3.1 Preferences s a consumer, you choose among many goods. You must allocate your money to buy abundle A of goods (market basket, or combination of goods). ● To explain consumer behavior, economistsassumethatconsumers have a set of tastes or preferences that they use to guide them in choosing between goods consumer chooses between bundles of goods by ranking them as to the pleasure the consumer A gets from consuming each. Properties of consumer preferences:completeness,transitivity and more is better. Completeness: This properly rules out the possibilitythat theconsumer cannot rank the bundles. ransitivityproperty eliminates the possibility ofcertain types of illogical behaviour. According T to the property if the consumer a≿b, b≿c then a≿c. ore is better (nonsatiation)property states thatall else the same, more of a commodity is M better than less of it. reference Maps P Each semester, Lisa decides how many pizzas and burritos to eat. The various bundles of pizzas and burritos she might consume are shown in panel a of Figure 3.1, with (individual-size) pizzas per semester,q1 on the horizontal axis and burritosper semester q2, on the vertical axis. ● At bundle e, Lisa consumes 25 pizzas and 15 burritos per semester. According to more-is-better property, all the bundles that lie above and to the right (area a) are preferred to bundle e. ● Lisa prefers point e to all the bundles that lie in area B rom panel A we do not know whether she prefers bundle e to bundles such as b (30 pizzas and F 10 burritos) in areaD, orc(15 pizzas and 25 burritos)in areaC. Indifference Curves uppose we asked Lisa to identify all the bundles that give her the same amount of pleasure as S consuming bundle e . Panel b of the figure above, shows curve I1 indicating all the bundles as much as she likes e. Thus the result is anindifferencecurve, the set of all bundles of goods that a consumer views as being equally desirable. I ndifference map or preference mapis a complete set of indifference curves that summarize a consumer’s taste. (i.e asking lisa to draw indifference curves through every possible bundle). ach indifference curve in an indifference map consists of bundles of goods that provide the E same utility but the level of utility differs from one curve to another. ● Refer to panel c from the figure above. All indifference maps must have five important properties: 1. Bundles on indifference curves farther from the origin are preferred to those on indifference curves closer to the origin.[ Becausemore is better property..] 2. Every bundle lies on an indifference curve.[becauseof the completeness property consumers can compare any bundle to another bundle.] 3 . Indifference curves cannot cross.[because indifferencecurves are transitive] 4. Indifference curves slope downward. 5. Indifference curves cannot be thick.[If curves werethick, they would include two bundles where one has strictly more of both goods.] 3.2 Utility tilityis the set of numerical values that reflectthe relative rankings of various bundles of U goods. The utility function is the relationship between utility and every possible bundle of goods. utility function U(x) assigns a numerical value to the bundle x, which might consist of certain A numbers of pizzas and burritos. One commonly used utility function is called thecobbdouglas utility function: whereUis the number of utils that the consumer receivesfrom consuming and q1 and q2. uppose that Lisa’s utility function is a Cobb-Douglas with a = 0.55. Then the amount of utility S that she gets from consuming pizzas and burritos is sing this function, we can determine whether she would be happier if she had Bundlexwith 16 U pizzas and 9 burritos or BundleYwith 13 of each. ● The utility she gets fromxis U(x) = 16 × 9= 12. The utility she gets from Y is U(x) = 13 × 13= 13. ● ● She prefers Y to X. he utility function is an economic concept that helps to think about consumer behaviour and T does not exist in real life. Ordinal Preferences nordinal measureis one that tells us the relativeranking of two things but does not tell us by A how much more one is valued than the other. Letter grades (A > B > C) rank performance but don’t showhow muchbetter A is than B. ultiple utility functions can represent the same preference ranking. For instance, if Bill prefers M Bundle X to Bundle Y, utility functions assigning 6 to X and 5 to Y, or 12 to X and 10 to Y, are both valid. The absolute utility numbers are not meaningful: only the relative ranking matters Willingness to Substitute Between Goods How willing a consumer is to trade one good for another depends on the slope of the consumer’s 𝑑𝑞 indifference curve,𝑑𝑞2 , at the consumer’sinitial bundle of goods. 