Download risk neutral

Infinitely repeated games
The concept of present value (see pp.14-18):
Profit today is more valuable than profit one year from today.
The present value of the future profit is
Profit
PV = ---------- ,
(1+i)
where i is the discounting factor, usually set equal to the
interest rate.
A firm that is believed to exist and earn profits infinitely into
the future has the present value of
PV   0 
1
(1  i)

2
(1  i)
2

3
(1  i)
3
 
(Such an expression is called an infinite series.)
A couple of useful facts about infinite series:
If the profits, π, are the same in each period and we
start counting with the present period, then



(1  i)

PV   


  
 
2
3
(1  i) (1  i)
(1  i )
i
i
If the first period in the series is “a year from now”,
then
PV 

(1  i )


(1  i )
2


(1  i )
3
  

i
Airline pricing game, revisited
Firm 2
Low price
High price
Low price
3, 3
8, 1
High price
1, 8
6, 6
Firm 1
What if this game is played repeatedly?
Firms use “trigger strategies” – strategies contingent on the
past play of a game (a certain action “triggers” a certain
response).
Suppose both firms are currently keeping their prices high.
An example of a trigger strategy:
“I will continue to play “HIGH” as long as you are playing
“HIGH”. Once you cheat by playing “LOW”, I will play
“LOW” in every period thereafter.”
If firm 1 continues to cooperate, its present value is
PVcoop   

(1  i)


(1  i)
2


(1  i)
3
 
Firms use “trigger strategies” – strategies contingent on the
past play of a game (a certain action “triggers” a certain
response).
Suppose both firms are currently keeping their prices high.
An example of a trigger strategy:
“I will continue to play “HIGH” as long as you are playing
“HIGH”. Once you cheat by playing “LOW”, I will play
“LOW” in every period thereafter.”
If firm 1 continues to cooperate, its present value is
PVcoop  6 
6
6
6
6







6

(1  i) (1  i) 2 (1  i)3
i
If firm 1 cheats, its present value is
PVcheat
3
3
3
3
 8


