the current account in an intertemporal equilibrium model

Simple two-period model Current account Adding government General equilibrium

The current account in an intertemporal equilibrium modelLecture 1, ECON 4330

Tord Krogh

January 10, 2012


Intro - what questions?

What determines current account deficits and surpluses?

How are they affected by fiscal and monetary policy?

Can deficits be sustained? For how long?

Will they self-correct or do they warrant policy changes?

How does the current account behave during business cycles?

Is a current account deficit a threat to employment?

Can a current account deficit force a country to devalue?

What are the effects of a Eurozone collapse?

Will not answer all these questions. But you will learn tools that help you understandthe mechanisms.


Intro - what approach?

This course will lead you through two very different modeling traditions.

The intertemporal approach: Intertemporal general equilibrium models, explicitoptimization over time, no nominal rigidities (in the models we consider).Representative consumers and producers. Countries treated as if they wereindividuals. Obstfeld and Rogoff

The traditional macro approach: Less focus on explicit optimization in micro,more focus on macro behavioral equations that seem to have empirical support.Nominal rigidities and unemployment problems. Rødseth


Simple two-period model

Our first model will be the small two-period model found in OR Chapter 1. We willassume that

The economy exists for two periods only

Small open economy. Outside world provides a world interest rate r

Endowment economy (production in each period is given)

Representative agent: All individuals are identical

Perfect foresight (no uncertainty)


Simple two-period model II

The representative agent consumes C1 in period 1 and C2 in period 2. Utility function:

U = u(C1) + βu(C2) (1)

Standard assumptions hold for u(C): u′(C) > 0, u′′(C) < 0 and limC→0 u′(C) =∞.

β ∈ (0, 1) is the discount factor. Determines how heavily future consumption isdiscounted.


Simple two-period model III

Income is given by Y1 and Y2. Goods only last for one period (cannot be stored).Under autarky, the budget constraints are:

C1 = Y1

C2 = Y2

With access to trade, we instead get

C1 +C2

1 + r= Y1 +

Y2

1 + r(2)


Simple two-period model IV

We assume that behavior is captured by utility maximization. Maximize (1) subject to(2). Alternatively, insert for C2 in (1) using the budget constraint:

maxC1

u(C1) + βu ([1 + r ](Y1 − C1) + Y2)

The first-order condition is

u′(C1) = β(1 + r)u′(C2) (3)

where C2 = [1 + r ](Y1 − C1) + Y2. (3) is the consumption Euler equation.Interpretation?


Simple two-period model V

We can also write the Euler equation as

βu′(C2)

u′(C1)=

1

1 + r(4)

i.e., MRS = relative price of consumption. In terms of C1 or C2?


Simple two-period model VI

This is the whole model (for now). Given income Y1 and Y2, and a world interest rater , the consumption path is determined by the Euler equation (plus the BC).


Simple two-period model VII

Main message I: The country will adjust its consumption path depending on the worldinterest rate.

β(1 + r) > 1⇒ C1 < C2

β(1 + r) < 1⇒ C1 > C2

β(1 + r) = 1⇒ C1 = C2

Main message II: Consumption growth rate can be different from output growth rate.


Illustration


Illustration II


Autarky interest rate

The autarky real interest rate, rA, is the equilibrium interest rate if there is no trade.Since that implies C1 = Y1 and C2 = Y2, we have rA defined by the first-ordercondition

βu′(Y2)

u′(Y1)=

1

1 + rA

If rA > r , then future consumption is cheaper in our country compared to the rest ofthe world. We will import current consumption (i.e. CA < 0) and export futureconsumption. How can we see that in the previous graph?


Defining the CA

What about the current account (CA)? In general, the CA is defined as:

CA = Trade account + Primary income account + Secondary income account

Trade account: Exports minus imports (trade balance)Primary income account: Payments for use of labor and financial resourcesSecondary income account: Foreign aid, remittances, etc.


Defining the CA II

Let CAt denote the current account surplus in a given period, and Bt be net foreignassets at the beginning of period t. Accumulation equation:

CAt = Bt+1 − Bt

Implicit here is the risk-free nature of foreign assets. A real life accumulation equationshould take into account possible revaluations.


