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Macroeconomic Analysis: handout, Miguel Lebre de Freitas ([email protected])

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7. The two-period endowment economy

Index:

7 Two-period endowment economy ........................................................................... 3

7.1 Introduction ........................................................................................ 3

7.2 Inter-temporal budget constraints ...................................................... 4

7.2.1 Assets ......................................................................................................... 4 7.2.2 Expenditure, income and the current account ............................................ 5 7.2.3 The household’ budget constraint .............................................................. 6 7.2.4 The government budget constraint ............................................................. 7 7.2.5 The economy’ budget constraint ................................................................ 7

7.3 Optimal Consumption ........................................................................ 8

7.3.1 Preferences ................................................................................................. 8 7.3.2 Euler equation ............................................................................................ 9 7.3.3 Optimal consumption ............................................................................... 10 7.3.4 What happens when the interest rate increase? ........................................ 11

7.4 The Ricardian equivalence ............................................................... 14

7.4.1 Government bonds are not wealth ........................................................... 14 7.4.2 The Ricardian equivalence ....................................................................... 15 7.4.3 Savings are impacted! .............................................................................. 16

7.5 Macroeconomic equilibrium ............................................................ 18

7.5.1 Expenditure, Income and Current Account functions.............................. 18 7.5.2 Open economy ......................................................................................... 19 7.5.3 Closed economy ....................................................................................... 19

7.6 Temporary versus permanent changes in income ............................ 21

7.6.1 Temporary output expansion ................................................................... 21 7.6.2 Anticipated (future) output expansion ..................................................... 23 Box 1 - Portugal, Greece, and the Euro ............................................................... 25 7.6.3 Permanent output expansion .................................................................... 26 7.6.4 Change in real wealth............................................................................... 28 Box 2 – Terms of trade changes .......................................................................... 28 Box 3 - Infinite horizon ........................................................................................ 29 Box 4 – Life cycle model ..................................................................................... 32 Box 5 – Overlapping generations ........................................................................ 33

7.7 Government expenditures ................................................................ 34

7.7.1 Temporary expansion of government consumption ................................ 34 7.7.2 Government consumption and twin deficits ............................................ 35 7.7.3 Anticipated government expenditures ..................................................... 36

7.8 Failure of the Ricardian equivalence ............................................... 36

7.8.1 The case with borrowing constraints ....................................................... 36 7.8.2 Inter-generational effects ......................................................................... 39 7.8.3 Distortionary taxation .............................................................................. 39 7.8.4 Numerical example .................................................................................. 42


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7.9 Summary .......................................................................................... 43

Appendix 1 - Consumption and the interest rate ......................................... 44

Review questions and exercises ................................................................... 48

Review questions ................................................................................................. 48 Exercises .............................................................................................................. 48


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7 Two-period endowment economy

7.1 Introduction

The main limitation of the one-period model is that it can only account for intra-

temporal decisions. Intra-temporal decisions refer to the problem of allocating resources

across different activities within a period. In real life many decisions are inter-temporal: they

refer to choices regarding the allocation of resources over time. For example, borrowing and

lending involve contracts that commit one of the parties to pay the other a given amount in

the future. By not accounting for the time dimension, the one-period economy is doomed to

ignore borrowing and lending, and hence to assume that all agents have their budgets

balanced. This is an obvious limitation, because governments, for instance often engage in

fiscal deficits. Another limitation of the one period model is that it cannot account for

changes in output that are temporary in nature. When households perceive income shocks to

be temporary, they may try to smooth consumption over time, using the financial system.

This behaviour impacts on key macroeconomic variables, such as the interest rate and the

external accounts.

In this note, we introduce the temporal dimension in the simplest possible manner.

We consider an economy with two periods, only. To keep things simple, we abstract from the

choice between consumption and leisure. More precisely, we assume that each period there is

an exogenous supply of the consumption good. We also assume that price mechanism ensures

the equality between demand and supply. In this economy, there is a representative

household, a government, and an external sector. Depending on the institutional setup under

consideration, the economy may be open or closed to capital flows.

In Section 2, we introduce the main accounting identities that will frame our two-

period model. In Section 3, we describe the microeconomics of consumer behaviour. In

Section 4, we integrate the household and the government budget constraint to illustrate a

controversial proposition in macroeconomics known as the Ricardian equivalence. In Section

5, we embed the consumption function in the macro-model to characterize the

macroeconomic equilibrium in the open and in the closed economies. In Section 6 we see

how the macroeconomic equilibrium changes with temporary and permanent changes in

output. In Section 7, we analyse the effects of changes in government expenditures. In


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Section 8, we show how different types of market failures may cause the timing of taxation to

matter. In Section 9, we summarize the main ideas.

7.2 Inter-temporal budget constraints

Consider a small economy with two periods: current (period 1) and future (period 2).

In this economy, the amount of output available in each period tQ is exogenous (endowment

economy). Along this handout, we consider three institutional units: the household sector, the

government sector, and the external sector.

7.2.1 Assets

In this economy, households can hold assets in the form of government bonds ( Gtd )

and/or foreign bonds ( *tb ):

* Gt t tb b d (1)

Assets are stock variables, in contrast to income, that is a flow. Flow variables are

defined relative to a given period. Stock variables are defined at a given moment in time. In

what follows, we assume that stock variables are defined at the end of each period. Thus, for

instance, tb refers to the household’ net financial position at the end of period t.

In what follows, we experiment with two possible institutional setups regarding the

availability of foreign assets: the case where agents are allowed to borrow or lend abroad

(open economy); and the case in which the economy is closed to capital flows

( 0* tb ).Whenever government and foreign bonds are both available, it is assumed that they

are perfect substitutes, implying that the domestic interest rate will be equal to the foreign

interest rate:

*t tr r . (2)

In the case of openness, it is assumed that the home economy is small, implying that

the foreign interest rate is exogenous. When the economy is closed, the domestic interest is

determined by the domestic demand and supply of loanable funds. The autarky interest rate

will be denoted as at tr r .


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7.2.2 Expenditure, income and the current account

Since there is no investment, Gross National Expenditure (or absorption), 1A , is equa

to the sum of private consumption, 1C , and government consumption, 1G :

1 1 1A C G (3)

The amount of output (GDP) each period tQ is exogenous in this model. Output can

either be consumed domestically of abroad. The equilibrium in the market for goods and

services requires:

t t t t t tQ A TB C G TB (4)

Where t t tTB Q A stands for the trade balance. Apart from domestic production,

agents in the home economy can be entitled with income from holdings of foreign assets.

National Income (GNI) therefore corresponds to the sum of domestic production tQ with

interest on foreign assets:

*1

*1 tttt brQY (5)

Remember that GNI refers to the income accruing to resident units, regardless of

whether it was produced inside or outside country borders. The income on foreign assets

captures the difference between the geographical criterium and the residence criterium.

In the Balance of Payments accounting, returns on foreign assets are recorded in the

Balance of Primary Income, which we denote for NFIA (Net Factor Incomes from Abroad).

The Current Account (CA) is therefore defined as1:

* *1 1t t t tCA r b TB (6)

Using (4), the Current Account can also be expressed as the difference between

National Income and National Expenditure:

1 In this note, international unilateral transfers (Secondary Income) are assumed equal to zero.


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t t tCA Y A (7)

7.2.3 The household’ budget constraint

Abstracting from valuation changes and unilateral transfers from abroad, the net

financial position of the private sector shall evolve over time according to:

tttttt CTQbrb 111 . (8)

The term tr refers to the interest rate on tb , and tT refers to a lump-sum tax. Another

way of writing (8) is:

1P P

t t t t t tb b Y T C S . (8a)

Where tttP

t QbrY 11 refers to the household’ gross income and PtS denotes for private

savings. Equation (8a) state the equality between household’ savings and net lending.

Since the consumer lives only two periods, no agent will be willing to lend to this

consumer beyond the end of period 2. This means that 2b cannot be negative (non-Ponzi

game condition). On the other hand, the consumer will not be willing to hold financial assets

beyond the end of period 2, because that would imply loss of lifetime consumption. Thus, she

will never choose a positive 2b . These two conditions together imply that the only possible

terminal condition in this model is 2 0b .

Using (8) for t=1 and t=2, and setting 02 b one obtains:

1

221100

1

21 1

11 r

TQTQrb

r

CC

(9)

Equation (9) is the intertemporal budget constraint of the household sector. It states

that the current value of household’ lifetime consumption must equal its lifetime wealth, net

of taxes. In the following, we denote the current value of the household’ life-time wealth with

the symbol 1 , that is:

1

2211001 1

1r

TQTQrb

(10)

Given the exogenous components of the household life-time wealth, the inter-

temporal budget constraint can be written is a shorter way, as:


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11

21 1

r

CC (9a)

Whit lifetime wealth being defined as in (10).

