lecture02_slides - Econ 121. Intermediate Microeconomics....

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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 2 Outline of the course I. Introduction II. Individual choice III. Competitive markets IV. Market failure Outline of the course I. Introduction II. Individual choice Budget constraint (Ch. 2) Preferences (Ch. 3) Utility (Ch. 4) Consumer problem (Ch. 5) III. Competitive markets IV. Market failure Consumer Theory Consumers choose the best consumption bundle they can afford. this is virtually the entire theory in a nutshell but this theory has many surprising consequences Two parts to theory: "can afford" - budget constraint "best" - according to consumers' preferences 3 Consumption bundle L goods in the economy, labeled ` D 1; 2; : : : ; L .x1; x2; : : : ; xL/ - how much of each good is consumed .p1; p2; : : : ; pL/ - prices of each good m - money the consumer has to spend Budget constraint: p1x1 C p2x2 C : : : C pLxL m all non-negative .x1; x2; : : : ; xL/ that satisfy this constraint make up the budget set of the consumer. See figure on blackboard. 5 Two goods theory works with more than two goods, but can't draw pictures. two goods enough to understand conceptual issues often think of good 2 (say) as a composite good, representing money to spend on other goods. budget constraint becomes: p1x1 C p2x2 m : money spent on good 1 (p1x1) plus the money spent on good 2 (p_2x_2) must be less than or equal to the amount of money available (m). 6 Budget line The set of consumption bundles that satisfy the budget constraint with equality: p1x1 C p2x2 D m ; which can also be written as x2 D budget line has slope of p1 p2 m p2 p1 x1 p2 m p2 and vertical intercept of m set x1 D 0 to find vertical intercept ( p ); set x2 D 0 to find horizontal 2 m intercept ( p ) 1 slope of budget line measures opportunity cost of good 1 - how much of good 2 you must give up in order to consume more of good 1 7 Changes in budget line increasing m makes parallel shift out increasing p1 makes budget line steeper increasing p2 makes budget line flatter just see how intercepts change multiplying all prices by t same as dividing income by t multiplying all prices and income by t does not change consumption possibilities 8 The numeraire can arbitrarily assign one price a value of 1 and measure other price (and income) relative to that useful when measuring relative prices; e.g., British pounds per dollar, 1987 dollars versus 1974 dollars, etc. the good whose price is normalized to one is called the numeraire good 9 Taxes, subsidies and rationing quantity tax - tax levied on units bought: p1 C t value tax - tax levied on dollars spent: p1 C p1. Also known as ad valorem tax subsidies - opposite of a tax p1 s or .1 /p1 lump sum tax or subsidy - amount of tax or subsidy is independent of the consumer's choices. Also called a head tax or a poll tax. 10 Example - food stamps before 1979 was an ad valorem subsidy on food paid a certain amount of money to get food stamps which were worth more than they cost some rationing component - could only buy a maximum amount of food stamps after 1979 got a straight lump-sum grant on food coupons. Not the same as a pure lump-sum grant since could only spend the coupons on food. 11 Preferences Preferences are relationships between bundles Binary relations: heavier than, taller than, etc. If a consumer would choose bundle .x1; x2/ when .y1; y2/ is available it is natural to say that bundle .x1; x2/ is preferred to .y1; y2/ by this consumer Preferences have to do with the entire bundle of goods, not with individual goods. 12 Notation .x1; x2/ .y1; y2/ means the x -bundle is strictly preferred to the y -bundle .x1; x2/ .y1; y2/ means the x -bundle is regarded as indifferent to the y -bundle .x1; x2/ .y1; y2/ means the x -bundle is at least as good as the y -bundle 13 Assumptions about preferences Complete - any two bundles can be compared: for all bundles x and y , either x y or y x. Reflexive - any bundle is at least as good as itself: for all x , x x. y and y z , then x z. Transitive - if x 14 Indifference curves graph the set of bundles that are indifferent to some bundle. See figure on blackboard. boundary of the upper contour set Note that indifference curves describing two distinct preference levels cannot cross. 15 upper contour set Examples of preferences perfect substitutes red pencils and blue pencils; quarts and gallons constant rate of trade-off between the two goods perfect complements always consumed together right shoes and left shoes; coffee and cream bads neutrals satiation or bliss point 16 Well-behaved preferences monotonicity - more of either good is better. Implies indifference curve has negative slope. convexity - averages are preferred to extremes. slope gets flatter as you move further to the right example of non-convex preferences. See figure on blackboard. 