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

## 3.3 Goods and preferences

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’.
preferences
A description of the relative values a person places on each possible outcome of a choice or decision they have to make.

We can think of both free time and total consumption spending as goods for Karim. Economists use the word ‘good’ to refer to anything that people care about, and would like to have more of. We will simplify the model by assuming he doesn’t care about anything else; in particular, he doesn’t care about the future, so is not interested in saving any of his income. We will also assume that his average spending cannot exceed his earnings (for example, he cannot borrow to increase his consumption).

Since Karim cares about both goods, his choice of working hours involves thinking about a trade-off: how much consumption is he willing to give up, in order to have more free time? To understand the decision Karim will make, we need to know his preferences—how much he values the two goods, relative to each other.

We illustrate his preferences using Figure 3.4, with free time on the horizontal axis and consumption on the vertical axis. Free time is defined as all the time that he does not spend working. Every point in the diagram represents a different combination of free time and consumption spending. Given his wage, many of these combinations will not be possible for Karim. But for the moment we will focus on which combinations he would prefer if he could have them.

We can assume that:

• For a given amount of consumption, he prefers a combination with more free time to one with less free time. Therefore, even though both A and B in Figure 3.4 correspond to €540 of consumption, Karim prefers A because it gives him more free time.
• Similarly, if two combinations both have 20 hours of free time, he prefers the one with a higher consumption.
• But compare points A and D in the table. Would Karim prefer D (low consumption, plenty of time) or A (higher consumption, less time)? One way to find out would be to ask him.
utility
A numerical indicator of the value that one places on an outcome. Outcomes with higher utility will be chosen in preference to lower valued ones when both are feasible.

Suppose he says he is indifferent between A and D, meaning he would feel equally satisfied with either outcome. We say that these two outcomes would give Karim the same utility. And we know that he prefers A to B, so B provides lower utility than A or D.

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250

Mapping Karim’s preferences.

Figure 3.4 Mapping Karim’s preferences.

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250

Karim prefers more free time to less free time

Combinations A and B both deliver €540 of consumption, but Karim will prefer A because it has more free time.

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250

Karim prefers more consumption to less

At combinations C and D, Karim has 20 hours of free time per day, but he prefers D because it gives him more consumption …

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250

Indifference

… but we don’t know whether Karim prefers A (with higher consumption) or E (more free time), so we ask him: he says he is indifferent.

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250

More combinations giving the same utility

Karim says that F is another combination that would give him the same utility as A and E.

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250

Constructing an indifference curve

By asking more questions, we discover that Karim is indifferent between all of these combinations between A and D.

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250

Constructing an indifference curve

These points are joined together to form an indifference curve.

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250

Other indifference curves

Indifference curves can be drawn through any point in the diagram, to show other points giving the same utility. We can construct other curves starting from B or C in the same way as before, by finding out which combinations give the same amount of utility.

In this model, we can think of utility as a measure of Karim’s overall living standards, taking into account that he cares about free time as well as consumption.

A systematic way to graph Karim’s preferences would be to start by plotting all of the combinations that give him the same utility as A and D. We could ask Karim another question: ‘Imagine that you could have the combination at A (15 hours of free time, €540). How much consumption, in euros, would you be willing to sacrifice for an extra hour of free time?’ Suppose that after due consideration, he answers ‘€94’. Then we know that he is indifferent between A and E (16 hours, €446). Then we could ask the same question about combination E, and so on until point D. Eventually we could draw up a table like the one in Figure 3.4. Karim is indifferent between A and E, between E and F, and so on, which means he is indifferent between all of the combinations from A to D.

indifference curve
A curve that joins together all the combinations of goods that provide a given level of utility to the individual.
consumer good
Any good that can be bought by consumers, including both short-lived goods and long-lived goods, which are called consumer durables.

The combinations in the table are plotted in Figure 3.4, and joined together to form a downward-sloping curve, called an indifference curve, which shows all of the combinations that provide equal utility or ‘satisfaction’.

