hat Uncertainty All the models considered so far have one thing in common. There is no uncertainty. This is a very restrictive assumption. Often in economic situations there is less than perfect information. Both production and consumption often involve unknown variables that affect the profits and utility of the When an agent makes a decision(about consumption for example) the utility they will receive may be uncertain. decision making under uncertainty be modelled? Random chance of this sort in economics is referred to as nature taking an action. Nature decides which state of the world occurs. For example, a consumer is considering whether or not to purchase car. Unfortunately cars can break down. If the consumer purchases the car. nature "decides"whether the car breaks down or not. These are the two possible states of the world. How he consumer compare the utility from purchasing the car with the utility from keeping their money? Market- What Nerty Lotteries Economists model this sort of decision problem as a choice between lotteries. One lottery involves buying the car(which costs 500)and then facing the uncertainty of a possible break down (which occurs 20% of the time). The other lottery involves keeping the money for sure. 1000 Buying the car Suppose a car is worth 1500 to the agent, and the cost of repairing after a break down is 100 How can a consumer compare these two lotteries c and n? Consumers have preferences over lotteries: Either n>c or c>n. Representing this preference relation as a utility function: Either un ue or ue > un ner preferences over bundles. An analogous argument for the case of preferences over lotteries can be lade. Preferences that satisfy certain consistency requirements can be represented by an expected utility function
Market — What Next? 1 Uncertainty • All the models considered so far have one thing in common. There is no uncertainty. • This is a very restrictive assumption. Often in economic situations there is less than perfect information. Both production and consumption often involve unknown variables that affect the profits and utility of the agents. • When an agent makes a decision (about consumption for example) the utility they will receive may be uncertain. • How can decision making under uncertainty be modelled? • Random chance of this sort in economics is referred to as nature taking an action. • Nature decides which state of the world occurs. For example, a consumer is considering whether or not to purchase a car. Unfortunately cars can break down. If the consumer purchases the car, nature “decides” whether the car breaks down or not. These are the two possible states of the world. • How can the consumer compare the utility from purchasing the car with the utility from keeping their money? Market — What Next? 2 Lotteries • Economists model this sort of decision problem as a choice between lotteries. • One lottery involves buying the car (which costs 500) and then facing the uncertainty of a possible break down (which occurs 20% of the time). The other lottery involves keeping the money for sure. ................................................................................................................................................................................. ................................................................................................................................................................................. u ................................................................................................................................................................... c 1000 0 • un • 500 • • • 0.2 0.8 1 Buying the car Keeping the money • Suppose a car is worth 1500 to the agent, and the cost of repairing after a break down is 1000. • How can a consumer compare these two lotteries c and n? Consumers have preferences over lotteries: Either n º c or c º n. Representing this preference relation as a utility function: Either un ≥ uc or uc ≥ un. • Recall consumer preferences over bundles. An analogous argument for the case of preferences over lotteries can be made. Preferences that satisfy certain consistency requirements can be represented by an expected utility function
hat Expected Utility The consistency requirements include reflexivity, completeness and transitivity as well as two others. A lottery r is a set of possible outcomes rI, .. In) and a set of probabilities for each outcome ip The representation theorem states that a preference relation over lotteries, > that satisfies these conditions can be represented by a ton-Neumann Morgenstern expected utility function > y÷→ pu(x)2∑q qiu(yi) The vNM expected utility for lottery r is U(r)=iipa u(ri)where u(ai)is the regular utility from outcome I, In the earlier example, suppose the utility from each outcome is simply the money payoff. The probability of each utcome is known. If the consumers preferences can be represented by such a VNM expected utility function then: U(c)=(0.8×1000+(0.2×0)andU(n)=(1×50)soU(c)>U(n) thus c>n In this example the consumer prefers the lottery c and will buy the car Market- What Nerty Risk aversion The consumer has taken the action involving uncertainty. Do they then like risk? Not necessarily A common assumption is that of risk aversion. A consumer is risk averse when their utility function is concave. Suppose there is just one good- money m. The consumer gets regular utility u(m) from money m (5) u(1)+u(9) u(1) In the above diagram the consumer prefers 5 for sure than 9 with probability half and I with probability half. The expected value of the latter lottery is 5. But u(5)2 u(1)+=u(9). This is because u is concave- risk aversion the earlier This is a case of
