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PS1+Solutions - Choice Sets and Budget Constraints 32 2.17...

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Unformatted text preview: Choice Sets and Budget Constraints 32 2.17 PolicyApplication: Tax Deductions and Tax Credits: In the U.S. income tax code, a number of ex- penditures are “deductible’i For most tax payers, the largest tax deduction comes from the portion of the income tax code that permits taxpayers to deduct home mortgage interest (on both a primary and a va- cation home). This means that taxpayers who use this deduction do not have to pay income tax on the portion of their income that is spent on paying interest on their home mortgage(s). For purposes of this exercise, assume that the entire yearly price of housing is interest expense. A: True or False: For someone whose marginal tax rate is 33%, this means that the government is subsidizing roughly one third of his interest/house payments. Answer: Consider someone who pays $10,000 per year in mortgage interest. When this person deducts $10,000, it means that he does not have to pay the 33% income tax on that amount. In other words, by deducting $10,000 in mortgage interest, the person reduces his tax obligation by $3,333.33. Thus, the government is returning 33 cents for every dollar in interest payments made — effectively causing the opportunity cost of paying $1 in home mortgage interest to be equal to 66.67 cents. So the statement is true. fi'sfianc 120,00 (a) Consider a household with an income of $200,000 who faces a tax rate of 40%, and suppose the price of a square foot of housing is $50 per year: With square fi)otage of housing on the horizontal axis and other consumption on the vertical, illustrate this household’s budget con- straint with and without tax deductibility (Assume in this and the remaining parts of the question that the tax rate cited fi2r a household applies to all of that household’s income.) Answer: As just demonstrated, the tax deductibility of home mortgage interest lowers the price of owner-occupied housing, and it does so in proportion to the size of the marginal income tax rate one faces. Panel (a) of Graph 2.16 illustrates this graphically for the case described in this part. With a 40 percent tax rate, the household could consume as much as 0.6(200,000)=120,000 in other goods if it consumed no housing. With a price of hous- ing of $50 per square foot, the price falls to (1 — 0.4)50 = 30 under tax deductibility. Thus, the budget rotates out to the solid budget in panel (a) of the graph. Without deductibil- ity, the consumer pays $50 per square foot — which makes 120,000/50=2,400 the biggest possible house she can afford. But with deductibility, the biggest house she can afford is 120,000/30=4,000 square feet. (ms. 033 7 454x clechch'm f tax «@061 \ ’\ i 260 ‘0. ’1‘) + " m 30 m“ J 1700 mm Graph 2.16: Tax Deductions versus Tax Credits (b) Repeat this for a household with income of$50,000 who faces a tax rate of10%. Answer: This is illustrated in panel (b). The household could consume as much as $45,000 in other consumption after paying taxes, and the deductib ility of house payments reduces the price of housing from $50 per square foot to (1 — 0.1)50 = $45 per square foot. This results in 33 Choice Sets and Budget Constraints the indicated rotation of the budget from the lower to the higher solid line in the graph. The rotation is smaller in magnitude because the impact of deductib ility on the after-tax price of housing is smaller. Without deductibility, the biggest affordable house is 45,000/50z900 square feet, while with deductibility the biggest possible house is 45,000/45=1,000 square feet. (c) An alternative way for the government to encourage home ownership would be to offer a tax credit instead of a tax deduction. A tax credit would allow all taxpayers to subtractafi'action k of their annual mortgage payments directly from the tax bill they would otherwise owe. (Note: Be careful — a tax credit is deducted from tax payments that are due, not from the taxable income.) For the households in (a) and (b), illustrate how this alters their budget if k = 0.25. Answer: This is illustrated in the two panels of Graph 2.16 — in panel (a) for the higher in- come household, and in panel (b) for the lower income household. By subsidizing housing through a credit rather than a deduction, the government has reduced the price of hous- ing by the same amount (k) for everyone. In the case of deductibility, the