# Market Demand, Elasticity, and Marginal Revenue

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 12

# Today's Agenda

• Market demand
• Elasticity
• Marginal revenue

# Today's Agenda

• Market demand
• Elasticity
• Marginal revenue

# Section

• Doing the math on elasticity

# Individual and Market Demand

Individual demand curve, $$x^*(p)$$: quantity demanded by a consumer at each possible price

Market demand curve, $$D(p)$$: quantity demanded by all consumers at each possible price

If there are $$N_C$$ consumers in a market:

\displaystyle D(p) = N_Cx^*(p)
\displaystyle D(p) = \sum_{i=1}^{N_C}{x_i^*(p)}

If all of those consumers are identical:

\displaystyle D(p) = \sum_{t=1}^T N_tx_t^*(p)

If there are $$T$$ different "types" of consumers, with type $$t$$ having $$N_t$$ identical consumers:

# Special Case: Cobb-Douglas

\Sigma
NOTATION AHEAD

STAY FOCUSED ON
ACTUAL ECONOMICS

## Special Case: Cobb-Douglas

u(x_1,x_2,...,x_n) = \alpha_1 \ln x_1 + \alpha_2 \ln x_2 + \cdots + \alpha_n \ln x_n

Suppose each consumer has the utility function

where the $$\alpha$$'s all sum to 1.

x_{k,i}^*(p_1,p_2,...,p_n) = \frac{\alpha_{k,i}\times m_i}{p_k}

We've shown before that if consumer $$i$$'s income is $$m$$, their demand for good $$k$$ is

quantity demanded of good $$k$$ by consumer $$i$$

consumer $$i$$'s preference weighting of good $$k$$

consumer $$i$$'s income

price of good $$k$$

There are 200 people, and they each have $$\alpha = \frac{1}{2}, m = 30$$

u_i(x_1,x_2) = \alpha_i \ln x_1 + (1-\alpha_i) \ln x_2

Suppose there are only two goods, and each consumer has the utility function

x_{1,i}^*(p_1) = \frac{\alpha_{i}\times m_i}{p_1}

So consumer $$i$$ will spend fraction $$\alpha_i$$ of their income $$m_i$$ on good 1:

x_1^*(p_1) = \frac{15}{p_1}
D_1(p_1) = 200 \times \frac{15}{p_1}

Market demand:

= \frac{3000}{p_1}

number of consumers

quantity demanded by each consumer

= \frac{3000}{p_1}

Note: total income is $$200 \times 30 = 6000$$, so this means the demand is the same "as if" there were one "representative agent" with $$\alpha = \frac{1}{2}, m = 6000$$

Individual demand:

Now suppose there are two types of consumers:

u_i(x_1,x_2) = \alpha_i \ln x_1 + (1-\alpha_i) \ln x_2

Again there are only two goods, and each consumer has the utility function

x_{1,i}^*(p_1) = \frac{\alpha_{i}\times m_i}{p_1}
x_{1,L}^*(p_1) = \frac{5}{p_1}
x_{1,H}^*(p_1) = \frac{30}{p_1}
D_1(p_1) = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +

100 low-income consumers who don't like this good: $$\alpha_L = \frac{1}{4}, m_L = 20$$

100 high-income consumers who do like this good:$$\alpha_H = \frac{3}{4}, m_H = 40$$

100\times \frac{30}{p_1}
= \frac{3500}{p_1}
100\times \frac{5}{p_1}

(demand from
low-income)

(demand from
high-income)

\Rightarrow

Market demand:

Individual demand:

Note: total income is $$100 \times 20 + 100 \times 40 = 6000$$, so this means the demand is the same "as if" there were one "representative agent" with $$\alpha = \frac{7}{12}, m = 6000$$

# Conundrum

In both cases, average income was 30 and average preference parameter $$\alpha$$ was $$\frac{1}{2}$$.

When everyone was identical, it was "as if"
there was a representative agent with all the money
with preference parameter $$\alpha = \frac{1}{2}$$.

When rich people had a higher $$\alpha$$, it was "as if"
there was a representative agent with all the money
with preference parameter $$\alpha = \frac{7}{12} > \frac{1}{2}$$.