7.2.4 The government budget constraint

The government debt accumulates over time according to:

tttGt

Gt TGrdd 11 1 (11)

The second term in the right-hand side of (11) is called the government’ primary

deficit. Another way of writing (11) is

1 1 1G G Gt t t t t t tS T G r d d d (11a)

Where GtS denotes for government’ savings. Government Savings are equal to the

primary balance minus interest payments on government debt. The symmetric of Government

Savings is the government overall deficit.

Using (11) for t=1 and t=2 and setting 02 Gd (same argument as before), the

government inter-temporal budget constraint becomes:

2 20 0 1

1 1

11 1

GT GT d r G

r r

(12)

This equation states that the present value of tax revenues must equal the present

value of government expenditures plus the initial debt.

7.2.5 The economy’ budget constraint

Consolidating the household and the government sectors, one obtains the economy’

budget constraint. Subtracting (11) from (8) and using (4), (1), and (2), this gives:

tttt TBbrb *

1*

1* 1 (13)

Another way of writing (13) is

tttttt TBbrbbCA *

1*

1*

1* (6a)


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This equation states the equality between the CA and the change in the Net

International Investment Position2.

Adding households’ and Government Savings (8a) and (11a), and using (1), we obtain

national savings:

tttttGt

Ptt GCYbbSSS

*1

* (14)

Note that in this model national savings (14) are equal to the Current Account (7)

because investment is equal to zero. In a model with investment, the Current Account would

be equal to national savings minus investment.

Combining (12) and (9), and using (1), one obtains the economy’ intertemporal

budget constraint:

*1

2211*

1

21

*00 11

1r

GCGC

r

QQrb

(15)

Equation (15) implies that the economy life-time National Expenditure must equal the

country’ wealth in present value terms. This intertemporal resource constraint represents the

consumption possibilities frontier of the economy. Using (4), another way of writing (15) is

01

1 *1

21

*0

*0

r

TBTBrb (15a)

This condition states that the present value of future trade balances plus the inherited

external assets must equal zero.

7.3 Optimal Consumption

7.3.1 Preferences

2 Remember that the NIPP is also influenced by valuation changes. The current model does not capture this effect.


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Assume that the life-time utility function of the representative consumer is time

additive3:

1

, 2121

CuCuCCU (16)

Where tCu are the utility levels generated by consumption in each period

(instantaneous utility), and denotes for the rate of time preference. The rate of time

preference is the extra units of utility in the future that the consumer would require in

exchange of one unit of utility today and remain indifferent. This parameter captures the

household’ degree of impatience. As most utility functions, specification (16) implies that the

household prefers a diversified consumption basket (across time) than to concentrate most

consumption in a given period.

In most of our discussion, we will refer to a particular case of (16), where the

instantaneous utility function is logarithmic:

1

lnln, 2

121

CCCCU (16a)

An important property of the logarithmic utility function is that the elasticity of

substitution between current and future consumption is equal to one. This may not be entirely

realistic, but it simplifies the calculations a lot. In the appendix, we discuss other possibilities.

7.3.2 Euler equation

The consumer problem is to maximize (16) subject to (9). The first order conditions

of this problem deliver:

3 We are abstracting from the eventual positive effect of government spending on household’s utility. We could easily account for such case postulating household preferences of the form 2121 ,, GGVCCUW .

Since these preferences are additively separable, government expenditures only impact on the utility level of households without changing the indifference map in terms of current and future consumption. Hence, such an extension would not change any of the conclusions in the main text.


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1

2

1 11'

'r

Cu

Cu (17)

This condition, known as the Euler equation, states that the marginal rate of

substitution between current consumption and future consumption shall be equal to the

opportunity cost of current consumption in terms of future consumption in the market. The

interest rate can be interpreted as the price (opportunity cost) of current consumption: when

the interest rate increases, more future consumption is foregone by the fact that the household

is consuming today.

Another view of (17) is that the optimal consumption pattern depends on the

relationship between the market interest rate and consumer’ degree of impatience: the optimal

consumption will be increasing or decreasing over time depending on whether the market

interest rate exceeds or falls short the rate of time preference.

When the Lifetime Utility is of the form (16a), the Euler equation simplifies to:

11

2 11 rC

C (17a)

7.3.3 Optimal consumption

The other condition that comes out of the optimization problem is the budget

constraint, (9), or simply (9a) because all terms in the household life-time wealth are

exogenous. Solving together (17a) and (9a), the optimal consumption in the current period is:

1 1

1

2C

(18)

From (18), private consumption depends on all exogenous variables that determine

the household’ lifetime wealth, (10) including current and future output and lifetime taxation.

Figure 1 illustrates the consumer problem. The endowment point is described by point

E-The consumer’ optimum corresponds to point C, where the inter-temporal budget

constraint (9) crosses the income expansion path (given by equation 16a). In this case the

household is a borrower, because the optimal consumption today is higher than its current

endowment (C is at the right-hand-side of E). As the figure illustrates, the opportunity of

borrowing and lending increases welfare relative to the point without financial trade.


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Figure 1 – Optimal Consumption

E

11 r

0 0

1 1

1 r b

Q T

2 2Q T

2C

2C

1C1C

C

Income expansion path (Euler)

1

As for a numerical example, consider a case where 1001 Q , 802 Q ,

*1 0 0r b , and no government. In this case, the household wealth will be

180180100 *11 r , implying an optimal consumption equal to 901 C . Note that

the optimal consumption pattern is horizontal ( 9021 CC ) because the international

interest rate is equal to the rate of time preference (eq. (17a)).

7.3.4 What happens when the interest rate increase?

A question that naturally arises is how current consumption responds to changes in

the interest rate.

Referring to previous numerical example, let’s stick with the case with 0 , but

allow the interest rate to be any value. In this case, the household lifetime wealth will be

*11 180100 r , implying an optimal consumption function of the form

*1 1 11 2 50 40 1C r . Thus, for instance, when the international interest rate is

0*1 r the optimal consumption will be 9021 CC . When instead 25.0*

1 r , the

consumer life-time wealth reduces to 1641 , and the optimal consumption pattern

becomes 821 C and 5.1022 C . Note that this result is in line with the Euler equation:

when the international interest rate is higher than the rate of time preference, the household

prefers to postpone consumption instead of choosing a horizontal consumption path.


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To further explore the relationship between current consumption and the interest rate,

consider again the consumer budget constraint, (9). In this equation, the interest rate appears

both on the left-hand side and in the right-hand side. In the left-hand side, it plays the role of

relative price of current consumption versus future consumption. On the right-hand side, it

enters as a discount factor that determines the present value of future income. This second

channel gives rise to a “wealth” effect that adds to the conventional substitution and income

effects.

Figure 2: Change in interest rate without wealth effects

C2

C1

A

B

A‘

New Income expansion path

C1A C

1B

Old incomeexpansion path

E1

C2A

C2B

Figure 3: Change in interest rate with wealth effects


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C2

C1

A

C


C1A


2AC

2CC

E

1CC0

B

A‘

1'1

To disentangle the different effects, assume for a moment that 022 TQ , so that

changes in the interest rate do not alter the value of household’ wealth (there is no future

income to discount). Figure 2 describes the effect of an increase in the interest rate in that

case: there is an upward rotation of the budget constraint. The decline in the relative price of

future consumption gives rise to an inter-temporal substitution effect, whereby the household

is induced to substitute current consumption for future consumption. This inter-temporal

substitution effect corresponds to a move from the original point A to a notional point A’, in

the new income expansion path. The inter-temporal substitution effect is negative, because

when the interest rate increases, current consumption declines. On the other hand, since

future consumption is now cheaper, the household purchasing power (or “real income”)

increases. Thus, there is a positive income effect, through which the household consumes

more today and more in the future4. This effect is illustrated in Figure 2 with the move from

the notional point A’ to point B, along the new income expansion path.

4 Note that we are discussing the “income effect” on 1C . As for 2C , substitution and income effects

reinforce each other, delivering an unambiguous positive relationship between 2C and the interest rate, holding

wealth constant.


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In sum, holding the right-hand side of (9a) (life-time wealth) constant, an increase in

the interest rate induces the household to decrease current consumption through a substitution

effect (future consumption becomes relatively cheaper) and to increase current consumption

through an income effect (the purchasing power of a given amount of wealth increases).

Whether the sum of these two effects is positive or negative, it depends on the utility

function. In the case illustrated in Figure 2, these two effects, from A to A’ and from A’ to B

exactly cancel out. This is because we are assuming a logarithmic utility function (16a).

However, this is not a general case (see Appendix 1 for other possibilities).

