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PROPERTIES OF UTILITY One way to begin an analysis of individuals choices that we say are characterized in a utility function is to state a basic set of postulates, or axioms, that characterize what we call “rational behavior o Completeness: If A and B are any two situations, the individual can always specify exactly one of the following possibilities: A is preferred to B B is preferred to A A and B are equally attractive here, people are not paralyzed by indecision --- this rules out that A is preferred to B, and, B is preferred to A

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Transitivity: If an individual reports A is preferred to B and that B is preferred to C, then A is preferred to C --- choices are internally consistent Continuity: If an individual reports A is preferred to B, the situations suitably close to A must also be preferred to B --- this helps us analyze relatively small changes in income and prices

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UTILITY Given the assumptions of completeness, transitivity, and continuity, it is possible to show formally that people are able to rank in order all possible situations from least desirable to most desirable This ranking we call “utility” after the inventor, Jeremy Bentham, a 19 th century

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The utility of situation A and situation B would be denoted U(A,B) Bentham suggested the utilitarian approach as “more is better” Therefore, if a person prefers situation A to situation B, then U(A) is greater than U(B)

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We could attach numbers to these utility rankings, but these numbers will not be unique Any set of numbers we arbitrarily assign that accurately reflects the original preference ordering will imply the same set of choices It is the same to say that U(A) = 5 and U(B) = 4, as U(A) = 1million, U(B) = 0.5

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The nonuniqueness of utility measures suggest that it is not possible to compare utilities between people So utility = U(q 1, q 2,..., q n ; other things) In finance we are usually dealing with intertemporal utility as U(Co, C1 s ), where Co = current consumption and C1 s = next period consumption, but that consumption can come in different states, s.

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Trades and Substitution Most economic activity involves voluntary trading between individuals When someone buys bread, they voluntarily give up something, usually money (in this age) They give up the money for something of greater value

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INDIFFERENCE CURVES Hence we come to “INDIFFERENCE CURVES” An indifference curve shows a set of consumption bundles among which the individual is indifferent – that is, the bundles all provide the same level of utility

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INDIFFERENCE CURVES Co C1 U C1 C1΄ Co Co΄ The bundle (Co,C1) has the same utility level as the bundle (Co΄,C1΄)

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Increasing Utility in the NE direction Co C1 U C1 C1΄ Co Co΄ Increasing utility

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Inconsistent preferences Co C1 U1U1 C1 C1΄ Co Co΄ U2U2 Intersecting indifference curves imply inconsistent preferences

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MRS – marginal rate of substitution Co C1 U1U1 C1 C1΄ Co Co΄ The negative slope of an indifference curve at some point is termed the marginal rate of substitution (MRS) MRS = -dCo/dC1, given U = U 1

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MRS tells us something about the trades a person voluntarily makes The MRS diminishes between bundles (Co,C1) and (Co΄,C1΄) The indifference curve becomes flatter or less steep --- one can trade less Co and get more C1 – either in a static sense or intertemporally

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Intertemporal substitution Co C1 U1U1 C1 C1΄ Co Co΄ With Co = current consumption and C1 = future consumption, MRS is intertemporal substitution --- a tradeoff in time for goods MRS = -dCo/dC1, given U = U 1

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Indifference curves are convex --- utility is a convex set Co C1 U1U1 C1 C1΄ Co Co΄ Utility set Utility frontier We can connect points on the frontier with a line – any point on the line is also in the set Any point on a linear combinatio n of point is within the set

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Nonconvexity – insconsistent preference Co C1 U1U1 C1 C1΄ Co Co΄ Utility set Utility frontier We can connect points on the frontier with a line – any point on the line is also in the set

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Examples U = Co 1/2 C1 1/2 = CoC1 We could square this Utility function and preserve the ordering of preference as CoC1 Suppose CoC1 = U = 100, then CoC1 = 10

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Find the MRS of CoC1 = 100, by transforming to C1 = 100/Co Then MRS = -dC1/dCo = 100/Co 2 Clearly MRS declines as Co increases CoC1 is actually a rectangular hyperbola as you recall from geometry, and therefore, the slope of the indifference curve is the same at all points

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MRS and Marginal Utility Let utility be in general, U = (Co,C1) We can totally differentiate this function as, dU = (∂U/∂C1)dC1 + (∂U/∂Co)dCo Along any particular indifference curve dU = 0 (same utility level)

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So manipulate dU = (∂U/∂C1)dC1 + (∂U/∂Co)dCo = 0 To get MRS =-dC1/dCo= (∂U/∂C1) / (∂U/∂Co) MRS = the ratio of the marginal utility of C1 to the marginal utility of Co

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Utility increasing at a decreasing rate If utility increases at a decreasing rate, then if utility is U(Co, C1) ∂U/∂Co > 0, but ∂ 2 U/∂Co 2 < 0 That is the first derivative of utility with respect to an argument, such as Co, is positive, but the second derivative of utility with respect to that same argument is negative

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This suggests that utility is increasing as the argument, say Co, increases, but the rate of increase is diminishing as Co increases

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Examples Suppose we develop the utility of wealth, say W as U(W) = Ln(W) --- a natural logarithmic utility function U΄= ∂U/∂W = 1/ W, and is > 0 for any value of wealth, W U΄΄= ∂ 2 U/∂W 2 = -1/ W 2 < 0

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Look at quadratic utility U(W) = a + bW + cW 2 for a, b, c being parameters U΄ = b + 2cW > 0 for all W > 0 U΄΄ = 2c, a constant If c 0 and U΄΄< 0

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And now exponential utility as U(W) = -e (-aW) = -EXP[-aW] For a being a positive constant U΄ = ae -aW > 0 and is a constant U΄΄ = -a 2 e (-aW) and is also a constant Because the dedikind, e, is a constant

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We use all these properties when we discuss Expected Utility, Risk Aversion, and applications to finance and insurance

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