government had made the price subsidy dependent on one’s tax rate — with those facing higher tax rates also getting a higher subsidy. The price of housing how falls from $50 to (1 — 0.25)50 = $37.50 — which makes the largest affordable house for the wealthier household 120,000/37.5=3,200 square feet and, for the poorer household, 45,000/37.5=1,200 square feet. Thus, the poorer household benefits more from the credit when k = 0.25 while the richer household benefits more from the deduction. ((1) Assuming that a tax deductibility program costs the same in lost tax revenues as a tax credit program, who would favor which program? Answer: People facing higher marginal tax rates would favor the deductib ility program while people facing lower marginal tax rates would favor the tax credit. B: Letx1 and x2 representsquare feet of housing and other consumption, and let the price of a square foot of housing be denoted p. (a) Suppose a household faces a tax rate t fi2r all income, and suppose the entire annual house payment a household makes is deductible. What is the household’s budget constraint? Answer: The budget constraint would be :62 = (1 — t)I — (1 — t) px1. (b) Now write down the budget constraint under a tax creditas described above. Answer: The budget constraint would nowbe x2 = (1 — t)I — (1 — k) pxl. Choice Sets in Labor and Financial Markets 66 3.19 Policy Application: The Earned Income Tax Credit: During the Clinton Administration, the EITC— orEaereTaxCredit, was expanded considerably The program provides a wage subsidy to low in- come families through the tax code in a way similar to this example: Suppose, as in the previous exercise, that you can earn $5 per hour: Under the EITC, the governmentsupplement’s your first$20 of daily eam— ings by 100% and the next $15 in daily earnings by 50%. For any daily income above $35, the government imposes a 20% tax. A: Suppose you have at most 8 hours of leisure time per day. (a) Illustrate your budget constraint (with daily leisure on the horizontal and daily consumption on the vertical axis) under this EI TC. Answer: The budget constraint is graphed in Graph 3.13. For the first 4 hours of labor, the take-home wage is $10 per hour because of the 100% subsidy. For the next 3 hours of labor, the take-home wage is $7.50 because of the 50% subsidy. Finally, for any work beyond 7 hours, the take home wage is $4 because of the 20% tax. Graph 3.13: EITC Budget Constraint (b) Suppose the government ends up paying a total of $25 per day to a particular worker under this program and collects no tax revenue. Identify the point on the budget constraint this worker has chosen. How much is he working per day? Answer: The worker would work for 6 hours. At a wage of $5, this would mean making $30 per day. But, for the first $20, the government adds $20, and for the next $10 hours, the government adds $5 — for a total EITC supplement of $25. Thus, the worker will have $55 in income for other consumption. This gives us A in the graph — leisure of 2 hours per day (because of 6 hours of work) and consumption of $55 per day. (c) Return to your graph of the same worker’s budget constraint under the AFDC program in exercise 3.18. Suppose that the government paid a total of $25 in daily AFDC benefits to this worker: How much is he working? Answer: The worker is working at most 1 hour. ((1) Discuss how the difference in trade-offs implicit in the E1 TC and AFDC programs could cause the same individual to make radically different choices in the labor market. Answer: Despite the government spending the same on the worker under AFDC and EITC, the worker might choose to not work much under AFDC and a lot under EITC. This is be- cause of the implicit large tax rate imposed on the worker under AFDC but not under EITC. B: More generally, consider an El TC program in which thefirstx dollars ofincome are subsidized at a rate2s; the next x dollars are subsidized at a rate s; and any earnings above 2x are taxed at a rate t. 85 Tastes and Indifference Curves 4.7 Everyday Application: Did 9/11 Change Tastes?: In another textbook, the argument is made that con- sumer tastes over “airline miles traveled” and “other goods” changed as a result of the tragic events of September 11, 2001. A: Below we will see how you might think of that argument as true or false depending on how you model tastes. (a) To see the reasoning behind the argument that tastes changed, draw a graph with “airline miles traveled” on the horizontal axis and “other goods” (denominated