Now, consider the possibility of wealth effects ( 022 TQ ). When the interest rate

increases, the present value of future income declines, causing lifetime wealth (10) to decline.

In Figure 3, this is represented by a leftward shift of the budget constraint. Departing from the

notional point B (same as in figure 2), the leftward shift of the budget constraint means that

the household is now poorer. Through this “wealth effect”, current consumption decreases

along the new income expansion path, from B to C. Because this “wealth effect” comes on

the top of two other effects that exactly cancel out, the total effect in the relationship between

consumption and interest rate (from A to C) is unambiguously negative. The “wealth effect”

is what causes the negative relation between current spending and the interest rate found in

the numerical example above.

7.4 The Ricardian equivalence

7.4.1 Government bonds are not wealth

The household wealth (10) depends on taxes. Taxes, however, are not collected for

nothing. Tax revenues are used to pay for government consumption and to service the

government debt. As long as the government is in shape with its inter-temporal budget

constraint (12)), perfectly informed households will know that any debt issued by the

government today will be matched by more taxes tomorrow. Being aware of this, forward

looking households will assess their lifetime wealth (10) taking into account the government

budget constraint, (12).

Substituting (12) in (10), one obtains an alternative view of the private sector wealth:

* * 2 21 0 0 1 1

1

11

Q Gb r Q G

r

(10a)


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This equation brings three novelties relative to (10):

- First, the household’ life-time wealth depends negatively on government

consumption in each period. Government expenditures do impact on

household wealth, because they come along with higher taxation, today or in

the future. From the household point of view, government expenditures act

like an income loss (“confiscation”).

- Second, what matters for wealth is the total amount of government spending

to be financed, not the timing when taxes are paid. The timing of taxation

does not affect private wealth.

- Third, government debt disappeared from the household’ inter-temporal

budget constraint, (10a). Since government bonds are no more than future

taxes, when the government issues government debt is only postponing the

timing of tax collection. When households buy government bonds, they are

just hedging against future taxation. For the private sector, government

bonds are not net wealth.

7.4.2 The Ricardian equivalence

Taking into account the fact that the government meets its intertemporal budget

constraint, optimal consumption in period 1 can be rewritten replacing (10a) in (18). This

gives:

* * 2 21 0 0 1 1 *

1

11

2 1

Q GC b r Q G

r

(18a)

This equation shows that what matters for consumption spending is the time profile of

government expenditures: the timing of taxation does not alter consumption choices.


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The irrelevance of government debt for consumption decisions is known as the

Ricardian Equivalence Theorem5. The theorem states that, for any given level of government

expenditures, differing taxes in time does not affect private wealth or consumption. In short,

deficits and taxes are equivalent avenues to finance government expenditures.

7.4.3 Savings are impacted!

The Ricardian equivalence states that the timing of taxation does not change the

private sector lifetime wealth and consumption. But it does not say that everything remains

the same. The composition of national savings between private and government savings is

affected.

To see this, consider first the following numerical example: 10021 QQ ,

021 GG , Gdb 0*0 , and 0* r . In this economy, output is constant over time, so the

household does not need to smooth anything. In case of no taxes, optimal consumption and

the current account will be 10021 CC , and 1 0CA . Now suppose that the government in

this economy launched a subsidy today, 201 T to be financed with a tax in period 2,

202 T . From (10) you see that private wealth does not change. In consequence,

consumption does not change.

But the following question arises: if a tax cut comes along with a higher disposable

income at the private sector, where is the extra income going to? First, note that household

disposable income is given by 1 1 1 1Y T Q T . When taxes are zero, the household

disposable income is 1 1 100PY Q and the household saving is 0111 CYS PP . Since in

this first scenario government savings are also zero, domestic savings are 0111 GP SSS .

When instead 201 T , the household disposable income becomes 1 1 100 20 120PY T .

Since the household knows that the extra disposable income will come at the cost of future

5 Barro, J, 1974. “Are Government Bonds Net Wealth,” Journal of Political Economy, 1974, volume 82, pages 1095-1117.


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taxes, he will save this amount, consuming only 1001 C . Hence, private savings will be

1 1 1 1 20P PS Y T C . At the end of period, the household asset holdings will be 201 b .

Symmetrically, the government needs to finance its deficit: 20111 TSd GG . Thus, the

household buys the government debt, and the country national savings remains unchanged.

Formally, using (18a) in (8a) the expression for private savings is as follows:

* * 2 21 1 1 1 1 0 0 1 0 0 1 1 *

1

11

2 1P P Q G

S Y T C Q r b T r b Q Gr

(19)

This expression reveals that private savings do depend on current taxes. A

postponement of taxation, with 2 1 11T T r , impacts on private savings:

111 TbS P .

Government savings are also impacted. From (11a):

tGt

Gt TdS .

Summing up, National Savings remain constant: 0*1111 bSSS GP .

All in all, the tax cut today only impacts on the profile of private and Government

savings without any impact on private consumption and national savings: the fact that

households have more disposable income today and less in the future translates into higher

private savings today, that exactly match the government financing needs that the tax cut

brought about.

Figure 4 provides an illustration of the Ricardian equivalence, departing from a

situation where the Government savings are initially zero. In the figure, the right panel shows

the National Savings (19), as a positive function of the interest rate, and the left panel shows

the private sector savings and the government deficit. We assume that initially private savings

and the government deficit are both equal to zero (point A). With the tax cut, the government

deficit increases to point B, and private savings increase exactly by the same amount. In the

right panel, nothing has changed.

Figure 4: Ricardian equivalence


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1 1,P GS S

1PS

SG1

S1

1S11 r 11 r

AB

A=B

7.5 Macroeconomic equilibrium

We now embed the consumption function (18a) in our simple micromodel, to see how

the equilibrium looks like in a closed and in an open economy.

7.5.1 Expenditure, Income and Current Account functions

National Expenditure is defined by (3). Given (18a), the National Expenditure in

period 1 as a function of the other parameters in the model will be:

1 11 1 1

1

2A C G G

(20)

Since the consumer life-time wealth 1 is a negative function of the interest rate,

National Expenditure is also a negative function of the interest rate. National expenditure is

represented with a negative slope in the left-panel of figure 5. In the same panel, we represent

National Income (5) by a vertical line (note that in the context of this model output is given,

and *or refers to last’ period interest rate, not to the current period interest rate).

The difference between National Income and National Expenditure is the Current

Account, (7):

* *1 1 1 1 1 0 0 1 1

1

2CA Y C G Q r b G

(21)

At the right-hand side of figure 5, the Current Account is represented with a positive

slope, mirroring the difference between National Expenditure and National Income.


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Remember that in this model, the Current Account is also equal to National Savings, because

there is no investment: 1 1 1 1P GS S S CA .

7.5.2 Open economy

In the figure, the equilibrium corresponding to the open economy is described by

point 1. Since the international interest rate is such that national expenditure exceeds national

income, there is a deficit in the current account.

Referring to our numerical example with 1001 Q and 802 Q , the expression for

the Current Account will be: 1 1 1 1 1 150 40 1CA Y C Q C r . In case the

international interest rate is 0*1 r , optimal consumption will be 901 C , implying 1 10CA .

When the international interest rate increases to 25.0*1 r , the optimal consumption today

becomes 821 C , implying 1 18CA . Then, in period 2, income will be

5.845.480*1

*122 BrQY , and the Current Account in period 2 will be

* *2 2 2 2 2 1 1 18CA S Y C TB r B .

Figure 5: National expenditure, National Income and the Current Account

11 r

1CA

11 r1Y

*11 r

1Q

1 1 1A C G

*11 r

1 0CA 1 0CA

11 ar 1‘

1 1

7.5.3 Closed economy

The key distinction between the open and the closed economy refers to the variables

that are exogenous and those that are endogenous: in the open economy, the international

interest rate is exogenous (we are considering a small open economy), and the Current


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Account is endogenous; in the closed economy, the Current Account is zero, and the

domestic interest rate is endogenous.

In figure 5, the autarky equilibrium corresponds to the point where domestic output

equals national expenditure. The autarky interest rate 1ar is such that *

1 0 0CA b in (21) and

in (10a). This gives:

2 21

1 1

1 1a Q Gr

Q G

(22)

To interpret equation (22), note that, in a closed economy there can be no

consumption smoothing in the aggregate: t t tC Q G , for t=1,2. Replacing this in the Euler

equation (17), we obtain the Autarky interest rate (22). In terms of the consumer problem

described in figure 1, the autarky interest rate corresponds to the slope of the indifference

curve at point E.