Of the three curves drawn in Figure 3.4, the one through A gives higher utility than the one through B. The curve through C gives the lowest utility of the three. To describe preferences, we don’t need to know the exact utility of each option; we only need to know which combinations provide more or less utility than others.

The curves we have drawn capture our typical assumptions about people’s preferences between two goods. In this model of Karim’s preferences, the goods are ‘consumption spending’ and ‘free time’. In other models, they will often be particular consumer goods such as food or clothing, and we refer to the person as a consumer. We typically assume the following:

• Indifference curves slope downward due to trade-offs: If you are indifferent between two combinations, the combination that has more of one good must have less of the other good.
• Higher indifference curves correspond to higher utility levels: As we move up and to the right in the diagram, further away from the origin, we move to combinations with more of both goods.
• Indifference curves are usually smooth: Small changes in the amounts of goods don’t cause big jumps in utility.
• Indifference curves do not cross: Work through the steps in Exercise 3.1 to understand why.
• As you move to the right along an indifference curve, it becomes flatter.
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.

To understand the last property in the list, we plot Karim’s indifference curves again in Figure 3.5. If he is at A, with 15 hours of free time and €540 of consumption, he would be willing to sacrifice €94 of consumption for an extra hour of free time, taking him to E (remember that he is indifferent between A and E). We say that his marginal rate of substitution (MRS) between consumption and free time at A is 94; it is the reduction in his level of consumption that would keep Karim’s utility constant following a one-hour increase of free time.

We have drawn the indifference curves as becoming gradually flatter because it seems reasonable to assume that the more free time and less consumption he has, the less willing he will be to sacrifice further consumption in return for free time, so his MRS will be lower. In Figure 3.5, we have calculated the MRS at each combination along the indifference curve. When Karim has more free time and less consumption, the MRS—the amount of spending he would give up to get an extra hour of free time—gradually falls.

Figure 3.5 The marginal rate of substitution.

Karim’s indifference curves

The diagram shows three indifference curves for Karim. The curve furthest to the left offers the lowest utility.

Point A

At A, he has 15 hours of free time and €540 to spend.

Karim is indifferent between A and E

He would be willing to move from A to E, giving up €94 for an extra hour of free time. His marginal rate of substitution is 94. The indifference curve is steep at A.

Karim is indifferent between H and D

At H, he is only willing to give up €32 for an extra hour of free time. His MRS is 32. As we move down the indifference curve, the MRS diminishes, because consumption spending becomes scarce relative to free time. The indifference curve becomes flatter.

All combinations with 15 hours of free time

Look at the combinations with 15 hours of free time. On the lowest curve, consumption is low. The curve is quite flat, so MRS is small. Karim would be willing to give up only a little consumption for an hour of free time. As we move up the vertical line, the indifference curves are steeper: the MRS increases.

All combinations with €282 of consumption

Now consider all the combinations with €282 of consumption. On the curve farthest to the left, free time is scarce, and the MRS is high. As we move to the right along the red line, he is less willing to give up consumption for free time. The MRS decreases—the indifference curves get flatter.

The MRS corresponds to the slope of the indifference curve, and it falls as we move to the right along the curve. If you think about moving from one point to another in Figure 3.5, the indifference curves get flatter if you increase the amount of free time, and steeper if you increase consumption. When free time is scarce relative to consumption, Karim is less willing to sacrifice an hour for more consumption spending: his MRS is high and his indifference curve is steep.

As the analysis in Figure 3.5 shows, if you move up the vertical line through 15 hours, the indifference curves get steeper: the MRS increases. For a given amount of free time, Karim is willing to give up more consumption for an additional hour when he has high consumption, compared to when his consumption is low (for example, if he is struggling to afford enough food). By the time you reach A, where his consumption is €540, the MRS is high; consumption is so plentiful here that he is willing to give up €94 for an extra hour of free time.

