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

Conclusion: Aggregating Preferences

You can model market demand as reflecting the preferences of a single representative agent. 

But...know that you're weighting the preferences of richer people more. 

Elasticity

Why Elasticity?

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)

Three Relationships

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)

Three Relationships

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}

Demand Elasticity:

\(\epsilon_{q,p} = \frac{\%\Delta q}{\%\Delta p}\)

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

Revenue

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: \(MR = p\)

Econ 50 | Spring 2022 | 12 | Market Demand

By Chris Makler

Econ 50 | Spring 2022 | 12 | Market Demand

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