Market demand

• 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 - graphically

• Market demand is the horizontal addition of individual demands.

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

Market demand - Example - Cobb Douglass

• $u = x_1^a x_2^b$

• $100$ 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?

Market demand - Example - Linear Demands

• 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:

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

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

Reservation price

• 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

Elasticity

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 $\epsilon$

Elasticity - Example for linear demand curve:

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

• Let demand curve be: $q = a − bp$

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

Elasticity - Iso-elastic demand curve:

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}

Elasticity classification

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

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

Revenue, price increase and elasticity

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 $\frac{dR}{dp} > 0$ ?

Revenue, price increase and elasticity

\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 $\frac{dR}{dp} > 0$.

Revenue, price increase and elasticity

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.