11 Market Demand – A Microeconomic Theory Workbook

11.1 Objective

We have developed the theory of individual consumer choice, which explains how consumers select the best bundle of goods they can afford. In this chapter, I will explain how to aggregate these individual choices to determine the total market demand. Following that, I will discuss some properties of market demand, such as its elasticity, which measures how responsive demand is to changes in price or income.

11.2 From Individual to Market Demand

Let $x_i^1(p_1, p_2, m_i)$ represent consumer $i$’s demand for good 1 (which depends on the price of good 1, the price of good 2, and consumer $i$’s income). Then the total market demand for good 1 is just the sum of the individual demands for all consumers:

\[\sum_{i=1}^n x_i^1 (p_1, p_2, m_i)\]

If we draw $p_1$ on the vertical axis and $x_1$ on the horizontal axis, summing the demand curve is summing horizontally: given any price, the quantity demanded is the sum of the quantity demanded by each consumer: $x^1 + x^2 + ... + x^n$.

11.3 Elasticity

Elasticity measures how much the demand for a product changes in response to a change in its price. Why is this important? Imagine you own a business. It is crucial to understand how many sales you might lose if you increase the price of your product by one dollar.

You could measure elasticity by the slope of the demand curve: $\frac{\Delta q}{\Delta p}$. The problem with this measure is that it depends on the units of measurement. It would become cumbersome to economists to specify units every time we talk about the elasticity of a certain good: for instance, when you increase the price of apples by $1, you might sell 10 fewer apples (slope of -10), which is also around 3.3 fewer pounds of apples (slope of -3.3), which is also around 0.08 fewer bushels of apples (slope of -0.08). A unit-free measure of responsiveness would be much more convenient:

Elasticity is a unit-free measure of the responsiveness of supply or demand to changes in price or income. So the price elasticity of demand is how responsive quantity demanded is to changes in the price of the good, and the formula is the percent change in quantity demanded divided by the percent change in price: \[\epsilon = \frac{\Delta q / q}{\Delta p / p}\]

So if the price elasticity of demand for apples is -1, that means that when the price of apples increases by 1%, then the quantity demanded for apples will decrease by 1%.

Rearranging the elasticity formula, we have:

\[\epsilon = \frac{p \Delta q}{q \Delta p}\]

And in terms of derivatives, that is:

\[\epsilon = \frac{p \ dq}{q \ dp}\]

If a good has an elasticity of demand greater than 1 in absolute value, it is said to have elastic demand: demand is very responsive to price changes (demand is stretchy like an elastic band). If the elasticity of demand is less than 1 in absolute value, it is said to have inelastic demand: demand is not very responsive to price changes. If it has an elasticity of exactly -1, it is said to have unit elastic demand.

The elasticity of demand for a good depends largely on how many close substitutes it has. For example, consider butter and margarine. If everyone regards butter and margarine as perfect substitutes, they must be sold at the same price if both are bought. Suppose butter is sold for $1 and margarine increases its price to $1.01. Demand for margarine would drop to zero because everyone would buy butter instead. When a good has many close substitutes, demand for it will be very elastic. Conversely, if a good does not have many close substitutes, demand for it will be very inelastic. Note: The fact that butter and margarine are often not sold at the same price indicates that many people do not regard them as perfect substitutes.

11.4 Classwork 11

A linear demand curve is given by $q = a - bp$. Use the formula $\epsilon = \frac{p}{q} \frac{dq}{dp}$ to show that the elasticity of demand is given by $\epsilon = \frac{-bp}{a - bp}$. What is the elasticity when $p$ is zero (the y-intercept)? What about when $q$ is zero (the x-intercept)? Show that $\epsilon = -1$ when $p = \frac{a}{2b}$.
Revenue is the amount of money a supplier makes from selling a good: it is $R = p q$, the price that the good is sold for times the number of units sold. If you increase the price of a good and find that quantity demanded doesn’t change very much, you can increase your revenue by increasing the price. But if you increase the price of a good and find that quantity demanded falls by a lot, you might decrease your revenue by increasing the price. For this question, take $\epsilon = -0.5$.
- For this elasticity, you will (circle one:) increase/decrease your revenue when you increase your price.
- Elasticity is the percent change in quantity demanded divided by the percent change in price. If $\epsilon = -0.5$ and price increases by 10%, how much will the quantity demanded increase by?
- Let $R_0$ be the old revenue before the price change and let $R_1$ be the new revenue after the price change. Then the percent change in revenue is $\frac{R_1 - R_0}{R_0} = \frac{p_1 q_1 - p_0 q_0}{p_0 q_0}$. Use your answers to the previous part of this question to substitute for $p_1$ and $q_1$ as functions of $p_0$ and $q_0$ to show that when the elasticity is -0.5 and price increases by 10%, we should expect that revenue will increase by 4.5%.
If revenue stayed constant for all changes in price, then demand for that good must be unit elastic everywhere: $\epsilon = -1$ and $pq = \bar{R}$ where $\bar{R}$ refers to a constant. If we solved for $q$, we’d get $q = \frac{\bar{R}}{p}$: that’s the demand curve with a constant elasticity of -1.
- Use the formula $\epsilon = \frac{p}{q} \frac{dq}{dp}$ to validate that this demand curve has a constant elasticity of demand of -1.
- In the workbook video, I told you that the general formula for demand with constant elasticity is $q = A p^\epsilon$. What are $A$ and $\epsilon$ in this case?
- Use R to draw this demand curve.