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International Macroeconomics ASpring 2010 - Prof. Ph. Bacchetta

Makeup Exam - Solutions

1. (10)What is the impact of an increase in the rate of time preference on the current account?

benchmark case: (1 +r) = 1 ) C1= C2= C : at consumption path

increase in the rate of time preference: " )# ) C2. In an SOE the agent can lend and borrow,therefore he can satisfy his impatience by borrowingC1 Y1 in the rst period and

repaying (1 + r)(C1 Y1) in the second period. Thus, he will run a current accountdecit in the rst period and match it by a current surplus in the second period inorder to satisfy intertemporal feasibility.

2. (10)Switzerland has experienced a dramatic reduction in its current account surplus in 2008.What explains this reduction?

The current account is dened by the following accounting relationship:

CAt= N Xt+N F It+T rt

Considering the Swiss current account statistics, it appears that foreign trade has slightlyincreased between 2007 and 2008 while services have remained roughly stable over thatsame time frame. Labor income as well as current transfers have not been subject toremarkable uctuations. The main driver of the reduction in the current account surplushas been the sharp reversal in investment income (from +21:47%to 24:5%).

3. (10)What is the consumption co-movement puzzle?

Basic setup: two countries, uncertainty and complete markets

C1+C

1 = Y1+Y

1

C2(a) +C

2(a) = Y2(a) +Y

2(a) Y2(a)W

C2(b) +C

2(b) = Y2(b) +Y

2(b) Y2(b)W

Consider the case where (1 +r) 6= 1 and assume a specic utility function: U(C) =ln(C)) U0(C) = 1=C

Solving for optimal consumption in state a we obtain

C2(a) =(a)(1 +r)

p(a) C1:

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The same result holds for Foreign, thus

C2(a) +C

2(a) =(a)(1 +r)

p(a) (C1+C

1):

By substituting into the equilibrium equation we get

Y2(a)W =

(a)(1 +r)

p(a) YW1

This impliesC2(a)

C1=

C2(a)

C1

=Y2(a)

W

YW1

;

meaning that the per capita consumption growth rate should be perfectly correlated acrosscountries if markets are complete. More generally, with nancial globalization and risk-sharing consumption co-movements should be higher than output co-movements.However, empirical evidence shows the exact opposite: corr(y; yw)> corr(c; cw).The two most advocated explanations to the consumption co-movement puzzle are the fol-

lowing: rst, markets are far from being complete, and second, there exists a home-biaspuzzle" for equities.

4. (10)Greece is on the brink of bankruptcy and there is some probability it will default on itssovereign debt. There is actually a probability p of a 20% haircut on the debt. There isa spread of 2000 basis points on this debt, while the risk-free corresponding interest rateis 5%. Compute p. (Remarks: i) the solution can be found without calculator; ii) thenumbers used in this question are ctitious)

haircut (1 z) = 20%

spread on debt rs rf = 20%

rf = 5%

) compute the probability of default p

1 +rf = (1 p)(1 +rs +pz(1 +rs)

= [1 p(1 z)](1 +rs)

Manipulating this expression, we nally obtain

p =

1

1 z 1 1 +rf

1 +rs =

1

0:20

1

1:05

1:25

=

4

5

5. (25)Consider the benchmark two-period model with consumption only. Assume that there isuncertainty on future output so that

Y2 = (Y1+ with probability 1/2

Y1 with probability 1/2

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Also assume that= 1=(1 +r) and that the utility function is

ln C1+E1[C2]

a) (15) What is the impact of" on the current account in this case?

maxC1;C2

ln C1+E1[C2]

subject to C1 = Y1 B1

CH2 = Y1++ (1 +r)(Y1 C1)

CL2 = Y1 + (1 +r)(Y1 C1)

Set up the Lagrangian replacing E1[C2]:

L= ln C1+f0:5[Y1++ (1 +r)(Y1 C1)] + 0:5[Y1 + (1 +r)(Y1 C1)]g

The rst order condition w.r.t. C1 reads

1

C1= [0:5(1 +r) + 0:5(1 +r)]

,C1 = 1

Plugging this result in the current account identity, we obtain

CA1= B1 = Y1 C1

= Y1 1:

Considering the same setup without second period uncertainty delivers exactly the sameresult for rst period consumption (C1= 1) and for the current account (CA1= Y1 1).

We can conclude that uncertainty, i.e. the presence of , has absolutely no impact onthe current account. This can be rationalized by the fact that households have linearpreferences in period 2, meaning that they are risk neutral in period 2.

Now assume that the utility function is ln C1+E1[ln C2].b) (10) What is the impact of" on the current account in this case?

maxC1;C2

ln C1+E1[ln C2]

subject to C1 = Y1 B1

CH

2 = Y1++ (1 +r)(Y1 C1)CL2 = Y1 + (1 +r)(Y1 C1)

We again set up the Lagrangian replacing E1[ln C2]:

L= ln C1+f0:5ln[Y1++ (1 +r)(Y1 C1)] + 0:5ln[Y1 + (1 +r)(Y1 C1)]g

The rst order condition w.r.t. C1 reads

1

C1= 0:5

(1 +r)

Y1++ (1 +r)(Y1 C1)+

(1 +r)

Y1 + (1 +r)(Y1 C1)

1

C1= 0:5 1Y1++ (1 +r)(Y1 C1)+

1

Y1 + (1 +r)(Y1 C1) (1)3

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If we had no uncertainty, we know that the household would smooth consumption (C1=C2) and that C1= Y1, thus implying C A1= 0.