Market — What Next? 3 Expected Utility • The consistency requirements include reflexivity, completeness and transitivity as well as two others. • A lottery x is a set of possible outcomes {x1, . . . , xn} and a set of probabilities for each outcome {p1, . . . , pn}. • The representation theorem states that a preference relation over lotteries, º, that satisfies these conditions can be represented by a von-Neumann Morgernstern expected utility function: x º y ⇐⇒ Xn i=1 piu(xi) ≥ Xm i=1 qiu(yi) • The vNM expected utility for lottery x is U(x) = Pn i=1 piu(xi) where u(xi) is the regular utility from outcome xi . • In the earlier example, suppose the utility from each outcome is simply the money payoff. The probability of each outcome is known. If the consumers preferences can be represented by such a vNM expected utility function then: U(c) = (0.8 × 1000) + (0.2 × 0) and U(n) = (1 × 500) so U(c) > U(n) thus c º n • In this example the consumer prefers the lottery c and will buy the car. Market — What Next? 4 Risk Aversion • The consumer has taken the action involving uncertainty. Do they then like risk? Not necessarily. • A common assumption is that of risk aversion. A consumer is risk averse when their utility function is concave. Suppose there is just one good — money m. The consumer gets regular utility u(m) from money m. . . . . . . . . . . ............. ... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. . . . . . . ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ....................................................................................................................................................................................................................................................................................................................................................................................................................... .............................................................................................................................................................................................. . ................................................................................................................................................................................................................................................................................................................................................ 0 u m u(m) • • • • 1 5 9 u(1) u(5) u(9) 1 2 u(1) + 1 2 u(9) • In the above diagram the consumer prefers 5 for sure than 9 with probability half and 1 with probability half. The expected value of the latter lottery is 5. But u(5) ≥ 1 2 u(1) + 1 2 u(9). This is because u is concave — risk aversion. • In the earlier example the consumer had u(m) = m. This is a case of risk neutrality
hat Asymmetric Informatio What happens when individuals have different information? This is called a situation of asymmetric information. The less informed individual is called the principal (P), the more informed is called the agent(A) Economists think about asymmetric information problems using"principal-agent"models. For example, a second-hand car salesman knows the quality of a particular used car, but the buyer does not. The buyer is the principal and the salesman is the agent. Asymmetric information is an important type of market failure. Typically the agent takes a hidden action or has hidden information. The principal tries to learn about the action or information whilst the agent tries to conceal or reveal the action or information. Market- What Nerty dverse selection Nature chooses how good a particular car is. It can be bad or good with equal probability. The principal(buyer) ffers a price. The agent(seller)can either accept or reject this price ○O Contract 200. The principal might thi lly likely for a car to be good or bad. Therefore they might offer a price 600. The agent, howeve the quality of the car. If the car is good, they will not sell it for 600. (It is worth 1000). If it is bad they will sell it, as it is worth 200 Only bad cars will be sold for a price below 1000. The principal knows this and offers a price 200 There is no equilibrium in which good cars are sold. Half the market collapses a serious failure
Market — What Next? 5 Asymmetric Information • What happens when individuals have different information? This is called a situation of asymmetric information. • The less informed individual is called the principal (P), the more informed is called the agent (A). • Economists think about asymmetric information problems using “principal-agent” models. • For example, a second-hand car salesman knows the quality of a particular used car, but the buyer does not. The buyer is the principal and the salesman is the agent. Asymmetric information is an important type of market failure. • Typically the agent takes a hidden action or has hidden information. The principal tries to learn about the action or information whilst the agent tries to conceal or reveal the action or information. • How does a market fail in the presence of asymmetric information? Market — What Next? 6 Adverse Selection • Nature chooses how good a particular car is. It can be bad or good with equal probability. The principal (buyer) offers a price. The agent (seller) can either accept or reject this price. