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Transcript
```What is game theory?
• “Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent, rational decision-makers. Game
theory provides general mathematical techniques for analyzing situations
in which two ore more individuals make decisions that will influence one
another’s welfare.” (-Myerson, Game Theory: Analysis of Conflict
Why use game theory?
• “Neo-classical” economic theory can be very limiting in the kinds of questions it can answer
• Traditional theories of economic and social behavior are not generally
useful for policy purposes
• With so much data available today, we can analyze markets and strategic
situations on very small scales, which would have been impossible before
the invention of powerful computers and the internet, but what should the
analysis look like?
The perfectly competitive market: Firm behavior
A representative firm receives a price p per unit of a good it sells, and it
c
costs C(q) = q 2 to produce q units. Then
2
• Total revenue of q units is pq
c
• Total costs are C(q) = q 2
2
• Profits are the difference between total revenue and total cost, π(q) =
c
pq − C(q) = pq − q 2
2
How do we find the profit-maximizing quantity, q ∗ , for which π(q ∗ ) ≥ π(q ′ ) for
any other q ′ the firm could pick?
The perfectly competitive market: Firm behavior
Let’s start by thinking about “searching” for a maximum, starting from
some arbitrary quantity q. Consider the change in profit from using some other
quantity, q ′ :
c c π(q) − π(q ′ ) = pq − q 2 − pq ′ − q ′2
2
2
c
′
′
′
= p(q − q ) − (q + q )(q − q )
2
′
Now, divide by q − q to get
π(q) − π(q ′ )
c
= p − (q + q ′ )
q − q′
2
This represents the gain/loss to switching from q to q ′ .
1
The perfectly competitive market: Firm behavior
We now take the limit as q ′ goes to q
lim
′
q →q
π(q) − π(q ′ )
c
= lim
p − (q + q ′ )
′
′
q →q
q−q
2
or
π ′ (q) = p − cq
So the change in profit π(q) when evaluated at the optimal quantity q equals
p − cq. If this is positive, it means that profits are increasing at q, so we should
pick a larger value. If this is negative, it means that profits are decreasing at
q, and we should pick a smaller value. Only when π ′ (q) = 0 do we know that
there are no better options than q “nearby”.
(Maximization in general)
(Maximization in general)
So for x∗ to solve
max f (x),
x
′
∗
it is necessary that f (x ) = 0; we call the equation f ′ (x∗ ) = 0 the first-order
necessary conditions.
For some problems, this approach doesn’t work. Why not?
In this class, we’ll have to maximize functions like
f (x) =
a 2
x − bx + c
2
2
a lot, with a, b > 0. In this case, the derivative is
f ′ (x) = ax − b
where the equation f ′ (x∗ ) = 0 has the solution x∗ = b/a. When is this actually
a maximizer?
(Common Derivatives and Differentiation Rules)
• [log(x)]′ =
√
1
1
, [xb ]′ = bxb−1 , [ x]′ = √ , [ex ]′ = ex
x
2 x
• Addition/Scalar Multiplication: Dx [af (x) + bg(x)] = af ′ (x) + bg ′ (x)
• Multiplication Rule: [f (x)g(x)]′ = f ′ (x)g(x) + f (x)g ′ (x)
• Chain Rule: f ((g(x))′ = f ′ (g(x))g ′ (x)
• Derivative Rule:
f (x)
g(x)
′
=
f ′ (x)g(x) − f (x)g ′ (x)
g(x)2
3
The perfectly competitive market: Firm behavior
Now we know that a necessary condition for q ∗ to be a maximum of π(q) is
that
π ′ (q ∗ ) = p − cq ∗ = 0
or
q∗ =
p
c
In other words, the firm’s supply curve is q S = p/c.
The perfectly competitive market: Firm behavior
The perfectly competitive market: Consumer behavior
There is a representative consumer who buy the good q along with another
good, we’ll call m, the numeraire or “money”. The consumer has a utility
function over bundles (q, m) of
u(q, m) = b log(q) + m
The consumer also faces a budget constraint, w = pq + m, so that total expenditures on the two goods cannot be more than w. What bundle should the
consumer pick?
The perfectly competitive market: Consumer behavior
Rather than use Lagrange multipliers, we can rewrite the budget constraint
as m = w − pq and substitute it into the utility function to get
max b log(q) + w − pq
q
4
which is unconstrained in q, making it a much easier problem to solve. Then
our necessary condition is
b
b
− p = 0 −→ q ∗ =
∗
q
p
giving us an demand curve, q D = b/p.