Comparing the autarky interest rate to the international interest rate, we can assess

whether a country has comparative advantages in borrowing or in lending. If, for instance the

autarky interest rate is higher than the international interest rate, this means that current

consumption is relatively more valuable at home than abroad. Thus, the country has

comparative advantages in future consumption. After openness, the country will export future

consumption and will import current consumption. If, in alternative, the international interest

rate is higher than the autarky interest rate, it will pay for the country to lend abroad,

engaging in a Current Account surplus.

Returning to our numerical example with 1001 Q and 802 Q , the closed economy

equilibrium requires 10011 QC and 8022 QC . Replacing this in the Euler equation,

we get 11 80 100 0.8ar . Note that the autarky interest rate in this case is lower than the

rate of time preference: reflecting the abundance of current consumption relative to future

consumption, the price of current consumption must fall the enough to induce the consumer

to optimally choose the spend more today than in the future.

A question that may arise is about the meaning of the “autarky” interest rate: after all,

if the economy is closed, which assets are being traded at that rate? In fact, the autarky

interest rate will apply to any financial transaction between domestic agents, ether between

private agents and the government, or between heterogeneous private agents.


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To see this, assume that the economy was composed by two consumers, A and B with

similar preferences but different endowments: 1 65AQ , 1 35BQ , 2 20AQ , 2 60BQ . Since

at the aggregate level 1001 Q and 802 Q , we already know that the domestic interest rate

will be 11 80 100 0.8ar . The question is how these quantities are allocated across

resident consumers in the first and in the second period. Given the domestic interest rate, it is

easy to see that 1 165 20 1 90A ar , 1 135 60 1 110B ar , implying

1 1 2 45A AC , 1 55BC , 2 1 11 36A A aC C r , and 2 44BC . This means that agent A

saves 1 1 1 65 45 20A A AS Q C in period 1, and dissaves 2 2 2 20 36 16A A AS Q C

in period 2. These savings by consumer A will meet the borrowing needs of consumer B

1 1 1 35 55 20B B BS Q C , and 2 2 2 60 44 16B B BS Q C . Note that there is only

one interest rate that makes these consumption patterns compatible, and this is exactly the

autarky interest rate6.

7.6 Temporary versus permanent changes in income

In this section, we examine how the macroeconomic equilibrium changes with

temporary and permanent changes in income. To abstract from other complications, we

assume for a moment that there is no government, and that endowments are revealed before

decisions regarding 1C are taken7.

7.6.1 Temporary output expansion

6 In alternative, you may interpret the closed economy as the world economy, and consumers A and B as the home country and the rest of the world respectively. In that case, the “autarky” interest rate will correspond to the equilibrium interest rate in the global economy.

7 The case in which decisions on current consumption are taken under uncertainty regarding future output are analysed in a different handout.


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Consider an open economy where initially TB=0. Then, assume that output expands

in the first period, but remains constant in the second period: 01 Q , 02 Q , implying

that 1 1 0Q (equation (10)). Since the household is now wealthier, he will react

consuming more. Since both current and future consumption are normal goods, the household

will consume more today and more in the future.

Formally, from (18) and (10), we see that the change in current consumption will be:

1

1

10

2

C

Q

The parameter 121 measures the increase in private consumption when

current disposable income increases by one unit and is labelled the marginal propensity to

consume on current income. Hence, for instance, when 0 , a unitary change in current

output implies an increase in current consumption by one half, only8. The ability to borrow or

lend abroad implies that consumption today is impacted less than proportionally.

Since current consumption expands by less than current income, the consumer is

saving for the future. Thus, the impact on the CA will be positive:

1

1

1 11 0

2 2

CA

Q

The Current Account acts therefore as a shock-absorber against temporary changes in

income. Note that the surplus in the CA is matched by an accumulation of external assets by

the household, 0* tt bb . The household is lending to smooth consumption.

Figure 6: Temporary output expansion

8 In this model the marginal propensity to consume out of current income is roughly one half because our household only lives two periods. Extending the model to larger horizons, the marginal propensity to consume out of current income becomes much smaller (see boxes 3 and 4).


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1CA

1 1 1A C G

11 'ar

1Q

11 r11 r1Y

1 0CA 1 0CA

1 'Y

*11 r

In figure 6, we examine the implications of a temporary output expansion, referring to

the locus of National Income, National Expenditure, and Current Account. In the left panel,

an increase in current output causes the National Income curve to shift to the right. Since the

household’ life-time wealth increases, there is also an increase in current consumption,

causing the National Expenditure curve to shift rightwards as well, although with a lower

magnitude. In the right panel, the curve describing the Current Account (and National

Savings) shifts to the right. If, as represented, the initial situation was that of external balance,

then with the temporary output expansion the current account would turn to positive,

reflecting the higher national savings.

If the economy was closed, the autarky interest rate should decline: intuitively, at the

initial interest rate the household would like to save to postpone some of the extra income

received this year. Since this is not possible in a closed economy, there will be an excess

supply of savings in the market for loanable funds. Thus, the interest rate must decline, to

induce agents to consume more today.

7.6.2 Anticipated (future) output expansion

Now consider the arrival of some new information that makes the consumer anticipate

a higher income in the future. By assumption, the new information becomes available before

current consumption is chosen.

Formally, consider the case with 01 Q and 02 Q , implying

1 2 11 0Q r . Given (18), In an open economy, the change in consumption will be:


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20/09/2021

1*

2 1

1 10

2 1

C

Q r

As before, the fact that the household life-time wealth has increased impacts

positively on current consumption, with a weight that is less than one, because the households

desires to smooth the gain along the two periods.

Because consumption today expands, the Current Account worsens:

1*

2 1

1 10

1 2

CA

Q r

The decline in private savings today is matched by a deficit in the Current Account,

implying that the private sector accumulates external liabilities: 0*11 bb . Now, the

household is borrowing to smooth consumption.

An interesting implication of this example refers to the causality between

consumption and future output: if one run a linear regression where output was a function of

past consumption, eventually one would find a significant positive correlation. Based on this

finding. one could be tempted to conclude that an expansion of current consumption causes

an expansion of future output. In light of the model, however, the causality runs in the

opposite direction: it is a future output expansion that is causing the expansion in current

consumption, not the other way around. Statistical precedence and causality are not

necessarily the same.

Figure 7: Anticipated output expansion

1CA

1 1 1A C G

*11 r

11 'ar

1Q

11 r1Y 11 r

1 0CA 1 0CA

*11 r


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In figure 7, we see that the Current Account schedule shifts leftwards (right panel).

The reason is that the higher future output causes the expansion of National Expenditure (left

panel). If the initial equilibrium was that of external balance, as represented, then the

anticipated output expansion turns the Current Account to a deficit.

In a closed economy, the autarky interest rate should increase, reflecting the relative

scarcity of current output: intuitively, the fact that future output is now higher induces

household to consume more today, trying to borrow. Since in a closed economy there is no

net lending from abroad, the interest rate must rise to balance the market for loanable funds.

Box 1 - Portugal, Greece, and the Euro

During the run up to the euro, some countries in the EU periphery, namely Portugal

and Greece, faced large Current Account deficits. These deficits have a natural interpretation

in terms of the model sketched out above9.

First, because Portugal and Greece were the two poorest members of the Euro Area,

economic agents presumed that EMU participation would deliver a positive impact on these

countries’ growth prospects. In light of (18), when the household anticipates a higher 2Q , the

optimal response is to borrow abroad to expand current consumption.

Second, the openness to capital flows implied a decline in the borrowing interest rate.

To capture this, assume that the domestic interest rate was equal to *11 11 rr , where

represents the wedge between the interest rate at which Portugal and Greece could borrow

and the foreign interest rate. Using (10) and (21), and abstracting from the government sector

and the initial international investment position, we get:

2

1 1 1 *1

1

2 1

QCA Q Q

r

(21a)

9 Blanchard, O., Giavazzi, 2002. Current account deficits in the euro area: the end of the Feldstein-Horioka Puzzle? Brookings Papers on Economic Activity, vol. 33(2), 147-210.


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The juncture in the 1980s and early 1990s was characterized by a positive

reflecting the initial restrictions to capital mobility. Along the 1990s, in the context of

financial integration vis-à-vis the European Union, capital controls were eliminated, and the

regulatory system for financial markets was harmonized, increasing transparency and

reducing uncertainty in cross-border lending. On the top of this, the fact that Portugal and

Greece joined the EMU in 1998 implied the virtual elimination of currency risk. In terms of

the model, one may take these developments as causing the term to decline to zero. This,

in turn, amplified the effect of 2Q on current consumption, exacerbating the impact of

overoptimistic expectations on the CA deficit.

7.6.3 Permanent output expansion

Now, consider the case in which the economy experiments a permanent increase in

GDP. In this case, the household lifetime wealth increases by much more than in the previous

two cases (note that a permanent increase in GDP is just the sum of the two changes

described above).

