Oligopoly

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 7

pollev.com/chrismakler

Markets and Market Failures

  • Econ 50, Econ 51 thus far: "Libertarian Paradise"
    • Markets are efficient
    • Interfering with markets is inefficient
  • Problems:
    • Markets can fail to produce an efficient result
    • Many important things that affect our lives are not produced in competitive markets -- and maybe couldn't be

Big Idea for the Rest of the Course:
Strategic Interactions between Small Numbers of Agents

  • Very realistic - lots of examples, from families to international diplomacy
  • Models of oligopoly
    • 1838: Cournot
    • 1883: Bertrand
    • 1929: Hotelling
    • 1934: Stackelberg
  • Mid-20th century: John Nash formalized and generalized the notion of strategic equilibrium among small numbers of agents
  • We will follow this intellectual history, by first analyzing oligopoly models and then generalizing to look at broader classes of problems

Competition

  • Lots of "small" firms selling basically the same thing (commodity goods)

Oligopoly

  • A few "medium" or "large" firms selling differentiated products
  • Firms face essentially horizontal demand curve
  • Firms face downward sloping demand curve
  • Interdependence:
    each firm's choice
    affects other firms
  • Independence:
    no individual firm's choice affects other firms

Review: Monopoly

Profit

The profit from \(q\) units of output

\pi(q) = r(q) - c(q)

PROFIT

REVENUE

COST

is the revenue from selling them

minus the cost of producing them.

Revenue

We will assume that the firm sells all units of the good for the same price, \(p\). (No "price discrimination")

r(q) = p(q) \times q

The revenue from \(q\) units of output

REVENUE

PRICE

QUANTITY

is the price at which each unit it sold

times the quantity (# of units sold).

The price the firm can charge may depend on the number of units it wants to sell: inverse demand \(p(q)\)

  • Usually downward-sloping: to sell more output, they need to drop their price
  • Special case: a price taker faces a horizontal inverse demand curve;
    can sell as much output as they like at some constant price \(p(q) = p\)

Notation Alert: q not y

q_i = \text{Quantity produced by firm }i
Q = \text{Total quantity produced}

Demand and Inverse Demand

Demand curve:

quantity as a function of price

Inverse demand curve:
price as a function of quantity

QUANTITY

PRICE

\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

\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

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 \(p\): \(dq \times p\)

Profit Maximization with Market Power

\pi(q) = r(q) - c(q)

Optimize by taking derivative and setting equal to zero:

\pi'(q) = r'(q) - c'(q) = 0
\Rightarrow r'(q) = c'(q)

Profit is total revenue minus total costs:

"Marginal revenue equals marginal cost"

Example:

p(q) = 20 - q \Rightarrow r(q) = 20q - q^2
c(q) = 64 + {1 \over 4}q^2

What is the profit-maximizing value of \(q\)?

p(q) = 20 - q \Rightarrow r(q) = 20q - q^2
c(q) = 64 + {1 \over 4}q^2

What is the profit-maximizing value of \(q\)?

\pi(q) = r(q) - c(q)

Multiply right-hand side by \(q/q\): 

\pi(q) = \left[{r(q) \over q} - {c(q) \over q}\right] \times q
= (AR - AC) \times q

Profit is total revenue minus total costs:

"Profit per unit times number of units"

AVERAGE PROFIT

Special simpler case: constant marginal cost, no fixed costs.

Simple case: linear demand, constant MC, no fixed costs

P(Q) = 14 - Q
c(Q) = 2Q
\text{No fixed costs }\Rightarrow MC = AC = 2

Baseline Example: Monopoly

P(Q) = 14 - Q
c(Q) = 2Q
\pi(Q) = P(Q)Q - c(Q)
= (14 - Q)Q - 2Q
= 14Q - Q^2 - 2Q
\text{total revenue}
\text{total cost}

14

2

units

$/unit

P(Q) = 14 - Q
MC = AC = 2
\text{No fixed costs }\Rightarrow MC = AC = 2

14

P

Q

Baseline Example: Monopoly

P(Q) = 14 - Q
c(Q) = 2Q
\pi(Q) = P(Q)Q - c(Q)
= (14 - Q)Q - 2Q
= 14Q - Q^2 - 2Q
\text{total revenue}
\text{total cost}