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ . ................................. ... ......... ................. .... . ................................. ... ......... ................. ... . ................................. ... ....... ..................... ... . . . . . . N P A Good Bad Contract Accept Reject • • • • • A good car is worth 1000. A bad car is worth 200. The principal might think it equally likely for a car to be good or bad. Therefore they might offer a price 600. The agent, however, knows the quality of the car. • If the car is good, they will not sell it for 600. (It is worth 1000). If it is bad they will sell it, as it is worth 200. • Only bad cars will be sold for a price below 1000. The principal knows this and offers a price 200. • There is no equilibrium in which good cars are sold. Half the market collapses — a serious failure
hat Moral hazard Consider a farmer trying to hire a worker. The farmer (principal)offers a contract which the worker(agent) accept or reject. If the worker joins the farm they can be lazy or work hard. The weather can be good or bad. If it is good, crop yields will be higher than if it is he farmer observes the yield but not the effort of the worker. They do not know whether the good (or bad) yield is due to effort or weather. Effort This is an example of moral hazard. Once given the job, the worker might have an incentive to do nothing and "blame it on the weather". The principal needs to give the worker the right incentives to work hard Notice the previous example was one of hidden information. Here, it is the action that is hidden from the principal. Again, there is market failure in the sense that a less than efficient amount of effort is provided. Market- What Nerty Contracts and incentives What are the solutions to these types of market failure? In the case of moral hazard the principal needs to draw up effective contract to induce the worker to put in high effort levels. The principal faces two constraints, (i) a participation constraint (the agent must accept the contract)and(ii)an incentive compatibility constraint (the agent must choose effort to maximise their payoff) In the case of adverse selection ble solution is a signal -a warranty in the car example. Signal Contract Of course, the agent needs to convince the principal that the signal is reliable. The salesman can't simply claim the car is good. In fact the warranty needs to be more costly to obtain for the bad car than it is for the good car. Signals can rescue markets from the serious collapses due to adverse selection good cars
Market — What Next? 7 Moral Hazard • Consider a farmer trying to hire a worker. The farmer (principal) offers a contract which the worker (agent) can accept or reject. If the worker joins the farm they can be lazy or work hard. • The weather can be good or bad. If it is good, crop yields will be higher than if it is bad. The farmer observes the yield but not the effort of the worker. They do not know whether the good (or bad) yield is due to effort or weather. ........................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................................ . ........................................................................................................................................................ ................................... ... ....... ..................... ... . ................................. ... ........ .................. .... . .................................. ... ........ .................. .... . ................................. ... ........ .................. ... . . . . . . P A1 A2 N Good Bad Effort Contract Accept Reject • • • • This is an example of moral hazard. Once given the job, the worker might have an incentive to do nothing and “blame it on the weather”. The principal needs to give the worker the right incentives to work hard. • Notice the previous example was one of hidden information. Here, it is the action that is hidden from the principal. Again, there is market failure in the sense that a less than efficient amount of effort is provided. Market — What Next? 8 Contracts and Incentives • What are the solutions to these types of market failure? In the case of moral hazard, the principal needs to draw up an effective contract to induce the worker to put in high effort levels. • The principal faces two constraints, (i) a participation constraint (the agent must accept the contract) and (ii) an incentive compatibility constraint (the agent must choose effort to maximise their payoff). • In the case of adverse selection one possible solution is a signal — a warranty in the car example. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ . ................................. ... ....... ..................... ... . ................................. ... ....... ..................... ... . ................................. ... ......... ................. .... . ................................. ... ......... ................. .... . . . . . . N A1 P A2 Good Bad Signal Contract Accept Reject • • • • • Of course, the agent needs to convince the principal that the signal is reliable. The salesman can’t simply claim the car is good. In fact the warranty needs to be more costly to obtain for the bad car than it is for the good car. • Signals can rescue markets from the serious collapses due to adverse selection — good cars can be sold!
