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## Advanced Micro Theory

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**Advanced Micro Theory**Preferences and Utility**Consumer Choice**• Postulate: an unproved and indemonstrable statement that should be taken for granted: used as an initial premise or underlying hypothesis in a process of reasoning • Consumer choice postulate: People choose from available options to maximize their well-being (utility).**Criticisms**• Criticisms • What about irrational consumers? • Can consumers make these internal calculations? • Irrelevant. We just want to successfully predict behavior. To do that, we assert that all consumers behave accordingly. • Refutation comes if theorems that derive from this postulate are inconsistent with the data. • That is, if behavior contradicts the implications of the model, then the theory is wrong.**Alternatives**• We could devise a hypothesis that postulates that consumers • act randomly • do what they think society wants them to do • But all behavior would be consistent with these assumptions, so no refutable implications (theoretical results) are possible… therefore, a theory based on such a hypothesis is useless.**Consumer Choice Model**• “People choose from available options to maximize their well-being (utility).” • “Available options” in the model will be handled by the budget constraint. • The budget constraint will provide decision-makers with MC of choices. • “Maximizing well-being” will be incorporated into the model via assumptions about preferences – which will then be used to build a utility function. • The preferences part of the model will provide decision-makers with the MB of choices.**Modeling Preferences**• Let bundle A = (x1, y1) and B = (x2, y2) where the goods are x and y. Y A y1 y2 B x1 x2 X**Varian’s Version**• Let bundle X = (x1, x2) and Y = (y1, y2) where the goods are x1 and x2 . So the goods listed on the axes and the quantities of each good in the first bundle are the using the same notation. X2 X x2 y2 Y x1 y1 X1**Varian’s Version**• He does this to be consistent with his advanced micro text. good 2 In that text, he uses vector notation and eliminates the subscripts by not noting quantities of each good on the axes. X Y good 1**Modeling Preferences**• IMO, students have invested so much math time with X and Y on the axes, that I want to leverage that. E.g. slope = • Also, with all the derivations coming up, we will have plenty of subscripts floating around that I hate to add an additional set with goods X1 and X2. Y A y1 y2 B x1 x2 X**Modeling Preferences**• Three choices: • And therefore Y A y1 y2 B x1 x2 X**Axioms of Preference**• Axiom: a proposition that is assumed without proof for the sake of studying the consequences that follow from it (dictionary.com). • These are based on ensuring logical consistency. • Completeness: • Any pair of bundles can be compared and ordered • Reflexivity: • A bundle cannot be strictly preferred to an identical bundle. • Transitivity • Not a logical imperative according to Varian, but preferences become intractable if people cannot choose between three bundles because • Continuity, next page**Continuity**• Preferences must be continuous • Rules out this situation: • The bundle with more X is always preferred. Holding X constant, more Y is better. • But, no matter how close C gets to A, • The utility function in this case must be discontinuous (i.e. there must be a vertical jump between A and C Y B Ub=15 Ua=10 Uc=20 A C X**Goods, Bads and Neutral Goods**• Goods are good (more is better) • Bads are bad, less is better • Neutrals mean nothing to the consumer • Some goods start out good, but then become bads if you consume too much**Possible Indifference Mappings Thus FarCharacterize the**Goods Y Y Y X X X Y Y Y X X X**And we have…**Both are good but become bad Y is a neutral good Two goods Y Y Y X X X X good and Y bad Y Y Y Two bads X good that becomes bad, Y good X X X**Perfect Compliments and Substitutes**Perfect Compliments: More is only better if you have more of the other Perfect Substitutes: Two goods, indifferent to trading off a constant amount of Y for X Y Y X X**Well-behaved Preferences**• If we want to avoid situations where demand curves are upward sloping or people spend all their money on one good, then we need well-behaved indifference curves. • Preferences must also be • Monotonic • Convex**Monotonic**• Monotonic: If bundle A is identical to B, except A has more of at lease one good, then • A.K.A, nonsatiationor “more-is-better” • Ceteris paribus, increasing the quantity of one good creates a bundle that is strictly preferred. • Indifference curves must be downward sloping. • Paired with transitivity, means indifference curves cannot cross.