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 )
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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
(Such an expression is called an infinite series.)
A couple of useful facts about infinite series:
If the profits, , are the same in each period and we start counting with the present period, then
If the first period in the series is a year from now, then
Airline pricing game, revisited
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
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
If firm 1 cheats, its present value is
Firm 1 will prefer to cooperate if
or
or
6 i+ 6 > 8 i + 3
If the rate of discounting is less than 66.7%, then both firms prefer to cooperate
3 > 2 i
i < 66.7%
Another example of a trigger strategy:
Quality choice by a firm.
The good can be purchased repeatedly.
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, qiis the probability that payoff i occurs, and q1 + q2 + + qn= 1.
Variance (a measure of risk):
Var = q1(x1 E[x])2 + q2(x2 E[x])2 + + qn(xn E[x])2
The standard deviation, , 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?
Why the difference?
The attitude to risk may vary.
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
Risk neutral
Risk loving
will choose B
indifferent; either A or B
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.
What can a risk averse individual do to reduce risk?
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 (0.580 + 0.540) = $120
If you buy one share of each of the two companies:
E[x] = (0.580 + 0.540) + (0.580 + 0.540) = $120
The variance:
If you invest in one company:
Var = 0.5 (160 120)2 + 0.5 (80 120)2 =
If you invest in both
Possible outcomes:
W/prob both stocks go up, x = $160
W/prob stock A goes up, stock B falls, x = $120
W/prob stock B goes up, stock A falls, x = $120
W/prob both stocks fall, x = $80
If the payoffs from two assets are negatively correlated, then diversification becomes even more attractive.
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 risk-neutral manner (to care only about expected values).
Summary:
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 ?
Normally, the rule wed apply would be P=MC.
Here, we replace P with its expected value, E(P).
b. What output should you produce to maximize expected profits?
c. What are your profits under each outcome and the expected profits?
You produce Q=37 which determines your cost,
TC = 237 + 2372 = 74 + 2738 = $ 2,812
If P = 120, your profit is = 12037 2812 = $ 1,628
(happens w/prob )
If P = 160, your profit is = 16037 2812 = $ 3,108
(happens w/prob )
Expected profit = 1628 + 3108 = $ 2,738
Looks like in one case we are underproducing and in the other case overproducing.
Wouldnt it be better to bet on the most likely outcome?
P = 160 MC = 2 + 4 Q
4 Q = 158
Q = 39.5and TC = 239.5 + 239.52 = $ 3,199.50
If P = 160, our profit = 16039.5 3199.50 = $ 3,120.50
If P = 120, our profit = 12039.5 3199.50 = $ 1,540.50
Expected profit = 1540.5 + 3120.5 = $ 2,725.50
Consumer search for the best price and implications for the firms 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 doesnt 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
How will the answer change if the best price found so far is $130?
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.
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)
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, 1,000good 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
P
2000
2000
P=$2000
1600
1600
P=$1000
1000
1000
800
800
Q
Q
1000
1000
Good carsLemons
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.
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 dont.
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,
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.
Ways to overcome the undesirable consequences of information asymmetry involved making the uninformed party better informed or reducing the amount at stake for them:
The last two deserve some discussion.
(To be continued.)