The same reasoning applies if you fix consumption and vary the amount of free time. If you move to the right along the horizontal line for €282, the MRS becomes lower at each indifference curve. As free time becomes more plentiful, Karim becomes less and less willing to give up consumption for more time.

We have said that the MRS corresponds to the slope of the indifference curve, but note that the MRS is a positive number, while the slope of the indifference curve is negative. To be precise, the MRS is equal to the absolute value of the slope.

### Exercise 3.1 Why indifference curves never cross

In this diagram, IC1 is an indifference curve joining all the combinations that give the same level of utility as A. Combination B is not on IC1.

1. Does combination B give higher or lower utility than combination A? How do you know?
2. Draw a sketch of the diagram, and add another indifference curve, IC2, that goes through B and crosses IC1. Label the point at which they cross as C.
3. Combinations B and C are both on IC2. What does that imply about their levels of utility?
4. Combinations C and A are both on IC1. What does that imply about their levels of utility?
5. According to your answers to 3 and 4, how do the levels of utility at combinations A and B compare?
6. Now compare your answers to 1 and 5, and explain how you know that indifference curves can never cross.

### Exercise 3.2 Your marginal rate of substitution

Imagine that you are offered a job at the end of your university course with a salary per hour (after taxes) of £12.50. Your future employer then says that you will work for 40 hours per week, leaving you with 128 hours of free time per week. You tell a friend: ‘at that wage, 40 hours is exactly what I would like.’

1. Draw a diagram with free time on the horizontal axis and weekly pay on the vertical axis, and plot the combination of hours and the wage corresponding to your job offer, calling it A. Assume you need about 10 hours a day for sleeping and eating, so you may want to draw the horizontal axis with 70 hours at the origin.
2. Now draw an indifference curve so that A represents the hours you would have chosen yourself.
3. Now imagine you were offered another job requiring 45 hours of work per week. Use the indifference curve you have drawn to estimate the level of weekly pay that would make you indifferent between this and the original offer.
4. Do the same for another job requiring 35 hours of work per week. What level of weekly pay would make you indifferent between this and the original offer?
5. Use your diagram to estimate your marginal rate of substitution between pay and free time at A.

### Question 3.4 Choose the correct answer(s)

Figure 3.4, shows Karim’s indifference curves for free time and consumption. Based on this information, read the following statements and choose the correct option(s).

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250
• Karim prefers C to B because at C he has more free time.
• Karim is indifferent between €540 of consumption with 15 hours of free time, and €250 of consumption with 20 hours of free time.
• Karim prefers D to C, because at D he has the same consumption and more free time.
• At G, Karim is willing to give up two hours of free time for €73 more consumption.
• The indifference curve through C is lower than that through B. Hence Karim prefers B to C.
• A, where Karim has €540 of consumption and 15 hours of free time, and D, where Karim has €250 of consumption with 20 hours of free time, are on the same indifference curve.
• At D, Karim has the same amount of free time but higher consumption.
• The opposite trade-off is true: going from G to D, Karim is willing to give up €73 of consumption for two extra hours of free time. Going from G to E, he is willing to give up two hours of free time for €123 of consumption.

### Question 3.5 Choose the correct answer(s)

What is the marginal rate of substitution (MRS)?

• The ratio of the amounts of the two goods at a point on the indifference curve
• The amount of one good that the consumer is willing to trade for one unit of the other
• The change in the consumer’s utility when one good is substituted for another
• The change in utility as you move down the indifference curve
• The marginal rate of substitution represents the ratio of the trade-off at the margin, in other words, how much of one good the consumer is willing to sacrifice for one extra unit of the other.
• This is the definition of the marginal rate of substitution.
• The MRS is the amount of one good that can be substituted for one unit of the other while keeping utility constant.
• Utility doesn’t change as you move down the indifference curve. The MRS is the absolute value of the slope of the indifference curve: the trade-off between two goods that keeps utility constant.

### Extension 3.3 Indifference curves, marginal changes, and the marginal rate of substitution

This extension investigates the mathematical properties of indifference curves and utility, using calculus throughout.

