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PS2+Solutions - Income and Substitution Effects in Consumer...

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Unformatted text preview: Income and Substitution Effects in Consumer Goods Markest 208 7.11 Policy Application: Substitution Effects and Social Security Cost of LivingAdjustments: In end—of— chapterexerciTlG,y0u investigated the government’s practice for adjusting social security income for seniors by insuring that the average senior can always afford to buy some average bundle of goods that remains fixed. To simplify the analysis, let us again assume that the average senior consumes only two different goods. A: Suppose that last year our average senior optimized at the average bundle A identified by the government, and begin by assuming that we denominate the units of x1 and 352 such that last year P1 = P2 = 1- (a) Suppose that p1 increases. On a graph with x1 on the horizontal and x2 on the vertical axis, illustrate the compensated budget and the bundle B that, given your sen ior’s tastes, would keep the senior just as well off at the new price. Answer: In panel (a) of Graph 7.14, bundle A lies on the original (solid line) budget. The price increase causes an inward rotation of that budget in the absence of compensation. To compensate the person so that he will be as happy as before, we have to raise income to the lower dashed line in the graph — the line that is tangent to B that lies on the indifference CUI'VB LlA . X . Graph 7.14: Hicks and Slutsky Social Security Compensation (b) In yourgraph, compare the level of income the senior requires to get to bundleB to the income required to get him back to bundle A. Answer: The income required (at the new prices) to get to A is represented by the second dashed line in panel (a) of the graph. (c) What determines the size of the difference in the income necessary to keep the sen ior just as well 0]?” when the price of good 1 increases as opposed to the income necessary for the senior to still be able to afford bundle A? Answer: The greater the substitutability of the two goods, the greater will be the difference between the two ways of compensating the person. This is illustrated across the three panels in Graph 7.14 where the degree of substitutability falls from left to right. (d) Under what condition will the two forms of compensation be identical to one another? Answer: The difference between the two compensation schemes disappears entirely in panel (c) of the graph when there is no substitutability between the goods (i. e. when they are per— fect complements). (e) You should recognize the move from A to B as a pure substitution efi‘ectas we have defined it in this chapter. Often this substitution effect is referred to as the Hicksian substitution effect — defined as the change in behavior when opportunity costs change but the consumer receives sufficient compensation to remain just as happy. Let B’ be the consumption bundle the av- erage senior would choose when compensated so as to be able to aflord the original bundle 209 Income and Substitution Effects in Consumer Goods Markest A. The movement from A to B’ is often called the Slutsky substitution effect 7 defined as the change in behavior when opportunity costs change but the consumer receives sufi‘icientcom- pensation to be able to afford to stay at the original consumption bundle. True or False: The government could save money by using Hicksian rather than Slutsky substitution principles to determine appropriate cost of living adjustments for social security recipients. Answer: The answer is true. The government in fact uses Slutsky compensation as it cal— culates cost of living adjustments — because it fixes a particular consumption bundle and then adjusts social security checks to make sure that seniors can still afford that bundle. For this reason, you wfll frequently hear proposals to adjust the way in which cost of liv— ing adjustments are calculated — with these proposals attempting to get closer to Hicksian compensation. (f) True or False: Hicksian and Slutsky compensation get closer to one another the smaller the price changes. Answer: This is true. Larger price changes result in larger substitution effects — and the dif— ference between Hicksian and Slutsky substitution is entirely due to the substitution effect. This is illustrated in the three panels of Graph 7.15 where, going from left to right, the size of the price change (as evidenced in the steepness of the slope of the compensated budget) decreases. (CA) X‘ X\ Graph 7.15: Hicks and Slutsky Social Security Compensation: Part II B: Now suppose that the tastes of the average senior can be captured by the Cobb-Douglas utility function u(x1,x2) = x1352, where 352 is a composite good (with price by definition equal to p2 = 1). Suppose the average senior currently receives social security income I (and no other income) and with it purchases bundle (35?, x?) (a) Determine (xix?) in terms of I and p1. Answer: Solving the usual maximization problem with budget constraint p1 x1 + x2 = I, we get I I xf‘z— and 3552—. 