The perfectly competitive market: Consumer behavior
Market Equilibrium
Definition 1. A price-quantity pair (p∗ , q ∗ ) is a price-taking equilibrium or
perfectly competitive equilibrium if
1. Firms maximize profits, taking the price as given.
2. Consumers maximize utility, taking the price as given.
3. Markets clear, so that supply equals demand, or q S = q D .
We’ve already solved for supply and demand above, so all that’s left to do
is equate them:
p
b
qS = = = qD
c
p
√
√ √
so that p∗ = bc and q ∗ = b/ c.
5
Market Equilibrium
(Wait, why models?)
In any economic model, we have a number of variables that are determined
within the model, (p, q), and other variables that are taken as given, and the
model is not meant to explain them (b, c).
• A variable is endogenous if its value is determined within the model. (p,
q)
• A variable is exogenous if its value is given, and we use it to solve for the
endogenous variables. (c)
• A model is good if the story it tells about how exogenous variables determine endogenous variables is convincing.
By using variation in what we observe, we can measure the magnitude and significance of different “forces” in the economic environment. Here, an economic
model is a positive explanation of how the world is: It is an attempt to describe
how the world really works.
Perfectly competitive markets in general
• A representative consumer chooses a bundle of goods (q, m) to maximize
its utility, v(q) + m, subject to the constraint pq + m = w, taking prices
as given.
• A representative firm chooses how much of the good q to supply to maximize its profits, π(q) = pq − C(q) where C(q) represents its total costs,
taking prices as given.
6
• The quantity demanded by the consumer equals the quantity supplied by
the firm, so that markets clear.
What are reasonable assumptions to make about what v(q) and C(q) look like?
The perfectly competitive market: Firm behavior in general
With the idea of a derivative, we know that
max pq − C(q)
q
has first-order necessary condition
p − C ′ (q ∗ ) = 0
or “price equals marginal cost”.
The perfectly competitive market: Consumer behavior in general
Similarly, the consumer solves
max v(q) − pq + w
q
which has first-order necessary condition
v ′ (q ∗ ) = p
or “marginal benefit equals price”.
The perfectly competitive market: Equilibrium in general
The combined conditions for the consumer and firm imply
C ′ (q ∗ ) = p∗ = v ′ (q ∗ )
so that price equals the marginal cost of providing the good to the firm, and
the marginal benefit of the last unit to the consumer.
The Welfare Theorem
Rather than letting the market determine the outcome, what if a “benevolent
social planner” decided q? How would the outcome be different?
Definition 2. The social planner’s problem is found by summing the payoffs
of all the agents, and maximizing the resulting social welfare function. The
outcome that maximizes the social welfare function is efficient.
Here, the social welfare function is the sum of profits and utility, giving us
W (q) = (pq − C(q)) + (v(q) + w − pq) = v(q) + w − C(q)
7
The Welfare Theorem
Now, maximizing over q gives
W ′ (q ∗ ) = v ′ (q ∗ ) − C ′ (q ∗ ) = 0
or “marginal benefit of the good equals the marginal cost of supplying it”,
exactly as in the price-taking equilibrium.
Theorem 3. In a perfectly competitive market, the outcome is efficient.
The Welfare Theorem
When does the welfare theorem fail?
1. Agents have price power
2. There are externalities: Agents’ actions have direct consequences for others that are not reflected through the price system
3. The government is intervening in ways that give agents strange incentives
4. Agents have different information about the value or cost of goods and
services
5. Agents have trouble finding one another quickly, leading to “search frictions”
6. There are “missing markets”
(Wait, why models?)
• Now we have another motivation for using a model: It provides us with
an argument about why perfectly competitive markets are desirable. In
other words, it is a normative explanation of how the world should be.
• In addition, while you may disagree with the conclusion that perfectly
competitive markets are desirable, at least you can argue with the assumptions of the model or provide an alternative model to illustrate your
point.
Price power: Monopoly
• Suppose there is a single firm in the market that knows it controls quantity
and price. How does it behave?
Suppose the inverse demand curve is p(q). Then a monopoly solves
max p(q)q − C(q)
q
with FONC
p′ (q ∗ )q ∗ + p(q ∗ ) − C ′ (q ∗ ) = 0
{z
}
| {z } |
<0
Price-MC
Is this efficient? This seems to solve the issue of studying market power. What
happens when there are, say, seven firms?
