2Chapter Outline The mean and the variance Chapter OverviewChapter OutlineThe mean and the varianceUncertainty and consumer behaviorRisk aversionConsumer searchUncertainty and the firmProducer searchProfit maximizationUncertainty and the marketAsymmetric informationSignaling and screeningAuctionsTypes of auctionsInformation structuresOptimal bidding strategies for risk-neutral biddersExpected revenues in alternative types of auctions
3Chapter OverviewIntroductionIn Chapter 11 we examined various pricing strategies that would permit firms with market power to enhance profits over charging a single, per-unit price.Most of our development of managerial economics has assumed that both consumers and firms were endowed with perfect information. Chapter 12 focuses on how imperfect information and uncertainty impacts:Consumers’ decisions and behaviorsFirms’ decisions and behaviorsMarkets efficiency and functioning
4Measuring Uncertain Outcomes The Mean and the VarianceMeasuring Uncertain OutcomesA variable that measures the outcome of an uncertain event is called a random variable.Probabilities can be attached to different values of a random variable that denote the chance that a value occurs.Information about uncertain outcomes can be summarized by the mean (or, expected value) and variance of a random variable.
5Measuring Uncertain Outcomes: Mean The Mean and the VarianceMeasuring Uncertain Outcomes: MeanThe mean of a random variable is the sum of the probabilities that different outcomes will occur multiplied by the resulting payoffs.If 𝑥 1 , 𝑥 2 , …, 𝑥 𝑛 denote the possible outcomes of the random variable and 𝑞 1 , 𝑞 2 , …, 𝑞 𝑛 the corresponding probabilities of the outcomes, then the mean of 𝑥 is:𝐸 𝑥 = 𝑞 1 𝑥 1 + 𝑞 2 𝑥 2 + …+ 𝑞 𝑛 𝑥 𝑛, where 𝑞 1 + 𝑞 2 ,+…+ 𝑞 𝑛 =1.The mean does not provide information about the risk associated with the random variable.
6Measuring Uncertain Outcomes: Variance and Standard Deviation The Mean and the VarianceMeasuring Uncertain Outcomes: Variance and Standard DeviationThe variance of a random variable is the sum of the probabilities that different outcomes will occur multiplied by the squared deviation from the mean of the resulting payoffs.If 𝑥 1 , 𝑥 2 , …, 𝑥 𝑛 denote the possible outcomes of the random variable, their corresponding probabilities are 𝑞 1 , 𝑞 2 , …, 𝑞 𝑛 , and the expected value of 𝑥 is 𝐸 𝑥 , then the variance of 𝑥 is:𝜎 2 = 𝑞 1 𝑥 1 −𝐸 𝑥 𝑞 2 𝑥 2 −𝐸 𝑥 …+ 𝑞 𝑛 𝑥 𝑛 −𝐸 𝑥 2The variance is a common measure of risk.The standard deviation is the positive square root of the variance: 𝜎= 𝜎 2 .
7Attitudes Toward Risk Attitudes toward risk differ among consumers. Uncertainty and Consumer BehaviorAttitudes Toward RiskAttitudes toward risk differ among consumers.A risk-averse consumer prefers a sure amount of $𝑀 to a risky prospect with an expected value of $𝑀.A risk-loving consumer prefers a risky prospect with an expected value of $𝑀 to a sure amount of $𝑀.A risk-neutral consumer is indifferent between a risky prospect with an expected value of $𝑀 and a sure amount of $𝑀.
8Managerial Decisions with Risk Averse Consumers: Product Quality Uncertainty and Consumer BehaviorManagerial Decisions with Risk Averse Consumers: Product QualityRisk analysis can used to examine situations where consumers are uncertain about product quality.Consider a consumer who regularly uses Brand X. If a new product enters the market, Brand Y, under what conditions will the consumer be willing to try the new product?Issues to overcome and consider:Relative certainty about Brand X.At equal prices among other things, a risk averse consumer will continue to purchase Brand X, since a risk averse consumer prefers the sure thing (Brand X) to a risky prospect (Brand Y).Two tactics can be employed to induce a risk averse consumer to try a new product:Lower the price of Brand Y.Try to convince consumer the new product’s quality is higher than the old product.
9Uncertainty and Consumer Behavior Consumer SearchTo identify the low-price seller from among many firms selling an identical product, consumers sometimes incur a cost, 𝑐, to obtain each price quote.After observing each price quote, a consumer faces must weigh the expected benefit from acquiring an additional price quote with the additional cost.