Formally, this case corresponds to 021 QQQ , implying

2 11 1

1 1

20

1 1

Q rQ Q

r r

.

In an open economy, the impact on consumption will be:

11

2

2

1*

1

*11

21

r

r

Q

C

QQ


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This derivative can be labelled as the marginal propensity to consume out of

permanent income10. When r , a permanent change in income delivers a one-to-one

increase in private consumption.

The impact on the CA (21) is:

1 2

* *1 1 1

* *1 1 1

21 11 0

2 21 1Q Q

CA r r

Q r r

In the particular case in which *1r , the impact of the permanent output change on

current consumption is one-to-one: the increase in current disposable income is fully matched

by an increase in current consumption, implying that the Current Account does not change at

all. Intuitively, when the change in income is permanent, there is nothing to smooth out, so no

external borrowing is needed. When *r , the impact on the Current Account will not be

exactly zero, but it will be very close to zero.

The effect of a permanent output expansion when *1r is described in figure 8. In

this diagram, we depict the National Income and National Expenditure schedules 11 .

Assuming that the initial equilibrium was of internal and external balance, the permanent

shock caused both National Income and National Expenditure to shift rightwards. The

horizontal shift of the National Expenditure schedule is equal to that of National Income,

implying that the economy remains in external balance.

Figure 8: Permanent output expansion ( *1r )

10 The flow equivalent to the household’ life-time wealth was coined by Milton Friedman as “permanent income”. More precisely, the permanent income is the constant level of income that would deliver the same life-time wealth as with a fluctuating income stream. In our two-period model, that will be

11Q Q r , implying 12 1Q r r . As you may check, the marginal propensity to consume

out of permanent income, 1C Q is the expression above. For more this model, see box 3 [Friedman, M.,

1957, A Theory of Consumption Function, Princeton University Press New Jersey].

11 The reason for not representing the current account schedule is to skip unnecessary details. As you may check in the exercises, the CA schedule rotates around the origin.


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1 1 1A C G

1Q

11 r1Y

*11 r

7.6.4 Change in real wealth

Another source of variation in the household lifetime wealth refers to valuation

changes altering the market price of a country net international investment position, *0b . In

real life, this may be a consequence of changes in the exchange rate, stock market valuations,

or changes in bond prices driven by credit risk assessments.

From (18), and using (10), the impact on private consumption will be:

*10*

0

11 0

2

Cr

b

From (21), the impact on the Current Account will be:

*0* *1

0 0*0

111 0

2 2

rCAr r

b

Thus, in case of a valuation change, there will be an increase in consumption and an

erosion of the current account.

Box 2 – Terms of trade changes

In our baseline model, we consider an homogeneous good that can be either

consumed, imported or exported. In the real World, final goods that countries export tend to


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be different from the basket of goods that are consumed. These differences are more relevant,

the more a country is engaged in inter-industry trade, and the more specialized it is in few

commodity exports. Inter-industry specialization makes countries more vulnerable to terms of

trade shocks, that is, to changes in prices that affect asymmetrically imports and exports.

To analyse the impact of terms of trade changes in our model, let’s assume that all

production is exported, and that all consumption is imported. Further assume that foreign

assets are denominated in units of the imported good. When this is so, the inter-temporal

budget constraint can be written as:

* * 2 2 21 0 0 1 1 1 *

1

11

TT Q Gb r TT Q G

r

(10c)

Where Q CTT P P is the relative price of output in terms of the imported good. This

equation reveals that terms of trade changes can be interpreted as changes in current or in

future output.

For instance, when terms of trade improve today, there is a positive wealth effect and

the current income rises relative to future income, just like in a temporary shock. In

consequence, there will be an increase in consumption and an improvement in the Current

Account (the savings schedule moves rightwards). In the literature, the positive relationship

between terms of trade, savings and the Current Account is known as the Harberger-Laursen-

Metzler effect’12

Box 3 - Infinite horizon

In this box, we extend the model to the case in which the consumer has an infinite

live. In order to make our calculations simple, we assume that the international interest rate is

constant over time. The consumer maximizes a life-time utility function of the form:

12 Harberger, A. 1950. “Currency Depreciation, Income and the Balance of Trade.” Journal of Political Economy 58: 47-60. Laursen, S. and L. Metzler, 1950. “Flexible Exchange Rate and the Theory of Employment.” Review of Economics and Statistics 32: 281-99.


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1

11

ln

tt

tCU

, (16b)

subject to the budget constraint:

* *3 32 2

1 0 1 1* 2 * 2* *.... 1 ....

1 11 1

C QC QC b r Q

r rr r

(9b)

By analogy with the two-period case, you may guess that the Euler equation for two

consecutive periods will be:

1

1 *

1r

C

C

t

t (17b)

Focusing on the case in which the interest rate is equal to the rate of time preference,

we obtain an optimal consumption that is constant over time:

1tC C t

Replacing this in the inter-temporal budget constraint, we get

12**1 ....1

1

1

1

rrC

Implying:

1*

*

1 1

r

rC (18b)

In this case, a permanent shock QQQQ ...321 causes a change in life-

time wealth equal to:

Qr

r

*

*

1

1

From (18b), the change in current consumption when the shock is permanent is

1 2

1

..

1Q Q

C

Q

.

As in the two-period case, current consumption responds on a one-to-one basis to the

permanent change in income. Hence, the Trade Balance is not impacted.


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Now consider a temporary shock, with 01 Q and 0...32 QQ . In this case,

the change in wealth is 11 Q and the change in consumption will be:

11 *

*

1

1

r

r

Q

C.

As in the two-period case, the marginal propensity to consume out of current income

is less than one. Since we are dealing with an infinite horizon, smoothness across an infinite

number of periods delivers a marginal propensity to consume on temporary shocks that is

very small compared to the two-period case. The other side of the same coin is that the Trade

Balance is more impacted.

As for a numerical example, suppose that the interest rate is 25.0* r and that

initially *0 0b and 100...321 QQQ , implying 50025.025.11 . Assuming that

the interest rate is equal to the rate of time preference, the consumption pattern will be flat

100...321 CCC and the Trade Balance will be always equal o zero. Now suppose that

current output fell to 801 Q , implying 4801 . If the economy had no access to external

finance, current consumption would fall to 1 80C and the interest rate should increase in

light of (17b). If however households are able to borrow from abroad, consumption will fall

to C=96, only. That will mean a marginal propensity to consume out of current income equal

to 4 20 0.2 0.25 1.25 . The economy will run a deficit in the CA equal to 16 in the first

period, giving rise to *1 16b . Then, in the periods that follow, interest payments will imply

a negative NFIA each year, amounting to * * 4r b . The negative NFIA will exactly match

the difference between production Q=100 and consumption C=96 (the trade balance). The

Current Account each year after period one will be equal to zero.

Another view of this model consists in defining “permanent income”, Q , as the

constant output flow that would be equivalent to the life-time flow of the endowment 13:

13 Friedman, M., 1957, A Theory of Consumption Function, Princeton University Press New Jersey.


32

20/09/2021

32

12 2* ** *.... ....

1 11 1

QQQ QQ Q

r rr r

Solving for Q , this gives

*32

1 2* * *...

1 1 1

QQrQ Q

r r r

, implying

*

* *1 0 *

11

rb r Q

r

.

Substituting in (18b), the optimal consumption becomes equal to:

* *1 0C r b Q .

Replacing this in the expressions for the TB and the CA, we get

* *1 1 1 1 0TB Q C Q Q r b

1 1CA Q Q

Thus, there will be a surplus in the Current Account whenever current output is higher

than the corresponding permanent level. Also note that valuation changes that alter the value

of *0b impact on consumption and thereby on the TB, without changing the Current Account.

Consumption smoothing in this case corresponds to consuming from initial assets *0b exactly

the income that these assets deliver each year, * *0 0b r , without changing the amount invested in

assets.

Box 4 – Life cycle model

The analysis above focuses on consumption smoothing across business cycles. An

alternative angle to interpret the consumer’ problem along its lifetime. In this case,


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consumption smoothing will mean accumulating wealth during the working phase to spend

out of the accumulated wealth during the retirement phase14.

To examine this case, consider an individual in the working age that accumulated 1tb

so far. This individual expects to live N years, and then to retire when aged H. Along the

remaining of his working life, H-t, he expects a constant annual income amounting to Q . To

keep the analysis simple, we stick to the case with 0* r .

Since the household’ remaining working life consists in tH periods, his lifetime

wealth at moment t will be QtHbtt )(1 . Then, because the individual is expected to

live N periods, his lifetime budget constraint will be 121 ... Nttt CCCC .

