, so if we join any two points on the curve, the line lies below the curve. The right-hand one illustrates a convex production function: its slope increases with \(x\), so a line joining any two points on the curve lies above the curve.

There are 4 diagrams. In each diagram, the horizontal axis shows the variable x, and the vertical axis shows the variable f(x). Diagram 1 shows an increasing and concave curve. The first derivative of f(x) is positive and the second derivative is negative. Diagram 2 shows an increasing and convex curve. The first and second derivatives of f(x) are both positive. Diagram 3 shows a decreasing and concave curve. The first and second derivatives of f(x) and both negative. Diagram 4 shows a decreasing and convex curve. The first derivative of f(x) is negative and the second derivative is positive.
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Figure E5.1 Concave and convex functions.

The bottom two are both decreasing functions—we draw decreasing functions to represent indifference curves, for example. The left-hand one is concave: as \(x\) increases, its slope decreases, getting more negative. Be careful here; the curve gets steeper because the absolute value of the slope increases, but the slope itself is decreasing. Hence \(f''(x)<0\): it is concave. The bottom right shows a convex decreasing function; this is the shape we expect most indifference curves to have. The curve gets flatter (its slope gets less negative) as \(x\) increases, which corresponds to a positive second derivative.

Joining two points on the curve with a straight line is a quick and easy way to distinguish between concave and convex functions. In Figure E5.1, the line lies below the curve for the two concave functions on the left, and above for the two convex functions on the right.

Economic and mathematical properties of production functions

Angela’s production function, shown in Figure 5.4, is similar to the grain production function in Section 1.6 and the olive oil production function in Extension 2.4. All of them show how the output of a product increases with the amount of labour used to produce it. If output rose in proportion to labour, the production function would be a straight line, with a constant slope. But in each of these examples, other inputs to production (land, or machinery) are fixed, so output rises less than proportionally to the labour input: the slope of the function gradually decreases as more labour is employed. In other words, these examples all have increasing and strictly concave production functions.

Figure E5.2a shows another increasing and strictly concave function, which we will suppose is the production function, \(y=g(h)\), of a farmer like Angela, where \(h \ge 0\) is daily hours of work, \(y\) is the number of bushels of grain produced, and \(g(0)=0\).

The function shown has a simple algebraic form that is often used in economic examples:

\[g(h)=ah^b \text{ where $a$ and $b$ are constants: } a>0, 0<b<1\]

In the figure, \(a=10\) and \(b=0.4\).

The figure shows that the function \(g(h)\) is increasing and strictly concave. We can verify this algebraically, using what we know about the constants, \(a\) and \(b\):

\[g'(h) = abh^{b-1}>0 \text{ and } g''(h)=ab(b-1)h^{b-2}<0 \text{ for all } h>0\]
In this diagram, the horizontal axis shows hours of work, denoted h, ranging from 0 to 24, and the vertical axis shows bushels of grain, denoted y, ranging from 0 to 40. Coordinates are (hours of work, bushels of grain). The production function is an upward-sloping concave curve with the equation y = 10 times h to the power of 0.4, which passes through the points (0, 0) and P (5, 19). The tangent line to point P shows the marginal product of labour, and the slope of the ray from (0, 0) to point P shows the average product of labour.
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Figure E5.2a A concave production function, \(y = 10h^{0.4}\).

marginal product
The marginal product of an input to production (for example, the marginal product of labour) is the additional amount of output produced in response to a 1-unit increase in the input.

The slope (derivative) of the production function tells us the marginal product of labour: that is, how much additional output is produced as more labour is employed. In other words, it is the rate at which output rises with increases in labour:

\[\text{MPL} = g'(h)\]

Here, we define the marginal product of labour as the rate of increase of output corresponding to a small (infinitesimal) increase in the input, although it is often interpreted as the rise in output when the labour input rises by one whole unit (one hour, or one worker, for example)—which is not quite the same.