2101 (b) Suppose that p1 is currently $1 and I is currently $2000. Then p1 increases to $2. How much will the government increase the social security check given how it is actually calculating cost of living adjustments? How will this change the senior’s behavior? Answer: The government compensates so as to make it possible for the senior to keep af— fording the same bundle as before. With the values p1 = 1 and I = 2000, xf‘ = x; = 1000. When the price of x1 goes to $2, this same bundle costs 2(1000) + 1000 = $3,000. Thus, the government is compensating the senior by increasing the social security check by $1,000. (7.53) Income and Substitution Effects in Consumer Goods Markest 210 With an income of $3,000, equations (7.53) then tell us that the senior will consume x1 = 3000/(2(2)) = 750 and x2 2 3000/2 = 1,500. Thus, even though the government makes it possible for the senior to consume bundle A again after the price change, the senior will substitute away from x1 because its opportunity cost is now higher. (c) How much would the government increase the social security check if it used Hicksian rather than Slutsky compensation? How would the senior’s behavior change? Answer: If the government used Hicksian compensation, it would first need to calculate the bundle B on the original indifference curve that would make the senior just as well off at the higher price as he was at A. At A, the senior gets utility uA = xf‘xé“ = 1000(1000) = 1, 000, 000. The government would then have to solve the problem pup 2x1 +x2 subject to )5le = 1,000,000. (7.54) . 1.. 2 Solving the first two first order conditions, we get x2 2 2x1. Substituting this into the con— straint and solving for x1, we get x1 : 707.1, and plugging this back into 352 2 2x1, we get x2 = 1414.2. This bundle B = (707.1,1414.2) costs 2(707.1) + 1414.2 2 2828.4. Thus, under Hicksian compensation, the government would increase the seniors social security check by $828.40 rather than $1,000. (d) Can you demonstrate mathematically that Hicksian and Slutsky compensation converge to one another as the price change gets small — and diverge from each other as the price change gets large? Answer: We start with p1 = 1 (and continue to assume p2 = 1).2 Then suppose p1 in— creases to p1 > 1 (or falls to p1 < 1). Slutsky compensation requires that we continue to be able to purchase A = (1000,1000) — so we have to make sure the senior has income of 1000p1 + 1000. Since the senior starts with an income of $2,000, this implies that Slutsky compensation is 1000p1 : 1000 2000 — 1000p1 1000 — 1000(p1 1). Hicksian compensation, on the other hand, requires we calculate the substitution effect to B as we did in the previous part for p1 = 2. Setting up the same problem but letting the new price of good 1 be denoted p1 rather than 2, we can calculate B = (xix?) = (1000/ p315 , 1000,9315 ). This bundle costs 1000 r r pl 05 + 1000,93“ = 200017290. (7.55) p 1 Given that the senior starts with $2000, this means that Hicksian compensation must be equal to 2000,9315 — 2000 = 2000(p‘f-5 — 1). The difference between Slutsky compensation and Hicksian compensation, which we will call D(p1) is then D001) _ 1000(p1 1) 2000(p(1)'5 1) _ 1000,91 1000 2000,7295 : 2000 , (7.56) = 1000 + 1000,91 (1 — 2pf0-J). As p1 approaches 1, the second term in the equation goes to —1000 — making the expres— sion go to zero; i.e. the difference between the two types of compensation goes to zero as the price increase (or decrease) gets small. In fact, it is easy to see that this difference reaches its lowest point when p1 = 1 and increases when p1 rises above 1 as well as when p1 falls below 1: Simply take the derivative of D(p1) which is dD(P1) 03101 Then note that dD/dp1 < 0 when 0 < p1 < 1, dD/dp1 = 0 when p1 = 1 and dD/dp1 > 0 when p1 > 1. This implies a U—shape for D(p1), with the U reaching its bottom at p1 = 1 when D(p1) = 0. Put into words, the difference between Slutsky and Hicks compensation is positive for any price not equal to the original price, with the difference increasing the greater the deviation in price from the original price. = 1000 [1 —2p1’°-5) + 1000,91 [pfl'SJ = 1000 [1 — Info-5). (7.57) 2We could start with any other price and change either p1 or p2 and the same logic will hold. 211 Income and Substitution Effects in Consumer Goods Markest (e) We know that Cobb-Douglas utility functions are part of the CBS family of utility functions — with the elasticity of substitution equal to 1. Without doing any math, can you estimate, for an increase in m above 1, the range of how much Slutsky compensation can exceed Hicksian compensation with tastes that lie within the CES family? (Hint: Consider the extreme cases of elasticities of snbsitntion. ) Answer: We know that if the two goods are perfect complements (with elasticity of substi— tution equal to 0), then there is no difference between the two compensation mechanisms (because, as we demonstrated in part A of the