17 implies decreasing MR for each good Marginal Rate of Substitution slope of the indifference curve changes as we move along the curve Let x2 D f .x1/ denote a certain indifference. Then, x2 MRS at .x1; x2/ D f .x1/ D x1 0 along an indifference curve natural sign is negative, since indifferent curves have negative slope measures how the consumer is willing to trade off consumption of good 1 for consumption of good 2. measures marginal willingness to pay (give up) not the same as how much you have to pay but how much you would be willing to pay 18 Utility Function summarizes preferences a utility function assigns a number to each bundle of goods so that more preferred goods get higher numbers that is, u.x1; x2 / > u.y1; y2/ if and only if .x1; x2/ .y1; y2/ only the ordering of bundles counts, so this is a theory of ordinal utility Utility functions are not unique If u.x1; x2 / is a utility that represents some preferences, and f . / is any increasing function, then f .u.x1; x2// represents the same preferences. Why? Constructing a utility function Easy task when preferences are monotonic. x2 IC'' IC' IC 45 degrees x1 Examples from utility to indifference curves from indifference curves to utility examples perfect substitutes - all that matters is total number of pencils, not their color, so u.x1; x2/ D x1 C x2 does the trick can use any monotonic transformation as well, such as log.x1 C x2/ perfect complements - all that matters is the minimum of the left and right shoes you have, so u.x1; x2/ D minfx1; x2g works quasilinear preferences - u.x1; x2/ D v.x1/ C x2, that is, indifference curves are vertically parallel. See figure on blackboard. Cobb-Douglas preferences b c utility has form u.x1; x2/ D x1 x2 convenient to take transformation f .u/ D u a 1 or x1 x2 a , where a D b=.b C c/ 1 bCc and write x1 b bCc x2 c bCc Marginal utility extra utility from extra consumption consumption of one of the goods, holding the other good fixed this is a derivative, but a special kind of derivative - a partial derivative just means you look at the derivative of u.x1; x2/ keeping x2 fixed treating it like a constant examples if u.x1; x2/ D x1 C x2, then M U1 D a 1 if u.x1; x2/ D x1 x2 a , then @u D 1: @x1 @u a 1 D ax1 1x2 M U1 D @x1 a : Marginal utility @u .x1; x2/ M U1 .x1; x2/ D @x1 note that MU depends on the choice of the utility function - not an ordinal concept however, MU is closely related to MRS, as we will see, which is ordinal Relationship between MU and MRS Fix a utility level k and consider the correspondent indifference curve: u.x1; x2/ D k This equation implicitly defines a function f .x1/ such that u.x1; x2/ D k if and only if x2 D f .x1/ Recall the definition of MRS: MRS.x1 ; x2/ D f 0.x1/ whenever u.x1; x2/ D k In principle, need to know f , which is only implicitly defined. MRS = slope of indifference curve Relationship between MU and MRS We have u.x1; x2/ D k along the indifference curve x2 D f .x1/ Thus, u.x1; f .x1// D k By the chain rule of calculus, for all x1: @u @u df d .x1 ; f .x1// C .x1; f .x1// .x1/ D u.x1; f .x1// D 0: @x1 @x2 dx1 dx1 Hence, @u @x1 @u @x2 df MRS D D dx1 D M U1 M U2 Optimal choice move along the budget line until upper contour set doesn't cross the budget set note that tangency occurs at optimal point. In symbols: MRS.x1; x2 / D p1 p2 MRS is negative because we assume preferences are monotone Exceptions: kinks and corner solutions. Warning: tangency is not a sufficient condition, unless indifference curves are convex. At this point, the indifference curve crosses the budget constraint transversely. This is not optimal because the budget constraint to the left of this intersection corresponds to indifference curves of higher utility Non-convex Indifference Curves Tangencies are not necessarily optimal Optimal choice If we know the utility function, the prices and the income we can calculate the consumer's optimal bundle. Two equations, two unknowns: MRS.x1; x2/ D p1x2 C p2x2 p1 p2 D m budget This always works when preferences are convex and optimum is interior. Optimal choice depends only on: preferences (not utility) price ratio MRS is an ordinal concept Why? Suppose v.x1 ; x2/ and u.x1; x2/ represent the same preference. Hence, v.x1; x2/ D f .u.x1; x2// where f is an increasing function. By the chain rule of calculus, @v @x1 @v @x2 @u f 0.u.x1; x2// @x1 MRS under v D D f 0 .u.x ; x / @u 1 2 @x2 D MRS under u Examples perfect substitutes: x1 D m=p1 if p1 < p2, and 0 otherwise perfect complements: x1 D m=.p1 C p2/ neutrals and bads: x1 D m=p1 discrete goods: good is either consumed or not; compare .1; .m p1/=p2/ with .0; m=p2/ concave preferences: similar to perfect substitutes. Note that tangency doesn't work. Cobb-Douglas preferences (Homework): x1 D am=p1 . Note that parameter a gives the budget share. Perfect Substitutes x2 IC m/p2 In this case, you will spend your entire budget on the item that is relatively cheaper BC m/p1 x1 Concave Indifference Curves x2 optimal x1 optimal Which is better, a commodity or an income tax? original budget constraint: p1x1 C p2x2 D m with tax: .p1 C t/x1 C p2x2 D m budget constraint rotates inward (due to commodity tax) optimal choice with tax: .p1 C t/x1 C p2x2 D m revenue raised is tx1 income tax that raises same amount of revenue yields the budget constraint: p1x1 C p2x2 D m tx1 budget constraint shifts inward (due to income tax) The new BC is parallel to the original this budget line passes through .x1 ; x2 /, so.... Income tax is at least as good! caveats only applies for one consumer income is exogenous - if income responds to taxes, problems no supply response, only looked at demand side Demand function optimal choice depends on prices and income when we hold income fixed, and examine dependence of optimal consumption on prices this is called the demand function when we hold prices fixed, and examine dependence of optimal consumption on income this is called the Engel function ...
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This note was uploaded on 03/10/2012 for the course ECON 121 at Yale.

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