School Of Distance Education MICROECONOMICS: THEORY

common information. Either different information or different abilities to process the sameinformation can cause subjective probabilities to vary among individuals.Regardless of the interpretation of probability, it is used in calculating two importantmeasures that help us describe and compare risky choices. One measure tells us the expectedvalue and the other the variability of the possible outcomes.Expected ValueThe expected value associated with an uncertain situation is a weighted average of thepayoffs or values associated with all possible outcomes. The probabilities of each outcomeare used as weights. Thus the expected value measures the central tendency—the payoff orvalue that we would expect on average.Our offshore oil exploration example had two possible outcomes: Success yields a payoff ofRs.40 per share, failure a payoff of Rs.20 per share. Denoting “probability of” by Pr, weexpress the expected value in this case asExpected value Pr (success)(Rs.40/share) Pr (failure)(Rs.20/share) (1/4)(Rs.40/share) (3/4)(Rs.20/share) Rs.25/shareMore generally, if there are two possible outcomes having payoffs X1 and X2 and if theprobabilities of each outcome are given by Pr1 and Pr2, then the expected value isE(X) Pr1X1 Pr2X2When there are n possible outcomes, the expected value becomesE(X) Pr1X1 Pr2X2 . PrnXnVariabilityVariability is the extent to which the possible outcomes of an uncertain situation differ. Nowwe can discuss why variability is important. Suppose you are choosing between two part-timesummer sales jobs that have the same expected income (Rs.1500). The first job is basedentirely on commission—the income earned depends on how much you sell. There are twoequally likely payoffs for this job: Rs.2000 for a successful sales effort and Rs.1000 for onethat is less successful. The second job is salaried. It is very likely (.99 probability) that youwill earn Rs.1510, but there is a .01 probability that the company will go out of business, inwhich case you would earn only Rs.510 in severance pay. The following table summarizesthese possible outcomes, their payoffs, and their probabilities.Table 1.1: Income from Sales JobsOutcome 1Outcome 2Expected

Income (Rs.)ProbabilityIncome (Rs.)ProbabilityIncome (Rs.)0.520000.5100015000.9915100.015101500Job 1:CommissionJob 2:FixedSalaryNote that these two jobs have the same expected income. For Job 1, expected income is0.5(Rs.2000) 0.5(Rs.1000) Rs.1500; for Job 2, it is 0 .99 (Rs.1510) 0.01(Rs.510) Rs.1500. However, the variability of the possible payoffs is different. We measure variabilityby recognizing that large differences between actual and expected pay offs (whether positiveor negative) imply greater risk. We call these differences deviations. Table 1.2 shows thedeviations of the possible income from the expected income from each job.Table 1.2: Deviations from Expected Income (Rs.)Outcome 1DeviationOutcome 2DeviationJob 120005001000-500Job 2151010510-990By themselves, deviations do not provide a measure of variability because they aresometimes positive and sometimes negative. We can see from the table 1.2 that the average ofthe probability weighted deviations is always 0. To get around this problem, we square eachdeviation, yielding numbers that are always positive. We then measure variability bycalculating the standard deviation: Square root of the weighted average of the squares of thedeviations of the payoffs associated with each outcome from their expected values. Thefollowing table shows the calculation of the standard deviation for our example.Table 1.3: Calculating Variance (Rs.)Outcome 1Deviation Outcome 2SquaredDeviationWeighted averageStandardSquaredDeviation SquaredDeviationJob 12000250,0001000250,000250,000500Job 21510100510980,100990099.50Note that the average of the squared deviations under Job 1 is given by0.5(Rs.250,000) 0.5(Rs.250,000) Rs.250,000