# How Demand Aggregates Preferences

Feel free to tune out the intermediate steps, but hang on to the econ...

## How market demand aggregates preferences

x_{k,i}^*(p_1,p_2,p_3,...,p_n) = \frac{\alpha_{k,i} m_i}{p_k}

If consumer $$i$$'s demand for good $$k$$ is

then the market demand for good $$k$$ is

\displaystyle = \sum_{i=1}^{N_C}\frac{\alpha_{k,i} m_i}{p_k}
\displaystyle = \frac{\alpha_k M}{p_k}
\displaystyle D_k(p_k) = \sum_{i=1}^{N_C}x_{k,i}^*(p_k)

where $$M = \sum m_i$$ is the total income of all consumers
and $$\alpha_k$$ is an "aggregate preference" parameter.

Conclusion: we can model demand from $$N_C$$ consumers with Cobb-Douglas preferences
"as if" they were a single consumer with "average" Cobb-Douglas preferences.

so what is $$\alpha_K$$?

\displaystyle \text{We want to get from }D(p) = \sum_{i=1}^{N_C}\frac{\alpha_{k,i} m_i}{p_k}\text{ to }D(p) = \frac {\alpha_k M}{p_k}

If everyone has the same income ($$m_i = \overline m$$ for all $$i$$), then demand simply aggregates preferences:

Let $$\overline m = M/N_C$$ be the average income. Then we can rewrite market demand as:

\displaystyle D_k(p_k) = \sum_{i=1}^{N_C}\frac{\alpha_{k,i} m_i}{p_k} \times \frac{M}{N_C \overline m}
\displaystyle = \frac{1}{N_C}\sum_{i=1}^{N_C}\left(\alpha_{k,i} \times \frac{ m_i}{\overline m}\right) \times \frac{M}{p}

$$\alpha_K$$

\displaystyle \alpha_k = \frac{1}{N_C}\sum_{i=1}^{N_C}\alpha_{k,i}

But if there is income inequality, $$\alpha_k$$ gives more weight to the prefs of those with higher income.

$$=1$$

Example: consider an economy in which rich consumers like a good more:

100 low-income people with $$\alpha_L = \frac{1}{4}, m_L = 20$$,

100 high-income people with $$\alpha_H = \frac{3}{4}, m_H = 40$$

\displaystyle \alpha = \frac{1}{N_C}\sum_{i=1}^{N_C}\left(\alpha_{i} \times \frac{ m_i}{\overline m}\right)
\displaystyle = \frac{1}{200}\left[100 \times \left(\frac{1}{4} \times \frac{20}{30}\right) + 100 \times \left(\frac{3}{4} \times \frac{40}{30}\right)\right]
\displaystyle = \frac{7}{12}

Average income is $$\overline m = 30$$, total income is $$M = 6000$$

So, market demand is

\displaystyle D(p) = \alpha \times \frac{M}{p} = \frac{7}{12} \times \frac{6000}{p} = \frac{3500}{p}

closer to $$\alpha_H$$ than $$\alpha_L$$

## Example, revisited

Source: trulia.com, 5/12/22

# Own-Price Elasticity

What is the effect of a 1% change
in the price of good 1 $$(p_1)$$ on the quantity demanded of good 1 $$(x_1^*)$$?

no change

perfectly inelastic

less than 1%

inelastic

exactly 1%

unit elastic

more than 1%

elastic

1. How does $$x_1^*$$ change with $$p_1$$?
• ​​Own-price elasticity
• Elastic vs. inelastic
x_1^*(p_1,p_2,m)

# Cross-Price Elasticity

What is the effect of an increase
in the price of good 2 $$(p_2)$$ on the quantity demanded of good 1 $$(x_1^*)$$?

no change

independent

decrease

complements

increase

substitutes

1. How does $$x_1^*$$ change with $$p_1$$?
• ​​Own-price elasticity
• Elastic vs. inelastic
2. How does $$x_1^*$$ change with $$p_2$$?
• ​​Cross-price elasticity
• Complements vs. substitutes
x_1^*(p_1,p_2,m)

# Income Elasticity

What is the effect of an increase
in income $$(m)$$ on the quantity demanded of good 1 $$(x_1^*)$$?