Consumption smoothing will then imply a constant consumption stream, that is:

QtHbtN

C tt

1

1 for Ht . The marginal propensity to consume out of current

income, tNtH depends inversely on the age of the household (t). When the

household reaches the retirement age, his marginal propensity to consume out of income

becomes zero ( 1

1

tt btN

C , for Ht ).

An implication of the life-cycle model is that aggregate consumption and savings

depend on demography: because individual savings depend on which phase of their lifetime

individuals are, the age structure of population matters for aggregate savings.

Box 5 – Overlapping generations

The simplest way to analyse the relationship between demography and aggregate

savings is to assume that the society consists in two groups of individuals, only: workers and

14 Modigliani, F., Brumberg, R., 1954. Utility analysis and the consumption function: an interpretation of cross section data. In Kurihara, K., (ed.), Post-Keynesian Economics. New Brunswick NJ,: Rutgers University Press. Ando, A., Modigliani, F., 1963. The life-cycle hypothesis of saving: aggregate implications and tests. American Economic Review, March.


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retirees. Returning to the two-period model, assume that all households are alike and that all

income is generated in the first period (working life): 11 Q and 02 Q . Further assume that

the rate of time preference and the interest rate are both equal to zero.

In light of that model, each household will optimally save 0.5 in the first period and

will dissave 0.5 in the second period. Now, consider the implications of different population

structures. If, in a given period, there are as many retirees as there are workers, then the

market for loanable funds will balance domestically, without any need for the economy to run

an external imbalance: for instance, if there are 1000 workers and 1000 retirees, the 500

savings of workers will exactly match the -500 savings of retirees, and aggregate saving will

be zero. If, however, the economy has 1000 workers and 800 retirees, there will be an excess

savings amounting to 1001 S , implying a CA surplus of equal amount. If the economy was

closed, the interest rate should fall.

7.7 Government expenditures

7.7.1 Temporary expansion of government consumption

We now examine the effects of an increase in government expenditures. In particular,

consider an increase in government’ consumption today, while future expenditures remain

unchanged: 01 G , with 02 G . In an open economy, the impact on consumption will be

(eq. (18a)):

1 1

10

2C G

From the consumer point of view, a temporary increase in government spending acts

like a decrease in current output: the household will find itself poorer and will decide to

consume less, through the wealth effect. Because of consumption smoothing, the fall in

private consumption in the first period is less than the increase in government spending. From

(20), we see that the net effect on National Expenditure is positive:

1

1

1 11 0

2 2

A

G

From (21), we see that the impact on the current account is:


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20/09/2021

1

1

1 11 0

2 2

CA

G

Graphically, the impact of an increase in government expenditures in period 1 is

similar to that described in Figure 7: the increase in government expenditures causes the

National Expenditure schedule to shift rightwards, and the CA schedule to shift leftwards. In

a closed economy, that would imply an increase in the interest rate.

7.7.2 Government consumption and twin deficits

Without knowing how the fiscal expansion is financed, we cannot tell about its impact

on private and government savings. That will depend on when taxes are raised. To see this,

we consider a numerical example, and two extreme cases: first, the case in which the

government spending is fully financed with current taxes. Second, the case in which the

government spending is financed with debt today and taxes tomorrow.

Assume that *1 0r and 101 G . In this case, the fall in private consumption

will be 51 C and the CA will deteriorate to 1 1 1 5 10 5CA C G . The

impact on consumption and on the current account is independent of how the government

expenditures are financed. This is the Ricardian equivalence.

Consider first the case in which the increase in government spending is matched by an

increase in current taxes: 101 TG . In this case, private disposable income decreases by

10. Since private consumption decreases by 5, only, private savings decrease by

5510111 CTS P : the household must borrow. From whom? Obviously, from

abroad: the CA imbalance generates a capital inflow, amounting to 5*11 bb . We

conclude that, when the increase in government spending is financed by an equal increase in

taxes, Government Savings remain unchanged and private savings decline (the fall in

consumption is less than the fall in disposable income). The external imbalance is matched by

an imbalance in the private sector: there is no “twin deficit”.

Now consider the case in which the increase in government spending is financed with

bond issuance. In this case, 101 G and 01 T . Since 51 C , private savings

increase: 51 PS . Note however that the increase in private savings is less than the

government deficit 101 GS . Thus, National Savings (Current Account) are 1 5S . The


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household buys government bonds amounting to 101 Gd and issues a foreign liability

amounting to 5*1 b , ending up with a net worth of 51 b . In this case, the government

deficit translates into an external deficit, generating a pattern of “twin deficits”.

7.7.3 Anticipated government expenditures

Assume now that the government announces an increase in future consumption:

01 G , 02 G . From the household’ point of view, that will be equivalent to an

anticipated drop in future output. In an open economy:

1 1*

2 2 1

1 10

2 1

C A

G G r

1*

2 1

1 10

2 1

CA

G r

The household, perceiving a contraction in its lifetime wealth, reduces consumption

today. If current taxes do not change, the household will save, buying foreign assets.

Graphically, when the government announces an increase in future expenditures,

National Expenditure contracts (shifts to the left), and the Current Account schedule shifts to

the right. If the economy is closed, the interest rate will fall; if the economy is open, the CA

will improve.

7.8 Failure of the Ricardian equivalence

In the real World, there are many reasons why the Ricardian equivalence does not

hold exactly. These, in turn, are related to the failure of various assumptions in the model

above. The main reasons why the Ricardian Equivalence fails in the real world are as follows:

(i) the private sector faces borrowing constraints; (ii) lump sum taxes are not available; (iii)

households have shorter lifetime horizons than governments. In the following, we address

these three cases.

7.8.1 The case with borrowing constraints


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Along this handout we assume that the household can use the financial system to

smooth consumption. In the real World, however, not all households have easy access to

credit. Households in general face no restrictions in savings, but they often have trouble in

obtaining banking credit. In this case, we say the households faces a borrowing constraint.

When a household would like to borrow but finds no credit, his current consumption

is determined by current disposable income, only:

1 1 1C Q T (18c)

In this case, the marginal propensity to consume on current income is equal to one. In

the literature, this consumer has been labelled as “hand-to-mouth” (HTM).

To see how the timing of taxation impacts on private consumption under borrowing

constraints, let’s assume that some fraction of households in our economy are HTM, while

the remaining fraction 1- are Ricardian (with access to financial markets)15.

Consider an open economy, with 1 2 0 0G G b . Assume that the government

increases taxes today to be financed by a negative tax (transfer) tomorrow, that is, 1 0T

and 21 *

1

01

TT

r

. As we know, such policy is irrelevant for Ricardian consumers.

However, it affects the choices of HTM consumers, because they are constrained in the

amount of current consumption they can buy. The aggregate consumption function is a

combination of (18a) and (18c):

21 1 1 1 *

1

11

2 1

QC Q T Q

r

(18d)

Thus, the impact of a budget-neutral tax increase on aggregate consumption will be

15 Focusing on the United States, John Campbell and Gregory Mankiw found that roughly one half of aggregate income accrues to constrained households. You may guess that in less advanced economies such proportion will be much higher [Campbell, J., Mankiw, G., 1989. Consumption, Income, and interest rates: reinterpreting the time series evidence. NBER Macroeconomics Manual 1989, Cambridge MA: the MIT press, pp. 185-216.


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011

1

21 TrTT

C

There is an effect on private consumption, corresponding to the fraction of

constrained households in the economy.

To see the implication on national savings, note that private savings consist only on

savings of Ricardian consumers:

21 1 1 1 *

1

11

2 1P Q

S Q T Qr

(19b)

Adding (11a), National Savings will be:

21 1 1 1 *

1

1 11

2 2 1

QS CA T Q

r

(21b)

The impact of the tax hike on National Savings is 1 1 1 0S CA T . The

current account must have a surplus: since production cannot change, the fall in consumption

of HTM households must be met by exports: 1 1 1 1 0CA Q C T .

In terms of financing, the tax increase generated a government surplus equal to

1 1 0GS T , that must be matched with somebody else’ deficit. In the case of Ricardian

consumers, savings will decline in the exact amount to borrow from the government,

1 11 0PS T . HTM consumers’, however, cannot borrow. Hence, the fraction

of the current tax increase finds no counterpart in lower private savings. The only way for the

government to use the remaining funds generated by the tax hike is buying abroad the amount

of bonds equivalent to the fall in consumption of constrained households, that is

*1 1 0b T .

Summing up, in the case of an open economy, a tax increase will cause HTM to

decrease consumption, giving rise to a contraction of National Expenditure and an

improvement in the Current Account. This, in turn, calls for a capital outflow, corresponding

to the part of the government surplus that is not financing Ricardian consumers. If the initial

situation was of external balance, a “twin surplus” will emerge.