1 conomists call the slope at a point on an indifference curve themarginal rate of substitution E (MRS). ● it is the maximum amount of one good that a consumer will sacrifice (trade) to obtain one more unit of another good isa’s MRS at Bundle e is the slope of the dashed line tangent to her indifference curve at e. This L slope is negative, indicating she is willing to give up some pizza for more burritos and vice versa. The relationship between the Marginal Rate of Substitution and Marginal Rate of Utility. e show that theMRSdepends on how much extra utilityLisa gets from a little more of each W good. ● The extra utility that a consumer gets from consuming the last unit of a good is the marginal utility. Given that Lisa’s utility function is𝑈(𝑞1 𝑞 ), the marginal utility she gets from a little more 2 pizza, holding the quantity of burritos fixed, is ∂𝑈 𝑀𝑈1 = ∂𝑞 (only differentiate𝑞1 andtreat𝑞2 as a constant) 1 ∂𝑈 Similarly, the marginal utility from more burritos is ∂𝑞 , where we hold the amount of pizza 2 constant. The slope of her indifference curve is the negative of the ratio of her marginal utilities. 𝑀𝑅𝑆 = − ∂𝑈 ∂𝑞1 ∂𝑈 ∂𝑞2 Diminishing Marginal Rate of Substitution he marginal rate of substitution (MRS)varies alonga convex indifference curve, becoming T smaller in absolute value as you move down and to the right. his reflects a diminishing MRS: Lisa will give up fewer burritos for one pizza as she moves T along the curve.The MRS approaches zero, becoming flatter as you move from Bundle f to e and then to g. Curvature of Indifference Curves The marginal rate of substitution varies along our typical convex indifference curve. ow the marginal rate of substitution varies along an indifference curve depends on the H underlying utility function. he indifference curves corresponding to these utility functions range between straight lines, T where theMRSis a constant, to right-angle indifferencecurves, where no substitution is possible. Convex indifference curves lie between these extremes. traight-Line Indifference Curve is one extreme casewhich occurs when two goods are S perfect substitutes: goods that a consumer is completely indifferent as to which to consume. ● Ex. Ben views Coca-Cola and Pepsi-Cola as perfect substitutes because he cannot taste any difference between them. ● The indifference curve for are parallel lines with slope of -1 and the MRS is also -1 at every point alone the indifference curves he slope of indifference curves for perfect substitutes can be any constant rate, not necessarily T just -1. ight angle indifference curve occurs when two goodsare perfect complements: goods that a R consumer is interested in consuming only in fixed proportions. x. Maureen doesn’t like apple pie,q1, by itself orvanilla ice cream, q2, by itself but she E loves apple pie with a scoop of vanilla ice cream on top. = 𝑚𝑖𝑛( 𝑖𝑞1, 𝑗𝑞2 ) where𝑖 = 𝑗 = 1and the min function says Her utility function is𝑈(𝑞1 , 𝑞2) that the utility equals the smaller of the two arguments𝑖𝑞1 or𝑖𝑞2. Her indifference curve is shown in panel b of figure 3.5 . ● i and 𝑗 are numbers that represent the fixed proportion in which the goods must be consumed. he marginal utility is zero for each good, because increasing that good while holding the other T one constant does not increase Maureen’s utility. ● We arbitrarily say that because she is unwilling to substitute more of one good for less of another. Convex Indifference Curvesshow that a consumer viewstwo goods as imperfect substitutes. Cobb-Douglas utility function results in convex indifference curves that approach but do not A touch the axes. 3.3 Budget Constraint or simplicity, we assume that each consumer has a fixed