CHAPTER 3 McGraw-Hill/Irwin RATIONAL CONSUMER CHOICE Copyright © 2006 The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline The opportunity set or budget constraint Budget shifts due to price or income changes Consumer preferences Completeness More-is-better Transitivity Convexity Indifference curves Trade-offs between goods The best feasible bundle Corner solution The utility function approach to consumer choice Generating Indifference Curves Algebraically Summary 3-2 An Overview Theory of rational consumer choice – underlies all individual purchasing decisions. Begins with the assumption that consumer enter the market with well-defined preferences. Given prices, their task is to allocate their incomes to best serve these preferences. 2 steps: describe various combinations of goods the consumer is able to buy (depend on Y & P) Select from among the feasible combinations the particular one that the consumer prefers to all others 3-3 The Opportunity Set or Budget Constraint Budget Constraint/ Budget Line The set of all bundles that exactly exhaust the consumer’s income at given prices. Assume only 2 goods ( food and shelter) are consumed, so we do not consider savings (all income are spent). Bundle A : 5 square yards/week of shelter & 7 pounds per week of food Bundle B: 3 sq yd/wk of shelter & 8 lb /wk of food 3-4 Two bundles of goods 3-5 The Budget Constraint or Budget Line Suppose consumer’s income, M = $100/wk (all spent on some combination of S & F) Price of shelter, Ps = $5/sq yd Price of food, Pf = $10/lb If spent all income on S could buy M/Ps = ($100/wk) ($5/sq yd) = 20 sq yd/wk (20,0). If spent all income on F could buy M/Pf = ($100/wk) ($10/lb) = 10 lb/wk (0, 10). 3-6 The Budget Constraint or Budget Line 3-7 Slope of budget line = - Ps / Pf = - 1 / 2 It is the rate at which food can be exchanged for shelter ( 1 lb of food can be exchanged for 2 sq yd of shelter). Feasible set or affordable set - Consumer can buy any bundles along the budget line and also bundles that lies within the budget triangle bounded by it and two axes (e.g. D ) Bundle that lie outside the budget triangle (e.g. E) are said to be infeasible or unaffordable 3-8 Budget constraint must satisfy the following equation: Ps S + Pf F = M (1) To represent the formula for a straight line, solve equation 1 for F in terms of S: F = M – Ps S Pf Pf slope intercept F = 10 – ½ S 3-9 Budget shifts due to Price or Income Changes Slope and position of BL are fully determined by the consumer’s income & prices. Assuming price of shelter increase from Ps1 = $5 to Ps2 = $10. Since both income & Pf unchanged, the vertical intercept of the BL stays the same. The ↑Ps rotates the BL inward. 3-10 The Effect of a Rise in the Price of Shelter 3-11 Income Changes The effect of a change in income is much like the effect of an equal proportional change in all prices. Assuming income decreased from $100/wk to $50/wk – the horizontal intercept of the BL falls from 20 sq yd/wk to 10 sq yd/wk, and the vertical intercept falls from 10 lb/wk to 5 lb/wk. There is a parallel shifts inward & slope of BL remain unchanged 3-12 The Effect of Decrease in Income 3-13 Consumer Preferences Preference ordering is a scheme that enables consumer rank different bundles of goods in terms of their desirability. For any two bundles of goods (e.g. A and B), the preference ordering enables the consumer to rank different bundles but not to make more precise quantitative statements about their desirability. E.g. consumer might be able to say that he prefers A to B but not that A provides twice as much satisfaction as B. 3-14 Properties of preferences Completeness if consumer able to rank all possible combinations of goods and services. More-Is-Better more of a good is preferred to less. Transitivity for any 3 bundles A, B, and C, if A preferred to B, and B preferred to C, then A is always preferred to C. Transitivity is a consistency property. Convexity mixtures of goods are preferable to extremes. Suppose consumer are