Partial Equilibrium

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 22

Fundamental Economic Questions

What, as a society, do we produce?

Who gets what?

How do we decide?

If you were an omniscient
"social planner" in charge of everything, how would you
make these decisions?

How do billions of people
coordinate their economic activities?

What does it mean to
"let the market decide"
what to produce?

Responding to Prices

Weeks 5-6: Consumer Theory

Firms face prices and
choose how much to produce

Consumers face prices and
choose how much to buy

Weeks 7-8: Theory of the Firm

Competitive Equilibrium

Consumers and producers are small relative to the market
(like an individual firefly)
and make one decision: how much to buy or sell at the market price.

Equilibrium occurs when
the market price is such that
the total quantity demanded
equals the total quantity supplied 

Equilibrium in General

Definition 1: a situation which economic forces are "balanced"

Definition 2: a situation which is
self-replicating: \(x = f(x)\)

Transition dynamics: excess demand and supply

Stability of Equilibria

All forces can be in balance in different ways.

Uniqueness of Equilibrium

Sometimes things can go multiple different ways...

Assumptions of
Perfect Competition

Perfect information

Homogeneous good

Lots of buyers and sellers

Free entry and exit

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)}

pollev.com/chrismakler

\text{Remember: }u(x_1,x_2) = x_1^\alpha x_2^{1 - \alpha} \Rightarrow x_1^*(p_1,p_2,m) = \frac{\alpha m}{p_1}
u(x_1,x_2) = x_1^{1 \over 4} x_2^{3 \over 4}

Check Your Understanding

Suppose there are \(N_C = 64\) identical consumers, each of whom has income \(m = 100\) and preferences which can be represented by the Cobb-Douglas utility function

What is the expression for market demand?

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

pollev.com/chrismakler

\text{Remember: }f(L,K) = \sqrt{LK} \Rightarrow q^*(w,r,p) = \frac{\overline K p}{2w}
f(L,K) = \sqrt{LK}

Check Your Understanding

Suppose there are \(N_F = 16\) identical firms, each of whom has a fixed level of capital at \(\overline K = 2\), pays wage rate \(w = 4\), and whose production function is

What is the expression for market supply?

Calculating Partial Equilibrium

\displaystyle \sum_{j=1}^{N_F}{q_j^*}
\displaystyle \sum_{i=1}^{N_C}{x_i^*}

Price \(p^*\) is an equilibrium price in a market if:

1. Consumer Optimization: each consumer \(i\) is consuming a quantity \(x_i^*(p^*)\) that solves their utility maximization problem.

2. Firm Optimization: each firm \(j\) is producing a quantity \(q_j^*(p^*)\) that solves their profit maximization problem.

3. Market Clearing: the total quantity demanded by all consumers equals the total quantity supplied by all firms.

p^* = \frac{MU_i(x_i^*)}{\lambda_i}
p^* = MC_j(q_j^*)
=

"Marginal benefit in dollars per unit of good 1"

\underbrace{S(p^*)}
\underbrace{D(p^*)}

\(N_C\) identical consumers, each of whom
has income \(m\) and the utility function

\(N_F\) identical firms produce good 1, each of which
has capital \(\overline K\) and the production function

d(p_1|p_2,m) = \frac{\alpha m}{p_1}
\displaystyle s(p_1|w,r) = \frac{\overline K p_1}{2w}
u(x_1,x_2) = x_1^\alpha x_2^{1-\alpha}
\displaystyle F(L,K) = \sqrt{LK}

1. Solve for the equilibrium price and quantity if \(\alpha = \frac{1}{4}, m = 100, N_C = 64, w = 4, \overline K = 2, N_F = 16\)

2. Solve for general formula for the equilibrium price and quantity.

Demand

Supply

D_1(p_1) = {1600 \over p_1}
S(p) = 4p
d(p_1|p_2,m) = \frac{\alpha m}{p_1}
\displaystyle s(p_1|w,r) = \frac{\overline K p_1}{2w}

2. Solve for general formula for the equilibrium price and quantity.

Demand

Supply

D_1(p_1) = N_C \times {\alpha m \over p_1}
S(p) = N_F \times {\overline K p \over 2w}
N_F \times {\overline K p \over 2w} = N_C \times {\alpha m \over p}
S(p) = D(p)
p^2 = {N_C \over N_F} \times {2\alpha mw \over \overline K}
p = \sqrt{{N_C \over N_F} \times {2\alpha mw \over \overline K}}
Q = N_F \times {\overline K \over 2w} \times p
= N_F \times {\overline K \over 2w} \times \sqrt{{N_C \over N_F} \times {2\alpha mw \over \overline K}}
=\sqrt{N_CN_F \times {\alpha m\overline K \over 2w}}
\displaystyle \sum_{j=1}^{N_F}{s^j(p)}
\displaystyle \sum_{i=1}^{N_C}{d^i(p)}