amount of money to spend now, so we F can use the terms budget and income interchangeably. Assume consumers spend their money on only two goods. If Lisa spends all her budget, Y, on pizza and burritos, then: Where p1q 1 is the amount spent on pizza and p2q 2 isthe amount spent on burritos. he equation above is herbudget line, orbudget constraint:the bundles of goods that can be T bought if a consumer’s entire budget is spent on those goods at given prices. n opportunity set consists of all the bundles a consumer can buy, including all the bundles A inside the budget constraint and on the budget constraint ● Exlisa could buy 10 burritos and 15 pizzas for $35is within the budget. he slope of the budget line is called themarginalrate of transformation(MRT). It is the T trade-off the market imposes on the consumer in terms of the amount of one good the consumer must give up to obtain more of the other good. ● The marginal rate oftransformationis the rate atwhich Lisa is able to trade burritos for pizzas in the marketplace when the prices she pays and her income are fixed. ● The marginal rate ofsubstitutionis the trade-offLisa wouldwantto make regardless of her income. he equation for the marginal rate of transformation (MRT) : the tradeoff rate between two T goods is as follows 𝑝1 𝑀𝑅𝑇 =− 𝑝 2 3.4 Constrained Consumer Choice Finding an Interior Solution Using Graphs o complete our analysis of consumer behavior, we have to determine the bundle of goods that T maximizes an individual’s well-being subject to the person’s budget constraint. ecause Lisa enjoys consuming two goods only, she spends her entire budget on them. That is, B she chooses a bundle on the budget constraint rather than inside her opportunity set, where she would have money left over after buying the two goods. ● Aninterior solutionoccurs when the optimal bundleincludes positive quantities of both goods, lying between the ends of the budget line. (mix of both goods) ● Acorner solutionoccurs when the consumer buys onlyone of the goods, placing the optimal bundle at one end of the budget line where it forms a corner with one of the axes. (all of one good, none of the other). undles that lie on indifference curves above the constraint, such as those on I3, are not in her B opportunity set (area A+B). Lisa prefers Bundlefon indifference curve to Bundleeon , she cannot afford to purchasef. ven though Lisa could buy a bundle inside the budget lineL, she does not want to, because E more is better than less. hus, the optimal bundle—theconsumer’s optimum—mustlie on the budget line and be on an T indifference curve that does not cross it. 2 ● Bundleeon indifference curve𝐼 is the optimumbundle or the indifference curve to touch the budget constraint but not cross it, it must betangentto the F budget constraint. The slope of the indifference curve is the marginal rate of substitution. The slope of the budget line is the marginal rate of transformation. And can be rearranged such that Thus, to find the interior solution, we can use either of two equivalent conditions: 1. Highest indifference curve rule: The optimal bundle is on the highest indifference curve that touches the constraint. 2. Tangency Rule: The optimal bundle is tangent to the budget line. Equivalently, MRS = 𝑈 𝑈 1 2 MRT and 𝑝1 = 𝑝2 he tangency rule only applies for interior solutions where the indifference curve has the usual T shape: It is a downward sloping, smooth curve that is convex to the origin. The highest indifference curve rule can always be used to find either interior or corner solutions. Finding Interior Solution Using Calculus e have just shown how to use a graphical approach to determine which affordable bundle gives W a consumer the highest possible level of utility. We now express this choice problem mathematically, and use calculus to find the optimal bundle. Lisa’s objective is to maximize her utility, U(q1,q2), subject to (s.t.) her budget constraint: I f we know that Lisa buys both goods, we can use the substitution method, the Lagrangian method, or a short-cut method. Budget constraint is given by𝑌 = 𝑝1 𝑞1 + 𝑝2 𝑞 , where Y is her income p1 and p2 are the prices 2 of goods ans and q1 and q2 are the