It introduces some important concepts that you will need in order to understand extensions in other units that use calculus: most importantly, what we mean by marginal changes in economics.

The Introduction to Mathematical Extensions explains briefly what we mean by calculus and describes the mathematical level, notation, and conventions used.

In Figure 3.4 (reproduced here as Figure E3.1) we mapped Karim’s preferences by finding points that gave him the same utility and joining them with indifference curves. We can compare different points on the figure and say which he prefers. But that doesn’t give us a complete description of his preferences: for example, if we took two points in the area between two indifference curves in Figure E3.1, we might not be able to tell which one he prefers.

A E F G H D
Hours of free time 15 16 17 18 19 20
Consumption (€) 540 446 376 323 282 250

Figure E3.1 Mapping Karim’s preferences.

### The utility function

utility function
A utility function is a mathematical representation of a person’s preferences for one or more goods. It gives a numerical value to the amount of utility the person obtains from each possible combination of goods.

To fully describe preferences we use a utility function, which tells us how a person’s ‘units of utility’ depend on the goods available. Karim only cares about two goods: free time and consumption. If he has $$t$$ hours of free time and $$c$$ units of consumption, his utility can be described by a function:

$u(t, c)$

Since both free time and consumption are goods—Karim would like to have as much of each as possible—the utility function must have the property such that increasing either $$t$$ or $$c$$ would increase $$u$$. We can work out how utility changes if $$t$$ increases by finding the partial derivative: that is, by differentiating the utility function with respect to $$t$$ while treating $$c$$ as a constant. Similarly we can find the partial derivative with respect to $$c$$, to determine how changes in $$c$$ affect utility. The partial derivatives are written as:

$\frac{\partial u}{\partial t} \text{ and } \frac{\partial u}{\partial c}$

Utility is increasing in $$t$$ and $$c$$ if:

$\frac{\partial u}{\partial t}>0; \frac{\partial u}{\partial c}>0$

We say that utility ‘depends positively’ on $$t$$ and $$c$$.

Karim’s utility function has two arguments. Just as a function of one variable may be represented graphically by a curve on a plane, a function of two variables may be represented by a surface in three-dimensional space (as we did in Extension 2.4). But three-dimensional diagrams can be awkward to handle, so economists usually analyse utility graphically using the same technique that is used to represent the three-dimensional space we live in: a contour map of the landscape. Contours are lines joining points of equal height above sea level. Similarly, indifference curves are the contours of the utility surface, joining points of equal utility.

Another example is weather maps, which use lines called isobars to join points of equal air pressure.

Karim’s utility function (which we used to draw Figure E3.1) is:

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

He has subsistence levels of consumption and free time: he needs $$t>6$$ and $$c>45$$ to get any utility at all. With this utility function, his utility at point E (t = 16, c = 446) is u = 40,100. You can check that the other points in the table below Figure E3.1 have the same level of utility (approximately, but not exactly because the consumption values in the table have been rounded to show whole numbers).

In many models, the actual numbers for the values of u are not meaningful. The purpose of the utility function is to capture preferences—to tell us which of two points has the higher level of utility. The level of utility doesn’t matter. We would get exactly the same indifference curves and preferences from (for example) the function $$u = 23(t-6)^2(c-45)$$.

#### Plotting Karim’s indifference curves

If you know that the utility function describing a person’s preferences is $$u(t, c)$$, then the equation of each indifference curve has the form:

$u(t, c) = u_0$

where $$u_0$$ is a constant corresponding to the level of utility on the curve. With Karim’s utility function, the equation of an indifference curve is:

$(t-6)^2(c-45) = u_0$

which can be rearranged to obtain:

$c = \frac{u_0}{(t-6)^2} + 45$

To draw an indifference curve, we can choose a value for $$u_0$$ and plot the graph of this relationship, with $$t$$ on the horizontal axis and $$c$$ on the vertical axis. To plot the indifference curves shown in Figure E3.1, we started by choosing $$u_0=40,100$$ to get the highest indifference curve shown, and then plotted two lower ones using $$u_0=21,000$$ and $$u_0=8,000$$.