   8 
2
3
(1  i) (1  i)
(1  i )
i
Firm 1 will prefer to cooperate if
PVcoop  PVcheat
or
6
3
6  8
i
i
or
6i+6>8i+3
3>2i
i < 66.7%
If the rate of discounting is less
than 66.7%, then both firms
prefer to cooperate
Another example of a trigger strategy:
Quality choice by a firm.
The good can be purchased repeatedly.
Firm
Low quality High quality
Don’t buy
0; 0
0; - 10
- 10; 10
1; 1
Consumer
Buy
A trigger strategy by consumers that can support the
mutually beneficial outcome:
“I will buy your product as long as you produce high
quality. Once you produce low quality, I will never buy
your product again”
Economics of incomplete information
Traditional microeconomic analysis deals with
economic agents making decisions under complete
information.
Examples of such assumptions:
• Consumers know the utility they get from a good;
• Firms know demand schedules;
• Firms know each others’ prices; and so on.
Real life is more complicated and less certain.
Comparing projects with uncertain outcomes
Every project has two characteristics, the expected value
and the degree of risk.
A certain outcome = no risk.
The expected value, or the mean:
Computed as the weighted sum of all possible payoffs
(“weighted” = multiplied by the probabilities of each
respective outcome):
E[x] = q1x1 + q2x2 + … + qnxn,
where xi is payoff i, qi is the probability that payoff i occurs,
and q1 + q2 + … + qn= 1.
Variance (a measure of risk):
The sum of (the probabilities of each outcome multiplied
by the squared differences between the value of the
random variable and its mean:
Var = q1(x1 – E[x])2 + q2(x2 – E[x])2 + … + qn(xn – E[x])2
The standard deviation, σ, is the square root of the
variance.
The larger the variance (or the standard deviation), the
riskier the project.
(For events occurring with certainty, Var = 0)
How much would you be willing to pay for a lottery
ticket that pays $100 with a 50% probability and
nothing with a 50% probability?
I personally would pay… $30
What is the expected value of this lottery?
E[x] = q1x1 + q2x2 = 0.5 ∙ 100 + 0.5 ∙ 0 = $50
Why the difference?
The attitude to risk may vary.
• Individuals who prefer less risk to more risk, all other
things being equal, are called “risk averse”. It is
believed that most people belong to this group.
• Those who don’t care about the degree of risk and
care only about the expected value are called “risk
neutral”.
• Individuals who prefer more risk to less risk, all other
things being equal, are called “risk preferring”, or “risk
loving”.
(Sort of an anomaly.)
In the example above, the person in question is . . .
risk averse.
“Risk premium” – the minimum reward that would
induce a risk averse person to accept risk while
preserving the same expected value.
Alternatively, risk premium is the maximum amount of
money an individual will be willing to pay to replace an
uncertain situation with a certainty situation that has the
same expected value.
Risk premium depends on the characteristics of the
lottery as well as on individual preferences.
In the example above, my risk premium is $20.
I will pay only $30 for a lottery; in other words, will trade certain $50
for this lottery only for a premium of 50 – 30 = $20
Project selection
Suppose we have two projects:
A: expected value = $1000, Var = 5000
B: expected value = $1000, Var = 1500
Which one will each type choose?
Risk averse –
will choose B
Risk neutral – indifferent; either A or B
Risk loving –
will choose A
What if the expected values differ as well?
A: expected value = $1200, Var = 5000
B: expected value = $1000, Var = 1500
Everyone prefers higher expected value to lower expected
value, all other things being equal.
Risk preferences of each type are the same as on the
previous slide.
Type of
Preferences
individual
Exp. value
Risk
Overall
Risk averse
A
B
It depends
Risk neutral
A
either
A
Risk loving
A
A
A
What can a risk averse individual do to reduce risk?
1. Be informed.
Useful information has value!!!
2. Diversify.
Diversification, or “spreading the risk”.
You are considering investing $100 into one or two
assets (stocks, for concreteness).
Each stock is worth $50 now, and you believe that within
the next three months its value can with equal probability
either increase to $80 or drop to $40.
For now, let us assume that what happens to one stock
is not correlated to what happens to the other one.
Expected value of the investment:
If you invest in the shares of only one of the companies:
E[x] = 2 (0.5·80 + 0.5·40) = $120
If you buy one share of each of the two companies:
E[x] = (0.5·80 + 0.5·40) + (0.5·80 + 0.5·40) = $120
The variance:
If you invest in one company:
Var = 0.5 (160 – 120)2 + 0.5 (80 – 120)2 =
= 0.5·1600 + 0.5·1600 = 1600
If you invest in both…
Possible outcomes:
W/prob ¼ both stocks go up, x = $160
W/prob ¼ stock A goes up, stock B falls, x = $120
W/prob ¼ stock B goes up, stock A falls, x = $120
W/prob ¼ both stocks fall, x = $80
Var = 0.25 (160 – 120)2 + 0.5 (120 – 120)2 + 0.25 (80 – 120)2 =
= 0.25·1600 + 0.25·1600 = 800
If the payoffs from two assets are negatively correlated,
then diversification becomes even more attractive.
Example: Two companies compete for a large
government contract. After the winner is announced, the
winner’s stock goes up and the loser’s stock falls.
When firms undertake many projects at the same time, it
is best for them to be risk neutral.
Moreover, shareholders WANT managers to act in a riskneutral manner (to care only about expected values).
Summary:
•Everyone prefers a higher expected value to a lower
expected value.
•Most individuals are risk averse. This means they prefer
less risk to more risk, provided the expected value stays