in dollars) on the ver- tical Draw one indifference curve from the map of indifference curves that representa typical consumer’s tastes (and that satisfli our usual assumptions.) Answer: This is illustrated in panel (a) of Graph 4.6 with the indifference curve labeled “pre- 9/ l l ”. (a) OHM! ConLflS Ollie! ConLJfl Graph 4.6: Tastes before and after 9/1 1 (b) Pick a bundle on the indifference curve on your graph and denote it A. Given the perception of increased risk, what do you think happened to the typical consumer’s MRS at this point after September 11, 2001 .9 Answer: The MRS tells us how much in “dollars of other goods” a consumer is willing to give up to travel one more mile by air. After 9/11, it would stand to reason that the typical consumer would give up fewer dollars for additional air travel than before. Thus, the slope of the indifference curve at A should become shallower — which implies that the MRS is falling in absolute value. (c) For a consumer who perceives a greater risk of air tran after September 11, 2001, what is likely to be the relationship of the indifference curves from the old indifference map to the indifference curves from the new indifference map at every bundle? Answer: The reasoning from (b) holds not just at A but at all bundles. Thus, we would expect the new indifference map to have indifference curves with shallower slopes at every bundle. ((1) Within the context of the model we have developed so far; does this imply that the typical consumer’s tastes fi2r air-travel have changed? Answer: Rationality (as we have defined it) rules out the possibility for indifference curves to cross. Thus, within the context of this model, it certainly seems that tastes must have changed. (e) Now suppose that we thought more comprehensively about the tastes of our consumer: In particular; suppose we add a third good that consumers care about— “air safety’i Imagine a 3-dimensional graph, with “air miles travele "on the horizontal axis and “other goods” on the vertical (as before) — and with “air safety” on the third axis coming out at you. Suppose “air safety” can be expressed as a value between 0 and 100, with 0 meaningcertain death when Tastes and Indifference Curves 86 one steps on an airplane and 100 meaning no risk at all. Suppose that before 9/1 1, consumers thought that air safety stood at 90. On the slice of your 3-dimensional graph that holds air safety constant at 90, illustrate the pre-9/11 indifference curve that passes through (xf‘, xf), the level of air miles traveled (x?) and other goods consumed (x?) before 9/11. Answer: This is illustrated in panel (b) of Graph 4.6 as the indifference curve labeled “air safety = 90”. (f) Suppose the events of 9/11 cause air safety to fall to 80. Illustrate your post-9/11 indifference curve through (xf, x?) on the slice that holds air safety constant at 80 but draw that slice on top of the one you just drew in (e). Answer: This is also done in panel (b) of the graph. (g) Explain that, while you could argue that our tastes changed in our original model, in a bigger sense you could also argue that our tastes did not change after 9/11, only our circumstances did. Answer: When we explicitly include air safety as something we value as consumers, we get indifference surfaces that lie in 3 dimensions. But since we don’t get to choose the level of air safety, we effectively operate on a 2-dimensional slice of that 3-dimensional indifference surface — the slice that corresponds to the current level of air safety. That slice looks just like any ordinary indifference curve in a 2-good model even though it comes from a 3-good model. When 9/11 changes the perceptions of air safety, outside circumstances are shifting us to a different portion of our 3-dimensional indifference surface — with that slice once again giving rise to indifference curves that look like the ones we ordinarily graph in a 2- good model. But when viewed from this perspective, the fact that the indifference curve that corresponds to more air safety crosses the indifference curve that corresponds to less air safety merely arises because we are graphing two different slices of a 3-dimensional surface in the same 2-dimensional space. While both curves then contain the bundle (xf‘, xf), they occur at different levels of x3. The pre-9/11 indifference curve really goes through bundle (xf‘, xf, 90) while the post-9/ 1 1 indifference curve really goes through bundle (xf‘ , xf, 80) — and the two therefore do not cross. Thus, when viewed