The standard deviation is therefore equal to the square root of Rs.250,000, or Rs.500.Likewise, the probability-weighted average of the squared deviations under Job 2 is0.99(Rs.100) 0.01(Rs.980,100) Rs.9900The standard deviation is the square root of Rs.9900, or Rs.99.50. Thus the second job ismuch less risky than the first; the standard deviation of the incomes is much lower.The concept of standard deviation applies equally well when there are many outcomes ratherthan just two. Suppose, for example, that the first summer job yields incomes ranging fromRs.1000 to Rs.2000 in increments of Rs.100 that are all equally likely. The second job yieldsincomes from Rs.1300 to Rs.1700 (again in increments of Rs.100) that is also equally likely.Expected UtilityConsider an agent trying to evaluate the utility associated to consuming the random variable.A random variable y is a map from Ω to R. which takes valuesy1 y(ω1); y2 y (ω2)The expected value of y is:py1 (1-p) y2;denoted Ey.Let the utility function of the agent be u(c). His utility when consuming y is: u(y1) withprobability p; and u(y2) with probability 1- p. We assume an agent evaluates the utility of arandom variable y by expected utility; that is,pu (y1) (1-p) u (y2)In other words, we assume that, when facing uncertainty, agents maximize expected utility. Alot of experiments document failures of this assumption in various circumstances. A lot oftheoretical work addresses the failure, postulating different (still optimizing) behavior on thepart of agents. Most economic theory still uses the assumption in approximation.The Von Neumann –Morgenstern TheoremIn their book The Theory of Games and Economic Behaviour, John von Neumann and OscarMorgenstern developed mathematical models for examining the economic behaviour ofindividuals under conditions of uncertainty. To understand these interactions, it was necessaryfirst to investigate the motives of the participants in such “games.” Because the hypothesis“that individuals make choices in uncertain situations based on expected utility seemedintuitively reasonable, the authors set out to show that this hypothesis could be derived frommore basic axioms of “rational” behaviour. The axioms represent an attempt by the authors togeneralize the foundations of the theory of individual choice to cover uncertain situations.

Although most of these axioms seem eminently reasonable at first glance, many importantquestions about their tenability have been raised.The von Neumann–Morgenstern utility indexTo begin, suppose that there are n possible prizes that an individual might win byparticipating in a lottery. Let these prizes be denoted by x1, x2, , xn and assume that thesehave been arranged in order of ascending desirability. Therefore, x1 is the least preferred prizefor the individual and xn is the most preferred prize. Now assign arbitrary utility numbers tothese two extreme prizes. For example, it is convenient to assignU(x1) 0,U (xn) 1,but any other pair of numbers would do equally well. Using these two values of utility, thepoint of the von Neumann–Morgenstern theorem is to show that a reasonable way exists toassign specific utility numbers to the other prizes available. Suppose that we choose any otherprize, say, xi. Consider the following experiment. Ask the individual to state the probability,say, πi, at which he or she would be indifferent between xi with certainty, and a gambleoffering prizes of xn with probability πi and x1 with probability (1- πi). It seems reasonable(although this is the most problematic assumption in the von Neumann–Morgensternapproach) that such a probability will exist: The individual will always be indifferent betweena gamble and a sure thing, provided that a high enough probability of winning the best prizeis offered. It also seems likely that πi will be higher the more desirable xi is; the better xi is,the better the chance of winning xn must be to get the individual to gamble. The probability πitherefore measures how desirable the prize xi is. In fact, the von Neumann–Morgensterntechnique is to define the utility of xi as the expected utility of the gamble that the individualconsiders equally desirable to xi:U(xi) πi . U(xn) (1- πi). U(x1)Because of our choice of scale in equation U(x1) 0, U (xn) 1, we haveU(xi) πi .1 (1 - πi).0 πi.By judiciously choosing the utility numbers to be assigned to the best and worst prizes, wehave been able to devise a scale under which the utility number attached to any other prize issimply the probability of winning the top prize in a gamble the individual regards asequivalent to the prize in question. This choice of utility numbers is arbitrary. Any other twonumbers could have been used to construct this utility scale, but our initial choice (EquationU(x1) 0, U (xn) 1) is a particularly convenient one.