the same. If one project offers a higher expected value
and a higher risk at the same time, then we need more
information to tell which of the two will a risk averse
person choose.
•Firms can be assumed to be risk neutral. They evaluate
projects based solely on their expected values.
Pricing and output decisions under uncertainty
Consider a modification of problem 4 on p.469.
You are the manager of a firm that sells soybeans in a perfectly
competitive market. Your cost function is C(Q) = 2Q +2Q2.
Due to production lags, you must make your output decision
prior to knowing what the market price is going to be. You
believe that there is a 25% chance the market price will be
$120 and a 75% chance it will be $160.
What is the optimal quantity of output ?
The good news:
All the rules we have learned before (MR=MC, etc.) still apply
but “expected” appear in them as needed.
Let’s do it step by step.
Normally, the rule we’d apply would be P=MC.
Here, we replace P with its expected value, E(P).
a. Calculate the expected market price.
E(P) = 0.25·120 + 0.75·160 = 30 + 120 = 150
b. What output should you produce to maximize expected
profits?
E(P) = MC
150 = 2 + 4Q
148 = 4Q
Q = 37
TC = 2Q +2Q2
therefore MC = 2 + 4Q
c. What are your profits under each outcome
and the expected profits?
You produce Q=37 which determines your cost,
TC = 2·37 + 2·372 = 74 + 2738 = $ 2,812
If P = 120, your profit is = 120·37 – 2812 = $ 1,628
(happens w/prob ¼ )
If P = 160, your profit is = 160·37 – 2812 = $ 3,108
(happens w/prob ¾ )
Expected profit = ¼ · 1628 + ¾ · 3108 = $ 2,738
Looks like in one case we are underproducing and in the other
case – overproducing.
Wouldn’t it be better to bet on the most likely outcome?
P = 160
MC = 2 + 4 Q
4 Q = 158
Q = 39.5
and TC = 2·39.5 + 2·39.52 = $ 3,199.50
If P = 160, our profit = 160·39.5 – 3199.50 = $ 3,120.50
If P = 120, our profit = 120·39.5 – 3199.50 = $ 1,540.50
Expected profit = ¼ · 1540.5 + ¾ · 3120.5 = $ 2,725.50
The same approach can be extended to the imperfectly
competitive market case.
Consider the following problem:
A firm with market power produces at constant marginal (and
average) cost of $1. There is a 50% chance of a recession and a
50% chance of an economic boom.
During a boom, the inverse demand for firm’s product will be
P = 10 – 0.5 Q
If there is a recession, the inverse demand will be P = 6 – 0.5 Q
The firm is risk neutral and must set output before demand is
known. How much output should it produce to maximize
expected profit?
Normally, we would look for the point where MR = MC.
This time, we will do E(MR) = MC.
There are two equally good ways to find expected marginal revenue:
1. Find expected demand, then expected marginal revenue:
Expected inverse demand:
E(P) = 0.5 (10 – 0.5 Q) + 0.5 (6 – 0.5 Q) = … = 8 – 0.5 Q
E(MR) = 8 – Q
2. Find the marginal revenue under each scenario, then find the
expected MR:
Boom:
P = 10 – 0.5 Q
MR = 10 – Q
Recession:
P = 6 – 0.5 Q
MR = 6 – Q
E(MR) = 0.5 (10 – Q) + 0.5 (6 – Q) = 8 – Q
The rest is trivial:
E(MR) = 8 – Q
MC = 1
E(MR) = MC
8–Q=1
Q=7
If boom, then
P = 10 – 0.5 Q = $6.50
Profit = (P – AC) Q = (6.50 – 1) · 7 = $38.50
If recession, then
P = 6 – 0.5 Q = $2.50
Profit = (P – AC) Q = (2.50 – 1) · 7 = $10.50
Expected profit = 0.5 · 38.50 + 0.5 · 10.50 = $24.50
Or directly:
“Expected price” given Q = 7 is E(P) = 8 – 0.5 Q = $4.50
Exp.profit = (E(P) – AC) Q = (4.50 – 1) · 7 = $24.50
Consumer search for the best price
and implications for the firm’s behavior
General idea:
A consumer samples several stores and obtains a price
quote from each.
The cost of obtaining each quote is the same.
The total number of stores is large, so “drawing” one of
them doesn’t affect the odds.
After several quotes, you can always return to the store
with the best price.
It makes sense to continue searching as long as the
(expected) benefit exceeds the cost of search.
Expected benefit:
Joe wants to buy a DVD player. He thinks one-third of
the stores charge $130 for a DVD player, one-third
charge $100, and one-third charge $85. He sampled
one store and the price was $100. What is the expected
benefit from sampling another store?
w/prob 1/3 next P = $85,
a $15 benefit
w/prob 1/3 next P = $100,
no benefit
w/prob 1/3 next P = $130,
since he can return to the
first store, no benefit
Exp.benefit = (1/3)·15 = $5
How will the answer change if the best price found
so far is $130?
w/prob 1/3 next P = $85,
a $45 benefit
w/prob 1/3 next P = $100,
a $30 benefit
w/prob 1/3 next P = $130,
no benefit
Exp.benefit = (1/3)·45 + (1/3)·30 = $25
In general, if you sample a store and the price is high, the
expected benefit from further search is greater
(it makes
more
sense to keep searching).
The lower the observed price, the more sense it makes to
stop searching and buy.
This principle holds even if the distribution of prices is not
known.
Cost/benefit of
continuing to
search
Exp.benefit of
another search
Cost of another search
Price observed
If observed P is at or
below this level, we stop
searching and buy
What happens if the cost of search increases?
Cost/benefit of
continuing to
search
Exp.benefit of
another search
Cost of another search
Price observed
Consumers are more likely to settle for higher prices.
What has happened with the advent of the Internet?
Search cost decreased;
As a result, consumers buy same goods at lower prices
Are all industries affected equally?
(Durable) consumer goods – very much so,
Groceries and expendable household items –
What has happened with the advent of the Internet?
Search cost decreased;
As a result, consumers buy same goods at lower prices
Are all industries affected equally?
(Durable) consumer goods – very much so,
Groceries and expendable household items – less;
