Williamson Macroeconomics Ch9 Solutions 681e3j

ECON 305 Tutorial 7 (Week 9) Questions for today: Ch.9 Problems 15, 7, 11, 12 MC113 Tutorial slides will be posted Thursday after 10:30am, on http://www.sfu.ca/~haiyunc/teaching.html.

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AQ1. Ch.9 Problem 1 A consumer's income in the current period is y = 100, and income in the future period is y0 = 120. He or she pays lump-sum taxes t = 20 in the current period and t0 = 10 in the future period. The real interest rate is 0.1 per period. (a) Determine the consumer's lifetime wealth. In present value , lifetime wealth is

we = y − t +

y0 − t0 120 − 10 = 100 − 20 + = 180. 1+r 1 + 0.1

(b) Suppose that current and future consumptions are perfect complements for the consumer and that he or she always wants to have equal consumption in the current and future periods. Draw the consumer's indierence curves. Indierence curves for perfect complements is L-shaped Utility function generating these curves has the form

U (c, c0 ) = min{αc, αc0 },

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α > 0.

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AQ1. Ch.9 Problem 1 c0 242 c = c0

198

115.18 110 94.24

E1

U1

80 94.24 120 115.18

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U2

E2

180

AQ1 (c) AQ1 (d) AQ1 (e) ECON 305 Tutorial 7 (Week 9)

220

c

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AQ1. Ch.9 Problem 1 (c) Determine what the consumer's optimal current-period and future-period consumptions are, and what optimal saving is, and show this in a diagram with the consumer's budget constraint and indierence curves. Is the consumer a lender or a borrower? Intertemporal budget constraint:

we = c + (1 + r)−1 c0

⇒

c0 = 0.91(180 − c).

(1)

Given the utility function, optimality occurs when

c = c0 .

(2)

Solving (1) and (2) for c, c0 , we get c = c0 = 94.24. Saving in current period is s = y − t − c = −14.24. Therefore the consumer is a borrower.

Draw

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AQ1. Ch.9 Problem 1 (d) Now suppose that instead of y = 100, the consumer has y = 140. Again, determine optimal consumption in the current and future periods and optimal saving, and show this in a diagram. Is the consumer a lender or a borrower? Present value wealth is now

we = 140 − 20 +

120 − 10 = 220. 1 + 0.1

Optimality condition (c = c0 ) and budget constraint (c0 = 0.91(220 − c)) together imply c = c0 = 115.18, s = 4.82. The consumer now is a lender.

Draw

(e) Explain the dierences in your results between parts (c) and (d). Consumer switches from borrower in (c) to lender in (d) due to a suciently large increase in the current period income.

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AQ2. Ch.9 Problem 2 An employer oers his or her employee the option of shifting x units of income from next year to this year. That is, the option is to reduce income next year by x units and increase income this year by x units. (a) Would the employee take this option (use a diagram)? Suppose t = t0 = 0. Initial lifetime wealth is

we0 = y + (1 + r)−1 y0 . Lifetime wealth after the shift is

we1 = y + x + (1 + r)−1 (y0 − x) = y + (1 + r ) −1 y0 + x 1 − (1 + r ) −1 | {z } | {z } =we0

>0

if r > 0

we1 > we0 whenever r > 0, and so the employee should take the option, provided that interest rate is positive.

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AQ2. Ch.9 Problem 2

c0 (1 + r)we1 (1 + r)we0

slope = −(1 + r)

y0 y0 − x we1 y

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y + x we0

ECON 305 Tutorial 7 (Week 9)

c

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AQ2. Ch.9 Problem 2 (b) Determine, using a diagram, how this shift in income will aect consumption this year and next year and saving this year. Explain your results. The shift in income creates a pure income eect (shifts budget constraint outwards in a parallel fashion). Thus, both c, c0 will increase. Saving will increase: c0 = y0 + (1 + r)s Since c0 increases while y0 decreases, s must increase for the equality to hold.

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AQ3. Ch.9 Problem 3 Consider the following eects of an increase in taxes for a consumer. (a) The consumer's taxes increase by ∆t in the current period. How does this aect current consumption, future consumption, and current saving?

c0

An increase in tax causes budget constraint to shift inwards.

(1 + r)we0

Consumption in both periods will drop. Saving also goes down:

(1 + r)we1 E1

y0

y − ∆t

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Consumption smoothing implies that c will decrease by an amount less than ∆t Recall s = y − t − c

E0

y we1

we0

c

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AQ3. Ch.9 Problem 3 (b) The consumer's taxes increase permanently, increasing by ∆t in the current and future periods. Using a diagram, determine how this aects current consumption, future consumption, and current saving. Explain the dierences between your results here and in part (a).

c0

A permanent increase in tax shifts budget constraint inwards.

