# hw3_key2 - Problems 1 Given information y=lﬂﬂ y'=]2...

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Unformatted text preview: Problems 1. Given information: y=lﬂﬂ y'=]2 I=ZD t'=]0 r=lll (a) To calculate wealth, we compute: w=y—r+yr_tl=so+ﬂ=tso +r LI 8’ (c) The consumer's optimal consumption bundle is at point A. Point A simultaneously solves: 12:42, and C 1+!“ c+ =c+i}.91c'=13i} Upon solving, we find that c = c' = 94.2. Savings is therefore given by: s=y—I—c=BD—9¢2=—14.2 The consumer is a borrower. In the figure above, the endowment point is EI and the consumer chooses A. (d) First—period income rises from IDE- to MD. We now recompute w = 22!]. Solving as in part (c), we ﬁnd that r: = c': 115.2, and s = 4.3. In the ﬁgure above, the endowment point is EI and the consumer chooses B. (e) In part {c}, the consumer is a borrower. In part [d], ﬁrst-period income increases and savings has consequently increased enough that the consumer is now a lender. 4. Temporary.r and Permanent Tar. Increases. (a) The increase in ﬁrst-period taxes induces a parallel leftward shift in the budget line. The original budget line passes through the initial endowment, E1. The new budget line passes through E. The consumer reduces both current and future consumption. In the ﬁgure below the consumer's optimum point moves from point A to point B. First-period consumption falls by less than the increase in taxes and so savings falls. 15’ {b} Next consider a permanent increase in taxes. A permanent tax increase adds a second tax increase to the ﬁrst tax increase, the current-period tax increase. The increase in second-period taxes induces a parallel downward shift in the budget line. The new budget line passes through E: in the ﬁgure above. The second part of the tax increase also reduces both ﬁrst-period and second— period consumption. The consumer moves from point B to point D. Because the second tax increase reduces ﬁrst-period consumption holding ﬁrst-period disposable income ﬁxed, savings must rise. Since the permanent tax increase is the sum of the two individual tax increases, the permanent tax increase reduces both ﬁrst-period and second-period consumption, but on net, savings ma}.r either rise, faJI, or remain unchanged. 5. A tax on interest income. (a) Initially, AB in the ﬁrst ﬁgure below depicts the consumer‘s budget constraint. The introduction ofthe tax results in a kink in the budget constraint, since the interest rate at which the consumer can lend, ril —!), is new smaller than the interest rate at which the consumer borrows, r. The kink occurs at the endowment, E. r.’ {b} The ﬁrst ﬁgure above shows the case of a consumer who was a borrower before the imposition of the tax. This consumer is unaffected by the introduction of the tax. The second ﬁgure above shows the case of a consumer who was a lender before the imposition of the tax. Initially the consumer chooses point G, and then chooses point H after the imposition ofthe tax. There is a substitution effect that results in an increase in ﬁrst—period consumption and a reduction in second—period consumption, and moves the consumer from point G to point J. Savings also fall from point G to point]. The income effect is the movement from point D to point B, and the income effect reduces both ﬁrst—period and second-period consumption, and increases savings. (in net, consumption must fall in period 2, but in period 1, consumption may rise or fall. The ﬁgures above show the case in which ﬁrst-period consumption increases, which is a case where the substitution effect dominates. E. Given information: y=2m y'=]5lZI 1:40 I'=SD rzﬂﬂﬁ (a) If the consumer could borrow and lend at the real interest rate, .r = 0.05, then the consumer’s lifetime budget constraint would be given by: c =}'—i‘+}I _r. {1+r} {1+r} Plugging in the numbeis from this problem, we obtain: r: + 0.956 = 255.2. In the ﬁgure below, the initial budget constraint is given by BEID. The budget constraint has a kink at the initial endowment point EL = (160,100), because the consumer cannot borrow, and therefore cannot consume more than 160 in the ﬁrst period. Because the consumer has perfect— complements preferences, the indifference curves are kinked at c = c'. C+ c‘ :b) TWith perfect-comple ments preferences, the consu mer picks point A in ﬁgure on the previous page. Plugging in c = 13' into the budget constraint and solving, we ﬁnd that c = c' = 130.? and so 3 = y — :t — c 2160—130? = 29.3. In this case, the fact that the consumer cannot borrow does not matter for the consumer‘s choice, as the consumer decides to be a lender. :c) When1= 2|] and t“ 2T], the consumer‘s lifetime wealth remains unchanged at 255.2. However, the budget constraint shifts to BEIF, ﬁgure on the previous page, with the new endowment point at E2 = [180,7'9). This change does not matter for the consumer's choice, again because he or she chooses to be a lender. Consumption is still 130\$, but now savings is s=y—t—c=lBD—l30.T=49.3. :d} Now ﬁrst-period income falls to 100. Wealth is now equal to w = 155.2. In the ﬁgure above, the budget constraint for the consumer is AE1D, so when the consumer chooses the point on his or her budget constraint that is on the highest indifference curve, any point on the line segment BE1 will do. Suppose that the consumer chooses the endowment point E], where r: = 61] and c' = 100. This implies that s = i}, and the consumer is credit—constrained in that he or she would like to borrow, but cannot. Now with the tax change, the budget constraint shifts to AEit}, with the endowment point E1 = (30,79). Thus the consumer can choose c = c‘ on the new budget constraint, and solving for consumption in each period using the budget constraint c+ [1.956 =155.2, we get I: = c' = Tr'9.5, and s = [1.5. Here, notice that ﬁrst—period consumption increased by almost the same amount as the tail. cut, although lifetime wealth remains unchanged at 155.2. Effectively, the budget constraint for the consumer is relaxed. Therefore, for tax cuts that leave lifetime wealth unchanged, lenders will not change their current consumption, but credit- constrained borrowers will increase current consumption. ...
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