UNIT IV: INFORMATION & WELFARE Decision under Uncertainty Externalities & Public Goods Review 7/26

Cartel Enforcement Consider a market in which two identical firms can produce a good with a marginal cost of $1 per unit. The market demand function is given by: P = 7 – Q Assume that the firms choose prices. If the two firms choose different prices, the one with the lower price gets all the customers; if they choose the same price, they split the market demand. What is the Nash Equilibrium of this game?

Cartel Enforcement Consider a market in which two identical firms can produce a good with a marginal cost of $1 per unit. The market demand function is given by: P = 7 – Q Now suppose that the firms compete repeatedly, and each firm attempts to maximize the discounted value of its profits (  < 1). What if this pair of Bertrand duopolists try to behave as a monopolist (w/2 plants)?

Cartel Enforcement What if a pair of Bertrand duopolists try to behave as a monopolist (w/2 plants)? P = 7 – Q; TC i = q i Monopoly Bertrand Duopoly  = TR – TCQ = q 1 + q 2 = PQ – QP b = MC = 1; Q b = 6 = (7-Q)Q - Q = 7Q - Q 2 - Q FOC: 7-2Q-1 = 0 => Q m = 3; P m = 4 w/2 plants: q 1 = q 2 = 1.5q 1 = q 2 = 3  1 =  2 = 4.5    =  2 = 0

Cartel Enforcement What if a pair of Bertrand duopolists try to behave as a monopolist (w/2 plants)? Promise: I’ll charge P m = 4, if you do. Threat: I’ll charge P b = 1, forever, if you deviate. 4.5 … 4.5 … 4.5 … 4.5 … 4.5 … 4.5 … 4.5 = 4.5/(1-  ) 4.5 … 4.5 … 4.5 … 9 … 0 … 0 … 0 If  is sufficiently high, the threat will be credible, and the pair of trigger strategies is a Nash equilibrium.  * = 0.5 Trigger Strategy Current gain from deviation = 4.5 Future gain from cooperation =  (4.5)/(1-  )

Decision under Uncertainty In UNIT I we assumed that consumers have perfect information about the possible options they face (their income and prices); and about the utility consequences of their choices (their preferences). Now, we will ask whether our model can be extended to deal with more realistic cases in which decisions are made without perfect information. We will also ask how imperfect (asymmetric) information affects market outcomes and their welfare consequences.

Decision under Uncertainty The Economics of Information: How can I maximize utility given incomplete info? How much info should I gather? We can distinguish between 2 sources of uncertainty: The behavior of other actors (strategic uncertainty) states of nature (natural uncertainty) –Will it rain? Or not? –Is there oil in the drilling hole? –Will the roulette wheel come up red? ( ) –Is the car a lemon?

Decision under Uncertainty Expected Value v. Expected Utility Risk Preferences Reducing Risk: Insurance Contingent Consumption Adverse Selection (and Moral Hazard)

Expected Value & Expected Utility Which would you prefer? A) chance of winning $30,000 or losing $5,000 B) Sure thing of $10,000 How much would you be willing to pay for the chance to win $2n if the head comes up on nth flip? 2(1/2) + 4(1/4) + … = … =

Expected Value & Expected Utility How much would you be willing to pay for the chance to win $2 n if a heads comes up on nth flip? Expected Value (EV): the sum of the value (V) of each possible state, weighted by the probability (  ) of that state occurring. On 1 flip:  (H) = ½ (2) + 4(1/4) + … = … =

Expected Value & Expected Utility How much would you be willing to pay for the chance to win $2 n if a heads comes up on nth flip? Expected Value (EV): the sum of the value (V) of each possible state, weighted by the probability (  ) of that state occurring. On 1 flip: EV =  (V)H = (½)2 + 4(1/4) + … = … =

Expected Value & Expected Utility How much would you be willing to pay for the chance to win $2 n if a heads comes up on nth flip? Expected Value (EV): the sum of the value (V) of each possible state, weighted by the probability (  ) of that state occurring. On nth flip: EV(H n ) = ½ n (2 n ) + 4(1/4 ) + … = … =

Expected Value & Expected Utility How much would you be willing to pay for the chance to win $2 n if a heads comes up on nth flip? Expected Value (EV): the sum of the value (V) of each possible state, weighted by the probability (  ) of that state occurring. On nth flip: EV(H n ) = ½ n (2n) + 4(1/4) + … = … = ½ ¼ 8 8 EV(H)=½(2)+(1/4)4+(1/8)8 Flip 1: Win $2 Flip 2: Win $4 Flip 3: Win $8 H T 1 1

Expected Value & Expected Utility How much would you be willing to pay for the chance to win $2 n if a heads comes up on nth flip? Expected Value (EV): the sum of the value (V) of each possible state, weighted by the probability (  ) of that state occurring. On n flips: EV(H)=(½)2+(1/4)4+(1/8)8+…=1+1+1+…= infinity So, you’d be willing to pay an awful lot? What’s going on here?