Defining the CA III

Relation to investment and saving?

Saving = Net investment in real capital + Net investment in financial assets

= Investment in real capital at home + CA

If I denotes real investment and S is saving:

CAt = St − It

A current account surplus means that you invest abroad


Defining the CA IV

Table: CA, saving and investment in per cent of GDP (2006)

Country CA S I T − GGermany 5.0 22.8 17.8 -1.6

Japan 3.9 28.0 24.1 -4.1UK -3.2 14.8 18.0 -2.7US -6.2 14.1 20.0 -2.6


CA in our simple model

In any period t we can a trade surplus Yt − Ct , while the primary income account isrtBt . The current account is therefore

CAt = Yt − Ct + rtBt

for t = 1, 2. Remember that we’ve assumed r1 = r2 = r .


CA in our simple model II

B1 is the NFA our economy starts out with. Assume that B1 = 0. Hence

CA1 = Y1 − C1

CA2 = Y2 − C2 + rB2


CA in our simple model III

B3 is the NFA our economy ends with. Since B3 = 0 (no need to save for the afterlife):

CA2 = −B2

= −CA1

What you save/borrow in period 1 will be consumed/paid back in period 2.


CA in our simple model IV

Inserting for CA1 and CA2:

Y2 + r(Y1 − C1)− C2 = −(Y1 − C1)

and by re-arranging we see how we got the budget constraint (2):

C1 +C2

1 + r= Y1 +

Y2

1 + r


Example

Assume that β(1 + r) = 1, which we know implies C1 = C2 = C∗ (completeconsumption smoothing). Consumption is given by (from the BC)

C∗ =(1 + r)Y1 + Y2

2 + r

while the current account is

CA1 = Y1 − C∗ =Y1 − Y2

2 + r

Implications:

The main determinant of CA is the difference between present and future income

CA1/Y1 depends only on the growth rate (Y2 − Y1)/Y1, not the absolute level ofincome


Example II

What if we have a CES utility function?

u(C) =1

1− 1/σC1−1/σ

The first derivative is u′(C) = C−1/σ , making the Euler equation:

β

(C2

C1

)−1/σ

=1

1 + r


Example III

You can show that that (homework)

C1 =(1 + r)Y1 + Y2

2 + r + {[β(1 + r)]σ − 1}

which makes the current account:

CA1 =Y1 − Y2 + {[β(1 + r)]σ − 1}Y1

2 + r + {[β(1 + r)]σ − 1}

Implications:

There are two motives for saving in period 1 (CA1 > 0): Consumption smoothing(if Y1 > Y2), or if the rate of return is high (β(1 + r) > 1).

CA1/Y1 is still independent of the absolute level of income


Example IV

Questions for yourself: What are the income and substion effects of r on C1? Is thenet effect unambiguously positive or negative? Review section 1.3.2 on your own.


Lessons so far

From this (very simple) model, we’ve learned that

The absolute level of income is not likely to be an important determinant of CA.At least not CA/Y .

Countries that expect high income growth should tend to have CA deficits (andvice versa)

Patient countries (β close to one) should tend to have CA surpluses


Gov’t and taxes

We will add government expenditure and taxes in a very simple manner. They chooseG1, T1, G2 and T2 with no special purpose.Private sector (the representative agent) will get a new budget constraint whichreplaces (2):

C1 +C2

1 + r= (Y1 − T1) +

Y2 − T2

1 + r(5)

Other than this, private behavior is unchanged as long as U is unaffected by G , or ifutility is separable in C and G :

U = u(C1) + v(G1) + β(u(C2) + v(G2))

[because in this case ∂U/∂Ct is still independent of G ]


Gov’t and taxes II

The government also faces a budget constraint:

G1 +G2

1 + r= T1 +

T2

1 + r

If you use this to insert for T2 in (28), we find that

C1 +C2

1 + r= (Y1 − G1) +

Y2 − G2

1 + r


Gov’t and taxes III

The effect of government expenditure is that

A temp.high level of G (G1 > G2) might produce a current account deficit(CA < 0) since it reduces current private disposable income relative to futureincome.

