Two goods, $ x_1 $ , $ x_2 $

$ n $ consumers, indexed by $ i $

Individual demands: $ x_{1,i}(p_1, p_2, m_i) $ and $ x_{2,i}(p_1, p_2, m_i) $

$ X_{1}(p_1, p_2, m_1, ..., m_n) = \sum_{i=1}^{n} x_{1,i}(p_1, p_2, m_i) $

$ X_{2}(p_1, p_2, m_1, ..., m_n) = \sum_{i=1}^{n} x_{2,i}(p_1, p_2, m_i) $

Market demand is the horizontal addition of individual demands.

Make sure you properly account for "zero demand" above highest "reservation price".

$ u = x_1^a x_2^b $

$ 100 $ consumers, indexed by $ i $

All individual have the same preferences, so same demand functions: $ x_{1,i}^* = \frac{ a }{ (a+b) } \frac{ m_i}{ p_1} $

$ \begin{aligned} X_1(p_1, p_2, m_1, ..., m_{100}) &= \sum_{i=1}^{100} \frac{ a }{ (a+b) } \frac{ m_i }{ p_1 } \\ ~ &= \frac{ a }{ (a+b) } \frac{ \sum_{i=1}^{100} m_i }{ p_1 } \\ ~ &= \frac{ a }{ (a+b) } \frac{ M }{ p_1 } \\ \end{aligned} $

- What if there are different types of consumers?

- Suppose that the market consists of two individuals with demand functions given by:

$ x_{adam}(p, m_a) = max(20-p,0) $

$ x_{bob}(p, m_b) = max(30-3p,0) $

- The aggregate demand function is thereby given by:

$ Q^d(p,m_a,m_b) = x_a(p,m_a)+x_b(p,m_b) $

$ Q^d(p,m_a,m_b) =\begin{cases} 50 - 4p & if ~ p < 10 \\ 20 - p & if ~ 10 \leq p \leq 20 \\ 0 & if ~ p > 20 \\ \end{cases} $

Some goods come in large discrete units

Reservation price is price that just makes a person indifferent

$ u (0 , m) = u(1, m − p^{*}_{a}) $

Add up demand curves to get aggregate demand curve

Measures responsiveness of demand to price

$$ \epsilon = \frac{p}{x} \frac{dx}{dp} $$

What does elasticity depend on? In general, how many and how close substitutes a good has.

What does it mean? Why is not just the derivative?

We care about the magnitude so we often use: $ | \epsilon | $ instead of $ \epsilon $

We will use q instead of $ x_1 $ or $ x_2 $

Let demand curve be: $ q = a − bp $

Price elasticity of demand: $ ε = −b \frac{p}{q} = \frac{−bp}{a−bp} $

Let demand curve be: $ Q = A p^{-b} $

Lets calculate the elasticity:

$$
\begin{aligned}
\epsilon &= \frac{p}{Q} \frac{dQ}{dp} \\\\
&= \frac{p}{ A p^{-b} } ~ A (-b p^{-b-1} ) \\\\
&= \frac{A}{A} (-b) \frac{p ~ p^{-b-1} }{ p^{-b} } \\\\
&= -b \\\\
\end{aligned}
$$

Elastic demand: absolute value of elasticity greater than $ 1 $

Inelastic demand: absolute value of elasticity less than $ 1 $

Demand curve: $ Q^d = Q(p) $

Revenue: $ R = p \times Q $

When does an increase in price turn into an increase in revenue?

That is, when $ \frac{dR}{dp} > 0 $ ?

$$
\begin{aligned}
\frac{dR}{dp} &= \frac{d(p Q(p))}{dp} \\\\
~ &= p \frac{dQ}{dp} + Q(p) \\\\
~ &= Q \big( \frac{p ~ dQ}{Q ~ dp} + 1 \big) \\\\
~ &= Q ( \epsilon + 1 ) \\\\
\end{aligned}
$$

So as long as $ \epsilon > -1 $, we will have that $ \frac{dR}{dp} > 0 $.

You can use the same math, to see that if $ \epsilon < -1 $, then $ \frac{dR}{dp} < 0 $.

Why does this matter? Consider the case of the linear demand where elasticity changes from 0 to -infinity as we increase price. Now we know that:

At a price level where demand is price-

**inelastic**(which happens at low prices), an**increase**in price will increase revenue. And,At a price level where demand is price-

**elastic**(which happens at high prices), a**decrease**in price will increase revenue.

In conclusion, the price level that maximises revenue must be when the price-elasticity equals 1. The point we call unit-elasticity price level.