indifferent between A (4, 0) and B (0, 4). If preferences are convex, consumer will prefer bundle C (2,2). Implies that consumer like balance in mix of consumption 3-15 goods. Generating Equally Preferred Bundles 3-16 An Indifference Curve Is a set of bundles that the consumer considers equally attractive. Bundles that lie above an IC are all preferred to the bundles that lie on it. Bundles that lie on an IC are all preferred to those that lie below it. 3-17 Indifference Map I1 < I2 < I3 < I4 3-18 Properties of Indifference curve Indifference curves are ubiquitous. Any bundle has an indifference curve passing through it. IC are downward sloping. An up-ward sloping IC would violate the more-is-better property. IC become less steep as we move downward and to the right IC cannot cross because they would have to violate at least one of the assumed properties of preference ordering. 3-19 Why Two Indifference Curves Do not Cross E is equally attractive as D D is equally attractive as F E is equally attractive as F (by the transitivity assumption) F is preferred to E (because more is better) 3-20 Trade-Offs between Goods An important property of a consumer’s preference is the rate at which they are willing to exchange or ‘trade off' one good for another. This is represented at any point on an IC by the MRS (slope of IC). MRS declines along the IC as we move downward to the right – implying that consumers are willing to give up goods that they already have a lot of to obtain more of those goods they have few. This is because the indifference curves are convex 3-21 The Marginal Rates of Substitution (MRS) More formally, IC may be expressed as a function, Y = Y (X) and the MRS at point A is defined as the absolute value of the derivative of the IC at that point: MRS = | dY (X) / dX | 3-22 Diminishing Marginal Rate of Substitution 3-23 The Best Affordable Bundle The indifference map tells how various bundles are ranked in order of preference. Budget line tells which bundles are affordable Using these two tools – consumer choose the most preferred or best affordable bundle 3-24 The Best Affordable Bundle Of the 5 labeled bundles – A, D, E, F and G G is the most preferred because it lies on the highest IC, however, it is not affordable. The best affordable bundle is F ( 6F : 8S) MRS = Ps/Pf (point of tangency). 3-25 The Best Affordable Bundle If MRS ≠ PF/PC then individuals can reallocate basket to increase utility If MRS > Ps/Pf Will increase shelter and decrease food until MRS = Ps/Pf If MRS < Ps/Pf Will increase food and decrease shelter until MRS = Ps/Pf 3-26 A Corner Solution A corner solution exists if a consumer buys in extremes, and buys all of one category of good and none of another. MRS is not necessarily equal to slope of budget line (MRS may be everywhere greater, or less, than the slope of the BL) Utility is maximized at a point on one axis where the budget constraint intersects the highest attainable indifference curve at zero consumption for one good with all income used for the other good. The best affordable bundle need not always occur at a point of tangency. 3-27 A Corner Solution At point A, MRS < Ps/Pf the best the consumer can do is to spend all his income on food. 3-28 A Corner Solution Frozen Yogurt (cups monthly) A U1 U2 U3 B A corner solution exists at point B. Ice Cream (cup/month) 3-29 Perfect Substitutes and Perfect Complements The shape of an indifference curve describes the willingness of a consumer to substitute one good to another Indifference curves with different shapes imply a different willingness to substitute Two extreme cases are Perfect substitutes Perfect complements 3-30 Perfect Substitutes 2 goods are perfect substitutes when the MRS of one good for the other is constant. Example 1: A person might consider apple juice and orange juice perfect substitutes They would always trade 1 glass of OJ for 1 glass of AJ Example 2: • Amy likes M&M, plain and peanut. For Amy the MRS between plain and peanut M&M’s does not vary with the