21 1 1 1 1 1
1
1 1
2 1
P Q
S Q C Q Q
r
(19c)
The expression for National Savings is therefore:
2
1 1 1 1
1 1
1 1
1 2 1
S CA Q Q Q
r
(21c)
In the case of an open economy, the Current Account deficit exactly matches the extra private consumption. The corresponding capital inflow is financing the government deficit. In the case of a closed economy, the autarky interest rate must rise, for private consumption to decrease (and private savings to increase) the enough to finance the government deficit.
government expenditures with taxes today or with debt: in light of the Ricardian equivalence, this will be irrelevant, because the consumer will perceive debt as future taxes.
In the real world, there are many reasons why the Ricardian equivalence fails: these include distortionary taxation, borrowing constraints and overlapping generations. In the real world, a tax cut has some effect in expanding private consumption. The failure of the Ricardian equivalence is consistent with the view that tax cuts often come along with twin deficits.
The fact that taxes are distortionary also points to the case that governments should smooth the marginal tax rate over time, instead of trying to get balanced budgets each moment in time: it is better to have a tax system in which marginal tax rates are constant over time, rather than a tax system in which marginal tax rates drift up and down, giving rise to inter-temporal distortions and extra deadweight losses, just to keep the government budget permanently balanced.
Appendix 1 - Consumption and the interest rate
When the utility function takes the form (16a), the income and substitution effect on current consumption exactly cancel out. This is not however a general case. It is rather an implication of the fact that we are using a logarithm utility function, where the elasticity of substitution between current consumption and future consumption is equal to one.
To see this, consider a more general utility function of the form
1
, 2 1 2
1
C C u
u C C
U , with
1 1
1
1 ln
1
C when
when C
C
u (16b)
In (16b), the coefficient measures the household’ relative risk aversion, and is assumed exogenous. This utility function is known and of Constant Relative Risk Aversion (CRRA).
Maximization of (16b) subject to (9a) delivers the following Euler equation:
1 1 1
1
2 r
C
C (17d)
Where 1 is the elasticity of inter-temporal substitution. When the elasticity of substitution is equal to one, we obtain (17a). Using the Euler equation in (9a) and solving for
C1 , one obtains
1 11 1 1 1
1
r
C (18f).
This is a more general formulation for optimal consumption than (18), as it allows for different elasticities of substitution. In (18f), we see that the optimal response of consumption to an increase in the interest rate (at a constant life-time wealth, 1) depends on the elasticity of inter-temporal substitution. When 1, the substitution effect is small, so the income effect dominates, and consumption becomes a positive function of the interest rate (holding wealth constant). In Figure A1, this is illustrated with a move from A to B’’. When 1, the elasticity of substitution is large and dominates the income effect, implying that current consumption decreases with the interest rate. In figure A1, this corresponds to the move from A to B’. In case 1 (as in the main text), then the substitution and income effect exactly cancel out, and consumption will not depend on the interest rate, except for wealth effects (move from A to B).
Figure A1: Optimal response to an increase in the interest rate under alternative elasticities of inter-temporal substitution: case without wealth effect
C2
C1 A
B
C1A
Old income expansion path
E1 C2A
B‘
B‘‘
0
New Income expansion path
Now, consider the possibility of the interest rate impacting on wealth 1 as well (that is, with Q2T2 0). In Figure A2 when the interest rate increases, the current value of the individual’ life-time wealth declines, from 1 to 1' moving the intertemporal budget leftwards. Through this wealth effect, current consumption declines. This wealth effect comes at the top of the substitution and income effects referred in Figure A1.
Thus, when 1 , the total effect of the increase in interest rate on current consumption is negative for sure (A to C or A to C’ in figure A2). When however, 1, there are two opposing movements: first, through the substitution and income effects, current consumption increases (from A to B”). Then, through the wealth effect the consumption declines along the income expansion path, from B” to C”. In figure A2, we describe a case where the total effect is positive: from A to C”, the increase in interest rate causes current consumption to increase. This is because the wealth effect in the figure is very small (point E is close to the horizontal axes). If future output was bigger, the endowment point would be more to the left, and the wealth effect would be larger, turning the relationship between consumption and interest rate negative, even with 1.
In sum, when the household’ current income is low and future income is high (mostly, the case of borrowers), the increase in the interest rate produces a strong wealth effect. With a strong negative wealth effect, one expects current consumption to depend negatively on the interest rate. However, when current income is high relative to future income (mostly, the case of lenders), the wealth effect becomes smaller, raising the possibility of a positive relationship between consumption and the interest rate, if and only if 1.
Empirically, most evidence has been supportive of a negative relationship between consumption and the interest rate, though not without controversy.