from this larger perspective, tastes have not changed, only circumstances have. B: Suppose an average traveler’s tastes can be described by the utility function u (x1 , x2 , x3) = x1x3 + x2, where x1 is miles traveled by air; x2 is “other consumption” and x3 is an index of air safety that ranges from 0 to 100. (a) Calculate the MRS of other goods fi2r airline miles — ie. the MRS that represents the slope of the indifference curves when x1 is on the horizontal and x2 is on the vertical axis. Answer: The MRS is 6u/6x1 __£ _ 6u/6x2 _ 1 _ (b) What happens to the MRS when air safety (xg ) falls from 90 to 80.? Answer: It changes from —90 to —80. MRS:— —x3. (4.10) (c) Is this consistent with your conclusions from partA? In the context of this model, have tastes changed? Answer: The change in the MRS as air safety falls is a decrease in absolute value — i.e. the slope of the indifference curve over x1 and x2 becomes shallower just as we concluded in part A. But we are representing tastes with exactly the same utility function as before — so tastes cannot have changed. ((1) Suppose that u(x1,x2,x3) = x1x2x3 instead. Does the MRS of other consumption for air miles traveled still change as air safety changes? Is this likely to be a good model of tastes fi2r analyzing what happened to consumer demand after 9/11 .9 Answer: The MRS now is 6u/6x1 =_x2x3 =_£ (411) MRS = — . au/axz x1x3 x1 87 Tastes and Indifference Curves Thus, the MRS for tastes represented by this utility function is unaffected by :63 — the level of air safety. This would imply that the two indifference curves in panel (b) of Graph 4.6 would lie on top of one another. If we think consumers felt differently about air travel after 9/1 1 than before, then this utility function would not be a good one to choose for analyzing changes in consumer behavior. (e) What ifu(x1,x2,x3) = xzxg +x1 .9 Answer: In this case, the MRS is au/axl = _i (412) MR = — . s 614/ 6x2 :63 This would imply that as :63 — air safety— falls, the MRS increases in absolute value; i.e. it would mean that a decrease in air safety would make us willing to spend more on additional air travel than what we were willing to spend before. It would thus result in a steeper rather than a shallower slope for indifference curves post-9/11. It seems unlikely that a typical consumer would respond in this way to changes in air safety. Different Types of Tastes 130 5.1 1 In this exercise, we are working with the concept of an elasticity of substitution. This concept was introduced in part B of the Chapter: Thus, this entire question relates to material from part B, but the A-part of the question can be done simply by knowing the formula for an elasticity of substitution while the B- part of the question requires further material from partB of the Chapten In Section 53.1, we defined the elasticity of substitution as %A (162 / x1) %AMRS . (5.21) A: Suppose you consume only apples and oranges. Last month, you consumed bundle A: (100,25) — 100 apples and 25 oranges, and you were willing to trade at most 4 apples for every orange. Tit/0 months ago, oranges were in season and you consumed B=(25,100) and were willing to trade at most4 oranges for 1 apple. Suppose your happiness was unchanged over the past two months. (a) On a graph with apples on the horizontal axis and oranges on the vertical, illustrate the in- difference curve on which you have been operating these past two months and label the MRS where you know it. Answer: This is illustrated in Graph 5.10. O/o no; «3 Ion App (ex 25 50 75 (oo Graph 5.10: Elasticity of Subsitution (b) Using the fi2rmula fi2r elasticity of substitution, estimate your elasticity of substitution of ap- ples for oranges. Answer: 1 his is (15/4)/4 (15/4)/4 a _ ((100/25) — (25/100))/(100/25) (—4— (—1/4))/(—4) = 1- (5.22) (c) Suppose we know that the elasticity of substitution is in fact the same at every bundle fi2r you and is equal to what you calculated in (b). Suppose the bundle C: (50,50) is another bundle that makes you justas happy as bundles A and B. What is the MRS at bundle C? Answer: Using B and C in the elasticity of substitution formula, setting 0 equal to 1 and letting the MRS at C be denoted by x , we get ((100/25) — (50/50))/(100/25) = 3/4 I: 1’ (5.23) (—4—x)/(—4) (4+x)/4 and solving this for x, we get x = —1 — i.e. the MRS at C is equal to —1. (d) Consider a bundle D = (25,25). If your tastes are homothetic, what is the MRS at bundle D .9 Answer: Since it, like bundle C, lies on the 45 degree line, homotheticity implies the MRS is again — 1. 