Timing of taxes and the size of deficits do not matter (Ricardian equivalence)

The first point is easily understood if you imagine that Y1 = Y2 = Y and assumeβ(1 + r) = 1. In the absence of government, we’ve seen that this gives CA1 = 0. WithG1 > G2, we get the same qualitative effect on the CA as when Y1 < Y2.


General equilibrium

So far we have assumed that r is exogenously determined by ’the outside world’. Letus instead assume that we have two countries (Home and Foreign) that trade goods.How will the equilibrium interest rate be determined?


General equilibrium II

For a given interest rate r , we know that C1 and C2 (Home’s consumption path) isdefined by the two equations

u′(C1) = β(1 + r)u′(C2)

C2 = (1 + r)(Y1 − C1) + Y2

More compactly, we have that

u′(C1) = β(1 + r)u′((1 + r)(Y1 − C1) + Y2)

defines C1(r), giving us the current account:

CA1(r) = Y1 − C1(r)


General equilibrium III

Foreign’s consumption path is determined by a similar set of conditions, giving us

u′(C∗1 ) = β∗(1 + r)u′((1 + r)(Y ∗1 − C∗1 ) + Y ∗2 )

which defines C∗1 (r), and the CA-equation

CA∗1 (r) = Y ∗1 − C∗1 (r)

Here it is assumed that Home and Foreign share the same utility function, except fordiscount factor.


General equilibrium IV

The only thing left is to determine the world interest rate r . We find that by imposingmarket clearing:

CA1(r) = −CA∗1 (r)

⇒ One country’s surplus must be met by a deficit somewhere else (do you hear that,Merkel?)


General equilibrium V

Remember that we defined the autarky real interest rate. For the two countries, theautarky rate defines what interest rate that would lead to a zero current account:

CA1(rA) = 0

CA∗1 (rA∗) = 0


General equilibrium VI


General equilibrium VII


General equilibrium VIII

Previous conclusion: rA > r should lead to CA1 < 0 since the internal price of futureconsumption is lower than the world market price. We see that comparing rA and rA∗

will help us determine the sign of CA.


General equilibrium IX


General equilibrium X

Analytical example: Return to the CES-case from before. Assume identical utilityfunctions and β = β∗. The current account functions are:

CA1 =Y1 − Y2 + {[β(1 + r)]σ − 1}Y1

2 + r + {[β(1 + r)]σ − 1}

CA∗1 =Y ∗1 − Y ∗2 + {[β(1 + r)]σ − 1}Y ∗1

2 + r + {[β(1 + r)]σ − 1}

Equlibrium condition is thatCA1 = −CA∗1

Which, after inserting for CA1 and CA∗1 , can be solved to yield

1 + r =1

β

(Y2 + Y ∗2Y1 + Y ∗1

)1/σ


General equilibrium XI

How do we interpret the last equation? First introduce the discount rate δ, defined as

β =1

1 + δ

Then let us take a log approximation of each side of the equilibrium condition. Sincelog(1 + x) ≈ x when x is small:

log(1 + r) ≈ r

log 1/β = log(1 + δ) ≈ δ

log

(Y2 + Y ∗2Y1 + Y ∗1

)1/σ

=1

σlog(1 + g) ≈

1

σg

where g = (Y2 + Y ∗2 )/(Y1 + Y ∗1 )− 1 is the global growth rate. Hence, the equlibriumcondition is approximately

r = δ +1

σg


General equilibrium XII

r = δ +1

σg

The world interest rate is thus a function of the discount rate (’patience’), the globalgrowth rate, and the elasticity of substitution (σ). The elasticity is usually found to bequite small (less than one). Hence a high global growth rate will push up the worldinterest rate. Why then such low real interest rates after 2001?


Next lecture

Wrap up chapter 1

Add investments

Discuss labor movements

the current account in an intertemporal equilibrium model

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