As explained in Extension 3.3, we use the calculus method to measure marginal changes in the mathematical extensions, while in the main sections, we usually measure them as the effect of an increase in one unit. In general, the rate of increase, \(g'(h)\), differs from the rise in output when \(h\) increases by one unit, which is \(g(h+1)-g(h)\)—unless \(h\) is measured in very small units.

Figure E5.2a shows the tangent to the production function at point P where \(h=5\) and \(y=19.04\). The slope of the tangent is 1.52—that is, \(g'(5)=1.52\). We say that when \(h=5\) the marginal product of labour is 1.52 bushels of grain per hour. When the production function is concave, the marginal product of labour diminishes as the labour input rises.

average product
The average product of an input is total output divided by the total amount of the input. For example, the average product of a worker (also known as labour productivity) is total output divided by the number of workers employed to produce it.

The marginal product is not the same as the average product. The average product of labour at \(h\) hours of work is the average amount produced over all the \(h\) hours of labour employed:

\[\text{APL} = \frac{g(h)}{h}\]

In Figure E5.2a, the APL at point P corresponds to the slope of the ray from the origin to P, which is 19.04/5 = 3.8.

We noted in the main part of this section that Angela’s APL diminishes with her hours of work. This property holds for any strictly concave production function. Intuitively, if an extra hour adds less to output than each of the hours previously worked (MPL is diminishing), then the average output over all hours worked must also go down (APL is diminishing).

Furthermore, if we differentiate the average product with respect to \(h\) using the quotient rule, we get:

\[\frac{d \text{APL}}{dh} = \frac{d}{dh} \left( \frac{g(h)}{h} \right) = \left( \frac{hg'(h) - g(h)}{h^2} \right)\]

So the property that the average product is diminishing is equivalent to the property that the marginal product is less than the average product:

\[\begin{align*} \frac{d \text{APL}}{dh} < 0 & \Rightarrow hg'(h) - g(h) < 0 \Rightarrow g'(h) < \frac{g(h)}{h} \end{align*}\]

Figure E5.2a shows this property: the slope of the tangent at P is less than the slope of the line between P and the origin. We can prove it directly for production functions of the form, \(g(h)=ah^b\), where \(a>0\) and \(0<b<1\):

\[g’(h)=abh^{b-1} < ah^{b-1}=\frac{g(h)}{h}\]

The feasible frontier and the marginal rate of transformation

If the technology is described by a production function, \(y=g(h)\), where \(y\) is output of grain, \(h\) is hours of work per day, and \(g\) is an increasing and strictly concave function, what can we say about the shape of the feasible frontier?

goods
Economists sometimes use this word in a very general way, to mean anything an individual cares about and would like to have more of. As well as goods that are sold in a market, it can include (for example) ‘free time’ or ‘clean air’.

To find the equation of Angela’s feasible frontier we rewrite the technology in terms of the two things that are goods for her: output and hours of free time, \(t\). Since \(h=24-t\), the feasible frontier is:

\[y=g(24-t)\]

Differentiating with respect to \(t\), using the composite function rule (sometimes called the chain rule):

\[\frac{dy}{dt}=g'(24-t)\frac{d}{dt}(24-t)=-g'(24-t)\]

Since \(g\) is an increasing function \((g'>0)\), we can deduce that the slope of the feasible frontier is negative. Furthermore, by differentiating again we can deduce that it is strictly concave:

\[\frac{d^2y}{dt^2}=-g''(24-t)\frac{d}{dt}(24-t)=g''(24-t)<0\]

As explained in Extension 3.4, the absolute value of the slope of the feasible frontier is the marginal rate of transformation—in this case, between free time and grain:

\[\text{MRT}=\left|\frac{dy}{dt}\right|=g'(24-t)\]

The MRT is equal to the marginal product of labour. The MRT increases along the frontier, as \(t\) rises and \(y\) falls:

\[\frac{d\text{MRT}}{dt}=-g''(24-t)>0\]