131 Different Types of Tastes (e) Suppose you are consuming 50 apples, you are willing to trade 4 apples fi2r one orange and you are just as happy as you were when you consumed at bundle D. How many oranges are you consuming (assuming the same elasticity of substitution)? Answer: Let the number of oranges be denoted y. Using the bundle (50, y) and D = (25,25) in the elasticity formula and setting it to l, we get ((50/y) - (25/25))/(50/y) = ((50/y) — 1)/(50/y) = 1 (-4- (-1))/(-4) (3/4) ' Solving this for y, we get y = 12.5. (f) Call the bundle you derived in part (e) E. If the elasticity is as it was before, at what bundle would you be just as happy as at E but would be willing to trade 4 oranges for 1 apple? Answer: If the elasticity is 1 from D to E and is again supposed to be 1 from D to this new bundle, there must be symmetry around the 45 degree line (as there was between A and B). At E = (50,125), the MRS is —1/4, and the necessary symmetry then means that MRS = —4 at (12.5,50). (5.24) _ _ —1l B: Suppose your tastes can besummarized by the utility function u(x1, x2) = (uxl p + (1 — 00x2 ) p. (a) In order for these tastes to contain an indifference curve such as the one containing bundle A that you graphed in A( a), what must be the value of p .9 What about a .9 Answer: The elasticity of substitution for the CES utility function can be written as a = 1/ (1+ p). Above, we determined that the elasticity of substitution in this problem is 1. Thus, 1 = 1/ (1 + p) which implies p = 0. Since our graph is symmetric around the 45 degree line, it must furthermore be true that a = 0.5 — i.e. x1 and :62 enter symmetrically into the utility function. (b) Suppose you were told that the same tastes can be represented by u(x1,x2) = xll’xg . In light of your answer above, is this possible? If so, what has to be true about 7 and 6 given the symmetry of the indifference curves on the two sides of the 45 degree line? Answer: Yes — it is possible because we determined that the elasticity of substitution is 1 everywhere, which is true for Cobb-Douglas utility functions of the form u(x1,x2) = xll’xg . The symmetry implies y = 6. (c) What exact value(s) do the exponents y and 6 take if the label on the indifference curve con- taining bundle A is 50? What if that label is2,500? What if the label is 6,250,000.? Answer: If the utility at A is 50, it means 507505 = 50. Since we just concluded in (a) that y = 6, this implies that y = 6 = 0.5. If the utility is 2,500, then 7 = 6 = l, and if the utility is 6,250,000, 7 = 6 = 2. ((1) Verify that bundles A, B and C (as defined in partA) indeed lie on the same indifference curve when tastes are represented by the three different utility functions you implicitly derived in B(c). Which of these utility functions is homogeneous of degree 1 .9 Which is homogeneous of degree 2.? Is the third utility function also homogeneous? Answer: The bundles are A=(100,25), B=(25,100) and C=(50,50). The following equations hold, verifying that these must be on the same indifference curve for each of the three utility functions: u(x1,x2) = x‘l’5xg'5, v(x1,x2) = xlxz and w(x1,x2) = u(100,25) = u(25,100) = u(50,50) = 50 v(100, 25) = v(25,100) = v(50,50) = 2,500 (5.25) w(100, 25) = w(25,100) = w(50,50) = 6,250,000. The following illustrate the homogeneity properties of the three functions: )0.5 = txpsxgs = mm’xfl u(tx1, txz) = (tx1)°'5(tx2 v(tx1, txz) = (tx1)(tx2) = tlexz = t2v(x1,x2) (526) w(tx1,tx2) = (th1)2(th2)2 = 4x§x§ = t4w(x1,x2)- Thus, u is homogeneous of degree 1, v is homogeneous of degree 2 and w is homogeneous of degree 4. Different Types of Tastes 132 (e) What values do each of these utility functions assign to the indifference curve that contains bundle D .9 Answer: Recall that D = (25,25). Thus, the three utility functions assign values of u(25,25) = 250-525"-5 = 25; v(25,25) = 25(25) = 625; and w(25,25) = 252 (252) = 390, 625. (f) True or False: Homogeneity of degree 1 implies that a doubling of goods in a consumption basket leads to the utility as measured by the homogeneous junction, whereas homo- geneity greater than 1 implies that a doubling of goods in a consumption bundle leads to more than “twice” the utility. Answer: This is true. Above, we showed an example of this. More generally, you can see this from the definition of a function that is homogeneous of degree k; i.e. u(tx1, txz) = tk u(x1,x2). Substituting k = 2, u(2x1,2x2) = 2k u(x1,x2). When k = 1 — i.e. when the utility function is homogeneous