10 Consumer Surplus – A Microeconomic Theory Workbook

10.1 Objective

By the end of this chapter, you will understand how consumer surplus represents the difference between what consumers are willing to pay for a good or service and what they actually pay. You will gain insight into how this concept not only measures consumer benefits but also plays a crucial role in economic welfare analysis.

10.2 Demand for a Discrete Good

If good 1 is a discrete good, then it is only available in integer amounts. Let’s think about how to describe the demand for good 1.

If $p_1$ is way too high, the consumer won’t want to buy a single unit.
If $p_1$ drops low enough, the consumer might buy one unit.
If $p_1$ drops lower still, the consumer might buy a second unit.
And so on.

So the demand for a discrete good is given by a list of reservation prices. For instance, if reservation prices are $(7, 4, 2, 1)$, that means that the consumer won’t buy any units of good 1 until the price falls to $7 or less, at which time the consumer would buy 1 unit. If the price fell to $4 or less, the consumer would be willing to buy 2 units in total. If the price fell to $2 or less, the consumer would be willing to buy 3 units, and if the price fell to $1 or less, the consumer would be willing to buy 4 units. Here’s the demand curve for good 1 in a figure (it’s a step function):

Let’s think of a utility function that could describe this kind of demand.

For simplicity, suppose that $u(0) = 0$: consuming zero units of good 1 generates no utility for the consumer.
How much utility would consuming one unit of good 1 yield? The consumer has a maximum willingness to pay of $7 for it, and at a price of $7, the consumer is just willing to buy it. So it must yield 7 units of utility for the consumer: $u(1) = r_1$.
How much utility would consuming 2 units yield? The consumer would be willing to pay a maximum of $7 for the first unit, and then a maximum of $4 for the second unit, so the consumer is indifferent between having $7 + 4 = 11$ dollars and having two units of the good. Therefore, $u(2) = 11$.
The same way, $u(3) = 7 + 4 + 2 = 13$, and $u(4) = 7 + 4 + 2 + 1 = 14$.

Notice that this utility function is also the area under the demand curve: $u(1)$ is the area under the demand curve out to $x_1 = 1$, $u(2)$ is the area under the demand curve out to $x_1 = 2$, and so on.

If $p_1 = 5$, the consumer would purchase 1 unit, and they would get $u(1) = 7$ from that unit. The total benefit, or consumer surplus, from purchasing that unit is the utility of $7 minus the amount paid (one unit at a price of $5 means $5 was paid). So the consumer surplus would be $7 - 5 = 2$ dollars.

Consumer surplus is the difference between the total amount that consumers are willing to pay for a good and the total amount they actually pay. It represents the benefit or excess satisfaction that consumers receive when they purchase a product for less than the highest price they are willing to pay. Graphically, it is the area between the demand curve and the price level, up to the quantity purchased.

10.3 Approximating a Continuous Demand

If $x_1$ is not a discrete good, demand might be something like this continuous curve:

In which case, the consumer surplus is still just the area under the demand curve and above the price that the consumer pays.

10.4 Interpreting a Change in Consumer Surplus

10.5 Classwork 10

Let demand for a good be given by $x_1 = 10 - p_1$. If $p_1$ falls from $3 to $2, what is the change to consumer surplus?
Consider the quasilinear utility function $u(x_1, x_2) = \ln(x_1) + x_2$. Let $p_2 = 1$ and find the change to consumer surplus from $p_1$ falling from 2 to 1. Start by finding the demand function for good 1 by finding the consumer’s choice (MRS = price ratio and budget constraint). You should find that $x_1 = \frac{p_2}{p_1}$. Remember the integral of $\frac{1}{x}$ is $\ln(x)$.