In a closed economy, current consumption cannot change. Thus, the interest rate must

fall. The intuition is as follows: departing from an autarky equilibrium, a tax hike will cause


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private disposable income to fall, and HTM consumers to consume less. The government

surplus generates an excess supply of loanable funds, depressing the interest rate. The later

must decrease the enough for Ricardian households to expand consumption the enough to

borrow from the government the exact amount of the government surplus16.

7.8.2 Inter-generational effects

Governments are infinitely lived, while households are not. In the real world, it may

be that governments engage in debt finance for long period of time, before rising taxes in the

distant future. The generation of old consumers today may find more or less irrelevant the

prospects of higher taxes in a distant future. In that case, they will interpret a lower taxation

today as an increase in their lifetime wealth.

In terms of our model, just assume that each household lives one period only, while

the government lives two periods. In that case, there is no consumption smoothing: if the

government levies a tax on the current generation to be paid by the next generation in the

future, then the disposable income of current generation increases and so will do current

consumption. Just as in the case of an HTM consumer, a tax cut will come along with higher

consumer spending and a deficit in the Trade Balance matched by the government deficit. In

period 2, the higher tax burden on the future generation will produce an opposite effect.

7.8.3 Distortionary taxation

Another reason why the Ricardian equivalence fails in the real world is that most

taxes are distortionary. Taxation impacts on relative prices, causing households to alter their

choices. When taxes are distortionary, the timing of taxation is not, in general, irrelevant.

To illustrate how distortionary taxation works on an endowment economy, assume

that taxes (transfers when negative) are proportional to consumer spending. Let the tax rates

16 See exercise 7.21.


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in period 1 and 2 be 1 and 2 . As for the remaining assumptions, just consider the simplest

case, with 021 GG , and Gdb 0*0 . The government budget constraint is now:

01

2211

r

CC

(12a)

The household inter-temporal budget constraint in this case is:

1

21

1

2211 11

11

r

QQ

r

CC

(9c)

Substituting (12a) in (9c), we see that the forward-looking household, being aware

that current taxes and future taxes are related, does not consider a change in taxes as affecting

his lifetime wealth directly17:

2 21 1 1

1 11 1

C QC Q

r r

(10b)

However, the slope of the inter-temporal budget constraint (9c) is affected by the

timing of taxation. The first order conditions of the problem of maximizing (16) subject to

(9c) imply:

12

1

1

2 11

11 r

C

C

(17c)

Since consumption taxes impact on the relative price of current consumption versus

future consumption, they show up in the new Euler equation, (17c)18. Replacing this in (9c),

delivers the optimal current consumption as follows:

17 In the closed economy, an indirect effect may however occur through a change in the interest rate.


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11

1 1

1

2

1

C (18e)

Equation (18e) reveals that consumption taxes impact on current consumption.

As for private savings, the following expression arises:

21 1 1 1 1 1

1

11

2 1P Q

S Q C Q Qr

(19c)

The expression for National Savings is therefore:

21 1 1 1

1 1

1 1

1 2 1

QS CA Q Q

r

(21c)

The impact of the tax cut in the savings schedules is described in figure 9. On the

right panel, the tax cut increases the government deficit, shifting the corresponding schedule

to the right. In contrast to the case with lump-sum taxation, the private savings schedule does

not shift to accommodate the higher government deficit (point B). Thus, National Savings

shift to the left, reflecting the expansion of private consumption and national expenditure.

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.

Figure 9: Tax cut under distortionary taxation

18 Note that the distortion only arises because the tax rate is not uniform along time: if 1 and 2 were

equal (to finance some positive amount of government expenditures), then there would be no change in relative prices of current vs future consumption, and hence no distortion in the consumption-saving decisions (see exercise on “tax smoothing”, below). This is an important, finding, because it implies that governments should try to smooth taxes over time, even if government expenditures vary over time. The argument only applies for temporary shifts in government spending: permanent increases in government spending should be met by permanent increases in tax rates. [Barro, R., 1979. On the determinants of the Public debt. Journal of Political Economy, 64: 93-110].


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1 1,P GS S

1PS1

GS

S1

1S11 r 11 r

A B

A

B

7.8.4 Numerical example

Consider an open economy starting out with: 10021 QQ , 021 GG , Gdb 0*0 ,

and 0* r . Suppose that the government launches a tax on consumption today

amounting to 25.01 , with the proceeds sent back to households in period 2. Will this

policy have any impact on consumption?

The equality between the MRS and the relative price of current versus future

consumption (Euler equation) becomes:

21

2

1

25.1

C

C

Substituting this in the household inter-temporal budget constraint, one obtains:

8020025.11

21

1

C .

Hence, current consumption is impacted. Using (4), we see that TB=20 and.

2 120C . The consumer is obviously worse off, because is no longer smoothing

consumption over time.

In this case, the household is not saving: the household disposable income in the first

period is 80201001111 CQY P , equal to consumption. Thus, 01 b . The

government, however, runs a surplus 20111 CS G , that needs to be matched by someone

else’ deficit. In this case, the government is financing the external sector: the negative

government debt is matched by an increase in the country NIIP: 201*1 db . All in all, the

household exports 20 units of the consumption good in exchange for foreign assets, and then


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issues a liability by the same amount to the government, hedging his position. Indirectly, the

government is lending abroad.

Of course, if the economy was closed, the government would not be able to use the

foreign economy to park its savings. In that case, the excess saving in the home economy

would drive the autarky interest rate downwards, inducing the household to dissave the

enough to match the government surplus. Using (18e), the closed economy case would

correspond to 1 1 11 1

1 5050 0

1 1 aCA Y C

r

, which solves for 11 2 3ar . The

optimal consumption is 1001 C , implying a tax revenue equal to 2511 C . The household

saving will be 1 1 1 1 1 25PS Y C C , matching exactly what the government wants to

lend. In the closed economy, the failure of the Ricardian equivalence materializes with a

change in the autarky interest rate.

7.9 Summary

The two-period model builds up on the observation that households prefer a stable as

opposed to highly variable pattern of consumption. Because income tends to drift up and

down, it is not current income that determines current consumption, but rather life-time

wealth (or its average, called permanent income). When deciding how much to consume,

households disentangle how much of their income has a permanent nature versus temporary.

When income changes are perceived to be temporary, the household may try to

smooth consumption, by borrowing or drawing down accumulated assets. The fact that

consumption responds little to income changes that are temporary implies that consumption

may be a source of stability: it will decline less than GDP in recessions and will expand less

than GDP in upturns. In contrast, the saving rate should rise during economic booms and fall

during recessions. When instead shocks are perceived to be permanent, consumption adjusts

strongly, and the Current Account remains stable.

In a closed economy, the Current Account cannot adjust to smooth consumption,

Hence, the relative price of current and future consumption – the autarky interest rate - must

change. Thus, for instance, when current income expands relative to future income, current

consumption becomes more abundant and the autarky interest rate will fall.


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From the household’ point of view, increases in government spending, because they

come along with higher taxes today or in the future, reduce lifetime wealth and thereby

consumer spending. A different question relates to differential impact of financing

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

, 2121

CuCuCCU , with

11

11ln

1

whenC

whenCCu (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:


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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

1C , one obtains

111111

1

rC (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


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C2

C1

A

B

C1A


E1

C2A

B‘

B‘‘

0


Now, consider the possibility of the interest rate impacting on wealth 1 as well (that

is, with 022 TQ ). 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


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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


C2A

B‘

B‘‘

0


1'1

E

C‘C

C‘‘


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Review questions and exercises

Review questions

7.1. (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”

7.2. 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?

7.3. Explain why the economy-wide saving rate varies with the share of working-age population in total population.

7.4. 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?

7.5. (Intertemporal budget constraint): Can a country starting out with *0 0b have

TB<0? What about perpetual CA deficits?

Exercises

7.6. (Temporary vs permanent income shocks): Consider a household who leaves only two periods and whose expected production pattern is 1Q and 2Q . Also assume that his

lifetime utility function is given by 21 lnln CCU and that the interest rate is 1r .

a) Plot the household inter-temporal budget constraint in a graph.


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b) Suppose first that 01 Q , and 02 Q . Is this household expected to be a saver or a borrower? Represent in a graph.

c) Suppose now that 01 Q , and 02 Q . 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 private savings in period 1 as a function of 1Q and 2Q .

f) Discuss the impact of a temporary output shock on current consumption and on savings. Represent in a graph.