hat Signalling, Pooling and Separating The following example illustrates some of the difficulties with signalling. An individual (the agent) is either clever (C)or stupid (S). They know this, but an employer (the principal) does They can either take a degree(D)or not(O). The employer observes whether they take a degree Nature decides whether the individual is clever. 50% of the time they are clever No degree Degree Not hire Degre Not hire r not. The dotted lines represent information set -the principal does not know which node they are at There may be a separating equilibrium or a pooling equilibrium. rket- what Nerty Auctions An important(and successful)application of game theory is to auctions. Auctions are mechanisms for selling goods, they differ from the market mechanism presented in earlier lectures There are many different types of auction available for modelling. Here are some: 1. English: (Or Japanese), ascending bids, last bidder left wins the object at the price they bid. 2. Dutch: Descending price. is the first to make a bid(stop the auction). They pay that price 3. First price sealed bid: Write bids down, put in sealed envelope. Highest bidder wins and pays that price. L Second price sealed bid: Same procedure. Highest bidder wins but pays price of second highest bidder. An auction can be used to sell one or many goods. The first and last above have the same strategic structure. The second and third have the same strategic structure. Why? All four auctions raise the same expected revenue. What is the optimal auction to sell a particular good? Optimal in the sense of efficiency - the highest valuation agent wins the object, and optimal in the sense of raising the highest revenue
Market — What Next? 9 Signalling, Pooling and Separating • The following example illustrates some of the difficulties with signalling. • An individual (the agent) is either clever (C) or stupid (S). They know this, but an employer (the principal) does not. They can either take a degree (D) or not (O). The employer observes whether they take a degree. • Nature decides whether the individual is clever. 50% of the time they are clever. . . ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ ....... ........................................................................................................................................ ........ ........ ........ ........ ........ ........ ........ ........ ........ ...... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ...... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ...... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ...... ...................................................................................... ...................................................................................... ...................................................................................... ...................................................................................... . ................................... ... ....... ..................... ... . .................................. ... ....... ..................... ... . ................................... ... ....... ..................... ... . ... ........ ................... .... ................................. . ... ....... ..................... ... ................................. . .................................. ... ....... ..................... ... . .................................. ... ....... ..................... ... . . . . . . . . . . . . . . N AC AS PD PD PO PO Hire Hire Hire Hire Not hire Not hire Not hire • Not hire • • • • • • • Degree Degree No degree No degree 0.5 0.5 • Notice that the payoffs are omitted, the decision of the employer is to hire or not. The dotted lines represent an information set — the principal does not know which node they are at. • There may be a separating equilibrium or a pooling equilibrium. Market — What Next? 10 Auctions • An important (and successful) application of game theory is to auctions. • Auctions are mechanisms for selling goods, they differ from the market mechanism presented in earlier lectures. • There are many different types of auction available for modelling. Here are some: 1. English: (Or Japanese), ascending bids, last bidder left wins the object at the price they bid. 2. Dutch: Descending price, winner is the first to make a bid (stop the auction). They pay that price. 3. First price sealed bid: Write bids down, put in sealed envelope. Highest bidder wins and pays that price. 4. Second price sealed bid: Same procedure. Highest bidder wins but pays price of second highest bidder. • An auction can be used to sell one or many goods. The first and last above have the same strategic structure. The second and third have the same strategic structure. Why? All four auctions raise the same expected revenue. • What is the optimal auction to sell a particular good? Optimal in the sense of efficiency — the highest valuation agent wins the object, and optimal in the sense of raising the highest revenue
hat Dynamic Games and Equilibrium Dynamic games can be represented in normal form. Consider the entry game of lecture 10. (-1,0 (12) (0.5) 2 There are two pure Nash equilibria-(DE, F) and E, DF). In fact, there are infinitely many mixed equilibria also- where the entrant plays DE and the monopolist plays F with at least probability half. Market- What Nerty Credibility However, working from the back of the game in extensive form, the entrant will always choose to enter given they now that if they do. the monopolist will not fight. The threat to fight is not credible. An equilibrium is subgame perfect if it is a Nash equilibrium of each subgame and of the whole game a subgame is simply a game starting at any particular node. Subgame perfection rules out incredible threats. The only subgame perfect equilibrium in this game is E, DFJ. This is a refinement of Nash equilibrium. Can the monopolist ever deter entry? If the game is repeated many (many)times then, yes fighting in the above game, collusion in oligopoly and even cooperation in the repeated prisoners'dilemma
Market — What Next? 11 Dynamic Games and Equilibrium • Dynamic games can be represented in normal form. Consider the entry game of lecture 10. .............................................................................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................... • .............................................................................................................................................................................................................................................................................................. • • • • E M Fight Don’t Fight Enter Don’t Enter (0, 5) (1, 2) (−1, 0) F DF E 0 −1 2 1 DE 5 0 5 0 • There are two pure Nash equilibria — {DE, F} and {E, DF}. In fact, there are infinitely many mixed equilibria also — where the entrant plays DE and the monopolist plays F with at least probability half. Market — What Next? 12 Credibility • However, working from the back of the game in extensive form, the entrant will always choose to enter given they know that if they do, the monopolist will not fight. The threat to fight is not credible. • An equilibrium is subgame perfect if it is a Nash equilibrium of each subgame and of the whole game. • A subgame is simply a game starting at any particular node. Subgame perfection rules out incredible threats. • The only subgame perfect equilibrium in this game is {E, DF}. This is a refinement of Nash equilibrium. • Can the monopolist ever deter entry? If the game is repeated many (many) times then, yes. • The folk theorems prove that in an infinitely repeated game almost anything is subgame perfect. For example, fighting in the above game, collusion in oligopoly and even cooperation in the repeated prisoners’ dilemma