Competitive Equilibrium
in the Short Run

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 13

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 1-4: Consumer Theory

Firms face prices and
choose how much to produce

Consumers face prices and
choose how much to buy

Weeks 5-6: 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

Today's Agenda

Part 1: Theory

Part 2: Worked Examples

Review: Market demand and supply

Finding the equilibrium price

Welfare analysis of equilibrium quantity

"Representative Agents"

Heterogeneous agents

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

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

Unit I: Demand for Good 1

Unit II: 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 \(\alpha = \frac{1}{4}, m = 100, N_C = 64, w = 4, \overline K = 2, N_F = 64\)

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

Demand

Supply

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:

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:

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:

\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}

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

Welfare Analysis:
Consumer and Producer Surplus

Is this the “right" price?

If you were an omniscient social planner, could you do "better"
than the price the market "chooses"?

The Social Planner's Problem

Suppose you were in charge of the economy.

How would you answer the fundamental economic questions about a particular good?

How to produce it?

Want to produce any given quantity Q
at the lowest possible cost

Who gets to consume it?

How much to produce?

Want to distribute any given quantity Q
to the people who value it the most

Want to choose the quantity Q*
to maximize total surplus 
(benefit to consumers minus costs of production)

Model 1: One Consumer, One Firm

FIRM

CONSUMER

Quasilinear utility function:

u(x_1,x_2) = 10x_1 - {1 \over 2}x_1^2 + x_2

Good 2 is "dollars spent on other goods"

p_2 = 1\text{, 1 util = 1 dollar}

Total benefit (in dollars)
from \(x_1\) units of good 1:

v(x_1) = 10x_1 - {1 \over 2}x_1^2

Total cost function:

c(q) = q + {1 \over 4}q^2

Note: variable costs only

GROSS CONSUMER'S SURPLUS

(total benefit, in dollars)

v(x_1) = 10x_1 - {1 \over 2}x_1^2

Marginal benefit,
in dollars per unit:

v'(x_1) = 10 - x_1

(also MRS, marginal willingness to pay)

TOTAL VARIABLE COST

(dollars)

c(q) = q + {1 \over 4}q^2

Marginal cost,
in dollars per unit:

c'(q) = 1 + {1 \over 2}q

What is the optimal quantity \(Q^*\) to produce and consume?

FIRM

CONSUMER

Total benefit:

v(Q) = 10Q - {1 \over 2}Q^2

Total cost:

c(Q) = Q + {1 \over 4}Q^2

Total welfare: 

Marginal welfare from producing another unit:

TOTAL WELFARE

(dollars)

Marginal welfare,
in dollars per unit:

Total benefit to consumers minus total cost to firms

Marginal benefit to consumers minus marginal cost to firms

How do competitive markets
solve this problem?

FIRM

CONSUMER

Maximize net consumer surplus

v(x_1) - e(x_1) = 10x_1 - {1 \over 2}x_1^2 - p_1x_1

Maximize profits

r(q) - c(q) = pq - \left[q + {1 \over 4}q^2\right]

FIRM

CONSUMER

Net benefit from buying \(Q\) units at price \(P\):

v(Q) - P \times Q

Net benefit from selling \(Q\) units at price \(P\):

P \times Q - c(Q)

Total welfare: 

Marginal welfare from producing another unit:

Welfare Effects of a Tax

D(p_C) = 10 - p_C
S(p_F) = 2p_F - 2

Solve for the equilibrium prices \(p_F\) and \(p_C\)
and quantity \(Q = D(p) = S(p)\),
as well as government revenue  if:

(1)\ \ \text{No tax: }p_C = p_F = p
(2)\ \ \text{Firms pay tax }t: p_C = p, \ \ \ p_F = p - t
(3)\ \ \text{Consumers pay tax }t: p_C = p + t, \ \ \ p_F = p
p_C = \text{price consumers pay}
p_F = \text{Price firms receive}

Elasticity and Tax Incidence

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?

Externalities and Pigovian Taxes

Econ 50 | Spring 2022 | 13 | Partial Equilibrium

By Chris Makler

Econ 50 | Spring 2022 | 13 | Partial Equilibrium

Bringing supply and demand together

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