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

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

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.

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.

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 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:

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

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.

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.