**Monotonic**• These still possible Y Y Y X X X**Indifference Curves Cannot Cross**Y A B C X**Convexity**• Convexity: People prefer more balanced bundles. • Let A = (x1, y1) and B = (x2, y2). • Define C = (tx1 + (1-t)x2, ty1 + (1-t)y2) • where 0 ≤ t ≤ 1 • then Y A y1 C t y1+ (1-t)y2 B y2 x1 tx1+ (1-t)x2x2 X**Convexity:Indifference Curves Bound Convex Sets**• Convexity: Bundles weakly preferred to those lying on an indifference curve bound a convex set. • Any bundle which is a weighted average of bundles on the indifference curve are weakly preferred to bundles lying on the curve. Y Y A y1 A y1 B B y2 y2 x1 x2 X x1 x2 X**Convex Preferences**• These still possible Y Y X X**Strict ConvexityIndifference Curves Bound Convex Sets**• Strictly Convex Preferences: • Any bundle which is a weighted average of bundles on the indifference curve are strictly preferred to bundles lying on the curve (weights 0 > t > 1). • Simple convexity allows for straight line segments of the indifference curve • Strict convexity does not, the curve must have an increasing slope as X increases. Y A y1 C ty1+ (1-t)y2 B y2 x1 tx1+ (1-t)x2x2 X**Convexity**• Intuition: people prefer balanced bundles of goods to bundles with a lot of one good and little of the other good. U=4 Y U=7 Along a straight line connecting the axis, Utilty will rise and then fall. U=10 U=7 U=4 X**Convexity: Intuition**• Which implies indifference curves bow towards the origin. U=4 Y U=7 U=10 U=7 U=4 X**Marginal Rate of Substitution**• The change in the consumption of the good on the Y axis necessary to maintain utility if the consumer increases consumption of the good on the X axis by one unit. • MRS = the slope of the indifference curve. • Although, , MRS is almost always defined as the abs value of the slope. • In this class,**MRS = MB**• The MRS describes, at any given point along the indifference curve, the consumer’s willingness to give up Y for one more X. • It is therefore the marginal willingness to pay for X • I.e. it is the marginal benefit of consuming X.**Digression: Cardinal Utility**• Utilitarians believed that utility, like temperature or height, was something that was measurable (Cardinal utility). • And that a unit of utility was the same for everyone so if we could find out how to measure it, we could redistribute to maximize social welfare. • The hope of some way of measuring utils did not survive long. • Early neoclassical economists (e.g. Marshall) still held the idea that for an individual, utility may be a cardinal measure. • Believed marginal utility was strictly decreasing. • Marshall’s demand curve was downward sloping for this reason. • He is the reason P is on the vertical axis. Diminishing willingness to pay reflected diminishing marginal utility. • Also believed that utility was additive, consumption of one good did not affect the MU from another (Uxy = 0).**Digression: Ordinal Utility**• Pareto (1906) first considered the idea that ordinal utility (ordering the desirability of different choices) might be the way to think of utility. • Work by Edgeworth, Fischer and Slutsky advanced the theory. • Hicks and Allen (1934) came up with the defining theory… that we still use today. • Pareto, Vilfredo (1906). "Manuale di economia politics, con una introduzione ulla scienza sociale". Societa Editrice Libraria. • Viner, Jacob. (1925a). "The utility concept in value theory and its critics". Journal of political economy Vol. 33, No. 4, pp. 369-387 • Hicks, John and Roy Allen. (1934). "A reconsideration of the theory of value". Economica Vol. 1, No. 1, pp. 52-76**The Utility Function**• A utility function is simply a way to mathematically represent preferences. • Utility is Ordinal: The ability to order bundles is all that matters. • The magnitude of the difference in utility is meaningless as the numbers reflecting utility are arbitrary. • No interpersonal comparisons are possible.**The Utility Function**• A function such that • Preference can be represented by a continuous U=U(A) so long as preferences are reflexive, complete, transitive, continuous • Note, monotonacity and convexity are not needed. • Monotonacity is always assumed because it makes the existence proof easier and more intuitive.**The Utility Function**• While we need preferences to be reflexive, complete, transitive, continuous for utility functions to exist. • We need monotonacity and convexity to make them well behaved. • By well behaved, we want unique solutions that are not extreme and are relatively stable. • We don’t want individuals spending all their income on one good or slight changes in price or income to drastically affect the optimal choice.