Expected Value & Expected Utility With examples such as these, David Bernoulli (1738) observed that rational agents often behave contrary to expected value maximization. Instead, they maximize: Expected Utility (EU): the sum of the utility of each possible state, weighted by the probability of that state occurring. EU =  1 (U(s 1 )) +  2 (U(s 2 )) + …  n (U(s n )) Where  is the probability of that state occurring. arise because utility will be a non-linear function of “wealth”.

Expected Value & Expected Utility With examples such as these, David Bernoulli (1738) observed that rational agents often behave contrary to expected value maximization. Instead, they maximize: Expected Utility (EU): the sum of the utility of each possible state, weighted by the probability of that state occurring. Rankings of expected values and expected utilities need not be the same! Differences arise because utility will be a non-linear function of “wealth” and will depend on endowments. * or “income” or “consumption” *

Expected Value & Expected Utility Diminishing Marginal Utility: The intrinsic worth of wealth increases with wealth, but at a diminishing rate W U U(15) U(10) U(5) von Neumann-Morgenstern Utility Indexes MU = ½W -½ U = W ½ MU = 1/W U = lnW For 2 states: EU =  (U(W i )) + (1-  )(U(W j )) MRS = (  /(1-  ))MU i /MU j

Risk Preferences A risk averse consumer will prefer a certain income to a risky income with the same expected value. 5 CE W U U(15) U(10) U(5).5U(5) +.5U(15) The chord represents the chance to win $5 or $15.

Risk Preferences A risk averse consumer will prefer a certain income to a risky income with the same expected value. 5 CE W U U(15) U(10) U(5).5U(5) +.5U(15) Certainty Equivalent (CE) of an equal chance of winning $5 and $15 Risk Premium = 10 – CE

Risk Preferences A risk loving consumer will prefer a risky income to a certain income with the same expected value. 5 CE W U U(15).5U(5) +.5U(15) U(5) U(10)

Risk Preferences A risk neutral consumer is indifferent between a risky income and a certain income with the same expected value. 5 CE W U U(15) U(10) U(5)

Risk Preferences A risk neutral consumer is indifferent between a risky income and a certain income with the same expected value. 5 CE W U U(15) U(10) U(5) Do any of these cases violate any of our assumptions about well-behaved preferences? Draw a set of indifference curves for each case.

Risk and Insurance A risk averse consumer will prefer a certain income to a risky income with the same expected value. Given the opportunity, therefore, she will attempt to smooth the variability of her wealth, by spreading (or diversifying) her risks across states. Insurance offers a way to buy wealth in the event of a low wealth (or “bad”) state, by transferring some wealth from the “good” to the “bad” state.

Risk and Insurance A risk averse consumer has wealth of $35,000, including a car worth $10,000. There is a 1/100 chance that the car will be stolen. So there is a 0.01 chance his wealth will be $25,000 and a 0.99 chance it will be $35,000. EW = 0.01(25000) (35000) Buying insurance can change this distribution.

Risk and Insurance If his car is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,000 (35,000 – 10) (25, ) Suppose he can by $1000 insurance at a premium of $1/100.  =.01 How much insurance will he buy?he buy? Not to scale

Risk and Insurance If his car is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,000 Suppose he can by $1000 insurance at a premium of $1/100.  =.01 How much insurance will he buy?he buy? ?

Risk and Insurance If his car is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,000 Given the chance to buy insurance at an “actuarily fair” price (i.e.,  =  ), a risk averse consumer will fully insure. Equalizing wealth across states. he buy? 34,900 Certainty Line

Risk and Insurance Insurance is a way to allocate wealth across possible states of the world. In essence, he is purchasing contingent claims on consumption (wealth) in the two states. So we can solve in the usual way: Eb Wb Wg Eg More generally: E = Endowment K = dollars of insurance  = premium ? Eg -  K Eb + K -  K Endowment

Contingent Consumption If his car is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,  k 25,000 + K -  K Now suppose the premium rises to $1.10/100 (  =.011). His vN-M Index: U = lnW How much insurance will he buy? Endowment

Contingent Consumption If his car is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,000 Slope(m) =  Wg/  Wb = -  K/(K-  K) = -  /(1-  )  = Pb 1-  = Pg Not to scale m = -Pb/Pg

Contingent Consumption If his car is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,000 Wg* Wb* What is his budget constraint? Not to scale m =

Contingent Consumption If his car is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,000 Wg* Wb* Budget Constraint: Wg = m(Wb) + Wg(int) Wg = -(.011/.989)Wb Not to scale m =

Contingent Consumption If his car is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,000 Wg* Wb* U = lnW EU =  (U(Wb)) + (1-  )(U(Wg)) MRS = (  /(1-  ))MUb/MUg = (.01/.99)(Wg/Wb) = P(Wb)/P(Wg) =  /(1-  ) Not to scale

Contingent Consumption If his can is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,000 Wg* Wb* MRS = (.01/.99)(Wg/Wb) Pb/Pg =  /(1-  ) MRS = Pb/Pg => Wb =.909Wg Wg = -(.011/.989)Wb Wg = $ Not to scale

Contingent Consumption If his can is stolen, his wealth will be $25,000; if it is not stolen, his wealth will be $35,000. Buying insurance is transferring wealth from the “good” to the “bad” state. $25,000 Wb Wg $35,000 Wg*=34925 Wb*=31743 Wg = $ So he pays $75 for $6818 of ins Not to scale

Contingent Consumption How would the answer change for a risk lover? Eb Wb Wg Eg A risk lover will maximize utility (reach her highest indifference curve) in a corner solution. In this case, remaining at the endowment.

