Econ100A Spring 1999Page 1 of 6Suggested Answers to Problem Set #21) Give as many reasons as you can as to why we believe that indifference curves are convex.a) Diminishing Marginal Rate of Substitution: If someonebegins with a bundle composed of a large quantity of X2and a small quantity of X1 (like point “a” in the graph tothe right), she is willing to give up a great amount of X2just to get one unit of good X1 and still be as happy as shewas before (i.e. still be on the same indifference curve).For this person, good X1 was relatively rare and she waswilling to give up a lot to consume more of it. The nexttime she goes to trade X2 for X1, however, she will notwant to give up as much X2 as before because good X1,while perhaps still rare, is less rare than before. In otherwords, this person’s marginal rate of substitution isdiminishing.She could continue trading like this, remaining on the same indifference curve, until she reaches a bundle whereshe has a lot of X1 and very little X2 (like point “c”). At that point she will be willing to give up only a very littleof X2 in order to get one more unit of X1.(NB: The discussion speaks of 1 unit of X1. In fact, we really want to think about infinitesimal increments of X1.)b) Indifference curves which are concave to theorigin result in optimal bundles which consist ofone only good (“corner solutions”). This is notusually observed. The graphs to the right showindifference curves (thin lines) convex to theorigin. Note the third case where bundlescomposed of only X1 or only X2 yield the samelevel of utility.c) If an indifference curve had both concave and convex regions, an interior solution would always occur in theconvex region. Hence, the local convex region of the indifference curve is the part we are interested in.2) What happens to a consumer’s equilibrium if all prices and income double? [Hint: What happens to the intercepts of the budget line?]Nothing happens to the consumer’s equilibrium because the budget constraint is unchanged when all prices andincome double. The new budget is no different from theoriginal budget. Original Budget: P1X1 + P2X2 = YNew Budget: (2 P1)X1 + (2P2)X2 = 2YP1X1 + P2X2 = Y Another way to see this is to note that the budget constraintintercepts have not changed: Y/Pi = 2Y/2Pi.(NB. We have focused on the budget constraint because preferences are not affected by a change in prices or income.)If the budget is unchanged, then the consumer’s utility maximization problem has not changed, so the outcome mustbe the same.X1(Quantity/Time) X2(Quantity/Time)1 Unit 1 UnitabcdI0X1X2X1X2X1X2Budget Constraint is both: P1X1 + P2X2 = Y (2P1)X1 + (2P2)X2 = 2YOptimal Bundle isunchangedX1(Quantity/Time) X2(Quantity/Time)Y/P1Y/P2Econ100A Spring 1999Page 2 of 63) In Larry’s state, a sales tax of 10% is applied to clothing but not to food. Show the effect of this tax on Larry’s choice betweenfood and clothing using indifference curves and budget lines.The price of clothing, Pc, has increased due to the ad valorem tax from Pc to (1 + 0.1)Pc or 1.1Pc, causing Larry’sbudget constraint to pivot inwards around the food axis intercept, as shown in the first graph below.Notice that his opportunity set has decreased, so Larry cannot be any better off.In the second graph we examine the changes in Larry’s choice between food and clothing. Under original prices heconsumes bundle “a.” The tax pivots the budget constraint inwards, and we can break the change into two stages.a) First he substitutes away from clothing and towards food because clothing is more expensive relative to food thanbefore. This is called the substitution effect and is shown in the graph by the movement from bundle “a” tobundle “b” – the optimal bundle under new prices which yields the same utility as the original bundle.b) Second there is an income effect due to the fact that Larry has less money after the tax. In the graph below theincome effect is shown by the movement from bundle “b” to bundle “c” – the optimal bundle under new pricesafter a parallel shift in of the budget constraint to reflect lower income.Notice that Larry is now on a lower indifference curve (I1 is lower than I0). He has been made worse off by the tax. Clothing(Quantity/Time)Y/PcY/1.1PcOriginal Budget ConstraintAfter-Tax Budget Constraint Food(Quantity/Time)Y/PcY/1.1PcY/PFEffect on Larry’s Budget Constraint Effect on Larry’s Optimal Consumption BundleI0I1abca to b: Substitution Effectb to c: Income EffectClothing(Quantity/Time) Food(Quantity/Time)Y/PFNB: the graph presents both food and clothing as normal goods. (You could show one of the two as inferior if you wished.) The intercept values also assumes that Larry bears the full incidence of the tax, while the direction of pivot only assumes he bears at least some of the tax. If Larry didn’t bear any of the tax this would be an uninteresting question.4) A consumer’s utility function is U = 10X2Y. [Thus, MUX = 20XY, MUY=10X2.] The price of X is PX = $10, the price of Y is PY =$5, and his income is Y = $150. What is his optimal consumption bundle? Show in a graph. 2510 51021020 102022−=−=⇒==−=−=⇒==XYYXXYYXMRTPPXYXXYMRSXMUXYMUa) At the Utility maximizing point MRS = MRT, hence: ****or 2X2Y XYMRTMRS ==⇒=b) Using the budget constraint, substituting in our Utilitymaximization result, prices, and income, we can solve for the optimum amount of each good: ********, 10 10150,5,10 , , 510XYYXIncomePPXYIncomeXXIncomeYPXPYXYX========+=+Graphically, this looks like:X (Quantity/Time) Y(Quantity/Time)3015I01010MRT = -2MRS = -2Y/XEcon100A Spring 1999Page 3 of 65) Derive the demand curve for Coke for a person who views Coke and Pepsi as perfect substitutes.The utility function for perfect substitutes is: U(C,P) = aC + bP , where C is quantity of coke per time period, Pis quantity of Pepsi per time period, and “a” and “b” are positive constants. If we assume a = b = 1, this reduces toU(C,P) = C + P.Then MRSCP = -MUC/MUP = -1/1 and the MRTCP = -PC/PP. Hence, MRSCP will only ever equal MRTCP if the MRTCPis equal to -1. Which is to say, we will only have an interior solution if MRTCP, by pure chance, is equal to a-1. Forall other price ratio values, we will have a corner solution, i.e. either spend all income on coke or spend no income oncoke. The demand
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