Partial Equilibrium

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 21

Friday Class:

 

Brooke Jenkins

 

SF District Attorney

 

Bishop Auditorium

 

5 participation points!!

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.

Assumptions of
Perfect Competition

Perfect information

Homogeneous good

Lots of buyers and sellers

Free entry and exit

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

Review: Market Demand

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

Review: 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 the Cobb-Douglas utility function

\(N_F\) identical firms produce good 1, each of which
has the Cobb-Douglas production function

Week 5: Demand for Good 1

Week 8: Supply of Good 1

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}

and income \(m\)

and capital fixed in the short run at \(\overline K\)

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

pollev.com/chrismakler

Suppose that instead of 16 firms, we had only 9 firms.

Then, we would expect the equilibrium price to _____ and the equilibrium quantity to _____.


(Hint: think about what happens to the market demand and supply curves.)

\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:

Effect of Government Policies

Effect of Government Policies

  • Taxes
    • Who pays the tax?
    • Is it based on quantity or value?
  • Subsidies
    • Like a negative tax
  • Price Controls
    • Price ceilings
    • Price floors
P = \text{“List price”}
P_F = \text{Price received by firms}
P_C = \text{Price paid by consumers}

If consumers pay the tax:

If firms pay the tax:

If they split it evenly:

Imposing a Tax

P + t
P
P
P - t
P + \tfrac{1}{2}t
P - \tfrac{1}{2}t
D(P_C) = 100 - 3P_C
S(P_F) = 2P_F

Equilibrium price and quantity with no tax.

Equilibrium quantity and prices
faced by consumers and firms if
 consumers pay a tax of t = 10.

Equilibrium quantity and prices
faced by consumers and firms if
 firms pay a tax of t = 10.

Tax burden for consumers:
the amount of the tax that results in an increase in the price paid by consumers,
relative to the equilibrium price

Tax burden for firms:
the amount of the tax that results in an decrease in the price received by firms,
relative to the equilibrium price

What is the burden in this case?

How does tax burden relate to the relative elasticities of demand and supply?

Elasticity and Tax Incidence

The equilibrium quantity, price paid by consumers, and price received by firms doesn't depend on who pays the tax

It does depend on the relative elasticity of demand and supply.

Doing the math on elasticity

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: is the market price "good"?