Figure A2: Change in interest rate with wealth effects
C2
C1 A
B
C1A
Old income expansion path
C2A
B‘
B‘‘
0
New Income expansion path
1 '
1
E C‘
C C‘‘
Review questions and exercises
Review questions
7.1. (Intertemporal budget constraint): Can a country starting out with b0* 0 have TB<0? What about perpetual CA deficits?
7.2. Following the findings of Kuznets in the late 1940s, it was recognized that any theory of aggregate consumption should be able to explain two empirical facts: that C/Y is smaller on average during boom periods and greater on average during slumps; that in the long run there is no tendency for the C/Y ratio to change. Are these facts consistent with the permanent income hypothesis?
7.3. (Interest rate determination in a closed economy): Referring to a 2-period economy with exogenous output, explain the following statement (http://andolfatto.blogspot.pt/2011/11/negative-real-interest-rates.html): “The decline in real rates that has taken place, especially since the beginning of 2011, is a troubling sign.
A negative 5-year rate implies that current output is now less valuable than future (6 year) output. In other words, (claims to) future output are now trading at a premium. This premium may be signaling an expected scarcity of future output. If so, then this is a bearish signal”
7.4. Explain why the economy-wide saving rate varies with the share of working-age population in total population.
7.5. Consider an economy where some fraction () of the population consists in “HTM”
consumers, and the remaining are “Ricardian” consumers (1-).
a) What are the implications of a small for the impact of a fiscal cut on private consumption?
b) Do you expect to be larger or smaller in the US, when compared to other economies (e.g, emerging)?
c) In any given country, do you expect to increase or to decrease during a financial crisis? What are the implications for the impact of taxation on private consumption and on the Current Account?
Exercises
7.6. (Optimal Consumption): Consider a household who leaves only two periods and whose expected production pattern is Q1 and Q2. Also assume that his lifetime utility function is given by U lnC1lnC2 and that the interest rate is r1.
a) Plot the household inter-temporal budget constraint in a graph.
b) Suppose first that Q1 0, andQ2 0. Is this household expected to be a saver or a borrower? Represent in a graph.
c) Suppose now thatQ1 0, and Q2 0. Is this household expected to be a saver or a borrower? Represent in a graph.
d) Discuss, with the help of a graph, the implications of an increase in the interest rate in cases b) and c). In particular, identify the wealth effect.
e) Obtain an expression for the household’ optimal savings in period 1 as a function of Q1 and Q2.
7.7. (Optimal Consumption): Consider a small open economy where the lifetime utility function of the representative consumer is
1 . 0 1 ln 1 ln 2
C C
U and the interest rate is
1 10%
r . Initially, assume that Q1 110 and Q2 110.
a) Plot the household’ inter-temporal budget constraint in a graph.
b) Find out the optimal consumption pattern. Is this household expected to be a saver or a borrower? Represent in a graph.
c) Examine the implications of a change in current output to Q1 68 . Is this household expected to be a saver or a borrower? Represent in a graph. Find out the consumer income and saving in period 2.
d) Consider in alternative that the negative shock affected future output, Q2 68. In this case, what would be time profile of consumption? Find out the consumer income and saving in period 2.
e) Finally, examine the implications of a permanent shock, so that Q1 Q2 68.
7.8. (Equilibrium: Closed vs open): Consider an endowment economy, where the preferences of the representative consumer are given by U lnC10.8lnC2. In this economy, current and future GDP are Q1 1125 and Q2 1350.
a) Find out the optimal saving in this economy as a function of the interest rate.
b) Find out the equilibrium interest rate assuming that the economy is closed to capital flows.
c) Suppose that the economy opens to international flows of capital and that the world interest rate is r* 25% . Describe the impact of trade openness in a graph and compute the current and future: (c1): consumption; (2) trade balance; (c3) GNI; (c4) current account; (c5) Net International investment position.
d) Departing for c), examine the implications of a fall in the interest rate to r*0%. Represent this in a graph. Will the country be better off or worse off?
7.9. (Infinite horizon) Consider an infinite horizon small open economy, where output is constant and equal to Q100 and households’ welfare is maximized when consumption is constant over time. Further assume that initially there are no external assets or liabilities and that the world interest rate is equal to r* 5%.
a) Find out the value of the country wealth 1.
b) Under the assumptions above, how much should private consumption be each year?
c) Now suppose that, because of an earthquake, current output in the first year happened to be Q1 79, only. Assuming that this shock was perceived to be transitory (that is, Q=100 in the following years), describe its impact (in the first year and thereafter) on: the optimal consumption path; the trade balance; NIIP;
NFIA, the current account?
d) Now suppose that the transitory shock above was the result of an increase in government expenditures and taxation because of a war (that is, Q100 but
1 21
1 T
G ). Assuming that in the following years GT 0, what would be impact on private consumption, trade balance, external debt, etc?
e) Returning to d), if the government decided to smooth the tax burden so as to make it equal every year, would that change the consumption path? And what about the trade balance, external liabilities, and so on?