question, the difference is due entirely to the substitution effect). Thus, one end of the range of how much Slutsky compensation can exceed Hicksian compensation is zero. The other extreme is the case of perfect substitutes. In that case, it is rational for the con— sumer to choose bundle A initially since the prices are identical and the indifference curve therefore lies on top of the budget line (making all bundles on the budget line optimal). But any deviation in price wfll result in a corner solution. Thus, if m increases, the consumer can remain just as well off as she was originally by simply not consuming 352. Thus, Hicksian compensation is zero while Slutsky compensation still aims to make bundle A affordable — i.e. Slutsky compensation is still 1000(p1 — 1) as we calculated in part (d). So in this extreme case, Slutsky compensation exceeds Hicksian compensation by 1000(p1 — 1). Depending on the elasticity of substitution, Slutsky compensation may therefore exceed Hicksian compensation by as little as 0 (when the elasticity is 0) to as much as 1000(p1 — 1) (when the elasticity is infinite). 249 Wealth and Substitution Effects in Labor and Capital Markets 8.10 Policy Application: Subsidizing Savings versus Taxing Borrowing In end-of-chapter exercise 6.10 we arme the interest rates for borrowing and saving are dififeren t. Part of the reason they might be different is because of government policy. A: Suppose banks are currently willing to lend and borrow at the same interest rate. Consider an individual who has income e1 now and eg in a future period, with the interest rate over that period equal to r. After considering the tradeofifs, the individual chooses to borrow on his future income rather than save. Suppose in this exercise that the individual ’s tastes are homothetic. (a) Illustrate the budget constraint for this individual — and indicate his optimal choice. Answer: This is illustrated in panel (a) of Graph 8.13. Burrowing Graph 8.13: Saving and Borrowing under Different Policies (b) Now suppose the government would like to encourage this individual to save for the future. One proposal might be to subsidize savings (through something like a 401K plan) — i. e. a policy that increases the in terest rate for saving without changing the in terest rate for borrow- ing Illustrate how this changes the budget constraint. Will this policy work to accomplish the government’s goal? Answer: This policy will work if the increase in the interest rate for savings is sufficiently high. In panel (b) of Graph 8.13, we illustrate the smallest possible increase in the inter— est rate for savings that is necessary to induce a change in the individual’s behavior. The increase in the interest rate for savings causes the slope of the budget to become steeper to the left of the bundle (e1,e2) where the individual saves rather than borrows. If it only becomes a little steeper, A is still optimal and no change in behavior takes place. But if it becomes sufficiently steeper (as graphed), B becomes optimal — which implies the individ— ual switches from borrowing to saving. (c) Another alternative would be to penalize borrowing by taxing the interest the banks collect from loans — thus raising the effective interest rate for borrowing Illustrate how this changes the budget. Will this policy cause the individual to borrow less? Can it cause him to start saving? Answer: An increase in the interest rate for borrowing makes the slope of the budget past (e1,e2) steeper. This causes a substitution effect in the direction of less borrowing (from A to B’ in panel (c) of Graph 8.13). But with homothetic tastes, we can be sure that a tangency will never happen on the solid portion of the budget to the left of (e1,e2) — because initially (in panel (a)), when the interest rate was r for both borrowing and saving, the individual choose A. When tastes are homothetic, this means that the MRS is equal to this slope along the ray from the origin — which never intersects the budget to the left of (e1,e2). Thus, the most this policy might do is to cause the individual to optimize at the kink point of the new budget — point B in panel (c) of the graph. Thus, if the interest rate for borrowing increases enough, the individual might choose to stop borrowing — but he will not start saving. Wealth and Substitution Effects in Labor and Capital Markets 250 (d) In reality, the government often does the opposite of these two policies: Savings (outside qual- ified retirement plans) are taxed while some forms of borrowing (in particular borrowing to buy a home) are subsidized. Suppose again that initially the interest rate for borrowingand saving is the same — and then suppose that the combination of taxes on savings (which low- ers the effective interest rate on savings) and subsidies for borrowing (which lowers the effec- tive interest rate for borrowing) reduce the interest rate to r’ < r equally for both saving and borrowing How will this individual respond to this combination of policies? Answer: This