quantities she consumes. Apple 4 Juice (glasses) 3 2 1 0 1 2 3 4 Orange Juice (glasses) 3-31 Two goods are perfect complements when the IC for the goods are shaped as right angles. Example 1: 1 left shoe and 1 right shoe- both must be used at the same time. Example 2: Peter is very choosy about his buttered popcorn. He tops every quart of popped corn with exactly one quarter cup of melted butter. Perfect Complements Left Shoes 4 3 2 1 0 1 2 3 4 Right Shoes 3-32 Measuring Consumer Preferences The theory of consumer behavior does not required assigning a numerical value to the level of satisfaction. There are two types of ranking Ordinal ranking Places market baskets in the order of most preferred to least preferred, but it does not indicate how much one market basket is preferred to another. Cardinal ranking Utility function describing the extent to which one market basket is preferred to another. 3-33 Utility Although ranking of market baskets are good (where we use indifference curve to describe graphically consumer preferences), sometimes numerical value are useful The concept is known as Utility A numerical score representing the satisfaction that a consumer gets from a given market basket. 3-34 Utility Utility function Formula that assigns a level of utility to individual market baskets If the utility function is U(F,C) = F + 2C It tells the level of satisfaction obtained from consuming F units of food and C units of clothing o A market basket with 8 units of food and 3 units of clothing gives a utility of 14 = 8 + 2(3) 3-35 Utility - Example Market Food Basket A 8 B 6 C 4 Clothing Utility 3 4 4 8 + 2(3) = 14 6 + 2(4) = 14 4 + 2(4) = 12 Consumer is indifferent between A & B because the utility is same (14) and prefers both to C because utility is smaller (12) 3-36 Utility Example If the utility function, U = FC Basket C 25 = 2.5(10) A 25 = 5(5) B 25 = 10(2.5) Clothing If the utility function, U (F, C) = 4FC C 100 = 4 (2.5)(10) A 100 = 4 (5) (5) B 100 = 4 (10) (2.5) 15 C 10 U3 = 100 A 5 B 0 5 10 U2 = 50 15 U1 = 25 Food 3-37 The utility function approach to consumer choice As consumption moves along an indifference curve: Additional utility derived from an increase in the consumption one good, food (F), must balance the loss of utility from the decrease in the consumption in the other good, clothing (C). Formally: 0 MUF(F) MUC(C) No change in total utility along an indifference curve. Trade off of one good to the other leaves the consumer just as well off. 3-38 Marginal Utility and Consumer Choice Rearranging: C / F MU F / MU C Since C / F MRS of F for C We can say MRS MUF/MU C 3-39 Marginal Utility and Consumer Choice When consumers maximize satisfaction: MRS PF/PC Since the MRS is also equal to the ratio of the marginal utility of consuming F and C MUF/MUC PF/PC 3-40 Marginal Utility and Consumer Choice Rearranging, gives the equation for utility maximization: MU F / PF MU C / PC Total utility is maximized when the budget is allocated so that the marginal utility per dollar of expenditure is the same for each good. This is referred to as the equal marginal principle. 3-41 Generating Indifference Curves Algebraically Using calculus to solve the consumer’s budget allocation problem. Let U (X, Y) be the consumer’s utility function M, Px and Py = income, price of X and price of Y Consumer’s allocation problem can be stated as: Maximize U(X, Y) subject to PxX + PyY = M X, Y (Eq.1) Find values of X and Y using method of Lagrangian Multipliers (technique used to maximize or minimize a function subject to one or more constraints) 3-42 The Method of Lagrangian Multipliers First transformed the constrained maximization problem in equation (1) into unconstrained maximization problem: Maximize £ = U(X,Y) - (PxX + PyY – M) X, Y, (Eq.2) Differentiate £ with respect to X, Y and and then equate the derivatives to zero. £ = U - Px = 0 X X (Eq.3) £ = U - Py = 0 Y Y (Eq.4) £ = M – PxX - PyY= 0 (Eq.5) 3-43 Next solve Eq.3-4 for X, Y and (to guarantee that the budget