quantities of goods1 and 2 Langrangian Method Another way to solve this constrained maximization is to use the Lagrangian method. The expression that corresponds to is : Whereλ (lamda) is the lagrange multiplier. he first-order conditions that determine the critical values q1, q2and λ for an interior T maximization: Equation (1): U1= λp1 Equation (2): U2=λp2 hese say:marginal utility of each good = lambda× its price. In other words, utility per dollar T spent is the same across goods. If we solve the first two equations above forλ,then equate these expressions, we find that eaningλ is the utility of income = the extra utilitythe consumer gets from having one more M dollar to spend. To solve using the langrain equation 1 . Write the lagrangian L 2. Satisfy the first three first order conditions 3. Equate the first two equations and solve for λ 4. Use the budget to solve for q1 and q2 Finding an Interior Solution Using a Short Cut o matter which method you use (graphical, substitution, or Lagrangian), the interior solution N always comes from two conditions: 1. Tangent Condition (MRS = MRT) [consumer’s trade-off = market trade-off] 𝑈 𝑃 2 2 𝑀𝑅𝑆 = − 𝑈1 = − 𝑃1 2. Budget Constraint 𝑌 = 𝑝1 𝑞1 + 𝑝2 𝑞 2 Finding corner solutions I ndifference curves have a property if the consumer has a constant elasticity of substitution (CES) utility function. ρ ρ ρ ρ 1 𝑝 𝑈(𝑞1 + 𝑞2 ) = (𝑞1 + 𝑞2 ) ,where𝑝 < 1. ρ ρ However, P =1, then the function is𝑈(𝑞1 , 𝑞2) = 𝑞1 + 𝑞2, which is a perfect substitute utility function. I t is a type of utility function used to describe preferences when goods are substitutes but not necessarily perfectly substitutes. Perfect Substitutes Utility Function ecause a perfect substitutes utility function has straight-line indifference curves that hit the B axes, the optimal bundle may be at a corner or in the interior of the budget line. To illustrate why, we consider Ben’s choice. Ben views Coca-Cola, q1, and Pepsi-Cola,q2 , as perfect substitutes, so his utility function is𝑈(𝑞1 , 𝑞2) = 𝑞1 + 𝑞2 I f the price of p1 < p2 , ben gets more extra utilityfrom the last dollar spent on a can of coke than pepsi. Since they are perfect substitutes he spends his entire income on coke𝑞1 = 𝑌/𝑝1 and buys no pepsi q2 = 0. he figure above illustrates Ben’s decision. Because Coke is less expensive than Pepsi, his T budget line,L, is flatter than his indifference curves,I1 , I2, and I3. e can afford to buy Bundleaon indifference curveor Bundlebon indifference curve , but he H cannot afford any bundle on because it is above his budget constraint everywhere. e can find Ben’s optimal bundle using the highest indifference curve rule: Ben’s optimal I2, that W touches the budget constraint. ● At bundle b, the slope of the indifference curve, MRS = -1, does not equal the slope of the budget line, MRT = -p1/p2 > -1 . Thus we cannotding the optimal bundle at a corner using the tangency rule. f the price of p1 > p2 ,then𝑞2 = 𝑌/𝑝2 and buys no coke q1 = 0. I f p1= p2 then both his budget line and his indifferencecurves have the same slope, -1 , so one of his indifference curves lies on top of the budget line. He is willing to buy any bundle on that budget line, in the interior or at either corner.𝑞1 + 𝑞1 = 𝑌/𝑝 Minimizing Expenditure arlier, we showed how Lisa chooses quantities of goods to maximize her utility subject to a E budget constraint. In a related or dual constrained minimization problem, Lisa wants to find that combination of goods that achieves a particular level of utility for the least expenditure. he rule for minimizing expenditure while achieving a given level of utility is to choose the T lowest expenditure such that the budget line touches—is tangent to—the relevant indifference curve. Lisa’s objective is to minimize her expenditure,e,subject to the constraint that she hold her utility constant atŪ = 𝑈( 𝑞1 , 𝑞2): he solution of this problem is an expression of the minimum expenditure as a function of the T prices and the specified utility level: his is the expenditure function: the relationship showing the minimal expenditures necessary to T achieve a specific utility level for a given set of prices.
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