combination of policies causes the budget to rotate counter—clockwise through the bundle (e1,e2) — with a new slope that is shallower. This is illustrated in panel (a) of Graph 8.14. Such a change in the budget gives rise to a substitution effect from A to B and a wealth effect from B to C — both pointing toward greater consumption now and thus an increase in borrowing. Graph 8.14: Saving and Borrowing under Different Policies: Part 11 (e) Suppose that, instead of taxing or subsidizing interest rates, the governmentsimply "savesfor" the individual by taking some of the individual’s current income el and putting it into the bank to collect interest for the future period. How will this change the individual’s behavior? Answer: This is illustrated in panel (b) of Graph 8.14. All that the policy does is to transfer some of e1 to eg — moving us from E1 to E2 in the graph. Since the interest rate is un— changed, the budget remains exactly as it was before. Thus, if A was optimal before, A is still optimal for the consumer — it just now means that consumer has to increase his bor— rowing by the amount of saving the government did for him. In other words, the consumer will undo what the government is doing on his behalf. (f) Now suppose that, instead of taking some of the person’s curren t incomeand saving itfor him, the governmentsimply raises the social security benefits (in the future period) without taking anythingaway from the person now What will the individual do? Answer: This policy essentially shifts e2 up — but it does not change the interest rate. Thus, it causes a parallel shift in the budget — giving rise to a pure wealth effect that is illustrated in panel (c) of Graph 8.14. As a result, the individual will increase the amount he borrows. B: Suppose your tastes can be captured by the utility function u(c1, c2) 2 cf” c970”). (a) Assuming you face a constant interest rate r for borrowing and saving, how much will you consume now and in thefuture (as afunction ofel, e2 and r.) Answer: You would solve the problem a (lid) . _ may; c1 c2 sublect to (1 + r)e1 + e2 — (1 + r)c1 + c2. (8.59) Solving the first two first order conditions, you get c2 = (1 + r)(1 — a)c1/a, and plugging this into the budget constraint, you can solve for c1. Plugging the solution back into c2 = (1 + r)(1 — a)c1/a you can then solve for c2. You should get 251 Wealth and Substitution Effects in Labor and Capital Markets c1: W and c2 =(1—oc)[(1+r)e1+egj. (8.60) (1+r) (b) For what values of a will you choose to borrow rather than save? Answer: In order for you to borrow, it must be that your optimal c1 is greater than your current income e1 — i.e. c1 > e1. Using the optimal c1 just derived above, this implies a[(1 + r)e1+egj > . 8.61 (1 + r) 61 ( ) Solving for a, we get that 1 __ a > fl. (8.62) e1 (1 -— r) + e2 (c) Suppose that a = 0.5, e1 2 100,000, e2 2 125,000 and r = 0.10. How much do you save or borrow? Answer: Using our solutions in equation (8.60), we get c1 = $106,818.18 and c2 = $117,500. Thus, you would borrow $6,818. (d) If the governmentcould come up with a “financial literacy" course that changes how you view the tradeofl between now and the future by impactinga, how much would this program have to change your a in order to get you to stop borrowing? Answer: Using equation (8.62), a = 0.4681 would be necessary in order for you to neither borrow nor save. For a greater than that, you would continue to borrow. (e) Suppose the “financial literacy" program had no impact on a. How much would the govern- ment have to raise the interest rate for saving (as described inA(b)) in order for you to become a saver? (Hint: You need to first determine c1 and c2 as a function of just r. You can then determine the utility you receive as afunction ofjust r — andyou will not switch to saving until r is sufficiently high to give you the same utilityyou get by borrowing) Answer: Using our answers from above, the utility you get from borrowing is u(106818.18, 117500) 2 106818.180'5 1175000'5 : 112,031.85. (8.63) Put differently, the label on the indifference curve that you end up at if you borrow is 1 12,031.85. This indifference curve, and the optimal bundle when you borrow, is illustrated as point A in Graph 8.15. We know that, as the interest rate for savings goes up, the budget to the left of (e1,e2) = (100000, 125000) increases. To find how much the government must subsidize the inter— est rate in order to cause you to switch from borrowing to saving, we have to determine the interest rate that makes this steeper budget tangent to the same indifference curve that contains A — i.e. the indifference curve with utility 112,031.85. With e1 2 100,000, e2 2 125,000 and a = 1/2, equation (8.60) tells us you wfll consume 100000(1 + r) + 125000 [100000(1 + r) + 125000] c1 = — and (22 = — (8.64) 2(1 + r) 2 for any given interest rate r. This can also be written 50,000(1 + r) + 62,500 01 = — and Cg = 50,000(1 + r) +62,500. (8.65) (1 + r) Using the result...
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