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by: Sheila O'Hara IV


Sheila O'Hara IV
GPA 3.9


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This 12 page Class Notes was uploaded by Sheila O'Hara IV on Thursday October 29, 2015. The Class Notes belongs to ECON 102A at University of California Riverside taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/231761/econ-102a-university-of-california-riverside in Economcs at University of California Riverside.




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Date Created: 10/29/15
Lecture note 4 Choice Optimal choice The best affordable bundle is located on the highest indifference curve that just touches the budget line Consider X2 Indifference curves Optimal choice x2 Xi the following gureFiglue 51 Optimal choice The choice 301 3 is an optimal choice for the consumer ie the best bundle that the consumer can afford Note at 301 3 indifference curve is tangent to the budget line Tangency condition doesn t hold in all cases What is always true is that at the optimal point the indifference curve can t cross the budget line X2 Indifference curves X2 Budget line Xi Let s consider some exceptionsFigure 52 Kinky tastes Here a tangent just is not de ned But this does not change the de nition of optimal bundle ie the best affordable bundle is located on the highest indifference curve that just touches the budget line Another possible case is often referred to as corner or boundary solution to the consumer s optimization X2 Indifference curves Budget line Xfquot X1 problemFigure 53 Boundary optimum Note that in this case the slope of the indifference curve and the slope of the budget line are different but the indifference curve still doesn t cross the budget line We will restrict ourselves to smooth indifference curves which would rule out first example Besides in most of the cases optimal choice would imply positive quantities of both goods Such cases are known as interior solutions Therefore if we have an interior optimum with smooth ie differentiable indifference curves the slope of the indifference curve and the slope of the budget line must be the same Otherwise the indifference curve would cross the budget line and we couldn t be at the optimal point Important Tangency condition is only a necessary condition for the optimal choice not a suf cient To see that it is not suf cient consider the following figure X2 Indifference curves Optimal bundles Nonoptimal bundle Budget line Xi Figure 54 More than one tangency Here three points satisfy tangency condition but only two of them are optimal Note that the middle choice is nonoptimal because the indifference curve crosses the budget line Implications of convexity assumption 0 In the case of convex preferences any point that satisfies the tangency condition must be an optimal point since convex indifference curves must curve away from the budget line they cannot bend back to touch it again 0 Convexity assumption does not guarantee uniqueness of the optimal choice if indifference curves have at spots there might be several points that satisfy optimality conditions 0 Strict convexity assumption guarantees that there Will be only one optimal choice on each budget line The condition that the MRS must be equal the slope of the budget line is obvious graphically Let s consider economic intuition behind this nding Recall that the slope of the budget line is In other words this is the rate of exchange that the market is offering to the consumer if you give up one unit of good 1 you can buy 2 units of good 2 Suppose that MRS 7E g Let MRS and let 2 1 MRS condition implies that the consumer is Willing to exchange 2 units of good 1 for 1 unit of good 2 But the market offers 1 unit of good 1 per one unit of good 2 Hence the consumer can be better off by trading Consumer demand Demand bundle is the optimal choice of goods 1 and 2 at a given set of prices and income Obviously as prices and income change the consumer s optimal choice also changes Demand function is the function that relates the optimal choice the quantities demanded to the different values of prices and incomes Notation 0 Demand function for good 1 is 761 191192771 0 Demand function for good 2 is 762 191192771 Special cases Perfect substitutes In case of perfect substitutes we have three possible cases Case 1 MRS is flatter than the budget line 7 consumer spends all his or her income on good 2 boundary solution 1 Case 2 MRS is steeper than the budget line 7 consumer spends all his or her income on good 1 boundary solution 2 Case 3 MRS is equal to the slope of the budget line i there is a number of optimal choices ie any amount of goods 1 and 2 that satis es the budget constraint is optimal Consider the following gure Where MRS1 X2 Indifference curves Slope 1 Optimal choice 39 gt14 X1 mp1 X1 Figure 55 Optimal choice With perfect substitutes So the demand function for good 1 is given by b 17 if p1 lt p2 because MRS is steeper than the slope of the budget line gt 017 if p1 p2 because MRS is equal to the slope of the budget line b 0 if p1 gt P2 