Summer 2016 Microeconomics 2 ECON1201. Nicole Liu Z

Transcription

1 Summer 2016 Microeconomics 2 ECON1201 Nicole Liu Z

2 BUDGET CONSTAINT THE BUDGET CONSTRAINT Consumption Bundle (x 1, x 2 ): A list of two numbers that tells us how much the consumer is choosing of good 1, and how much the consumer is choosing to consume of good 2 Budget Constraint: Requires that the amount spent on consumption does not exceed the amount available to spend (i.e. bundles that are just affordable) m p! x! + p! x! Budget Set: The set of all affordable consumption bundles at given prices and income Budget Line: The boundary of the budget set consisting of all bundles that consumer can just afford. If the consumer is spending all of his income, this expression will hold with equality: PROPERTIES OF THE BUDGET SET x! = m p! p! p! x! Slope: Measures how much good 2 the consumer must give up to get an extra unit of good 1 Intercepts: Represent the bundles the consumer could purchase if he spent all his income on one good INCREASING INCOME An increase in income causes a parallel shift outward of the budget line. It increases the intercepts but does not change the slope of the line INCREASING PRICE If good 1 becomes more expensive, the budget line becomes steeper

3 PREFERENCES PREFERENCES Preferences: The ranking of all possible bundles Strictly Preferred: If a consumer strictly prefers bundle (x!, x! ) to another bundle (y!, y! ), if means that he would always choose (x!, x! ) over (y!, y! ) A B Indifference: A consumer is indifferent between bundle (x! x! ) and (y!, y! ) if he would just be satisfied, according to his own preferences, consumer either bundle A ~ B Weakly Preferred: A consumer weakly prefers (x!, x! ) over (y!, y! ) if he prefers or is indifferent between the two bundles A B Note: Weakly prefers x!, x! to vs. prefers to (x!, x! ) ASSUMPTIONS ABOUT PREFERENCES 1. COMPLETENESS Completeness: Consumer can rank any two bundles A B or A B Very realistic assumption about individual preferences even when a person doesn t know how to ran two alternatives, he is most likely either: - Indifferent between them - Doesn t have the experience with them to rank them, but would be able to rank them if he did 2. REFLEXIVITY Reflexivity: Any bundle is as good as itself A A Example of reflexivity violation: An apple is preferred to an apple Not many realistic examples of preferences violating reflexivity 3. TRANSITIVITY Transitivity: If A B and B C, it must be that A C Transitivity similarly implies that if A ~ B, and B ~ C, then A ~ C Usually realistic Example of transitivity violation: preferences in which the consumer has a ranking, but cannot tell small differences apart E.g. a consumer prefers his coffee with as much sugar as possible, but can t tell differences of ½ spoon or less apart C! ~ C! ~ C!! but C! C!

4 INDIFFERENCE CURVES Indifference Curves: A graphical representation of a consumer s preferences ASSUMPTIONS OF PREFERENCES ON INDIFFERENCE CURVE Completeness: means you can construct an indifference curve through every bundle Transitivity: means indifference curves cannot cross X and Y are on the same curve, so X ~ Y Y and Z are on the same curve, so Y ~ Z If preferences are transitive, it must be that X ~ Z But X and Z must then lie on the same indifference curve, and they don t CONCEPTS OF THE INDIFFERENCE CURVE Monotonicity: More is always better Monotonicity implies that higher indifference curves correspond to more preferred bundles Therefore, the direction of preference is away from the origin Monotonicity implies that the MRS must be negative Convexity: A set is convex if, for any two points in the set, all weighted averages of those two points are also in the set A consumer has convex preferences if the set of bundles is weakly preferred to any bundle in a convex set A consumer with convex prefers (weakly) prefers averages to extremes Strictly Convex Preferences: Weighted averages are strictly preferred to extremes WELL-BEHAVED PREFERENCES Assumptions about well-behaved preferences Monotonicity: More is better than less. This implies that indifference curves have a negative slope Convexity: Averages are preferred to extremes MARGINAL RATE OF SUBSTITUTION Marginal Rate of Substitution: Measures the slope of the indifference curve. This represents how much a consumer is willing to give up good 2 to acquire more of good 1 MRS = Diminishing Marginal Rate of Substitution: The more you have of one good, the more you are willing to give up some of it in exchange for the other good If preferences are monotonic and strictly convex, this implies diminishing MRS Indifference curves get flatter (MRS falls in absolute value) as x! is increased

