# A Model of the Current Account

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1 A Model of the Current Account Costas Arkolakis teaching assistant: Yijia Lu Economics 407, Yale January 2011

2 Model Assumptions 2 periods. A small open economy Consumers: Representative consumer Period 1: allocates income to consumption or bonds (saving) Consumption: C 1, C 2 Bonds B 0 (initial savings), B 1, B 2. Given interest r 0, r 1 Endoment Economy: Q 1, Q 2 available to consumer Equilibrium: World interest rate. Impose no saving in last period B 2 = 0 (happens in equilibrium) Normalize the price of the good to 1, in each period

3 Consumer Budget Constraints BC 1st period: BC 2ns period: No saving second period B 2 = 0 C 1 + B 1 B 0 = r 0 B 0 + Q 1 C 2 + B 2 B 1 = r 1 B 1 + Q 2 Combine the budget constraints and B 2 = 0 C 1 + C r 1 = (1 + r 0 ) B 0 + Q 1 + Q r 1

4 Consumer Utility U (C 1, C 2 ) Consumer maximizes utility U (C 1, C 2 ) subject to (s.t.) budget constraint C 1 + C 2 = (1 + r 0 ) B 0 + Q 1 + Q r r 1 If B 0 0, one choice is the basket C 1 = Q 1, C 2 = Q 2

5 Consumer [ gure for BC and Indi erence curves about here] In equilibrium U 1 (C 1, C 2 ) U 2 (C 1, C 2 ) = 1 + r 1 Equilibrium in the world market r 1 = r

6 Equilibrium In equilibrium U 1 (C 1, C 2 ) U 2 (C 1, C 2 ) = 1 + r 1 C 1 + C r 1 = (1 + r 0 ) B 0 + Q 1 + Q r 1 r 1 = r Equilibrium in the world market r 1 = r

7 Trade Balance In equilibrium (Q 1 C 1 ) (Q 2 C 2 ) 1 + r = (1 + r 0 ) B 0 =) TB 1 TB r = (1 + r 0 ) B 0 The Model will predict a behavior for the trade balance over the two periods. If the country starts as debtor, B 0 > 0, it requires to repay debt and thus TB 1 > 0 or TB 2 > 0 or both. (i.e. the rm has to be a net exporter to repay the debt)

8 Current Account We can rewrite the BC in terms of the current account Thus, CA 1 = r 0 B {z} 0 + TB {z} 1 net investment income net exports CA 2 = r B 1 + TB 2 TB 2 TB r = (1 + r 0 ) B 0 =) (CA 1 r 0 B 0 ) (CA 2 r B 1 ) 1 + r = (1 + r 0 ) B 0 =) (CA 1 ) CA r + r B r = B 0 =)

9 Figure 2.3 about here Current Account

10 Current Account Thus, TB 2 TB r = (1 + r 0 ) B 0 =) (CA 1 r 0 B 0 ) (CA 2 r B 1 ) 1 + r = (1 + r 0 ) B 0 =) (CA 1 ) CA r + r B r = B 0 =) But also CA 1 = B 1 B 0, (change in net investmestment position -accumulate debt or credit). so that (1 + r ) (CA 1 ) CA 2 + r B 1 r B 0 = B 0 =) CA 1 CA 2 = B 0

11 Current Account Imbalances Can a country run a perpertual CA de cit Use transversality condition, CA 1 CA 2 = B 0 Finite lifetime cannot happen. In nite yes, make sure debt does not grow faster than your economy

12 Temporary vs Permanent Shocks Temporary shock vs permanent shock Temporary: parallel shift of BC in one period. Smooth consumption in two periods (see FOCs) CA de cit in rst period. Surplus in second Permanent. Consumption smoothing, CA might not be a ected Conclusion. Temporary shocks, larger swings in CA.

13 Twin De cits: Fiscal & Current Account De cits Conjecture that an important determinant of CA is scal surplus (a ects government and thus total savings) Correlation: scal de cits various times coincide with CA de cits E.g. Reagan tax cuts caused large de cits, same time CA turned negative E.g.2 Obama stimulus plan, also at at time where de cit is very large Figure: US Saving and Investment in Percent of GNP. Source: Schmitt-Groche and Uribe 2010

14 Ricardian Equivalence Assume the existence of a Government Government has assets B g 0,Bg 1, Bg 2 and purchases goods G 1, G 2 Faces constraints G 1 + (B g 1 B g 0 ) = r 0B g 0 + T 1 G 2 + (B g 2 B g 1 ) = r 1B g 1 + T 2 LHS is spending. RHS is purchases. No Ponzi B g 2 B g 2 = 0 0 in equilibrium

15 Modeling the Government Assume the existence of a Government Government has assets B g 0,Bg 1, Bg 2 and purchases goods G 1, G 2 Faces constraints G 1 + (B g 1 B g 0 ) = r 0B g 0 + T 1 G 2 + (B g 2 B g 1 ) = r 1B g 1 + T 2 LHS is spending. RHS is purchases. No Ponzi B g 2 B g 2 = 0 0 in equilibrium

16 Government and Household Budget Constraint Comnbining Equations we have Gov. BC And household budget constraint G 1 + G r 1 = (1 + r 0 ) B g 0 + T 1 + T r 1 C 1 + T 1 + B p 1 B p 0 = r 0 B p 0 + Q 1 C 2 + T 2 + B p 2 B p 1 = r 1 B p 1 + Q 2 where household has to pay taxes and B p 2 = 0.Combining the two C 1 + C r 1 = (1 + r 0 ) (B g 0 + Bp 0 ) + Q 1 T 1 + Q 2 T r 1

17 Combining All the Constraints Combining the above equations C 1 + G 1 + C 2 + G r 1 = Q 1 + Q r 1 LHS is present discounted value of domestic absorption RHS is present discounted value of production Notice that taxes are not there. So that the timing of the taxes may not matter As long as G 1,G 2 are given and gov. intertemporal budget constaint is satis ed.

18 Private and Government Saving Assume G 1, G 2 are given Government saving Private saving S g 1 = r 0B g 0 + T 1 G 1 =) S g 1 = T 1 Total saving is S p 1 = Q 1 + r 0 B p 0 T 1 + C 1 =) S p 1 = T 1 S 1 = S g 1 + S p 1

19 Ricardian Equivalence Total saving is S 1 = S g 1 + S p 1 National savings is una ected by the timing of taxes: Ricardian equivalence Implies CA 1 = S 1 I 1 = 0 Changes in scal de cit may induce o seting increases in private savings (leaving total savings and CA constant) Households internalize government s problem, adjust savings/consumption rationally

20 If Ricardian Equivalence holds what is the cause of twin de cits? Reagan time: Government savings plummeted but private savings did not increase as much National Savings and the CA plummeted Some of the premises of the theory seem to not hold in this example Maybe more government expenditure rather than less taxation Ricardian equivalence does not hold!? Maybe both

21 Discussion of the Lucas article If all the countries have the same technologies Cobb-Douglas prod function Y = Ak β l 1 β Income per capita =) y = A kl Marginal product of capital =) r = βa β, k :capital, l :labor kl β 1 =) r = βa 1/β (y) β 1 β New investment should occur in poor countries Quite the opposite, capital ows to/among rich countries What is the explanation? Human capital? Externalities of human capital?

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