of degree 1, this implies u(2x1,2x2) = 2u(x1,x2) — a doubling of goods leads to a doubling of utility assigned to the bundle. More generally, a doubling of goods leads to 2k times as much utility assigned to the new bundle — and 2 is greater than 2 when k > 1 (and less than 2 when k < 1.) (g) Demonstrate that the MRS is unchanged regardless of which of the three utility functions derived in B(c) is used. Answer: The MRS of aCobb-Douglas utility function u(x1,x2) = xll’xg is MRS = —(yx2)/ (6x1) which reduces to —x2 /x1 when 7 = 6 which is the case for all three of the utility functions above. Thus, the MRS is the same for the three functions. (h) Can you think of representing these tastes with a utiliyfitnction that assigns the value of 1 00 to the indifference curve containing bundle A and 75 to the indifference curve containing bundle D .9 Is the utility function you derin homogeneous? Answer: The function u(x1,x2) = xg'5xg'5 + 50 would work. This function is not homoge- neous (but it is homothetic). (i) True or False: Homothetic tastes can always be represented by functions that are homoge- neous of degree k (where k is greater than zero), but even functions that are not homogeneous can represent tastes that are homothetic. Answer: This is true. We showed in the text that MRS(tx1, txz) = MRS(x1,x2) for homo- geneous functions — thus, for homogeneous functions, the MRS is constant along any ray from the origin, the definition of homothetic tastes. At the same time, we just saw in the an- swer to the previous part an example of a non-homogeneous function that still represents homothetic tastes. (j) True or False: The marginal rate ofsubstitution is homogeneous of degree 0 if and only if the underlying tastes are homothetic. Answer: For any set of homothetic tastes, the MRS is constant along rays from the origin; i.e. MRS(tx1, txz) = MRS(x1,x2). Thus, for homothetic tastes, the MRS is indeed homo- geneous of degree 0. But MRS(tx1, txz) = MRS(x1,x2) defines homotheticity — so non- homothetic tastes will not have this property, which implies their MRS is not homogeneous of degree zero. The statement is therefore true. Doing the “Best” We Can 176 6.16 Policy Application: Cost of Li vingAdjustments of S ocial Security Benefits: Social Security payments to the elderly are adjusted every year in the fi2llowing way: The government has in the past determined some average bundle of goods consumed by an average elderly person. Each year; the government then takes a look at changes in the prices of all the goods in that bundle and raises social security payments by the percentage required to allow the hypothetical elderly person to continue consuming thatsame bundle. This is referred to as a “cost of living adjustment” or COLA. A: Consider the impact on an average senior’s budget constraint as cost of living adjustments are put in place. Analyze this in a 2-good model where the goods are simply x1 and x2. (a) Begin by drawing such a budget constraint in a graph where you indicate the “average bundle” the governmenthas identified as A and assume that initially this average bundle is indeed the one our average senior would have chosen from his budget. Answer: This is done in panel (a) of Graph 6.22 with the initial budget line going from I / p2 on the vertical to I / p1 on the horizontal and with uA tangent to that budget line at A. I/?, Graph 6.22: Social Security COLAs (b) Suppose the prices of both goods went up by exactly the same proportion. After the govern- ment implements the COLA, has anything changed fi2r the average senior? Is behavior likely to change? Answer: If both prices go up by exactly the same proportion, the slope of the budget line —p1/p2 does not change. Rather, in the absence of a COLA adjustment, the budget would simply shift inward in a parallel way. But if the government will give enough additional money to the senior to allow him to again consume A after the increase in prices, then the government shifts the budget right back through A. Since the slope has not changed, the new budget lies exactly on top of the old. Thus, the senior will not change his behavior — A will still be optimal. (c) Now suppose that the price of x1 went up but the price of :62 stayed the same. Illustrate how the government will change the average senior’s budget constraint when it calculates and passes along the COLA. Will the senior alter his behavior? Is he better off, worse off or not affected? Answer: This is also illustrated in panel (a) of the graph. The