7.10. (Life-cycle) Consider an individual consumer that starts her working life at the age of 25, without any wealth. Her life expectancy is 85 years and the retirement occurs at the age of 65. Further assume that in this economy is open and the interest rate and the rate of time preference are both equal to zero.
a) Consider the problem starting at time t=25. If she expects an annual income amounting to 3 until retirement, how much would she save and consume each year?
b) Assume that, in the year t=35, her income fell to 2. Quantify the impact of such change in the consumption and saving patterns, assuming that: (b1) the shock lasted for one year only; (b2) the shock was permanent. Discuss.
7.11. (Life-cycle): Consider an open economy, where people work 40 years earning 1000 units of output per year, and where the average retirement duration is 40 years. Further assume that in this economy the interest rate and the rate of time preference are both equal to zero. If half of population was working and the other half where retirees, how much would be the aggregate saving rate in this economy? And what if the proportion of retirees was 80-20%? Would it make a difference if the economy was closed to capital flows?
7.12. (Demography and interest rate) Consider a country where consumers live two periods and are all alike. The inter-temporal utility function of each individual consumer is of the form U lnC1lnC2. Each consumer produces 2 units of output when young and zero when old.
a) Find out the individual optimal consumption, current savings, and future savings as functions of the interest rate [A: C2=1+r)].
b) Suppose that that population in this country has been constant along time and equal to 500 young and 500 olds each year. In any particular year, how much should be total production, total savings by young and total savings by old? Compute the equilibrium (autarky) interest rate in this case [A: r=0%].
c) Suppose now that in year 1 there is a fall in the birth rate, so that in period 2 there will be 400 workers, only. How much will be current and next-period output in that case?
In this case, how much should be the interest rate in period 1? [A: r=-20%].
7.13. (Sudden stop; debt maturity). Consider a small economy open to international capital flows. The lifetime utility function of the representative consumer is given by
ln 1.25
lnC1 C2
U . The economy initial Net International Investment position is
*
0 100
b . The international interest rate is constant at r* 0 . It is also known that the country’ GDP in each period are given by Q1 200and Q2 350.
a) Assume first that all initial liabilities mature at the end of period 1. Compute the present value of the country total wealth and find out the optimal consumption pattern. Describe the equilibrium in a graph.
b) Given your answer above, compute the implied (current and future) values of: GNP;
Trade Balance; Current account; Net international investment position as a percentage of GDP.
c) Now suppose that in period 1, the country was no longer able to borrow in international markets. Describe the impact of the “sudden stop” in consumption, interest rate an in the current account. Compare with b) using a graph describing the country current account as a function of the interest rate.
d) As alternative, examine now the case where the initial Net International Asset Position matured at the end of period 2. Quantify the implications of a sudden stop in this case. Compare with c) and conclude.
7.14. (Terms of trade) Consider a small economy where the life-time utility function of the representative consumer is given by U lnC1
lnC2 1.2
, where C refers to an imported good. Production in this economy is fully exported and given by Q1 264 and2 242
Q . This economy is open to capital flows and the international interest rate is 1
.
* 0
r . The initial net international financial position is zero. The country terms of trade, given by TT PQ PC , are expected to remain constant at TT1 TT2 1.
a) Write down the country inter-temporal budget constraint in units of the imported good.
b) Compute the optimal consumption pattern, as well as the corresponding trade balance, and current account [A:TB1=0].
c) Now suppose that the country suffers a temporary terms of trade deterioration, so that 8
.
1 0
TT . Examine the impact of this change on current and future consumption, income, savings, trade balance and current account [A: TB1=-24; Y2=239.6].
d) Now consider that the terms of trade deterioration was permanent, that is 8
.
2 0
1 TT
TT . What would be the impact on the current account? Discuss [TB1=0].
7.15. (Fiscal policy, Lump-Sum Taxes). Consider an economy where the preferences of the representative consumer are given by U C1C2. this economy, GDP is constant at
1 100
2 Q
Q .
a) Assume first that the economy is closed to capital flows and there is no government.