When the production function is \(g(h)=ah^b\), the equation of the feasible frontier is \(y=a(24-t)^b\). Differentiating (and remembering that \(a>0\) and \(0<b<1\)):

\[\begin{align*} \frac{dy}{dt}&=-ab(24-t)^{b-1}<0;\ \frac{d^2y}{dt^2}=ab(b-1)(24-t)^{b-2} < 0 \end{align*}\]

So the feasible frontier is decreasing and concave in \(t\), and the MRT is \(ab(24-t)^{b-1}\). Figure E5.2b shows the feasible frontier corresponding to the example in Figure E5.2a, where \(a=10\) and \(b=0.4\). The feasible frontier is the mirror image of the production function.

In this diagram, the horizontal axis shows hours of free time, denoted t, ranging from 0 to 24, and the vertical axis shows bushels of grain, denoted y, ranging from 0 to 40. Coordinates are (hours of free time, bushels of grain). The feasible frontier is a downward-sloping concave curve with the equation y = 10 times t to the power of 0.4, which passes through the points (0, 35.7), (19, 19), and (24, 0).
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Figure E5.2b A concave feasible frontier, \(y = 10(24 - h)^{0.4}\).

Angela’s preferences: Quasi-linearity

convex preferences
A person whose indifference curves have a convex shape—they get flatter as you move along the curve to the right of the diagram—is said to have convex preferences. This typical shape arises because when someone has more of one good (relative to another) they are willing to give up more of it in exchange for a unit of the other good: their marginal rate of substitution falls along the curve.

We have said that we generally expect indifference curves to be decreasing and convex. This is what we assume for Karim in Extension 3.3, and for Angela in the main part of this section. We say that Angela, like Karim, has convex preferences. The absolute value of the slope of the indifference curve—the marginal rate of substitution—falls as the amount of the good on the horizontal axis increases; in other words (as seems plausible) the more they have of that good, the more willing they are to trade it for the other. If we write the equation of the indifference curve so that the good on one axis is a function of the good on the other axis, it is a convex function.

quasi-linear, quasi-linear function
A utility function is said to be quasi-linear if it depends linearly on the amount of one good, and non-linearly on another. The marginal rate of substitution between the two goods then depends only on the non-linear variable.

But Angela’s preferences for consumption and free time also have a special property: her indifference curves are vertical shifts of each other. This property is known as quasi-linearity. It implies, as explained in the main part of this section, that her MRS between grain and free time depends on the amount of free time she has, but does not change if she receives more or less grain. The assumption that preferences are quasi-linear is less plausible in general, but it is a useful simplification that is used in some models to help us focus on particular aspects of a problem.

Let \(t\) be Angela’s daily hours of free time, and \(c\) the number of bushels of grain that she consumes per day. Quasi-linearity is illustrated in Figure 5.3b, reproduced below as Figure E5.3. Remember that the MRS at any point is the slope of the indifference curve. The figure shows that all the tangents at \(t=18\) are parallel to each other. So if Angela has 18 hours of free time, her MRS is the same however much grain she consumes.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. There are five downward-sloping, convex indifference curves that do not cross. Point Z lies on the second highest indifference curve at 12 hours of free time. The vertical distance between the second and third highest indifference curves at 12 hours of free time is 17 bushels, and it’s the same as the vertical distance between the same curves at 18 hours of free time.
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Figure E5.3 Angela’s MRS depends on her amount of free time but not on the amount of grain.

Quasi-linear preferences can be represented by a utility function of the form:

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

where \(v\) is an increasing function of \(t\): \(v'(t) \gt 0\) because Angela prefers more free time to less. A utility function like this is called quasi-linear because utility is linear in \(c\). We now show that this utility function has the required property.