Adverse Selection Consider the market for drivers insurance: “Good” drivers have accidents with prob = 0.2 “Bad” = 0.8 Good and bad drivers are equally distributed in population. At the actuarially fair price of $0.50/$1 coverage: –for good drivers price is too high -> don’t insure –for bad too low -> insure Bad drivers are “selected in”; good are “selected out” What price would an actuarially fair insurance company charge?

Adverse Selection Consider the market for drivers insurance: “Good” drivers have accidents with prob = 0.2 “Bad” = 0.8 Good and bad drivers are equally distributed in population. At the actuarially fair price of $0.50/$1 coverage: –for good drivers price is too high -> don’t insure –for bad too low -> insure Bad drivers are “selected in”; good are “selected out” Driver quality is a hidden characteristic Asymmetric Information

Acquiring a Company BUYER represents Company A (the Acquirer), which is currently considering make a tender offer to acquire Company T (the Target) from SELLER. BUYER and SELLER are going to be meeting to negotiate a price. Company T is privately held, so its true value is known only to SELLER. Whatever the value, Company T is worth 50% more in the hands of the acquiring company, due to improved management and corporate synergies. BUYER only knows that its value is somewhere between 0 and 100 ($/share), with all values equally likely. Source: M. Bazerman

Acquiring a Company What offer should Buyer make?

Acquiring a Company $ Source:Bazerman, BU MBA Students Similar results from MIT Master’s Candidates CPA; CEOs. Offers

Acquiring a Company OFFER VALUE ACCEPT OR VALUE GAIN OR TO SELLER REJECTTO BUYER LOSS (O) (s) (3/2 s = b) (b - O) $60 $0 A $0 $ A A A A A R R - -

Acquiring a Company The key to the problem is the asymmetric information structure of the game. SELLER knows the true value of the company (s). BUYER knows only the upper and lower limits (0 < s < 100). Therefore, buyer must form an expectation on s (s'). BUYER also knows that the company is worth 50% more under the new management, i.e., b' = 3/2 s'. BUYER makes an offer (O). The expected payoff of the offer, EP(O), is the difference between the offer and the expected value of the company in the hands of BUYER: EP(O) = b‘ – O = 3/2s‘ – O.

Acquiring a Company BUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted: EP(O) = b‘ – O = 3/2s‘ – O. O = s' + .Seller accepts if O > s. Now consider this: Buyer has formed her expectation based on very little information. If Buyer offers O and Seller accepts, this considerably increases Buyer’s information, so she can now update her expectation on s. How should Buyer update her expectation, conditioned on the new information that s < O?

Acquiring a Company BUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted: EP(O) = b‘ – O = 3/2s‘ – O. O = s' + .Seller accepts if O > s. Let’s say BUYER offers $50. If SELLER accepts, BUYER knows that s cannot be greater than (or equal to) 50, that is: 0 < s < 50. Since all values are equally likely, s''/(s < O) = 25. The expected value of the company to BUYER (b'' = 3/2s'' = 37.50), which is less than the 50 she just offered to pay. (EP(O) = ) When SELLER accepts, BUYER gets a sinking feeling in the pit of her stomach. THE WINNER’S CURSE!

Acquiring a Company BUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted: EP(O) = b‘ – O = 3/2s‘ – O. O = s' + . Seller accepts if O > s. Generally: EP(O) = O - ¼s' (-  ). EP is negative for all values of O. THE WINNER’S CURSE!

Acquiring a Company The high level of uncertainty swamps the potential gains available, such that value is often left on the table, i.e., on average the outcome is inefficient. Under these particular conditions, BUYER should not make an offer. SELLER has an incentive to reveal some information to BUYER, because if BUYER can reduce the uncertainty, she may make an offer that leaves both players better off.

Adverse Selection Lemons (Akerlof 1970): Buyers of used cars can’t distinguish between high and low quality cars (lemons); the price of used cars reflects this uncertainty; and the price is lower than high quality cars are worth. Thus owners of high quality cars won’t choose to sell their cars at the market price; eventually, only (mostly) lemons will be sold on the used car market. Sellers of high-quality products can use means to certify their value: Appraisals; audits; “reputable” agents; brand names.

Moral Hazard Buying insurance may make drivers take more risks. Measures to prevent damage or theft are costly, so drivers may decide to avoid these costs, e.g., “why lock the car, if I’m insured against theft?” If insurance companies cannot monitor driver’s habits, they will respond by charging higher prices to all, so good drivers leave the market …. The result is an inefficient allocation of insurance and a net loss to society, b/c the price of insurance does not reflect the true social cost.

Next Time 7/28Externalities and Public Goods Pindyck and Rubenfeld, Ch 18. Besanko, Ch. 17.