Let us consider whether the optimal consumption choice associated with the problem

without uncertainty represents a solution in the presence of uncertainty.ReplacingC1 byY1, we get

1

Y1= 0:5

1

Y1++ (1 +r)(Y1 Y1)+

1

Y1 + (1 +r)(Y1 Y1)

1

Y1= 0:5

1

Y1++

1

Y1

Further simplifying, we obtain

1 = Y21

Y21

2;

which is impossible given the requirement >0. Herewith we have shown that if we setC1 = Y1, then the l.h.s. of (1) is less than its right side. Because the left side of (1) isdecreasing in C1 whereas the right hand side is increasing in C1, it must be the case thatthe optimal level of consumption in period 1 satises

C1< Y1:

From this condition it follows that

CA1 = Y1 C1 > 0:

Households use the current account as a vehicle to save to avoid having to cut consumption

in the bad state of the world. With convex marginal utility of consumption in period 2the prospect of consuming Y1+ or Y1 with equal probability increases the expectedmarginal utility of consumption in that period.

6. (30)Consider the following problem for a representative consumer who lives in a small openeconomy that can borrow and lend at a constant real interest rate r. The representativeagent maximizes

U =1Xt=0

t log(Ct)

subject to the intratemporal constraint

Bt+1= (1 +r)Bt+Yt Ct

and the transversality condition

limt!1

Bt+1(1 +r)t

= 0:

The variableYt represents a constant, exogenous level of output. Assume that(1+ r) = 1.

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a) (10) First assume that agents in the economy expect Yt to remain constant at a levelY. Derive the optimal consumption path and the optimal current account in this case.

L=1

Xt=0tflog(Ct) +t[(1 +r)Bt+Y Ct Bt+1]g

The rst order conditions read:

@L

@Ct:

1

Ct=t

@L

@Bt+1: t= (1 +r)t+1

@L

@t: Bt+1= (1 +r)Bt+Yt Ct

From the Euler equation Ct+1 = (1 + r)Ct we obtain that consumption is at, i.e.Ct+1= Ct= C.

In order to determine the optimal consumption path, we need to derive the intertem-poral budget constraint:

1Xt=0

1

1 +r

tCt= (1 +r)Bt+

1Xt=0

1

1 +r

tY:

With Ct+1= Ct= C, the above equation rewrites:1 +r

r

C = (1 +r)Bt+

1 +r

r

Y

C = rBt+Y

The optimal current account in this case is described by:

CAt= Bt+1 Bt = Y C+rBt

= Y (rBt+Y) +rBt

= 0:

b) (10) Now assume that the economy learns that at time 0 output will increase perma-nently fromY toY+ , with >0. Derive the optimal consumption path and the optimalcurrent account in this case.

Y " permanently to Y +:

Consumption increases permanently by :

C= rBt+Y +:

The current account is still equal to zero because the increase in consumption is oset bythe increase in the endowment:

CAt= Bt+1 Bt = Y + C+rBt

= Y + (rBt+Y +) +rBt

= 0:

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c) (10) Now assume that the economy learns that at time 0 output will increase temporarily,i.e.

t=

(Y + for t = 0

Y for t >0

with >0. Derive the optimal consumption path and the optimal current account in thiscase.

Y " temporarily to Y +:

To derive the optimal consumption path in this case we need to go back to the intertem-poral budget constraint:

1Xt=0

1

1 +r

tC= (1 +r)Bt+Y +| {z }

t=0

+1Xt=1

1

1 +r

tY

| {z }t>0:

Simplifying: 1 +r

r

C = (1 +r)Bt+Y ++

1 +r

r

Y Y

C = rBt+Y + r

1 +r

) consumption increases by r1+r

(= annuity value of the increase in output).The behavior of the current account changes between t = 0 and t >0:

t= 0 : Y " )B1 = (1 +r)B0+Y + C

= (1 +r)B0+Y + rBt+Y + r1 +r

= B0+

1 +r

)CA0= B1 B0 =

1 +r >0

t= 1 :Y = )B2 = (1 +r)B1+Y + C

= (1 +r) B0+

1 +r+Y rBt+Y + r

1 +r

= B0+

1 +r= B1

)C A1= B2 B1 = 0

net foreign assets rise in period 1: the economy runs a CA surplus (intuition: savesome of the temporary output increase to consumer more in future periods)

from period 2 onward net foreign assets are constant, therefore C A= 0

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7. (25)Consider the following problem of a representative consumer who lives in a small openeconomy that can borrow and lend at an interest rater. The consumers problem is

MaxC1;C2

U= log C1+log C2

subject to

C1 = Y B1 G1

C2 = Y+ (1 +r)B1 G2

where Y is a constant, exogenous level of output. The government sets consumption taxrates1 and2 to balance its budget:

1C1= G1 and 2C2= G2:

Assume that(1 +r) = 1.a) (10) Show thatC1(1 +1) =C2(1 +2).

L= log C1+log C2+

W (1 +1)C1

1 +21 +r

C2

@L

@C1:

1

C1=(1 +1) (2)

@L

@C2:

C2=

1 +21 +r

(3)

@L

@

: (1 +1)C1+1 +2

1 +r

C2= 2 +r

1 +rY (4)

Combine (2) and (3) and obtain:

C1(1 +1) =C2(1 +2):

b) (8) Compute the value ofC1 andC2 as a function of1 and2.

To compute C1 and C2 use the intertemporal budget constraint and the result obtainedin (a):

(1 +1)C1+1 +2

1 +rC2 =

2 +r

1 +rY

2 +r1 +r

(1 +1)C1 =2 +r

1 +r

Y

C1 = Y

1 +1

C2 = Y

1 +2

c) (7) Characterize the behavior of the current account in periods 1 and 2.

CA1 = Y C1 G1= Y (1 +1)C1= 0

CA1 = Y C2 G2+rB1 = Y (1 +2)C2+rB1= 0

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