Find out the optimal consumption path and the domestic interest rate [A: 100;100;0].
b) Now assume that the government set G1T1G2 T2 20. Find out the optimal consumption path and the domestic interest rate. How much will be private and government savings in this case? [A: 80; 80; 0; 0; 0].
c) Suppose that, departing from (c), the government decided to eliminate all taxes today and clear all government debt in period 2. In that case, how much will be private consumption, private savings, government savings and the domestic interest rate in period 1? Who would be holding the government debt? [A: 80; 20; -20; 0].
d) Consider now the case in which the government policy is given by:G1 40, G2 0 and T1 20. Assuming that all debts clear at the end of period 2, find out the optimal private consumption, private savings, the domestic interest rate and government savings. Compare with b and discuss. [A: 60; 20; 2/3; -20].
e) Departing from d) would a shift of all taxation to period 1 (that is T1 40) change the optimal private consumption and the interest rate? What about private savings? [A:
60; 2/3; 0].
f) Assume now that the economy was open to capital flows and that both the private sector and the government could borrow or lend any amount of output at the international interest rate r* 0 . Examine in this case the implications of the following fiscal policy: G1 G2 20 and T1 0: in particular, find out the optimal private consumption, private savings, government savings, trade balance in period 1. . In this case, who would be holding the government debt? Compare to c). [A; 80; 20; - 20; 0].
g) Examine the implications of an anticipation of government expenditures to period 1:
1 40
G , G2 0 and T1 20. Assuming that all debts clear at the end of period 2, find out the optimal private consumption, private savings, government savings and trade balance. In this case, who would be holding the government debt? Compare to d). [A: 80; 0; -20; -20].
h) Departing from g) would a shift in taxation to period 1 (that is T1 40) change the optimal private consumption and the trade balance? What about private savings? Who would be holding whose debt in this case? [A: 80; -20; -20].
7.16. (Government, Lump Sum Taxes): Consider an endowment economy closed to private capital flows. The lifetime utility function of the representative consumer is given by U lnC1
lnC2 1.125
. In this economy, GDP is constant at Q2 Q1 750.a) (Equilibrium interest rate): Compute the private consumption and the equilibrium interest rate in this economy, assuming that government expenditures are equal to
1 250
2 1
2 G T T
G . Explain the intuition.
b) (Anticipated shock): Now suppose that the government announces an increase in future spending to G2 T2 350. Explain why the equilibrium real interest rate changes the way it does.
c) (Tax cut) Departing from (b), suppose that the government decided to reduce taxes today by T1 100, with government bonds being sold domestically). Would this policy change the optimal consumption pattern? Quantify.
d) (Non-equivalence) Suppose instead that the tax cut in period 1 was financed by international borrowing at a zero-interest rate, so that the CA turned negative (TB1 100). Quantify the impact of this policy on private consumption in period 1 and in period 2, as well as on the interest rate. Would consumers be better off?
Explain the intuition.
7.17. (Distortionary taxation): Consider an endowment economy where the lifetime utility function of the representative consumer is given by U C1C2. In this economy, GDP is constant at Q2 Q1 100. The economy is open to capital flows, being r* 0.
a) Find out the optimal path of consumption and private savings, as well as of the CA.
b) Suppose now that the government launched a lump sum tax today, T1 20, which proceeds are returned to consumers, as a lump-sum transfer, in period 2. Describe the impact of this policy on: private wealth; pattern of consumption and private savings;
national savings; CA. [CA=0].
c) Assume that, instead of lump-sum, the tax was proportional to private consumption, and returned in the form of a subsidy in period 2, also proportional to private consumption. Assuming that the government inter-temporal budget constraint,
2 0
2 1
1C C
, was met: find out the optimal consumption pattern. Considering in particular the case with 10.25, find out the implied consumption levels in period 1 and 2, private savings, national savings and the CA. Represent in a graph, comparing to a). Is the consumer better off? [CA=20]
d) Repeat exercise (c), assuming instead that the economy was closed to capital flows.
[r=0%].
7.18. (Tax smoothing). Consider an open economy where the lifetime utility function of the representative consumer is given by U lnC1lnC2. In this economy, Q1 110,
2 90
Q , G1 25, G2 15, and Q2 75. In this economy, taxes are proportional to private consumption. Further assume that the international interest rate is r1* 0.
a) Find out the household’ and the government budget constraints.
b) Find out the optimal consumption in period 1 ad in period 2 as a function of the tax rates 1 and 2.
c) Suppose the government sets the tax rates so as to maximize the households’ utility, given the level of expenditures. Find out: (c1) the optimal tax rates 1 and 2; (c2) the implied consumption levels; (c3) private savings each period; (c4) government savings; (c5) the current account; (c6) the trade in assets in period 1.
7.19. (Tax increase, heterogeneous agents): Consider a economy where the utility function of the representative consumer is equal to U C1C2 and each consumer is endowed with Q1Q2 80. In this economy, there are two consumers. One is Ricardian,