increase in p1 causes the ro- tation in the budget to the dashed budget line, and the COLA adjustment shifts that steeper budget up to bundle A so that the person can still afford A. But in the process, the new bud- get cuts the original indifference curve in away that makes new bundles above uA available. These bundles are more preferred by the senior — so the senior will optimize at a new bun- dle like B which contains less of x1 and more of x2. As a result, the senior moves to a higher indifference curve and is therefore better off. (d) How would your answers change if the price of 162 increased and the price of x1 stayed the same? 177 Doing the “Best” We Can Answer: This is illustrated in panel (b) of Graph 6.22. Now the increase in p2 causes the rotation of the budget to the dashed budget line, and the COLA adjustment then pushes this line outward until it once again goes through A. In the process, new bundles that lie above uA become available to the right of A, causing the person to optimize at a new bundle like C. Thus, the senior will reduce consumption of :62 and increase consumption of x1 — and will end up on a higher indifference curve at C. Thus, the senior is better off as a result of the price increase and COLA adjustment. (e) Suppose the government’s goal in paying COLAs to senior citizens is to insure that seniors become neither better nor worse ojffi'om price changes. Is the current policy successful if all price changes come in the form ofgeneral “inflation”7 i. e. i fall prices always change together by the same proportion? What if inflation hits some categories ofgoods more than others? Answer: The policy is successful when inflation is of a general kind and affects all prices by the same proportion. But if prices change at different rates, the COLA adjustment is too large to keep seniors just as happy as before. (f) If you could “choose” your tastes under this system, would you choose tastes fi2r which goods are highly substitutable or would you choose tastes fi2r which goods are highly complemen- tary? Answer: Seniors benefit from COLA adjustments to the extent to which prices change dif- ferentially and the extent to which they are willing to substitute between goods. Consider, for instance, the impact of a higher price p1 (like the one graphed in panel (a)) in the case of the two indifference curves uf‘ and u; in panel (c) of the graph. The indifference curve u? treats x1 and :62 as perfect complements — and as a result, the steeper new budget line with the COLA adjustment does not cut the indifference curve because the indifference curve has a sharp corner at A. Since the person with this indifference curve does not view x1 and :62 as in any way substitutable, he continues to buy A on the new budget. He therefore does not become any better off as a result of the COLA adjustment. Compare this to the individ- ual with indifference curve uf‘. This indifference curve is one that treats x1 and :62 as quite substitutable — and as a result, a large number of “better” bundles become available in the new COLA budget. These are indicated in the graph as the shaded area between uf‘ and the new budget constraint. Thus, individuals who view goods as more substitutable ben- efit more from the way the government adjusts social security checks in response to price changes. B: Suppose the average senior has tastes that can be captured by the utility function u(x1,x2) = —p (x1 (a) Suppose the average senior has income from all sources equal to $40, 000 per year and suppose _ —1lp p ) . that prices are given by p1 and p2. How much will our senior consume ofxl and x2 .9 (Hint: It may be easiest to simply use what you know about the MRS of CES utility functions to solve this problem.) _ _ —1l Answer: For the general CES utility function v(x1,x2) = (axl p + 0x2 ) p, the text derives the MRS to be MRS = —(a/fi)(x2 /x1)(P+1). In the utility function u of this problem, a and [3 are implicitly set to 1. Thus, MRS = —(x2 /x1)(P+1). At an optimum, MRS = —p1/p2, or p+1 —(x—2) =—fl. (6.70) 161 P2 Solving for 162, we get 1I(p+1) x2 = x1. (6.71) Substituting this into the budget constraint p1 x1 + pgxz = 40000 and solving for x1, we then get (after some tricky manipulation of exponents) 40000 x1 = )1/(p+1)' (6.72) 171 + (175171 Doing the “Best” We Can 178 Substituting this back into equation (6.71), we can then also solve for :62 (after some more tricky exponents) as 40000 )1l(p+l) . (6.73) x2 = 172 + (pf 172 (b) If p1 = p2 = 1 initially, how much of each good will