Travel fares – affected a lot;
Industrial shipping rates – less. Why?
Insurance rates, phone rates – also affected a lot
Socially optimal risk sharing
If individuals are risk averse while firms are risk neutral, what
is the optimal risk sharing between consumers and firms?
Getting out of risk has more value for consumers than for
firms.
Therefore there is room for mutually beneficial exchange,
where consumers reward the firm for accepting some of their
risk for them.
Example: insurance industry.
Insurance companies are able to make money selling
insurance because they may
- have better information about the odds than their clients;
- differ from clients in their attitude to risk;
- diversify.
The larger the group of the insured, the
the variance, hence the
lower
smaller
the risk.
Incomplete asymmetric information
“Asymmetric” in this case points at the fact that one party
is less informed than the other.
The following analogy may be helpful:
A card game may be played under different rules:
• All cards are dealt face up – complete information
• Some (or all) cards are dealt face down – incomplete
symmetric information.
• Some (or all) cards are dealt face down but player can look at
some of his own cards – incomplete asymmetric information.
An example of asymmetric information:
Sellers know product quality, buyers do not.
The only way for buyers to find out the true value is to try
the product. (“Experience goods”)
Under certainty, the rule for rational behavior is:
Buy if
Value > P,
where “value” (a.k.a. “utility”) stands for the subjective
value the buyer gets from the product.
Under uncertainty, it becomes
Buy if
Exp. Value > P (for a risk neutral consumer)
or
Buy if
Exp. Value – risk premium > P
(for a risk averse consumer)
Say the product has a value of $100 for a consumer if it
works as expected/promised.
The consumer believes that there is
- a 90% chance the product will deliver services;
- a 10% chance it will break down immediately (value = 0).
Up to what price will a risk neutral consumer pay for the
product?
Max P = Exp Value = 0.9·100 + 0.1· 0 = $90
What happens if the consumer is risk averse?
Max P = $90 – risk premium < $90
If buyers’ subjective valuations of the good are below the
producer’s cost of making it, the market breaks down –
nothing will be sold.
Both buyers and sellers are hurt by that.
Example:
The market for “lemons” (Akerlof, 1973) analyzes the
market for “lemons”, or cars with hidden defects.
Asymmetric information is reflected in the fact that the
quality of cars in the market is known to sellers but not
to buyers.
There are two types of used cars offered for sale in the
market, 1,000 good cars and 1,000 “lemons”.
The number of potential buyers exceeds the number of
cars available (a case of “sellers’ market”).
All buyers are identical – each of them will pay up to
$1,000 for a lemon and $2,000 for a good car.
(Those numbers are also called “reservation prices”.)
The sellers’ “reservation price” (the lowest price they
would agree to sell for) is $800 for a lemon and $1600
for a good car.
Case 1. Symmetric complete information – the true
quality of each car is known to both parties.
We have two separate markets:
P
2000
P
2000
P=$2000
1600
1600
P=$1000
1000
800
1000
800
Q
1000
Q
1000
Good cars
Lemons
All cars are sold.
Case 2. Symmetric incomplete information – the true
quality of a particular car is not known to anybody.
Each car is either a good one (with a 50% probability) or
a lemon (with a 50% probability).
Neither buyers nor sellers can tell one from another.
For simplicity, we are going to assume both sides are risk
neutral. Therefore they base their reservation prices on
expected values.
For sellers, EV = 0.5 ∙ 1600 + 0.5 ∙ 800 = $1,200
For buyers, EV = 0.5 ∙ 2000 + 0.5 ∙ 1000 = $1,500
P
2000
1500
1200
1000
800
Q
1000
2000
Equilibrium price = $1500
Equilibrium quantity = 2000
Case 3. Asymmetric incomplete information – sellers
know the quality, buyers don’t.
For buyers, the situation is the same as in the previous
case – they will pay $1500 for any car.
Sellers, however, can tell good cars from lemons, and
their reservation price is different for each category.
P
2000
1500
1200
1000
800
Q
1000
2000
As a result, only lemons are sold.
This is an example of adverse selection, or a situation
when poor quality products drive high quality products
out of the market.
Adverse selection prevents markets from operating
efficiently and is detrimental for both buyers and sellers.
After buyers realize that no good cars are being traded,
their EV drops to $1000.
What happens to the market price?
It also decreases to $1000.
Asymmetric information does not necessarily result in
adverse selection. For instance, if sellers’ reservation
price for a good car is $1200, then efficiency is restored.
See below.
P
2000
1500
1200
1000
800
Q
1000
2000
A similar example:
Adverse selection in the health insurance market.
• An individual knows his probability of accident,
illness, etc. better than the insurance company.
• Insurance companies know only the composition of
the population. If they offer a uniform insurance
contract and price it based on the average degree of
risk, then it is attractive only for the high-risk
individuals. Low-risk individuals don’t buy insurance,
and the average probability of accident/illness
exceeds the initial estimate.
Ways to overcome the undesirable consequences of
information asymmetry involved making the
uninformed party better informed or reducing the
amount at stake for them:
• laws protecting consumers;
• consumer reports;
• “screening”;
• “signaling”.
The last two deserve some discussion.
(To be continued….)