In Extension 3.3, we prove that for any utility function the MRS is the ratio of the marginal utilities:

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

Applying this formula to the case of the quasi-linear utility function, \(\frac{\partial u}{\partial t}=v'(t)\) and \(\frac{\partial u}{\partial c}=1\), so:

\[\text{MRS} = v'(t)\]

Hence Angela’s MRS between consumption and free time depends on how much free time, \(t\), she has, but not on her consumption, \(c\).

The same result can be obtained directly, without using the general formula. The equation of an indifference curve is \(v(t) + c = u_0\) where \(u_0\) is a constant. Rearranging this equation, we can write \(c\) in terms of \(t\):

\[c=u_0-v(t)\]

Hence the slope of an indifference curve is \(\frac{dc}{dt}=-v’(t)\), and the MRS is the absolute value of the slope, \(v'(t)\).

Writing the indifference curves this way also demonstrates that they are vertical shifts of each other. The vertical distance between an indifference curve corresponding to utility level, \(u_0\), and another with higher utility, \(u_1\), is \((u_1-v(t))-(u_0-v(t))=u_1-u_0\), which doesn’t depend on \(t\).

Convex quasi-linear preferences

In Figure E5.3, the indifference curves are both quasi-linear, and convex—they have the usual property of diminishing MRS, flattening as you move to the right. Since the MRS is \(v'(t)\), it must be the case that \(v’(t)\) falls at \(t\) increases. In other words, quasi-linear preferences are also convex if:

\[v''(t) < 0\]

Somewhat confusingly, this means that \(v(t)\) must be a concave function. As noted above, we can rearrange the indifference curves to write \(c\) as a function of \(t\): \(c=u_0-v(t)\). Then:

\[\frac{dc}{dt}=-v'(t) \text { and }\frac{d^2c}{dt^2}=- v''(t)\]

Hence the indifference curves are convex if and only if \(v(t)\) is concave: \(v''(t) > 0\).

Why do we use the assumption that preferences are quasi-linear?

Using a quasi-linear utility function means that we are making a restrictive assumption about preferences, which would not be plausible in all economic models. But it has a very useful implication. Because utility is of the form \('c + \text{something}'\), it is measured in the same units as consumption. Angela values \(t\) hours of free time as much as \(v(t)\) bushels of grain.

This is particularly useful for a model of a worker who values free time, and also all the goods she can buy with the income she earns. Imagine that, rather than simply consuming grain, Angela can sell grain on the market and purchase other food or clothes—or anything else—with the proceeds. Then everything other than free time can be valued in the same units: money income. If we model her preferences as quasi-linear, we can use money income to measure overall gains and loss of utility resulting from changes in consumption, free time, or both.

Quasi-linearity: An example

An example of a quasi-linear utility function is:

\[u(t,\ c) = c+ \beta t^\alpha\]

where \(\beta\) and \(\alpha\) are positive constants and \(\alpha\lt1\). It has the general form, \(v(t) +c\), with \(v(t)= \beta t^\alpha\). To demonstrate that it is a quasi-linear utility function as described above, we must show that the function, \(v\), is increasing and concave in \(t\). This is easily done:

\[v'(t)= \alpha \beta t^{\alpha -1}\]

which is positive because \(\beta\) and \(\alpha\) are positive, and

\[v''(t)= (\alpha -1)\alpha \beta t^{\alpha -2}\]

which is negative because \(\beta > 0\) and \(0\lt\alpha < 1\).

Alternatively, we can start with the equation of an indifference curve in the form \(c=u_0-\beta t^\alpha\), and differentiate to find \(\frac{dc}{dt}\) and \(\frac{d^2c}{dt^2}\).

Exercise E5.1 Quasi-linear utility functions

For each utility function, determine (by taking derivatives) whether it is quasi-linear, and calculate the MRS.

  1. \(u(t,c) = c + bln(t)\) 
  2. \(u(t, c) = c + \frac{1}{(1 + t)^{b}} \text{ for } b \gt 1\) 
  3. \(u(t,c) = (ac^2 + (1-a)t^2)^{1/2}\) 

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.