5 TYPES OF INDIFFERENCE CURVES PERFECT SUBSTITUTES Perfect Substitutes: The consumer is willing to exchange one good for the other at a constant rate MRS is constant PERFECT COMPLEMENTS Perfect Complements: The consumer wants the goods in a fixed proportion Indifference curves have a kink where the proportions are perfect BADS Bad: A bad is a good the consumer doesn t like A BAD AND A GOOD (UPWARD SLOPING) If chocolate is a good, and broccoli is a bad, more chocolate makes him happier Must give him more broccoli to return him to the same indifference curve Positive MRS TWO BADS (DOWNWARD SLOPING) If both goods are bads, indifference curves are downward sloping More boiled liver makes him sadder, must take away some of his broccoli to keen him on the same indifference curve Negative MRS

6 UTILITY Utility Function: A way to represent preferences mathematically A utility function u(x!, x! ) takes an argument the bundle (x!, x! ) and assigns to it a number such that: More preferred bundles get higher numbers than less preferred bundles Bundles that are as good as each other get the same number Cardinal Utility: The magnitude of u(x!, x! ) matters there is something to understand about the consumer s preferences from the utility difference between two bundles Ordinal Utility: The magnitude u(x!, x! ) does not matter all that matters is the order INDIFFERENCE CURVES AND UTILITY An indifference curve connects all the bundles that the consumer is indifference between (i.e. have the same level of utility) PERFECT SUBSTITUTES Perfect Substitutes: The consumer is willing to exchange the goods at a constant rate u x!, x! = ax! + bx! = k x! = k b a b x! MRS = a b PERFECT COMPLEMENTS Perfect Complements: The consumer wants to always consume the goods in a fixed proportion If the consumer already has a bundle with the desired proportions, increasing one good without increasing the other does not affect utility u x!, x! = min ax!, bx! Line of Desired Proportions: Kinks in the indifference curve lie along the line of desired proportions COBB-DOUGLAS UTILITY ax! = bx! Cobb-Douglas Utility: The values of a and b will change slightly the shape of the indifference curve Cobb-Douglas utility will correspond to a well-behaved preference for all positive values of a and b QUASILINEAR UTILITY u x!, x! = x!! x!! Quasilinear Utility: Useful for representing preferences between one particular good and everything else u x!, x! = v x! + x! where δx x! > 0 and δ! v x!! < 0 If x! is a particular good and x! is money is spend on everything else, it is reasonable that: Utility is increasing in x!, but at a decreasing rate Utility is increasing in x!, money, at a constant rate Indifference curves are vertically shifted versions of each other

7 DERIVING THE MRS Starting from a bundle (x!, x! ) we want to make a change (, ) that keeps utility constant Magnitude of change in utility due to a small change in x 1 Magnitude of change in utility due to a small change in x 2 du = du x!, x! du = du x!, x! Total Change in Utility: The total change in utility after changing x! and x! must be 0 (a movement along the indifference curve) Solve for the MRS du = du x!, x! + du x!, x! = MU! + MU! = 0 δu(x!, x! ) MRS = = MU! δx =! MU! δu(x!, x! ) MONOTONIC TRANSFORMATIONS If a utility function u(x!, x! ) represents a given set of preferences, then any monotonic transformation of u will also represent the same preferences f(u) is a monotonic transformation of u if and only if it is an increasing function of u for all values u can take A monotonic transformation is a function f(u) of the original utility function u, such that: u! > u! implies f u! > f u! for the domain of u Implications of Monotonic Transformations on the MRS Two utility functions that represent the same preferences must have the same MRS o But the same MRS does not necessarily imply same preferences