#### Calculating Karim’s marginal rate of substitution

Karim’s marginal rate of substitution (MRS)—his willingness to trade consumption for an extra hour of free time—depends on the combination of goods, $$(t, c)$$, that he has. It is given by the absolute value of the slope of the indifference curve $$u(t, c)=u_0$$ through that point.

How can we calculate the slope of the indifference curve $$u(t, c)=u_0$$?

With the particular utility function we used for Karim, this is easy to do using calculus. As before, we rewrite the equation of an indifference curve to give $$c$$ in terms of $$t$$:

$c = \frac{u_0}{(t-6)^2} + 45$

Then we differentiate to find the slope of the curve:

$\frac{dc}{dt} = \frac{-2u_0}{(t-6)^3}$

This gives us an expression for the slope in terms of $$t$$ and $$u_0$$ (the level of utility at the point). But it is often more useful to write it in terms of $$t$$ and $$c$$, by substituting for $$u_0$$ using the equation of the indifference curve:

$\frac{dc}{dt} = \frac{-2(t-6)^2(c-45)}{(t-6)^3} = \frac{-2(c-45)}{(t-6)}$

This gives us a negative number (for $$c>45$$ and $$t>6$$)—remember, indifference curves slope downward. The MRS is the absolute value of the slope:

$\text{MRS} = \frac{2(c-45)}{(t-6)}$

This expression can be used to calculate Karim’s MRS at any combination $$(t, c)$$ of free time and consumption. For example, the MRS at point A ($$t = 15, c = 540$$) is:

$\text{MRS at A} = \frac{2 \times (540-45)}{15-6} = 110$

The value of the MRS at A is not identical to the one we obtained in Figure 3.5. There, we worked in whole numbers, and defined the MRS as the amount of consumption Karim would trade for an increase of one whole hour in free time. At point A, this corresponds to the slope of the straight line from A to E. But here we have used calculus to find the MRS at point A using the exact slope of the curve at that point—which is slightly different. Either way, however, the MRS is measured in the same units: Karim’s MRS is measured in euros per hour.

### Derivatives and marginal changes

marginal change
When two variables, x and y, are related to each other, the effect of a marginal change is the change in y that occurs in response to a small increase in x. If y is a continuous function of x, the marginal change in y is the rate of change of y with respect to x: that is, the derivative of the function.

The marginal rate of substitution measures a rate of change, usually described by economists as the effect of a marginal change. It is one of many examples of marginal changes you will encounter. We often model a relationship between two variables, $$x$$ and $$y$$, as a function $$y=f(x)$$, and we want to measure how fast $$y$$ changes as $$x$$ increases: that is, the rate of change of $$y$$ with respect to $$x$$. We describe this as the marginal change in $$y$$ in response to a marginal change in $$x$$.

When we draw a graph of the function, the effect of a marginal change is measured by the slope of the line or curve. If the function is a curve, the slope is different at different values of $$x$$. The MRS is the slope of the indifference curve.

### Marginal changes

We use two ways of defining and measuring marginal changes in a relationship between two variables, $$x$$ and $$y$$:

• If $$x$$ is measured in whole numbers only, the marginal change in $$y$$ is defined as the change in $$y$$ when $$x$$ increases by one unit. This is the interpretation we use in the main sections of the book.
• If $$y$$ is a continuous function of $$x$$, the marginal change in $$y$$ is measured by the derivative of $$y$$ with respect to $$x$$. This calculus method is used in the mathematical extensions.

Remember that a derivative captures the effect of an infinitesimal change in $$x$$. So if the units are small, these two methods give almost the same answer: if $$y=f(x)$$, $$\frac{df}{dx}\approx f(x+1)-f(x)$$.

#### Marginal utility

marginal utility
The additional utility resulting from a one-unit increase in the amount of a good.

