Consumer Theory and X1

: Hal R. Varian. Intermediate Microeconomics, A Modern Approach. W. W. Norton & Company, Inc. 1 BUDGET CONSTRAINT Consumer theory —- how consumers buy their goods? Economists assume: consumers choose the best bundle of goods they can afford. Two aspects: —-Consumers choose the most preferred goods. —-They are limited by economic condition. The Budget Constraint Consumption bundles: ( , ): a list of numbers of goods and services. X = (x1, x2, †¦, xn,) In the case of two goods: good 1 and good 2. Bundle of goods: X = (x1, x2) Prices of goods: (p1, p2),The amount of money the consumer has to spend: m. The consumer’s affordable consumption bundles, (x1, x2) satisfy p1x1 + p2x2 ? m. —-The budget set of the consumer ( ) . good 2 m/p2 O m/p1 good 1 Two Goods Are Often Enough Composite good —-take x2 as everything else, the dollars spent on other goods. For example, x1: consumption of milk in quarts per month. The budget constraint will take t he form p1x1 + x2 ? m. The case of n goods Budget constraint: p1x1 + p2x2+†¦+ pnxn ? m. Properties of the Budget Set Budget line( ): p1x1 + p2x2 = m. Vertical intercept: m/p2Horizontal intercept: m/p1. Slope: – p1/p2 Economic interpretation of slope: For the bundle (x1, x2): p1x1 + p2x2 = m. After a change in bundle (? x1, ? x2): p1(x1+? x1) + p2(x2+? x2) = m. good 2 x2 ?x2 ?x1 O x1 good 1 Subtracting the first equation from the second gives p1? x1 + p2? x2 = 0. This gives The number of good 2 the consumer must give up when he increases his consumption of good 1 by 1 unit, and keeps the money spent unchanged. Opportunity cost of consuming good 1—- in order to consume more of good 1 you have to give up some consumption of good 2.Budget Line Changes How the budget line changes when prices and incomes change? Change in income Change in m results in a parallel shift of the budget line. Intercepts m/p2 and m/p1 will change. Slope – p1/p2 keeps unchanged. good 2 m/p2 O m/p1 good 1 Changes in prices Increasing p1 will not change the vertical intercept, but p1/p2 will become larger. good 2 m/p2 O m/p1 good 1 What happens to the budget line when we change the prices of good 1 and good 2 at the same time? Proportionally: (tp1)x1 + (tp2)x2 = m.What happens to the budget line when we change the prices of good 1 and good 2 and the consumers’ income at the same time? good 2 m/p2 O m/p1 good 1 Proportionally: (tp1)x1 + (tp2)x2 = tm. Some observations: If one price declines and all others stay the same, the consumer must be at least as well-off. If the consumer’s income increases and all prices remain the same, the consumer must be at least as well-off as at the lower income A perfectly balanced inflation cannot change anybody’s optimal choice. 2 PREFERENCES Consumer Preferences( Consumer ranks consumption bundles by his satisfaction from use of goods, irrelevant to the prices. The case of two goods Given any two consumption bund les, X=(x1, x2) and Y=(y1, y2), the consumer can rank them in one of three possible ways: (x1, x2) is strictly better than (y1, y2); (y1, y2) is strictly better than (x1, x2); (x1, x2) and (y1, y2) are indifferent. Two basic relations: [pic]: strictly preferred( ), (x1, x2) [pic] (y1, y2): the consumer strictly prefers (x1, x2) to (y1, y2). ~ : indifferent ( ) (x1, x2) ~ (y1, y2). he consumer is indifferent between (x1, x2) and (y1, y2). A composite relation: [pic]: weakly preferred ( ) (x1, x2) [pic](y1, y2): the consumer prefers (x1, x2) to (y1, y2) or is indifferent between (x1, x2) and (y1, y2). Assumptions about Preferences Axioms about consumer preference (weakly preference): Complete( ). Given any X-bundle and any Y-bundle, consumer can say that (x1, x2)[pic](y1, y2), or (y1, y2)[pic](x1, x2). Reflexive( ). Consumer should say that any bundle is at least as good as itself: (x1, x2)[pic](x1, x2). Transitive ( ).If a consumer feels that (x1, x2)[pic](y1, y2) and (y1, y2)[pic](z 1, z2) then he feels that (x1, x2)[pic](z1, z2). Indifference Curves Weakly preferred set: all of the consumption bundles that are weakly preferred to (x1, x2). Indifference curves( ): —-The boundary of weakly preferred set; Good 2 x2 O x1 Good 1 Further assumptions Well-behaved preferences( ): Monotonicity ( )—- more is better. If that x1 ( y1, x2 ( y2 and that x1 ( y1 , x2 ( y2 at least one hold, then (x1, x2) [pic] (y1, y2) —-indifference curves have negative slope.A indifference curve is the set of bundles for which the consumer is just indifferent to (x1, x2). Good 2 O Good 1 Convexity ( )—- averages are preferred to extremes. If (x1, x2) and (y1, y2) are indifferent, then the bundle ([pic]x1+[pic]y1, [pic]x2+[pic]y2) is strictly preferred to (x1, x2) and (y1, y2). —-indifference curves are convex. Good 2 O Good 1 Examples of preferences Perfect Substitutes( ) The consumer is willing to substitute one good for the other at a constant rate. Goo d 2 O Good 1Perfect Complements( ) Goods that are always consumed together in fixed proportions. Good 2 O Good 1 Discrete Goods( ) x1 : a discrete good that is only available in integer amounts. Suppose that x2 is money to be spent on other goods. Good 2 O Good 1 The Marginal Rate of Substitution Marginal rate of substitution (MRS, ): slope of an indifference curve. —- measures the rate at which the consumer is just willing to substitute one good for the other. MRS = [pic] Note: MRS is a negative number. Good 2 (x2O (x1 Good 1 The other form of MRS MRS =[pic] Good 2 x2 O x1 Good 1 Behavior of the Marginal Rate of Substitution Describe the indifference curves by the MRS. Perfect substitutes: the marginal rate of substitution is constant. Perfect complements: the MRS is either 0 or infinity, and nothing in between. In general case: Monotonicity: indifference curves must have a negative slope, i. e. negative MRS. Convex: the marginal rate of substitution decreases as we increase x1, —-diminishing MRS. ———————– [pic]