**Revisiting Monotonacity**• As all indifference curves are strictly downward sloping, they will all cross a 45 deg line. y d x**Revisiting Monotonacity**• Monotonacity, Weak and Strong • We will assume strong, so no thick indifference curves! Weak Monotonacity Strong Monotonacity U(d) U(d) d d**Establishing Monotonocity**• We need to demonstrate that the indifference curve is downward sloping. • Say U0 = U(x, y) • Solve for y = Y(x, U0), making the implicit function, U = U(x, y), explicit. • Calculate dy/dx**Example**• Say we have U = x2*y • Solve to get: • y = U/x2 • dy/dx = -2U/x3 • Also, U = x2*y, • So dy/dx = -2y/x • For all U and all x > 0, dy/dx < 0 and nonsatiation holds. • However, it may not be possible to solve for Y explicitly (e.g. U = 6y5 – 3xy + 7x3)**dy/dx via Implicit Differentiation**• We start with an implicit function (identity) defining an indifference curve. To hold when x changes, y must change too. https://www.khanacademy.org/math/calculus/differential-calculus/implicit_differentiation/v/implicit-differentiation-1**Example**• Start with U = x2*y • And • So monotonacity holds as -2y/x < 0 for all x,y >0**Intermediate Micro Version**• Take the total differential of U = U(x, y) • This derivation requires dividing by dx, which bothers some people, but everyone teaches it this way (e.g. Chiang and Wainwright, p. 375)**Transformations**• It sometimes makes life computationally easier to transform a utility function. • Start with utility function that is well behaved (satisfies all the axioms of preference). • We can transform that function with no loss of information so long as the relationship between any bundles A and B is unchanged.**Positive Monotonic Transformations**• Two functions with identical ordinal properties are called Positive Monotonic Transformations of one another. • Both functions will create identically SHAPED indifference curves (although the utility value associated with each curve will differ).**Order preserving transformations**• U = U(x,y), original utility function • UT = UT (U(x,y)), transformation function • UT = UT (U(x,y)) is a positive monotonic transformation if UT ‘(U) >0 for all U.**Convexity**• That is, MRS diminishes as x increase and y decreases. It is about curvature. This Not this Y Y MRS = 5 MRS = 5 MRS = 2 MRS = 2 MRS = 5 MRS = 1 MRS = 2 X X**Digression on indifference curves.**Indifference curves are often thought of as level curves projected onto the base plane U=U(x, y) U Y This utility function is strictly concave as drawn X**Indifference Curves are Level Curves**• Level Curves • Are a slice of the utility function at some U = U0 • Even if the utility function is not concave (as drawn above), but only strictly quasi concave, these level curves bound convex sets • Convex sets and level curves • Any line connecting two points on the same level curve lies within the set • So bundles with more balance than two bundles lying on an indifference curve will provide more utility (the utility function will be “above” any line connecting two points on an indifference curve.**Convexity**• Convexity • DOES NOT IN ANY WAY indicate that the utility function is convex as opposed to concave. • Convex functions and convex sets are two different concepts. • “Strictly” • Strictly quasi–concave utility function means the utility function has no flat spots and it’s level sets are strictly convex • Strictly convex level curves means the indifference curves have no straight line segments • “Strictly” required to ensure just one optimum**Convexity of Preferences Implies Indifference Curves Bound**Convex Sets This will hold if the utility function is Strictly Quasi Concave U=U(x, y) U Y Utility Function Not Concave, but Strictly Quasi Concave as the level sets bound convex sets Any point on one of these dotted lines (exclusive of end points), provides more utility than the end points X**Convexity**• Convexity of preferences will hold if: • dy/dx increases along all indifference curves (it gets less negative, closer to zero) • That is, either: • The level sets are strictly convex • The utility function is strictly quasi-concave**Convexity (level curves)**• dy/dx increases along all indifference curves • We can use the explicit equation for an indifference curve, y=Y(x, U0) and find to demonstrate convexity. • That is, while negative, the slope is getting larger as x increases (closer to zero).

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