(1 + r)we0

Consumption in both periods will drop. Saving is ambiguous.

E0

y0 E2

y0 − ∆t

y − ∆t

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y

we0

c

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AQ4. Ch.9 Problem 4 Suppose that the government introduces a tax on interest earnings. That is, borrowers face a real interest rate of r before and after the tax is introduced, but lenders receive an interest rate of (1 − x)r on their savings, where x is the tax rate. Therefore, we are looking at the eect of having x increase from zero to some value greater than zero, with r assumed to remain constant. (a) Show the eects of the increase in the tax rate on the consumer's lifetime budget constraint. c0

(1 + r)we

slope = −(1 + r)

(1 + (1 − x )r )y + y0

y0

slope = −(1 + (1 − x)r)

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Budget constraint for borrowers doesn't change.

E

y

Tax on interest income causes lender's budget constraint to pivot down around initial endowment.

we

c

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AQ4. Ch.9 Problem 4 (b) How does the increase in the tax rate aect the optimal choice of consumption (in the current and future periods) and saving for the consumer? Show how income and substitution eects matter for your answer, and show how it matters whether the consumer is initially a borrower or a lender. c0

Consumer is unaected if initially a borrower.

(1 + r)we

If initially a lender, (1 + (1 − x )r )y + y0

A

and post-policy consumption bundle is at point C,

B

C y0

then there will be substitution eect (A → B) and income eect (B → C).

E

y

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we

c

ECON 305 Tutorial 7 (Week 9)

c0 decreases as a result, but eect on c (hence s) is ambiguous and depends on shape of utility

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AQ5. Ch.9 Problem 5 A consumer's income in the current and future periods are y and y0 , respectively. He pays taxes t and t0 in the two periods. The consumer can borrow and lend at the real interest rate r. This consumer faces a constraint on how much he can borrow. That is, the consumer cannot borrow more than x, where x < we − y + t, with we denoting lifetime wealth. Use diagrams to determine the eects on the consumer's current consumption, future consumption, and savings of a change in x, and explain your results. c0

Saving is unaected. There is a borrowing constraint B¯ . Budget line becomes vertical at B¯ .

(1 + r)we

y0 − t0

If borrowing constraint is binding, IC touches budget constraint at the kink.

E

x y − t B¯

we

we − y + t

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c

If borrowing constraint not binding, the usual tangency condition applies.

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AQ6. Ch.9 Problem 7 Suppose that all consumers are identical, and also assume that the real interest rate r is xed. Suppose that the government wants to collect a given amount of tax revenue R, in present value . Assume that the government has two options: (i) a proportional tax of s per unit of savings, in that the tax collected per consumer is s(y − c); (ii) a proportional tax u on consumption in the current and future periods, so that the present value of the toal tax collected per 0 consumer is uc + 1uc +r . Note that the tax rate s could be positive or negative. For example if consumers borrow, then s would need to be less than zero for the government to collect tax revenue. Show that option (ii) is preferable to option (i) if the government wishes to make consumers as well o as possible, and explain why this is so. [Hint: Show that the consumption bundle that consumers choose under option (i) could have been chosen under option (ii), but was not.] The hint suggests us to appeal to the weak axiom of revealed preference (WARP).

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AQ6. Ch.9 Problem 7 Quick review of WARP c0

Suppose there are two consumption bundles A and B. Both are aordable under red budget constraint, where A is observed to be chosen.

A B

B is aordable under blue budget constraint and observed to be chosen, while A is not aordable. c

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We conclude that A must be preferred to B, because when both A and B are aordable, A is chosen over B.

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AQ6. Ch.9 Problem 7 In this question, the two tax schemes induce two dierent budget constraints. Let (c1 , c10 ) be the consumption bundle chosen under scheme that taxes savings, and (c2 , c20 ) the bundle chosen under scheme that taxes consumptions. We need to show that (c1 , c10 ) is aordable under the budget constraint induced by consumption taxes; and since it's not chosen, it must be inferior to (c2 , c20 ). Therefore, the tax scheme that induces (c2 , c20 ) is preferred from the consumer's perspective.

