Utility Functions and Indifference Curves

What is Utility?

When we consume a good, we assume that the good will have some impact on our total utility. Utility is a fundamental measure that helps economists model how consumers make decisions. An assumed rule in economics is that consumers will always act rationally, which translates to the assumption that consumers will always attempt to maximize their own utility.

It is important to note that utility doesn’t have specified units and even the face value of utility doesn’t have any meaning. What does an apple providing 5 utility units even mean? What is valuable, however, is that utility can be compared; if an apple provides 5 utility units and an orange provides 3 utility units, then we prefer apples to oranges.

As a very simple example, say Anne has 6 dollars and she can choose to buy any combination of goods A and B. If good A costs 2 dollars and provides 5 utility units per unit of A consumed, while good B costs 3 dollars and provides 6 utility units per unit of B consumed, then Anne will buy 3 units of good A, since that maximizes her utility.

In economics, however, our models are a little more complex than that. Typically, utility is the product of the consumption of many goods; typically having a lot of one good but not another does not provide much utility. In addition, consumption of one good faces diminishing marginal returns, i.e. holding all things equal, the consumption of one additional unit of a good will provide less utility than the utility received from the previous unit. Intuitively, imagine Bob is very hungry and decides to eat slices of pizza. The first slice of pizza will bring Bob the most utility, but the 8th slice will be much less satisfying to eat.

Utility Functions

A consumer’s utility is determined by the amount of consumption from all the goods they consume. Typically, utility functions are multivariate: they take in multiple inputs (which represent the different amounts of consumption for each good, which we call a consumption bundle), and output one value, the utility. Today, we’ll only look at the case where consumers can only choose between 2 goods \(x_1\) and \(x_2\). Hence, a utility function can be represented by: \(u(x_1,x_2)\).

With that in mind, let’s start graphing some utility functions!

Cobb-Douglas Utility Function

Consider the following utility function across \(x_1\) and \(x_2\):

\[u(x_1, x_2)=x_1^{\alpha}x_2^{1-\alpha}\quad\text{where } 0<\alpha<1\]

This is known as the Cobb-Douglas utility function. To visualize this function, we’ll need a 3D plot.

[Following image is an interactive 3D plot of Utility increasing with X1 and X2]

Examining the Utility Function

There are 2 rules that utility functions generally follow:

• Non-negative marginal utility: the consumption of a good will not decrease the utility. Economists generally assume that ‘more is better.’ If the consumption of a good decreased utility, then we would consume less of a good.

• Diminishing marginal returns: all else equal, as consumption increases the marginal utility derived from each additional unit declines.

Non-negative Marginal Utility

Say we are currently consuming 2 units of \(x_1\) and \(x_2\) each with \(\alpha = \frac{1}{2}\), providing \(u(2,2)=2^{0.5}2^{0.5}=2\) utility units. One additional unit of \(x_1\) will provide me a higher point of utility: we can verify this result both graphically and numerically: \(u(3,2)=3^{0.5}2^{0.5}\approx2.45\). Indeed, consuming one more unit of a good should increase our utility!

Marginal Utility and the Law of Diminishing Returns

Now let’s check for the second result: diminishing marginal returns. From above, we know that holding the consumption of \(x_2\) constant at 2, going from 2 to 3 units of \(x_1\) increases our utility by \(2.45-2=0.45\). Going from 3 to 4 units of \(x_1\) brings our utility to \(u(4,2)=4^{0.5}2^{0.5}\approx 2.83\), an increase of \(2.83-2.45=0.38\) utility units.

Using calculus, we can more formally define the marginal utility of a good. Since marginal utility is the change in utility that one additional unit of consumption provides (holding all others constant), the marginal utility with respect to \(x_1\) is its partial derivative: \(\frac{\partial u}{\partial x_1}\). In our case:

\[\begin{split} \begin{aligned} \textrm{Marginal Utility of } x_1: &\quad\frac{\partial u}{\partial x_1} = \frac{1}{2}x_1^{-0.5}x_2^{0.5} \\ \textrm{Marginal Utility of } x_2: &\quad\frac{\partial u}{\partial x_2} = \frac{1}{2}x_1^{0.5}x_2^{-0.5} \end{aligned} \end{split}\]

Or, more generally,

\[\begin{split}\begin{aligned} \textrm{Marginal Utility of } x_1: &\quad\frac{\partial u}{\partial x_1} = \alpha x_1^{\alpha-1}x_2^{1-\alpha} \\ \textrm{Marginal Utility of } x_2: &\quad\frac{\partial u}{\partial x_2} = (1-\alpha) x_1^{\alpha}x_2^{-\alpha} \end{aligned}\end{split}\]

With marginal utility defined, note that both conditions can be explained using the marginal utility function \(\frac{\partial u}{\partial x}\):

• Non-negative marginal utility: \(\frac{\partial u}{\partial x} \geq 0\)

• Diminishing marginal returns: \(\frac{\partial^2 u}{\partial x^2} < 0\)

Indifference Curves

Although the utility function above in 3D is cool, you’ll typically find utility graphs to be in 2D with \(x_1\) and \(x_2\) as the axis (eliminating the utility axis).

To represent utility levels, we plot a set of indifference curves on the 2D graph. An indifference curve satisfies the property in which any point on the curve has the exact same amount of utility, so that consumers are indifferent to any point on the curve. In our 3D plot, any point on the indifference curve has the exact same height, which represents the value of utility. If you’re familar with contour plots, you can also think of indifference curves as following the same idea.

[Following image is s graph of a series of nested convex Indifference Curves for different amounts of X1 and X2 ]