the senior consume? Does your answer depend on the elasticity of substitution? Answer: Substituting p1 = p2 = 1 into equations (6.72) and (6.73), we get 40000 40000 x1 = — = — =20,000 and 1+(1P1)1/(1+P) 2 (6 74) 40000 40000 ' x2 — = 20,000. = 1+(1P1)1/(1+P) = 2 Thus, the solution does not depend on p and therefore does not depend on the elasticity of substitution. (c) Now suppose that the price of x1 increases to p1 = 1.25. How much does the government have to increase the senior’s social security payment in order fi2r the senior to still be able to purchase the same bundle as he purchased prior to the price change? Answer: Before the price change, the senior purchases (20000,20000). To be able to afford this bundle after the price change, he must have income of 1.25(20000) + 1(20000) = 45000, or $5000 more than he had before. Thus, the COLA adjustment is $5000. ((1) Assuming the government adjusts the social security payment to allow the senior to continue to purchase the same bundle as before the price increase, how much x1 and :62 will the senior actually end up buying if p = 0.? Answer: The new income is now $45000. Thus, when p = 0 and income is raised to $45000, equations (6.72) and (6.73) reduce to 45000 45000 45000 _ 45000 —= and x =———. (6.75) pl+(pgpl)1/(o+1) 2p1 2 p2+(ptllp2)ll(0+l) 2p2 x1: Thus, when p1 = 1.25 and p2 = 1, we get — 40000 — 18 000 and x — 40°00 —22 500 (6 76) ' 2(1.25) ' ’ 2 ' 2(1) ' ’ ' ' 351 (e) How does your answer change if p = —0.5 and if p = —0.95? What happens as p approaches — 1 .9 Answer: When p = —0.5 and income again is set to $45000, equations (6.72) and (6.73) give x1 = 16,000 and xz = 25,000, (6.77) and when p = —0.95, x1 = 511 and xz =44,361. (6.78) Thus, the smaller p gets, the more the person will deviate from the original bundle. And as p approaches —1, x1 approaches 0 while :62 approaches 45,000. (f) How does your answer change when p = 1 and when p = 10? What happens as p approaches infinity? Answer: Again using equations (6.72) and (6.73), when p = 1, x1 = 19,003 and xz = 21,246. (6.79) and when p = 10, 179 Doing the “Best” We Can x1 = 19,819 and x2 =20,225. (6.80) Thus, the larger p gets, the less the person will deviate from his original bundle. And as p approaches infinity, x1 goes to 20,000, as does x2; i.e. as p approaches infinity, the change in behavior disappears. (g) Can you come to a conclusion about the relationship between how much a senior benefits from the way the government calculates COLAs and the elasticity of substitution that the se- nior’s tastes exhibit? Can you explain intuitively how this makes sense, particularly in light ofyour answer toA(f)? Answer: Above we noticed that as p gets smaller; the person deviates more from his original consumption bundle when prices change and COLAs are implemented. When p1 increases from 1 to 1.25, he does not change his consumption when p is very large but reduces his con- sumption from 20,000 to 19,003 when p = 1, to 18,000 when p = 0, to 16,000 when p = —0.5, to 511 when p = —0.95 and finally to 0 as p approaches zero. The elasticity of substitution is a = 1/ (1 + p) — which implies that, as p gets smaller, the elasticity of substitution increases. Thus, our results above say that, as the elasticity of substitution gets larger; the person devi- ates more from his original consumption bundle. This is exactly what our intuition told us in panel (c) of Graph 6.22 in our answer to A(f). (h) Finally, show how COLAs affect consumption decisions by seniors under general inflation that raises all prices simultaneously and in proportion to one another as, fi2r instance, when both p1 and p2 increase from 1 to 1.25 simultaneously Answer: When both p1 and p2 increase to 1.25, the COLA has to be sufficient for the senior to once again be able to afford (20000,20000) at the new prices. This implies the senior needs an income of 1.25(20000) + 1.25 (20000) = 50000, which is 10000 more than the 40000 he has before the inflation. Thus, the COLA has to be $10,000. Replacing 40000 with 50000 in equations (6.72) and (6.73) and replacing p1 with 1.25p1 and p2 with 1.25p2, we get x 50000 1 = — 1.25p1 + ((1.25p2)!’(1.25p1))1l(p+l) _ 50000 _ 40000 (6.81) _ 1I(p+1) ‘ 1/(p+1)' 1-25(p1 + (175171] ) 171 + (175171] and x 50000 2 = — 1.25p2 +((1.25p1)p(1.25p2))1l(p+1) _ 50000 _ 40000 (6.82) _ 1I(p+1) ' 1/(p+1)’ 1.25(p2+(pfp2) ) p2+(p‘1’p2) both of which are identical to what we had before the inflation and COLA increase. ...
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