10Uncertainty and Consumer Behavior Consumer SearchSuppose that three-quarters of stores in a market charge $100 and one-quarter charge $40.A consumer observing a price of $40 should stop searching since there is no price below $40.What should a risk-neutral consumer do after observing a price of $100, if search occurs with free recall and with replacement?One-quarter of the time the consumer will save $100−$40=$60.Three-quarters of the time the consumer will save nothing.The expected benefit from an additional search is: 𝐸𝐵= 1 4 $100−$ $100−$100 =$15.A consumer should search for a lower price as long as the expected benefits for an additional search are greater than the cost of an additional search.
11Optimal Search Strategy Uncertainty and Consumer BehaviorOptimal Search StrategyExpectedbenefitsand costs𝐸𝐵𝐸𝐵 𝑅 =𝑐𝑐$𝑐Reservation price:Price at which a consumeris indifferent betweenpurchasing at that price andsearching for a lower price.PriceAcceptance Price Region𝑅Rejection Price Region
12Consumer’s Search Rule Uncertainty and Consumer BehaviorConsumer’s Search RuleThe optimal search rule is such that the consumer rejects prices above the reservation price, 𝑅, and accepts prices below the reservation price. Stated differently, the optimal search strategy is to search for a better price when the price charged by a firm is above the reservation price and stop searching when a price below the reservation price is found.
13Increasing Cost of Search Uncertainty and Consumer BehaviorIncreasing Cost of SearchExpectedbenefitsand costs𝐸𝐵$𝑐 ∗𝑐 ∗Due toIncreasein searchcosts.$𝑐𝑐Price𝑅𝑅 ∗
14Manager’s Risk Attitudes Uncertainty and the FirmManager’s Risk AttitudesWhile manager must understand the impact of uncertainty on consumer behavior, uncertainty also impacts the manager’s input and output decisions.Manager’s risk profiles:Risk averse: a manager who prefers a risky project with a lower expected value if the risk is lower than a project with a higher expected value.Risk loving: manager who prefers a risky project with higher expected value and higher risk to one with lower expected value and lower risk.Risk neutral: manager interested in maximizing expected profits; the variance of profits does not impact a risk-neutral manager’s decisions.
15Manager’s Risk Attitudes In Action: Problem Uncertainty and the FirmManager’s Risk Attitudes In Action: ProblemA risk-averse manager is considering two projects. The first project involves expanding the market for bologna; the second involves expanding the market for caviar. There is a 10 percent chance of recession and a 90 percent chance of an economic boom. The following table summarizes the profits under the different scenarios. Which project should manager undertake, and why?aProjectBoom(90%)Recession (10%)MeanStandard DeviationBologna-$10,000$12,000-$7,800$6,600Caviar20,000-8,00017,2008,400Joint10,0004,0009,4001,800Safe (T-Bill)3,000
16Manager’s Risk Attitudes In Action: Answer Uncertainty and the FirmManager’s Risk Attitudes In Action: AnswerManagers should not invest in T-BillsThe joint project is assured of making at least $4,000, which is greater than $3,000 under the T-Bill scenario.Since the expected returns of the bologna project are negative, neither a risk-neutral nor a risk-averse manager would choose to undertake this project.The manager should adopt either the caviar project or the joint project. Which project will depend on his or her risk preferences.ProjectBoom(90%)Recession (10%)MeanStandard DeviationBologna-$10,000$12,000-$7,800$6,600Caviar20,000-8,00017,2008,400Joint10,0004,0009,4001,800Safe (T-Bill)3,000
17Manager’s Risk Attitudes and Diversification Uncertainty and the FirmManager’s Risk Attitudes and DiversificationNotice from the previous problem that by investing in multiple projects, the manager may be able to reduce risk.The process of potentially reducing risk by investing in multiple projects is called diversification.Whether it is optimal to diversify depends on a manager’s risk preferences and the incentives provided to the manager to avoid risk.
18Uncertainty and the Firm Producer SearchWhen producers are uncertain about the prices of inputs, an optimizing firm will use optimal search strategies.These strategies mimic consumer search previously developed.