Expected utility maximizationIn line with the choice of scale and origin represented by Equation U(x1) 0, U (xn) 1,suppose that probability πi has been assigned to represent the utility of every prize xi. Noticein particular that π1 0, πn 1, and that the other utility values range between these extremes.Using these utility numbers, we can show that a “rational” individual will choose amonggambles based on their expected “utilities” (that is, based on the expected value of these vonNeumann–Morgenstern utility index numbers).As an example, consider two gambles. One gamble offers x2, with probability q, and x3, withprobability (1- q). The other offers x5, with probability t , and x6, with probability (1-t). Wewant to show that this person will choose gamble 1 if and only if the expected utility ofgamble 1 exceeds that of gamble 2. Now for the gambles:expected utility (1) q.U(x2) (1-q).U(x3)expected utility(2) t.U(x5) (1-t).U(x6)Substituting the utility index numbers (that is, π2 is the “utility” of x2, and so forth) givesexpected utility (1) q. π2 (1-q). Π3expected utility(2) t. Π5 (1-t). π6We wish to show that the individual will prefer gamble 1 to gamble 2 if and only ifq. π2 (1 q). Π3 t. Π5 (1-t). π6.To show this, recall the definitions of the utility index. The individual is indifferent betweenx2 and a gamble promising x1 with probability (1- π2) and xn with probability π2. We can usethis fact to substitute gambles involving only x1 and xn for all utilities in Equation expectedutility (1) q. π2 (1-q). Π3 and expected utility (2) t. Π5 (1-t). π6. (even though theindividual is indifferent between these, the assumption that this substitution can be madeimplicitly assumes that people can see through complex lottery combinations). After a bit ofmessy algebra, we can conclude that gamble 1 is equivalent to a gamble promising xn withprobability q π2 (1-q) π3, and gamble 2 is equivalent to a gamble promising xn withprobability t π2 (1-q) π3, and gamble 2 is equivalent to a gamble promising xn withprobability t π5 (1-t) π6. The individual will presumably prefer the gamble with the higherprobability of winning the best prize. Consequently, he or she will choose gamble 1 if andonly ifq π2 (1-q) π3 t π5 (1-t) π6.But this is precisely what we wanted to show. Consequently, we have proved that anindividual will choose the gamble that provides the highest level of expected (von Neumann–

Morgenstern) utility. We now make considerable use of this result, which can be summarizedas follows:Expected utility maximization: If individuals obey the von Neumann–Morgenstern axioms ofbehaviour in uncertain situations, they will act as if they choose the option that maximizes theexpected value of their von Neumann–Morgenstern utility index.Expected Utility HypothesisIn economics, game theory, and decision theory, the expected utility hypothesis—concerningpeople’s preferences with regard to choices that have uncertain outcomes (gambles) —statesthat the subjective value associated with an individual's gamble is the statistical expectationof that individual's valuations of the outcomes of that gamble, where these valuations maydiffer from the Rupees value of those outcomes. The introduction of St. PetersburgParadox by Daniel Bernoulli in 1738 is considered the beginnings of the hypothesis. Thishypothesis has proven useful to explain some popular choices that seem to contradictthe expected value criterion (which takes into account only the sizes of the payouts and theprobabilities of occurrence), such as occur in the contexts of gambling and insurance.The von Neumann–Morgenstern utility theorem provides necessary and sufficient conditionsunder which the expected utility hypothesis holds. From relatively early on, it was acceptedthat some of these conditions would be violated by real decision-makers in practice but thatthe conditions could be interpreted nonetheless as ‘axioms’ of rational choice.Until the mid-twentieth century, the standard term for the expected utility was the moralexpectation, contrasted with “mathematical expectation” for the expected value. Bernoulli came across expected utility by playing the St Petersburg paradox. This paradoxinvolves you flipping a coin until you get to heads. The number of times it took you to get toheads is what you put as an exponent to 2 and receives that in rupees amounts. This gamehelped to understand what people were willing to pay versus what people were expected togain from this game.Formula for Expected UtilityWhen the entity whose value affects a person's utility takes on one of a set of discrete values,the formula for expected utility, which is assumed to be maximized, isE [u(x)] p1.u(x1) p2. u (x2) .where the left side is the subjective valuation of the gamble as a whole, x1 is the ith possibleoutcome, u(xi) is its valuation, and pi is its probability. There could be either a finite set ofpossible values xi in which case the right side of this equation has a finite number of terms;