Another example of a marginal change is marginal utility—the rate of change in utility as the amount of a good increases. Using the calculus approach, it is given by the partial derivative: for Karim, $$\frac{\partial u}{\partial t}$$ is the marginal utility of free time, and $$\frac{\partial u}{\partial c}$$ is the marginal utility of consumption. Marginal utilities are closely related to the MRS, as explained below.

### The marginal rate of substitution: A general formula

We calculated Karim’s MRS by rearranging the particular equation for his indifference curves and differentiating to find their slopes. For a general utility function $$u(t, c)$$ we can find the slope of the indifference curve using the marginal utilities, $$\frac{\partial u}{\partial t}$$ and $$\frac{\partial u}{\partial c}$$.

We apply a technique called implicit differentiation, which is useful in many economic models. Here, the method involves considering how consumption would need to change if free time increased by a small amount, in order to keep utility constant.

Starting from a point on the indifference curve $$u(t, c)=u_0$$, suppose both $$t$$ and $$c$$ change by small amounts, $$\Delta t$$ and $$\Delta c$$. The small increments formula for functions of two variables gives an approximation to the change in utility $$\Delta u$$, expressing it as the sum of a ‘free time effect’ and a ‘consumption effect’:

$\Delta u \approx \frac{\partial u}{\partial t} \Delta t + \frac{\partial u}{\partial c} \Delta c$

If the small changes, $$\Delta t$$ and $$\Delta c$$, are chosen so that we move to a point on the same indifference curve, utility does not change. In that case $$\Delta u=0$$, which implies that:

$\frac{\partial u}{\partial t} \Delta t + \frac{\partial u}{\partial c} \Delta c \approx 0$

Rearranging:

$\frac{\Delta c}{\Delta t} \approx - \frac{\partial u}{\partial t} \bigg/ \frac{\partial u}{\partial c}$

The changes $$\Delta t$$ and $$\Delta c$$ together produce a small movement along an indifference curve. So if we now take the limit as $$\Delta t \rightarrow 0$$, the left-hand side approaches the slope of the curve and the approximation becomes an equation.

Therefore the slope of the indifference curve through any point $$(t, c)$$ is given by the formula:

$\frac{dc}{dt} = -\frac{\partial u}{\partial t} \bigg/ \frac{\partial u}{\partial c}$

The right-hand side of this equation is negative, provided that both marginal utilities are positive (increasing either free time or consumption increases utility). This formula shows that for any utility function with positive marginal utilities, the indifference curves slope downward, as in the diagram. The marginal rate of substitution (MRS) is the absolute value of the slope:

$\text{MRS} = \frac{\partial u}{\partial t} \bigg/ \frac{\partial c}{\partial c}$

or, in words,

$\text{marginal rate of substitution} = \frac{\text{marginal utility of free time}}{\text{marginal utility of consumption}}$

Applying this formula to Karim’s utility function $$u=(t-6)^2(c-45)$$, the marginal utilities are:

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

So the MRS is given by:

$\text{MRS} = \frac{2(t-6)(c-45)}{(t-6)^2} = \frac{2(c-45)}{(t-6)}$

which is what we obtained directly before.

### Convex preferences

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.

Each indifference curve in Figure E3.1 becomes flatter as you move along it to the right: in other words, Karim’s MRS falls along the indifference curve as $$t$$ increases. You can infer this from the expression $$\text{MRS}=\frac{2(c-45)}{(t-6)}$$: as we move down an indifference curve, $$t$$ in the denominator increases and $$c$$ in the numerator decreases, so the MRS becomes smaller.

This property of Karim’s preferences is known as diminishing marginal rate of substitution and is usually assumed when we draw indifference curves with two goods. The more you have of a good, the more willing you are to trade it for the other.

Another way to describe this property is to say that Karim’s indifference curves are convex. A curve is said to be convex if, when you draw a straight line between any two points on the curve, the line lies above the curve. Since Karim’s indifference curves have this shape, we say that he has convex preferences.