19Profit Maximization and Uncertainty Uncertainty and the FirmProfit Maximization and UncertaintyThe basic principles of profit maximization can be modified to deal with uncertainty.If demand (hence, revenue) is uncertain and the manager is risk neutral, then the manager will want to maximize expected profits by producing the output where the expected marginal revenue equals marginal cost:𝐸 𝑀𝑅 =𝑀𝐶
20Profit Maximization and Uncertainty In Action: Problem Uncertainty and the FirmProfit Maximization and Uncertainty In Action: ProblemAppleway Industries produces apple juice and sells it in a competitive market. The firm’s manager must determine how much juice to produce before he knows what the market (competitive) price will be. Economists estimate that there is a 30 percent chance the market price will be $2 per gallon and a 70 percent chance it will be $1 per gallon when the juice hits the market. If the firm’s cost function is 𝐶= 𝑄 2 , how much juice should be produced to maximize expected profits? What are the expected profits of Appleway Industries?
21Profit Maximization and Uncertainty In Action: Answer Uncertainty and the FirmProfit Maximization and Uncertainty In Action: AnswerAppleway Industries’ profits are𝜋=𝑝𝑄−200− 𝑄 2Since price is uncertain, the firm’s revenues and profit are uncertain. To maximize expected profits, the manager equates expected price with marginal cost.𝐸 𝑝 =𝑀𝐶The expected price is: 𝐸 𝑝 =0.3×$2+0.7×$1=$1.30.Therefore, manager should produce output where $1.30=0.001𝑄⟹𝑄=1,300 gallons.Expected profits are $645.
22Asymmetric Information Uncertainty and the MarketAsymmetric InformationUncertainty can profoundly impact markets abilities to efficiently allocate resources.Some markets are characterized by individuals who have better information than others.Implication: Those individuals with the least information may choose not to participate in a market.When some people have better information than others in a market, the information people have is called asymmetric information.There are two specific manifestations related to asymmetric information in markets:Adverse selectionMoral hazard
23Asymmetric Information: Adverse Selection Uncertainty and the MarketAsymmetric Information: Adverse SelectionAdverse selection refers to situations where individuals have hidden characteristics and in which a selection process results in a pool of individuals with undesirable characteristics.In this context, a hidden characteristic is something that one party to a transaction knows about itself but which are unknown by the other party.
24Asymmetric Information: Moral Hazard Uncertainty and the MarketAsymmetric Information: Moral HazardMoral hazard refers to a situation where one party to a contract takes a hidden action that benefits his or her at the expense of another party.In this context, a hidden action is an action taken by one party in a relationship that cannot be observed by the other party.One way to mitigate the moral hazard problem is an incentive contract.
25Uncertainty and the Market SignalingAnother way to mitigate the problem of moral hazard is signaling, which is an attempt by an informed party to send an observable indicator of his or her hidden characteristics to an uninformed party.For signaling to be effective it must be:observable by the uninformed party.a reliable indicator of the unobservable characteristic(s) and difficult for parties with other characteristics to easily mimic.
26Uncertainty and the Market ScreeningA final way to mitigate the moral hazard problem is by screening, which is an attempt by an uninformed party to sort individuals according to their characteristics.Screening may be achieved through a self-selection device.A self-selection device is a mechanism in which informed parties are presented with a set of options, and the options they choose reveal their hidden characteristics to an uninformed party.
27AuctionsTypes of AuctionsAn auction is a mechanism where potential buyers compete for the right to own a good, service, or, more generally, anything of value.Sellers participating in an auction offer an item for sale, and wish to obtain the highest price.Buyers participating in an auction seek to obtain the item at the lowest possible price.Bidders’ risk preferences can affect bidding strategies and the expected revenue a seller receives.Four basic auction types:English (ascending-bid)First-price, sealed-bidSecond-price, sealed-bidDutch (descending-bid)
28Differences Among Auctions Types The timing of bidder decisions (simultaneously or sequentially)The amount the winner is required to pay.
29AuctionsEnglish AuctionAn English auction is an ascending sequential-bid auction in which bidders observe the bids of others and decide whether or not to increase the bid. The auction ends when a single bidder remains; this bidder obtains the item and pays the auctioneer the amount of the bid.Bidders continually obtain information about one another’s bids.Bidder who values the item the most will win.
30First-Price, Sealed-Bid Auction AuctionsFirst-Price, Sealed-Bid AuctionA first-price, sealed-bid auction is a simultaneous-move auction in which bidders simultaneously submit bids to an auctioneer. The auctioneer awards the item to the highest bidder, who pays the amount bid.Bidders obtain no information about one another’s bids.Bidder who values the item the most will win.