or there could be an infinite set of discrete values, in which case the right side has an infinitenumber of terms. When x can take on any of a continuous range of values, the expectedutility is given byE [u(x)] ʃ- u(x1) f(x) dx,Where, f(x) is the probability density function of x.Expected value and choice under riskIn the presence of risky outcomes, a human decision maker does not always choose theoption with higher expected value investments. For example, suppose there is a choicebetween a guaranteed payment of Rs.1.00, and a gamble in which the probability of getting aRs.100 payment is 1 in 80 and the alternative, far more likely outcome (79 out of 80) isreceiving Rs.0. The expected value of the first alternative is Rs.1.00 and the expected value ofthe second alternative is Rs.1.25. According to expected value theory, people should choosethe Rs.100-or-nothing gamble; however, as stressed by expected utility theory, some peopleare risk averse enough to prefer the sure thing, despite its lower expected value. People withless risk aversion would choose the riskier, higher-expected-value gamble. This is precedencefor utility theory.Bernoulli’s formulationNicolas Bernoulli described the St. Petersburg paradox (involving infinite expected values) in1713, prompting two Swiss mathematicians to develop expected utility theory as a solution.The theory can also more accurately describe more realistic scenarios (where expected valuesare finite) than expected value alone. In 1728, Gabriel Cramer, in a letter to Nicolas Bernoulli,wrote, “the mathematicians estimate money in proportion to its quantity, and men of goodsense in proportion to the usage that they may make of it.” In 1738, Nicolas’ cousin Daniel Bernoulli , published the canonical 18th Century descriptionof this solution in Specimen theoriae novae de mensura sortis or Exposition of a New Theoryon the Measurement of Risk. Daniel Bernoulli proposed that a nonlinear function of utility ofan outcome should be used instead of the expected value of an outcome, accounting for riskaversion, where the risk premium is higher for low-probability events than the differencebetween the payout level of a particular outcome and its expected value. Bernoulli furtherproposed that it was not the goal of the gambler to maximize his expected gain but to insteadmaximize the logarithm of his gain. Bernoulli’s paper was the first formalization of marginalutility, which has broad application in economics in addition to expected utility theory. He

used this concept to formalize the idea that the same amount of additional money was lessuseful to an already-wealthy person than it would be to a poor person.St. Petersburg ParadoxThe St. Petersburg paradox (named after the journal in which Bernoulli’s paper was published)arises when there is no upper bound on the potential rewards from very low probabilityevents. Because some probability distribution functions have an infinite expected value , anexpected-wealth maximizing person would pay an arbitrarily large finite amount to take thisgamble. In real life, people do not do this. Bernoulli proposed a solution to this paradox in hispaper: the utility function used in real life means that the expected utility of the gamble isfinite, even if it’s expected value is infinite. (Thus he hypothesized diminishing marginalutility of increasingly larger amounts of money.) It has also been resolved differently by othereconomists by proposing that very low probability events are neglected, by taking intoaccount the finite resources of the participants, or by noting that one simply cannot buy thatwhich is not sold (and that sellers would not produce a lottery whose expected loss to themwere unacceptable).The game:Flip a fair coin until the first head appearsThe payoff: If the first head appears on the kth flip, you get Rs.2k Using an expected valuerule, you should be willing to pay at least the expected value of the payoff from playing thegame. Then the expected Payoff for the St. Petersburg Game is thatRecall the expected payoff will be the probability weighted sum of the possible outcomes.Note: The tosses are independent; a tail on the previous toss does not influence the outcomeof the subsequent toss. Head has a ½ or 50% chance of occurring on any sing

eco1c01-microeconomics: theory and applications-i microeconomics: theory and applications-i (eco1c01) study material i semester core course ma economics (2019 admission onwards) university of calicut school of distance education calicut university- p.o, malappuram- 673635, kerala. 190301