We can check this algebraically for Karim’s utility function $$u=(t-6)^2(c-45)$$, by again writing the indifference curve as a function of $$t$$:

$c = \frac{u_0}{(t-6)^2} + 45$

Differentiating with respect to $$t$$ holding $$u_0$$ constant:

$\frac{dc}{dt} = \frac{-2u_0}{(t-6)^3} \ \text{and so MRS =} \ \frac{2u_0}{(t-6)^3}$

Differentiating again:

$\frac{d \text{MRS}}{dt} = \frac{-6u_0}{(t-6)^4} < 0$

and so MRS is diminishing as $$t$$ increases along the curve.

A person whose preferences are convex always prefers mixtures of goods to extremes of either good. If we draw a straight line between two points on the same indifference curve, then each point on the line is a ‘mixture’ (that is, a weighted average) of the two end points. When the indifference curves are convex, all points on the line between the end points give higher utility than the end points.

### The Cobb–Douglas utility function

Suppose that Karim has a different utility function:

$u (t, c)= t^\alpha c^\beta$

where $$\alpha$$ and $$\beta$$ are positive constants. This function has some very convenient mathematical properties. It is called a Cobb–Douglas function after the two people who introduced it into economics.

To draw Karim’s indifference curves, we again rearrange the expression above to find an equation for $$c$$ in terms of $$t$$, at a particular level of utility $$u_0$$:

$c=\left(\frac{u_0}{t^\alpha}\right)^\frac{1}{\beta}$

To find the marginal utilities of free time and consumption, we must find the partial derivatives of the utility function. Differentiating $$u$$ with respect to $$t$$, the marginal utility of free time is:

$\frac{\partial u}{\partial t} = \alpha t^{\alpha -1} c^\beta$

We know from the utility function that $$t^{\alpha -1} c^\beta = u/t$$, which gives us a simpler expression for the marginal utility of free time:

$\frac{\partial u}{\partial t} = \frac{\alpha u}{t}$

Similarly, the marginal utility of consumption is:

$\frac{\partial u}{\partial c} = \beta t^{\alpha} c^{\beta-1} = \frac{\beta u}{c}$

When $$t$$ and $$c$$ are positive, so is $$u$$. Hence the assumption that $$\alpha$$ is also positive implies that $$\partial u/\partial t \gt 0$$. Similarly, $$\beta\gt 0$$ implies that $$\partial u/\partial c \gt 0$$. In other words, the assumption that both $$\alpha$$ and $$\beta$$ are positive ensures that ‘goods are good’: Karim’s utility rises as free time or consumption increases.

### Exercise E3.1 Comparing indifference curves

Karim’s friend has the utility function, $$u(t, c) = (t-6)(c-45)^2$$.

1. Plot the indifference curves corresponding to utility levels of 40,100, 21,000, and 8,000, with consumption spending (in euros) on the vertical axis and hours of free time per day on the horizontal axis. Compare the shape with those of Karim’s (in Figure E3.1).
2. Use calculus to derive an expression for the MRS corresponding to this utility function. How does it compare to Karim’s MRS? Explain this result, by comparing the utility functions of Karim and his friend.

### Exercise E3.2 Cobb–Douglas preferences

Suppose Karim has the utility function, $$u(t, c) = t^2 c^3$$.

1. Plot Karim’s indifference curves corresponding to utility levels of 40,100, 21,000, and 8,000, with consumption in euros on the vertical axis and hours of free time on the horizontal axis.
2. Derive an expression for Karim’s MRS and describe how it changes as hours of free time increases.
3. Does Karim have convex preferences? Explain your answer, referring to Questions 1 and 2.
4. Karim’s friend has the utility function, $$u(t, c) = t^{0.4} c^{0.6}$$. Show that this utility function represents the same preferences as Karim’s.

Read more: Sections 14.2 (for the small increments formula) and 15.1 (for contours and implicit differentiation) of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.