7.7. (Temporary vs permanent income shocks): Consider a small economy open to international capital flows, where the expected production pattern is 1101 Q and

1102 Q . Also assume that the lifetime utility function of the representative consumer is

1.01

lnln 2

1

CCU

and the world interest rate is %10* r .

a) Plot the household’ inter-temporal budget constraint in a graph.

b) Find out the optimal consumption pattern, as well as the implied trade balance and current accounts.

c) Examine the implications of a change in current GDP to 681 Q , namely on consumption, the trade balance, income in each period and the current account. Would the economy be better off if it was closed?

d) Consider in alternative that the negative shock affected future output to 1 110Q , and

682 Q . In this case, what would be the effects?

e) Finally, examine the implications of a permanent shock, so that 6821 QQ .

7.8. (Closed vs open): Consider an endowment economy, where the preferences of the representative consumer are given by 21 ln8.0ln CCU . In this economy, current and

future GDP are 11251 Q and 13502 Q .

a) Find out the equilibrium interest rate assuming that the economy is closed to capital flows.

b) Suppose that the economy opens to international flows of capital and that the world interest rate is %25* r . 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.

c) Departing for c), examine the implications of a fall in the interest rate to %0* r . 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 100Q 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 %5* r .


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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 791 Q , 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, 100Q but

2111 TG ). Assuming that in the following years 0 TG , 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 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 economy, where people work 40 years earning 1000 units of output per year, and where the average retirement duration is 25. If there are 4 million workers and 1 million retirees, how much will be the aggregate saving rate in this economy? And what if the proportion of retirees was 50-50% ? 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 1 2ln lnU C C . 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 the current generation is giving birth to 400 newborns, only. How much will be current and next-period output in that case? In this case, how much should be the interest rate? [A: r=-20%].


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7.13. (Sudden Stop, heterogeneity): Consider a small economy open to international capital flows. The lifetime utility function of the representative consumer is given by

2.1lnln 21 CCU . The economy initial Net International Investment position is *0 40b . External liabilities mature each year. The international interest rate is constant

at 20.0* r . It is also known that the country’ GDP in each period are given by 881 Q and 8.1362 Q .

a) (a1) Find out the optimal consumption function. (a2) represent in a graph the national income schedule and the national expenditure schedule as a function of the interest rate.

b) Describe the macroeconomic equilibrium, assuming that the economy is open. (b1) describe the macroeconomic equilibrium in the YA graph. (b2) Revisit the consumer problem to describe the optimal consumption path in that equilibrium. Is this result intuitive? Compute the implied (current and future) values of: (b3) GNP; (b4) Trade Balance; (b5): Current account; (b6) Net international investment position as a percentage of GDP. [A: 84,84]

c) [No roll over] Now suppose that, in period 1, the consumer was no longer able to roll over its debt in international markets (that is *

1 0b ). (c1) How much would

be the admissible current account and absorption? (c2) Describe the adjustment process in the YA graph. (c3) What would happen to the domestic interest rate? (c4) would it make a difference if the initial debt matured at the end of period 2 instead? (c5) Revisit the consumer problem to describe the possible consumption path. [A: 40, 40, 310.4%]

d) [Heterogeneity] Finally, assume that the home economy was composed by two agents, both price takers, with endowments and initial debt distributed as follows:

*0 0 40Ab b , 1 46AQ ¸ 1 42BQ , 2 94.8AQ , 2 42BQ . Find out the agent’s

savings and consumption (d1) in the equilibrium with unlimited access to external finance; (d2) in the equilibrium with no roll over.

7.14. (Terms of trade) Consider a small economy where the life-time utility function of the representative consumer is given by 2.1lnln 21 CCU , where C refers to an imported good. Production in this economy is fully exported and given by 2641 Q and

2422 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 CQ PPTT , are expected to remain constant at 121 TTTT .

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 8.01 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].


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d) Now consider that the terms of trade deterioration was permanent, that is 8.021 TTTT . 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 21CCU . this economy, GDP is constant at

10012 QQ .

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 202211 TGTG . 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: 401 G , 02 G

and 201 T . 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 401 T ) 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 0* r . Examine in this case the implications of the following fiscal policy: 2021 GG and 01 T : 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: 401 G , 02 G and 201 T . 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 401 T ) 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 125.1lnln 21 CCU . In this economy, GDP is constant at 75012 QQ .


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a) (Equilibrium interest rate): Compute the private consumption and the equilibrium interest rate in this economy, assuming that government expenditures are equal to

2501212 TTGG . Explain the intuition.

b) (Anticipated shock): Now suppose that the government announces an increase in future spending to 35022 TG . 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 1001 T , 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 ( 1001 TB ). 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 21CCU . In this economy, GDP is

constant at 10012 QQ . The economy is open to capital flows, being 0* r .

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, 201 T , 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,

02211 CC , was met: find out the optimal consumption pattern. Considering in

particular the case with 25.01 , 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 21 lnln CCU . In this economy, 1 110Q ,

2 90Q , 1 25G , 2 15G , and 752 Q . In this economy, taxes are proportional to

private consumption. Further assume that the international interest rate is 0*1 r .

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 .


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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.

d) If the economy was instead closed to the international trade of assets, would he choice of the tax rate in each period matter? To which extent?

7.19. (Tax increase, heterogeneous agents): Consider a economy where the utility function of the representative consumer is equal to 21CCU and each consumer is

endowed with 1 2 80Q Q . In this economy, there are two consumers. One is Ricardian,

and the other is not allowed to consume in excess of disposable income. Assume that there are no initial assets, and 1 2 0G G .

a) Assume that the economy is open to capital flows, with 0*1 r . Examine the

implications of a tax 1 10T on each consumer, followed by a transfer of

2 110 1T r in period 2. In particular, find out the impact on (a1) consumption of

Ricardian consumers (R) as well as on non-Ricardian (HTM) [80, 70]; (a2) savings of R and HTM; (a3) current account in period 1, (a4) Who will buy debt of whom? [A: Government buys 10 of Ricardian debt and 10 of foreign assets].

b) Describe the impact of the policy change on the welfare of R-consumers and HTM-consumers.

c) Assume now that the economy was instead closed. c1) Find out the saving functions of R-consumers and of HTM-consumers; (c2) find out the expression of private savings; (c3) find out the expression of national savings; (c4) what will be the autarky interest rate when 1 10T and 2 110 1T r [A: r=-20%]? (c5) how much will be the

consumption levels of Ricardian and HTM? [A: 90, 70]. (c6) Who will buy debt of whom? [A: government buys 20 of Ricardian debt].

7.20. (Tax increase, heterogeneous agents) Consider a two-period endowment economy, open to capital flows, where the preferences of the representative consumer are

21

lnln

1.25

CU C . The international interest rate is *

11 1.25r . Assume that 2 450Q

but, because of a temporary contraction, output today fell down to 1 360Q . Initially

there is no government.

a) (Optimal consumption) Find out: (a1) the consumer’s lifetime wealth; (a2) the optimal consumption in periods 1 and 2. Explain the intuition regarding the pattern of consumption. (a3) the trade balance in period 1. (a5) The current account in period 2. (a6) At the end of period 1, which agent will buy debt from which other agent?

b) (Taxes and transfers) Departing from (a), suppose that the government launched a transfer in period 1 amounting to 1 40T , to be financed with future taxes,

2 0T . What will be impact on: (b1) private wealth; (b2) Future taxes; (b3)

private savings in period 1; (b4) government savings. (b5) Which proposition is being illustrated here? (b6) At the end of period 1, which agent will buy debt from which other agent?


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c) (Borrowing constraints) Departing from (a), assume that in this economy 50% of consumers were constrained on borrowing. In that case, how much would be: (c1) Private consumption? (c2) Trade balance in period 1. If the government launched a transfer in period 1 amounting to 1 40T , to be financed with future taxes,

2 0T , would that have an impact on (c3) private consumption and on the (c4)

trade balance? Quantify. (c5) Why should the government engage in such a policy?

7.21. (Two-country endowment economy, optimal capital controls) Consider a country (Arcadia) where consumers live two periods and are all alike. The inter-temporal utility function of each individual consumer is of the form 21 lnln CCU . In this economy,

production is constant in periods 1 and 2: 100021 QQ .

a) Describe the equilibrium in this economy, assuming that it is closed to international trade of assets (A: r=0%).

b) Now consider a second country (Begonia), with the following production pattern: 10001 Q and 8002 Q . Without capital movements, what will its market interest

rate in period 1? (A: r=-20%).

c) Now suppose that in period 1 the two countries open to capital flows. With the help of a graph describing the current account schedules, find out the world interest rate and the global imbalance [A: -10%; 55].

d) Now assume that Arcadia imposed capital controls so as to maximize its own inter-temporal utility, given the Begonia demand for Arcadian bonds (the Begonia’ CA schedule). Find out the optimal international interest rate, as well as the implied current account balance in period 1 and the domestic interest rate. [A: -13.59%; -6.9%; -37].

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