31Second-Price, Sealed-Bid Auction AuctionsSecond-Price, Sealed-Bid AuctionA second-price, sealed-bid auction is a simultaneous-move auction in which bidders simultaneously submit bids to an auctioneer. The auctioneer awards the item to the highest bidder, who pays the amount bid by the second-highest bidder.Bidders obtain no information about one another’s bids.Bidder who values the item the most will win, but pays the second-highest bid.
32AuctionsDutch AuctionA Dutch auction is a descending sequential-bid auction in which the auctioneer beings with a high asking price and gradually reduces the asking price until one bidder announces a willingness to pay that price for the item.Bidders obtain no information about one another’s bids throughout the auction process.Bidder who values the item the most will win and pay the amount of his or her bid.
33Strategic Equivalence of Dutch and First-Price Auctions The Dutch and first-price, sealed-bid auctions are strategically equivalent; that is, the optimal bids by participants are identical for both types of auctions.
34Information Structures AuctionsInformation StructuresWhile the four auction types differ with respect to the information bidders have about the bids of other bidders, bidders also have different information structures about the value of their own bids.Perfect informationIndependent private valuesAffiliated (or correlated) value estimatesSpecial case: common-value auctions
35Optimal Bidding Strategies for Risk-Neutral Bidders AuctionsOptimal Bidding Strategies for Risk-Neutral BiddersAn optimal bidding strategy for risk-neutral bidders is a strategy that maximizes a bidder’s expected profit.Optimal bids depends on thetype of auction.information available to bidders at the time of bidding.
36Strategies for Independent Private Value Auctions With independent private values, bidders know his or her own values prior to the auction start.English auctionRemain active until the price exceeds his or her own valuation of the object.Second-price, sealed-bid auctionBid his or her own valuation of the item. This is a dominant strategy.First-price, sealed-bid auction (strategically equivalent to the Dutch auction)Bid less than his or her valuation of the item. If there are 𝑛 bidders who all perceive valuations to be evenly (or uniformly) distributed between a lowest and highest possible valuations, 𝐿 and 𝐻, respectively, then the optimal bid, 𝑏, for a player whose own valuation is 𝑣 is:𝑏=𝑣− 𝑣−𝐿 𝑛
37Strategies for Independent Private Value Auctions In Action: Problem Consider an auction where bidders have independent private values. Each bidder perceives that valuations are evenly distributed between $1 and $10. Sam knows his own valuation is $2. Determine Sam’s optimal bidding strategy in:A first-price, sealed-bid auction with two bidders.A Dutch auction with three bidders.A second-price, sealed-bid auction with 20 bidders.
38Strategies for Independent Private Value Auctions In Action: Answer Sam’s optimal bid in a first-price, sealed-bid auction with two bidders is 𝑏=2− 2−1 2 =$1.50.Sam’s optimal bid in a Dutch auction with three bidders is 2− 2−1 3 =$1.67.Sam’s optimal bid in a second-price, sealed-bid auction with 20 bidders is to bid his true valuation, which is $2.00.
39Strategies for Correlated Values Auctions Bidders do not know their own valuations for an item, nor others’ valuations.Implication: makes bidders vulnerable to the winner’s curse, which is the “bad news” conveyed to the winner that his or her estimate of the item’s value exceeds the estimates of all other bidders.To avoid the winner’s curve in a common-value auction, a bidder should revise downward his or her private estimate of the value to account for this fact.The auction process may reveal information about how much the other bidders value the object.The winner’s curse is most pronounced in sealed-bid auctions since bidders don’t learn about other player’s valuation.English auction, in contrast, provides bidders with information. Therefore, bidders may have to revise up their initial bids.
40Expected Revenues in Alternative Auction Types AuctionsExpected Revenues in Alternative Auction TypesComparison of expected revenue in auctions with risk-neutral biddersInformation structureExpected revenuesIndependent private valuesEnglish=Second-price = First-Price = DutchAffiliated value estimatesEnglish > Second-price > First-price = Dutch
41ConclusionInformation plays an important role in how economic agents make decisions.When information is costly to acquire, consumers will continue to search for price information as long as the observed price is greater than the consumer’s reservation price.When there is uncertainty surrounding the price a firm can charge, a firm maximizes profit at the point where the expected marginal revenue equals marginal cost.Many items are sold via auctionsEnglish auctionFirst-price, sealed bid auctionSecond-price, sealed bid auctionDutch auction