8
Some Puzzles
What is missing in a perfectly competitive model of a market?
1. Why do firms offer quantity discounting?
2. Why are insurance or mortgages available to some people, but not others,
even when the others are willing to pay a higher price? (i.e., markets fail
to clear)
3. Why are fastfood restaurants usually located right next to each other,
rather than uniformly distributed around town?
4. Why do we need lawyers and contracts?
5. The most recent financial crisis largely concerned whether particular sheets
of paper were valuable or not. How is that possible?
6. What is going on when people “haggle”?
7. Suppose college has no impact on your productivity as a worker (for many
students I’ve seen, this is certainly true). Why do so many people go?
8. Why do firms exist?
Game Theory
Game theory is the study of strategic behavior through mathematical models. Its goal is to provide a comprehensive approach to explaining and predicting
the behavior of agents who act deliberately. Game theory also gives us substantially more ability to study how endogenous variables are determined by allowing
us to look at the actual “mechanics” of markets: Contracts, bidding, bargaining,
etc. Generally, we assume
• Agents are self-regarding, in that their preferences over outcomes do not
depend on the preferences of the other agents per se, or “directly”
• Agents are rational, meaning that they have complete, transitive preferences over all outcomes (this is the real definition of “rationality” in
economics).
Game Theory
A game is composed of
• Players : Those agents who take actions
• Actions : The choices which players can select
• Payoffs : The numerical value that players associate with different outcomes of the game, which are allowed to depend on each player’s action
• Timing : A description of which players take actions when
• Information : A description of what players know, and when they know it
9
Game Theory
• We begin with simultaneous-move games of complete information, in which
all players make their decisions at the same time and know everything
• We then add timing, giving us dynamic games of complete information,
where players make decisions in sequence, and cannot revisit their earlier
choices.
• Finally, we add incomplete or imperfect information to get Bayesian games
(think poker).
Prisoners’ Dilemma
• There are two burglars, who have been captured in the process of committing a crime. They have been very careful, and actually do not even
know each other’s real name. The district attorney tells them: “If you
both remain silent, I have enough evidence to send each of you to jail for
two years. However, if one of you confesses and the other tells the truth,
I will give the confessor a lighter sentence, sending him to jail for only
one year, while I prosecute the other aggressively and send him to jail for
five years. If both of you confess, there won’t be a trial, and you both get
three years.”
Prisoners’ Dilemma
Row
Silent
Confess
Column
Silent
-2,-2
-1,-5
Confess
-5,-1
-3,-3
What is the best outcome for both burglars (what would a social planner pick)?
What do you think the burglars do, and why? What other economic situations
have similar incentives?
Battle of the Sexes
• Two people have decided to go on a date. The two options are a Football
game, and the Ballet. The male prefers football, while the female prefers
ballet. They discuss which option they will pick, but both happen to
forget which they decided on. Worse, they both forgot their smart phones
at work, and the two events are about to begin. Both prefer to be together
rather than apart.
10
Battle of the Sexes
Male
Football
Ballet
Female
Football
2,1
0,0
Ballet
0,0
1,2
What is the best outcome for the couple (what would a social planner pick)?
What do you think they do, and why? What if we made the payoffs to Ballet
(10, 20)? What other economic situations have similar incentives?
Matching Pennies
• You are waiting for a plane with a friend. Both of you have plenty of
pocket change, so you propose the following game: You both secretly pick
Heads or Tails. If both coins are heads, you get both coins. If both coins
are tails, your friend gets both coins.
Matching Pennies
You
Tails
Friend
1,-1
-1,1
Tails
-1,1
1,-1
What is the best outcome for the couple (what would a social planner pick)?
What do you think they do, and why? What happens if we made the payoff to
(Tails, Tails) equal to (10, −1)?
The Strategic Form
The matrix of players/actions/payoffs that we’ve been using to describe
games is very helpful, since it summarizes all of the relevant information from
a game theory perspective. We call it the strategic form.
Row Player
U
D
Column Player
L
urow (U, L), ucolumn (U, L)
urow (D, L), ucolumn (D, L)
R
urow (U, R), ucolumn (U, R)
urow (D, L), ucolumn (D, L)
So you can think of game theory as a generalization of a regular economics
model where consumers have preferences over bundles of goods or firms have
preferences over quantities produced, to a setting where agents have preferences
over how the other agents act.
11
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