Market Demand & Supply

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 20

Individual and Market Demand

Individual demand curve, \(d^i(p)\): quantity demanded by consumer \(i\) at each possible price

Market demand sums across all consumers:

\displaystyle D(p) = N_Cd(p)
\displaystyle D(p) = \sum_{i=1}^{N_C}{d^i(p)}

If all of those consumers are identical and demand the same amount \(d(p)\):

There are \(N_C\) consumers, indexed with superscript \(i \in \{1, 2, 3, ..., N_C\}\).

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

Market demand sums across all consumers:

\displaystyle D(p) = \sum_{i=1}^{N_C}{d^i(p)}

Individual and Market Supply

Firm supply curve, \(s^j(p)\): quantity supplied by firm \(j\) at each possible price

Market supply sums across all firms:

\displaystyle S(p) = N_Fs(p)
\displaystyle S(p) = \sum_{j=1}^{N_F}{s^j(p)}

If all of those firms are identical and supply the same amount \(s(p)\):

There are \(N_F\) competitive firms, indexed with superscript \(j \in \{1, 2, 3, ..., N_F\}\).

Market supply curve, \(S(p)\): quantity supplied by all firms at each possible price

How Demand
Aggregates Preferences

\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 \(\alpha = \frac{7}{12} > \frac{1}{2}\).

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

What does Trulia say?

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.