decrease

good 1 is inferior

increase

good 1 is normal

1. How does $$x_1^*$$ change with $$p_1$$?
• ​​Own-price elasticity
• Elastic vs. inelastic
2. How does $$x_1^*$$ change with $$p_2$$?
• ​​Cross-price elasticity
• Complements vs. substitutes
3. How does $$x_1^*$$ change with $$m$$?
• ​​Income elasticity
• Normal vs. inferior goods
x_1^*(p_1,p_2,m)

# Three Relationships

## Demand Elasticity: $$\epsilon = \frac{\%\Delta Q}{\%\Delta P}$$

\text{Perfectly elastic}
|\epsilon| = \infty
\text{Elastic}
|\epsilon| >1
\text{Unit Elastic}
|\epsilon| = 1
\text{Inelastic}
|\epsilon| < 1
\text{Perfectly Inelastic}
|\epsilon| = 0

[poll question...]

# Calculating Elasticities in General

\epsilon_{y,x}
= \frac{\% \Delta y}{\% \Delta x}
= \frac{\Delta y / y}{\Delta x / x}
= \frac{\Delta y}{\Delta x}\times \frac{x}{y}
\text{Suppose }y \text{ is a function of }x.
\text{In the limit, as }\Delta x \rightarrow 0:
\epsilon_{y,x} = \frac{dy}{dx}\times \frac{x}{y}

## $$\epsilon_{q,p} = \frac{\%\Delta q}{\%\Delta p}$$

\epsilon_{q,p} = \frac{dq}{dp}\times \frac{p}{q}

# Demand and Inverse Demand

Demand curve:

quantity as a function of price

Inverse demand curve:
price as a function of quantity

QUANTITY

PRICE

Demand

Inverse Demand

q(p) = 20 - p
q(p) = 20 - 2p
p(q) =
p(q) =
\text{revenue} = r(q) = p(q) \times q
\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p(q)
\displaystyle{\text{average revenue} = \frac{r(q)}{q} = p(q)}

If the firm wants to sell $$q$$ units, it sells all units at the same price $$p(q)$$

Since all units are sold for $$p$$, the average revenue per unit is just $$p$$.

By the product rule...
let's delve into this...

# Total, Average, and Marginal Revenue

Demand

Inverse Demand

q(p) = 20 - p
q(p) = 20 - 2p
p(q) = 20-q
p(q) = 10 - \tfrac{1}{2}q

Revenue

r(q) =
MR(q)=
AR(q)=
r(q) =
MR(q)=
AR(q)=

[poll question...]

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p
dr = dp \times q + dq \times p
p
p(q)
q

The total revenue is the price times quantity (area of the rectangle)

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p
dr = dp \times q + dq \times p
p
p(q)
q
dp
dq

Note: $$MR < 0$$ if

dq \times p
{dq \over q} < {dp \over p}
\% \Delta q < \% \Delta p
|\epsilon| < 1
dp \times q
<

The total revenue is the price times quantity (area of the rectangle)

If the firm wants to sell $$dq$$ more units, it needs to drop its price by $$dp$$

Revenue loss from lower price on existing sales of $$q$$: $$dp \times q$$

Revenue gain from additional sales at $$p$$: $$dq \times p$$

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

## Marginal Revenue and Elasticity

= \left[{dp \over dq} \times {q \over p}\right] \times p + p
= {p \over {dq \over dp} \times {p \over q}} + p
= - {p \over |\epsilon_{q,p}|} + p

(multiply first term by $$p/p$$)

(simplify)

(since $$\epsilon < 0$$)

Notes

Elastic demand: $$MR > 0$$

Inelastic demand: $$MR < 0$$

In general: the more elastic demand is, the less one needs to lower ones price to sell more goods, so the closer $$MR$$ is to $$p$$.

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

## Marginal Revenue for Perfectly Elastic Demand

= \left[{dp \over dq} \times {q \over p}\right] \times p + p
= {p \over {dq \over dp} \times {p \over q}} + p
= - {p \over |\epsilon_{q,p}|} + p

(multiply first term by $$p/p$$)

(simplify)

(since $$\epsilon < 0$$)

Note

Perfectly elastic demand: $$MR = p$$

By Chris Makler

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