because MRS is atter than the slope of the budget line Note the these results are consistent with common sense if two goods are perfect substitutes then a consumer will purchase the cheaper one Perfect complements X2 Indifference curves Optimal choice Budget line Consider the case Of perfect complements Figure 56 Optimal choice With perfect complements In terms of our shoes example consumer would always buy shoes in pairs ie where am am It is easy to derive demand function for both goods We know that consumer always purchase the same amount of good 1 and good 2 no matter what the prices Then we can write x1 x2 at Budget constraint p1xp2xm 75191 192 m m 7 P1 P2 its just if the consumer were spending all of her money on a single good that has a price of p p1 p2 Concave preferences Indifference curves Nonoptimai choice Optimal choice Z X1 Consider the case of concave preferencesFigure 58 Optimal choice with concave preferences Important X on the gure above is not an optimal choice Why Because there exists a higher indifference curve 7 the one that de nes optimal Choice Z Note that for strictly concave preferences the optimal Choice is always going to be a boundary Choice ie when consumer spends all his income on one of the goods Recall that concave preferences mean that the consumer does not want to consume both of the goods together In this case the consumer would spend all income on one or the other Demand Recall Demand function gives the optimal amounts of each of the goods as a function of the prices and income faced by the consumer 961 961001710277 2 2 P171027 7 where the left hand side of each equation stands for the quantity demanded while the right hand side of each equation is the function that relates the prices and income to that quantity We are interested in how the demand for a good Changes as prices and income Change Comparative statics describes how a Choice responds to Changes in the economic environment In case of the consumer there are only two things that affect the optimal Choice 1 prices and 2 income Normal and inferior goods Here we will study how a consumerls demand for a good Changes as his income Changes So we will hold prices constant and examine only the Change in demand due to the income Change Recall that an increase in income shifts the budget line outward in a parallel fashion Normal good demand for it increases when income increases and demand decreases when income de creases ie for a normal good the quantity demanded always Changes in the same way as income Changes A1 0 Amgt The Change in demand as income Changes for normal goods is described on the following gure X2 Indifference curves Optimal Choices Budget lines X1 Figure 61 Normal goods Inferior good an increase of income results in a reduction in the consumption ie quantity demanded decreases as income increases X 2 Indifference curves X1 Figure 62 An inferior good Important whether a good is inferior or not depends on the income level that we are examining it might be the case that very poor people consume more of low quality goods as their income increases However after a point the consumption of these low quality goods would probably decline as income continue to increase Income offer curves and engel curves Income offer curve can be obtained by connecting together the demanded bundles that we get at different levels of income The income offer curves are also known as the income expansion path Engel curve is a graph of the demand for one of the goods as a function of income with all prices being held constant Income Engel curve Indifference curves X1 X1 A Income offer curve B Engel curve Figure 63 How demand changes as income changes Examples Perfect substitutes Consider the case where MRS is steeper than the slope of the budget line 7 In this case7 the consumer spends all of his income on good 1 When prices are xed increase in income will result in the increase of consumption of good 1 only Therefore7 income offer curve is just the horizontal axis From aloove7 we know that demand for good 1 in this case is m 11 i gt m 1131 P1 X2 m Indifference curves TypicaI budget Iine Income offer curve Slope x 1 A Income offercurve B Eng then the Engel curve is just a straight line with a slope ofp1Figure 6394 PerfeCt subStitUteS Perfect complements Since the consumer will always consume the same amount of each good7 the income offer curve is the diagonal here we assume proportion 1 to 1 line through the origin and the Engel curve is a straight line with a slope of X2 m Indifference curves Income offer Engel curve SIope p1 p2 Budget lines X1 X1 A Income offer curve B Engel curve M p2Fig ure 65 Perfect complements To see that the slope of the Engel curve is p1 192 recall the demand for perfect complements a Which can also be written as m p1 192 3 CobbDouglas preferences If u 331 32 m f m o the Cobb Douglas demand for good 1 has the form am P1 331 Which is linear function of m if p1 is xed The same holds for good 2 Since the demand functions for both goods are linear functions of income the income expansion paths are just X2 m