Important Note: Three Kinds of “=" Signs

1. Mathematical Identity: holds by definition

2. Optimization condition: holds when an agent is optimizing 

3. Equilibrium condition: holds when a system is in equilibrium

=
MRS = {MU_1 \over MU_2}
MRS = {p_1 \over p_2}

What happens if not everyone is identical?

Two consumers:

\text{Adam: }u_A(x_1^A,x_2^A) = 8 \ln x_1^A + x_2
\text{Eve: }u_E(x_1^E,x_2^E) = 4 \ln x_1^E + x_2^E
\frac{8}{x_1^A} = p_1
\frac{4}{x_1^E} = p_1
D_1(p) = d_1^A(p_1) + d_1^E(p_1)
d_1^A(p_1) = \frac{8}{p_1}
= \frac{8}{p_1} + \frac{4}{p_1}
= \frac{12}{p_1}

Consumer Optimization: Each consumer sets MRS = price ratio

Market Demand: Sum up individual demands for all people:

MRS = \frac{8}{x_1^A}
MRS = \frac{4}{x_1^E}
d_1^E(p_1) = \frac{4}{p_1}

This leads to the individual demand functions:

D_1(p) = d_1^A(p_1) + d_1^E(p_1)
= \frac{8}{p_1} + \frac{4}{p_1}
= \frac{12}{p_1}

Market Demand: Sum up individual demands for all people:

pollev.com/chrismakler

Suppose that Adam's preferences were instead given by

\(u^A(x_1,x_2) = 16 \ln x_1 + x_2\)

If Eve's preferences were still

\(u^E(x_1,x_2) = 4 \ln x_1 + x_2\)

what would the market demand be?

p = 2q_S
p = 4q_T
S(p) = s_S(p) + s_T(p)

Two firms: Subway's has \(\overline K = 2\), Togo's has \(\overline K =1\), both pay wage rate \(w = 4\).

s_S(p) = \frac{1}{2}p
s_T(p) = \frac{1}{4}p
=\frac{1}{2}p + \frac{1}{4}p
=\frac{3}{4}p
c_S(q_S) = q_S^2 + 2r
c_T(q_T) = 2q_T^2 + r
f(L) = \sqrt{2\overline KL}
MC = \frac{wq}{\overline K}
c(q) = \frac{wq^2}{2\overline K} + r\overline K

Firm Optimization: Each firm sets P = MC

Market Supply: Sum up individual supply for all firms:

\text{Subway}: f(L) = 2\sqrt{L}
\text{Togo's: }f(L) = \sqrt{2L}

Solving for \(q_S\) and \(q_T\) gives us the firms' individual supply functions:

S(p) = s_S(p) + s_T(p)
=\frac{1}{2}p + \frac{1}{4}p
=\frac{3}{4}p

Market Supply: Sum up individual supply for all firms:

Let's bring our consumers and firms together!

\text{Adam: }u(x_1,x_2) = 8 \ln x_1 + x_2
\text{Subway}: f(L) = 2\sqrt{L}
\text{Eve: }u(x_1,x_2) = 4 \ln x_1 + x_2
\text{Togo's: }f(L) = \sqrt{2L}

1. Consumer Optimization: each consumer \(i\) is consuming a quantity \(x_i^*(p^*)\) that solves their utility maximization problem.

2. Firm Optimization: each firm \(j\) is producing a quantity \(q_j^*(p^*)\) that solves their profit maximization problem.

3. Market Clearing: the total quantity demanded by all consumers equals the total quantity supplied by all firms.

D(p) = \frac{12}{p}
S(p) =\frac{3}{4}p
\frac{12}{p}
\frac{3}{4}p
=
p^* = 4
x_A^* = 2, x_B^* = 1
q_S^* = 2, q_T^* = 1

Note: if we go back to the individual demand
and supply functions, we get:

Endogenizing the Price

Previously: agents took the price
"as given" (exogenous) - it was determined outside the model

Today: we endogenized 
the market price by analyzing the model where it's determined

Next time: how do government policies (taxes and subsidies) affect this price?