14

2

units

$/unit

P(Q) = 14 - Q
MC = AC = 2
\text{No fixed costs }\Rightarrow MC = AC = 2

14

P

Q

Profit

Baseline Example: Monopoly

P(Q) = 14 - Q
c(Q) = 2Q
\pi(Q) = P(Q)Q - c(Q)
= (14 - Q)Q - 2Q
= 14Q - Q^2 - 2Q
\pi'(Q) = 14 - 2Q - 2 = 0
Q^* = 6
\text{total revenue}
\text{total cost}
\text{marginal revenue}
\text{marginal cost}
P^* = 14 - 6 = 8
\pi^* = 8\times 6 - 2 \times 6 = 36

14

8

2

6

Q

P

P(Q) = 14 - Q
MR(Q) = 14 - 2Q
MC = AC = 2

36

IMPORTANT

  • Don't look at the homework yet, I'm updating it
  • Refers to a bunch of stuff from game theory because we usually do that first
  • Mostly has to do with Thursday's model anyway

Duopoly

Quantity Duopoly

  • Two firms ("duo" in duopoly)
  • Each chooses how much to produce
  • Market price depends on
    the total amount produced
  • Each firm faces a residual demand curve 
    based on the other firm's choice

Quantity Leadership Model

  • Firm 1 chooses how much to produce  first \(q_1\),
    at a cost of $2 per unit
  • Firm 2 observes firm 1's choice,
    and then chooses how much to produce \(q_2\),
    also at a cost of $2 per unit
  • The market price is determined by \(P(q_1,q_2) = 14 - (q_1+q_2)\)
  • Profits are realized
    • \(\pi_1(q_1,q_2) = (14 - [q_1 + q_2]) \times q_1 - 2q_1\)
    • \(\pi_2(q_1,q_2) = (14 - [q_1 + q_2]) \times q_2 - 2q_2\)

How will firm 2 react to firm 1's quantity?

P(q_1,q_2) = 14 - (q_1+q_2)
c_2(q_2) = 2q_2
\pi_2(q_2) = P(q_1 + q_2)q_2 - c_2(q_2)
= (14 - q_1 - q_2)q_2 - 2q_2
= 14q_2 - q_1q_2 - q_2^2 - 2q_2
\pi'_2(q_2) = 14-q_1 - 2q_2 - 2 = 0
q_2^*(q_1) = 6-\frac{1}{2}q_1
\text{total revenue}
\text{total cost}
\text{marginal revenue}
\text{marginal cost}

2

P

P(q_2|q_1) = 14-q_1 - q_2
MR(Q) = 14-q_1 - 2q_2
MC(q_2) = 2
6-\frac{1}{2}q_1
14-q_1
q_2

"Firm 2's Residual Demand Curve"

Firm 2's "reaction function"

How will firm 1 choose, knowing firm 2's reaction?

P(q_1,q_2) = 14 - (q_1+q_2)
c_1(q_1) = 2q_1
q_2^*(q_1) = 6-\frac{1}{2}q_1

Firm 2's "reaction function"

\pi_1(q_1|r_2(q_1)) = P(q_1 + r_2(q_1))q_1 - c_1(q_1)
= (14 - q_1 - [6 - {1 \over 2}q_1])q_1 - 2q_1
= 14q_1 - q_1^2 - [6q_1 - {1 \over 2}q_1^2] - 2q_1
\pi'_1(q_1|r_2(q_1)) = 14 - 2q_1 - (6 - q_1) - 2 = 0
q_1^* = 6

Stackelberg Equilibrium

Firm 2's strategy: whatever \(q_1\) firm 1 produces, produce \(6 -{1 \over 2} q_1\) 

Firm 1's strategy: produce 6 units of output

Given what the other firm is doing,
does either firm have any incentive
to change its strategy?

In equilibrium, firm 2 produces 3 units.
Why don't we just say that its strategy is "produce 3 units of output"?

Next Time

  • Cournot: simultaneous move quantity game
  • Repeated Cournot & collusion

Econ 51 | Fall 2022 | 7 | Oligopoly

By Chris Makler

Econ 51 | Fall 2022 | 7 | Oligopoly

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