Income Indifference curves Budget mes Slope p1a x1 X1 A Income offer curve B EngeI curve straight lines through the originFig re 66 CObb DouglaS Homothetic preferences Real Engel curves do not have to be straight lines 0 Luxury good the demand for it goes up by a greater proportion than income o Necessary good the demand for it goes up by a lesser proportion than income Homothetic preferences if the consumer prefers mag to 241ny then the consumer prefers t lyt g to tylytyg for any positive value oft x2 m Indifference curves EngeI curve Budget Ines Income offer curve X1 X1 A Income offer curve B EngeI curve Figure 67 Homothetic preferences Examples Perfect substitutes u 17 2 1 x2 Consider u374 3 4 7 and u173 1 3 4 Then7 374 gt 173 This is also true for 2537254 versus 25172537 ie if t 10 30740 gt 10730 or if t 01 03704 gt 01703 Perfect complements7 ux172 min 17M Consider u775 min 77 5 5 and u479 min 479 4 Clearly7 77 5 gt 479 Let t 27 then 2577255 gt 2547259 gt 14710 gt 8718 CobbDouglas7 ux172 xlxg Consider u173 13 3 versus u37 2 3 2 6 Obviously7 37 2 gt 173 Let t 37 then 2537252 gt 2517253 976 gt 379 Therefore7 perfect substitutes7 perfect complements7 and Cobb Douglas are all homothetic preferences If consumer has homothetic preferences then the income offer curves are all straight likes through the origin Homothetic preferences imply that When income is scaled up or down by any amount if gt 07 the demanded bundle scales up or down by the same amount This implies that the Engel curves are straight lines as well multiplying the income by 2 increases the demand for each good by 2 Ordinary goods and giffen goods Suppose that we decrease the price of good 1 and hold the price of good 2 and the money income xed Then What can happen to the quantity demanded of good 1 Ordinary good X2 Indifference curves Xi Figure 69 An ordinary good The demand for an ordinary good increases when its price decreases In the gure the optimal choice of good 1 moves to the right Giffen good X2 Indifference curves Optimal choices Price decrease 4 Reduction X1 in demand for good 1 Figure 610 A Giffen good The reduction in the price of good 1 has freed up some extra money to be spent on other things Sometimes this might lead to a reduction of consumption of good 1 Even though money income remains constant a change in the price of a good will change purchasing power and thereby change demand However most goods are ordinary goods when their price increases the demand for them declines The price o er curve and the demand curve Suppose that we let the price of good 1 change while we hold p2 and income xed By connecting together the optimal points we can construct the price offer curve 10 X2 P1 Indifference 50 curves Price offer 40 curve 30 20 10 X1 12 X1 A Price offer curve B Demand curve Figure 611 The price offer curve and demand curve If we draw the optimal choices of good 1 against its price we can draw demand curve Demand curve is a plot of the demand function 331 191192771 holding p2 and in xed at some values When the price of a good increases the demand for that good Will decrease for ordinal goods A i lt 0 A191 which says that demand curves usually have a negative slope Notc in the case of Giffen goods the demand for a good may decrease when its price decreases Thus it is possible but not likely to have a demand curve with a positive slope Examples Perfect substitutes X2 P1 Indifference curves Demand curve Price offer 1 curve P1 P2 l I I I 1 x1 mp1 mpr x1 A Price offer curve B Demand curve Figure 612 Perfect substitutes Recall the demand for good 1 when goods are substitutes 1 to1 is m gt E if 131 lt pg 1131 gt 071 if P1 P2 gt 0 if 131 gt 132 Perfect complements We know that whatever the prices are7 a consumer will demand the same amount of goods 1 and 2 Thus the offer curve will be a diagonal line X p 2 Indifference Price 1 curves offer Demand curve X1 A Price offer curve B Demand curve Figure 613 Perfect complements Recall the demand for good 1 is the same as for good 2 m 1 7 P1 P2 if you x p2 and m at some values and vary only 131 you will get demand curve on the gure above Substitutes and complements Before we have seen peTfect substitutes and peTfect complements What if the goods are imperfect substitutes or complements Substitutes Consider pencils and pens These goods are7 to some degree7 a substitute for each other Although they are not perfect substitutesl Complements Consider a pair of shoes and a pair of socks While right and left shoes are always consumed together7 shoes and socks are usually consumed together The de nition of substitutes and complements depends on how the demand for good 1 changes as the price of good 2 changes o If the demand for good 1 goes up when the price of good 2 goes up7 then we say that good 1 is a substitute for good 2 Formally7 good 1 is a substitute for good 2 if A gt 0 AP2 ie7 when good 2 gets more expensive the consumer switches to consuming good 1 the consumer substitutes away from the more expensive good to the less expensive good 12


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