constraint is satisfied). Optimal X and Y values can be obtained by dividing Eq.3 by Eq.4 to get; U/X = Px = Px (Eq.6) U/ Y Py Py Eq. 6 implies optimal values of X and Y must satisfy MRS = Px/Py. The terms U/X and U/ Y is marginal utility (MU) of X and Y. Rearranging Eq.6 in the form; U/X = U/ Y Px Py MUx = MUy Px Py (Eq.7) 3-44 An Example Suppose that U (X, Y) = XY and that M = 40, Px = 4 and Py = 2. Constrained maximization problem; Maximize £ = XY - (4X + 2Y – 40) (Eq.1) X, Y, First-order conditions for a maximum of £ are given; £ = (XY) - 4 = Y - 4 = 0 (Eq.2) X X £ = (XY) - 2 = X - 2 = 0 Y Y (Eq.3) £ = 40 – 4X – 2Y = 0 (Eq.4) Divide Eq. 2 by Eq.3 and solve for Y, get Y = 2X. Substitute Y = 2X into Eq.4 and solve for X (X =5). Y = 2X = 10. Thus (5, 10) is the utility-maximizing bundle. 3-45 An Alternative Method Solve the budget constraint for Y in terms of X and substitute in the utility function. Then maximize by taking first derivative with respect to X and equate to zero. Suppose U (X, Y) = XY, with M= 40,Px = 4 and Py=2. Budget constraint, 4X + 2Y = 40, which solves for Y = 20 – 2X. Substitute this into utility function, U(XY) = X (20 – 2X) = 20X – 2X2 Take first derivative of U with respect to X and equate with zero, dU = 20 – 4X = 0 dX Solve for X = 5. Substitute X into budget constraint to get the optimal value of Y = 10 3-46 The Optimal Bundle when U=XY, Px=4, Py=2, and M=40 3-47 Example Question 1 1. Sally consumes two goods, X and Y. Her utility function is given by the expression U = 3 XY2. The current market price for Y is $5, while the market price for X is $10. (Note MUy= 6XY and MUx = 3Y2). Income = $500. a. Determine the X, Y combination which maximizes Sally’s utility given her budget constraint. b. Calculate the impact on Sally’s optimum market basket of an increase in the price of X to 15. What would happen to her utility due to price increase? 3-48 a. To maximize utility, MRS = Px / Py MRS = MU x / MU y Since MU x = 3Y2 and MU y = 6XY MRS = 3Y2 = Y 6XY 2X Px / Py = 10 / 5 = 2 To determine exact quantities, substitute Y = 4X into I = Px X + Py Y 500 = 10 X + 5 (4X) 500 = 30 X X = 16.67 Y = 4 (16.67) Y = 66.67 MRS = Px / Py Y = 2 2X Y = 4X 3-49 b. Calculate the impact on Sally’s optimum market basket of an increase in the price of X to 15. What would happen to her utility due to price increase? Solution: MRS remains Y / 2X, and Px / Py becomes 15 / 5 = 3 Equating MRS to Px / Py we get Y = 3 Y = 6X 2X Substitute Y = 6X into I = Px X + Py Y 500 = 15 X + 5 Y 500 = 15 X + 5(6X) 500 = 45 X X = 11.11 and Y = 66.67 Utility before price change: U = 3 (16.67)(66.67)2 = 222,289 Utility after price change: U = 3 (11.11)(66.67)2 = 148,148 3-50 Example 2 An individual consumes products X and Y and spends $25 per time period. The prices of the two goods are $3 per unit for X and $2 per unit Y. The consumer in this case has a utility function expressed as: U (X, Y) = 0.5 XY, MUx = 0.5 Y, MUy = 0.5 X a. Express the budget equation mathematically I = Px X + Py Y 25 = 3X + 2 Y 3-51 b. Determine the values of X and Y that will maximize utility in the consumption of X and Y. In equilibrium, utility is maximized when MU x / Px = MU y/ Py 0.5 Y / 3 = 0.5 X / 2 2Y = 3 X Y = (3/2) X On the budget line , 25 = 3X + 2 Y 25 = 3X + 2 (3 X) 2 25 = 6X X = 4.17 units, Y = 6.25 units c. Determine the total utility that will be generated. U (X, Y) = 0.5 X Y = 0.5 (4.17) (6.25) U = 13.03 3-52 Summary Theory of consumer choice rests on the assumption that people behave rationally in an attempt to maximize satisfaction by purchasing a particular combination of goods & services. 2 parts: consumer’s preferences & budget line. 4 properties of consumer’s preferences, indifference curves and MRS. Budget line – definition, slope and the effect changes in prices and income. Consumer maximization condition, MRS = ratio of prices. Utility function approach & cardinal vs ordinal utility Using calculus to maximize utility. 3-53

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