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AQ6. Ch.9 Problem 7 Budget constraint under savings tax is

c1 +

c10 y0 = y+ − s ( y − c1 ) . 1+r 1+r

(1)

Budget constraint under consumption taxes is

c20 = y+ 1+r c0 (1 + u) c2 + 2 = y+ 1+r c2 +

⇒

c0 y0 − u c2 + 2 1+r 1+r

(2)

y0 . 1+r

(3)

Since tax revenue is the same under both schemes, c0 R = s(y − c1 ) = u c2 + 2 . 1+r

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(4)

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AQ6. Ch.9 Problem 7 From (4), it follows that

−1 c0 u = R c2 + 2 1+r

−1 c20 c0 c2 + 2 1+r 1+r −1 y0 y0 = y+ y+ −R 1+r 1+r

⇒

1+u =

eq. (2) & (4)

=======⇒

R + c2 +

Evaluate (c1 , c10 ) in budget constraint under consumption tax (3):

c10 (1 + u) c1 + Q y+ 1+r − 1 0 c y0 y0 1 y+ c1 + = y+ y+ −R 1 + r 1 + r 1 + r

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y0 1+r y0 1+r

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AQ6. Ch.9 Problem 7 Therefore, (c1 , c10 ) is exactly aordable under the consumption taxes scheme. But (c2 , c20 ) is chosen over (c1 , c10 ). We thus conclude that consumption taxes is preferred to tax on savings. Despite the mathematics, the intuition is simple:

Tax on savings changes the slope of the budget line, thus creating income and substitution eects. The consumption taxes in this question are equivalent to a lump-sum tax, which creates a pure income eect, and hence lead to no distortion in consumer's behavior.

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AQ7. Ch.9 Problem 11 Suppose a consumer has income y in the current period, income y0 in the future period, and faces proportional taxes on consumption in the current future periods. There are no lump-sum taxes. That is, if consumption is c in the current period and c0 in the future period, the consumer pays a tax sc in the current period and s0 c0 in the future period where s, s0 are tax rates. The government wishes to collect total tax revenue in the current and future periods, which has a present value of R. Now suppose that he government reduces s and increases s0 , in such a way that it continues to collect the same present value of tax revenue R from the consumer, given the consumer's optimal choices of current- and future-period consumptions. (a) Write down the lifetime budget constraint of the consumer. Lifetime budget constraint is

(1 + s)c +

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y0 (1 + s0 )c0 = y+ . 1+r 1+r

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AQ7. Ch.9 Problem 11 (b) Show that lifetime wealth is the same for the consumer, before and after the change in tax rates. Lifetime wealth is y + (1 + r)−1 y0 , which is independent of s, s0 . (c) What eect, if any, does the change in tax rates have on the consumer's choice of current and future consumptions, and on savings? Does Ricardian equivalence hold here? Explain why or why not. The change will in general aect consumption and saving decisions. From the budget constraint, relative price of consumption is (1 + r)(1 + s)/(1 + s0 ),

(1 + s)c +

(1 + s0 )c0 y0 = y+ . 1+r 1+r

Changes in s, s0 alters the slope of the budget line, thus creating distortions in consumer's decisions. So Ricardian equivalence does not hold here.

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AQ8. Ch.9 Problem 12 Suppose in our two-period model of the economy that the government, instead of borrowing in the current period, runs a government loan program. That is, loans are made to consumers at the market real interest rate r, with the aggregate quantity of loans made in the current period denoted by L. Government loans are nanced by lump-sum taxes on consumers in the current period, and we assume that government spending is zero in the current and future periods. In the future period, when the government loans are repaid by consumers, the government rebates this amount as lump-sum transfers (negative taxes) to consumers. (a) Write down the government's current-period budget constraint and its future-period budget constraint. Current period: T = L; future period: T 0 = −(1 + r)L. (b) Determine the present-value budget constraint of the government. PV budget constraint:

L + (−L) = T +

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T0 1+r

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AQ8. Ch.9 Problem 12 (c) Write down the lifetime budget constraint of a consumer. Consumer's budget constraint:

c+

c0 = (y + ` − t) + 1+r



y0 − (1 + r)` − t0 1+r



where the lower case `, t, t0 are loan and taxes for an individual consumer. (d) Show that the size of the government loan program (i.e., the quantity L) has no eect on current consumption or future consumption for each individual consumer and that there is no eect on the equilibrium real interest rate. Explain this result. Loan size does not change the PV of taxes, and does not aect interest rate. Therefore, no change in consumptions.

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MC113 MC solutions: 15:

B

C

C

D

A

610:

A

B

B

B

C

1113:

B

B

A

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