# A Two-Period Model of the Current Account Obstfeld and Rogo, Chapter 1

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1 A Two-Period Model of the Current Account Obstfeld and Rogo, Chapter 1

2 1 Small Open Endowment Economy 1.1 Consumption Optimization problem maximize U i 1 = u c i 1 + u c i 2 < 1 subject to the budget constraint c i 1 + ci r = yi 1 + yi r :

3 Euler equation u 0 c i 1 = (1 + r) u 0 c i 2 When (1 + r) = 1; c i 1 = ci 2 = ci Use the budget constraint to solve for consumption c i = yi 1 (1 + r) + yi r

4 1.2 Current Account Increase in claims on ROW (rest of world) Equivalently, increase in net foreign assets CA t = B t+1 B t = Y t + r t B t C t GNP t = Y t + r t B t GDP t = Y t Current account in period 1: assume B 1 = 0 CA 1 = B 2 B 1 = Y 1 + r 1 B 1 C 1 = Y 1 C 1 = B 2

5 Current account in period 2: note B 3 = 0 since there are only two periods CA 2 = B 3 B 2 = Y 2 + r 2 B 2 C 2 = Y 2 + r 2 (Y 1 C 1 ) C 2 = B 2 Sum current accounts in periods 1 and 2 to yield intertemporal budget constraint CA 1 + CA 2 = (Y 1 C 1 ) (1 + r 2 ) + Y 2 C 2 = 0

6 1.3 Current Account as Intertemporal Trade Graph optimization problem with future consumption on verticle axis and current consumption on horizontal Slope of budget constraint is (1 + r) Autarky: country must consume endowment Trade can increase welfare since CA 1 = Y 1 C 1 ; particularly for a country with unequal current and future income A current account de cit today must be o set by a current account surplus in the future

7 Comparative advantage A country has a comparative advantage in the good it produces relatively cheaply in autarky In autarky, consumption equals production Solve for the autarkic interest rate, which yields goods market equilibrium (Y i = C i ) using the Euler equation u 0 (Y 2 ) u 0 (Y 1 ) = 1 = current consumption price of future consumption 1 + ra

8 If r < r A; domestic price of future consumption is relatively high, implying that the domestic country has a comparative advantage in current consumption Comparative advantage requires that the domestic country export the good for which its autarkic price is relatively low, that is export current consumption Domestic country will run a current account surplus in period 1 and a current account de cit in period 2

9 Consider an exogenous increase in Y 1 1 u 0 (Y 1 ) falls and 1+rA rises as current consumption becomes relatively cheaper compared to future consumtpion in autarky export even more current consumption, so period 1 current account surplus increases

10 Welfare and the world interest rate Assume that Y 2 > Y 1 for home and that r > r A Implies that the domestic price of future consumption is relatively low such that the domestic country has a comparative advantage in future consumption and a period-1 current account de cit as import current consumption Let world interest rate rise such that countries become more similar and comparative advantage shrinks Use graph to show that the domestic country becomes worse o

11 E ect of temporary versus permanent output change on current account Assume (1 + r) = 1 If Y 1 = Y 2, then there is no current account imbalance If Y 1 > Y 2 ; current account surplus in period 1 and de cit in period 2

12 Government consumption Now, agents have only Y 1 G 1 and Y 2 G 2 available for consumption in periods 1 and 2 An increase in G 1 ; reduces output available for consumption in period 1 and creates a period-1 current account de cit An increase in both G 1 and G 2 (permanent increase in G) has no current account e ect

13 2 Small Open Production Economy with Investment 2.1 Current Account with Investment Investment with no depreciation I t = K t+1 K t Change in total domestic wealth B t+1 + K t+1 (B t + K t ) = Y t + r t B t C t G t

14 Current account is increase in net foreign assets CA t = B t+1 B t = Y t + r t B t (C t + I t + G t ) = income - expenditure National Savings is de ned as S t = Y t + r t B t C t G t Current account is di erence between savings and investment CA t = S t I t

15 2.2 Intertemporal Budget Constraint Assume B 1 = B 3 = 0 Current account in period 1 B 2 = Y 1 (C 1 + I 1 + G 1 ) Current account in period 2 B 2 = Y 2 + rb 2 (C 2 + I 2 + G 2 ) B 2 = C 2 + I 2 + G 2 Y r

16 Equating the two expressions for B 2 yields the country s intertemporal budget constraint C 1 + I 1 + G 1 + C 2 + I 2 + G r = Y 1 + Y r

17 2.3 Optimal Consumption and Investment Production function Y t = F (K t ) Intertemporal budget constraint solved for C 2 C 2 = (1 + r) [F (K 1 ) C 1 G 1 I 1 ] + F (K 2 ) I 2 G 2 Substitute using K 2 = K 1 + I 1 K 2 = I 2 C 2 = (1 + r) [F (K 1 ) C 1 G 1 I 1 ]+F (K 1 + I 1 )+K 1 +I 1 G 2

18 Maximize present value of utility with respect to C 1 and I 1 u (C 1 )+u f(1 + r) [F (K 1 ) C 1 G 1 I 1 ] + F (K 1 + I 1 ) + K 1 + I 1 G 2 g First order condition with respect to C 1 is the standard Euler equation u 0 (C 1 ) = (1 + r) u 0 (C 2 ) Derivative with respect to I 1 u 0 (C 2 ) h (1 + r) + F 0 (K 2 ) + 1 i = 0 F 0 (K 2 ) = r Investment is independent of preferences toward consumption and savings

19 Can borrow at world interest rate to equate marginal product of capital with the interest rate

20 2.4 Production Possibility Fontier with Investment Two goods are future and current consumption PPF in autarchy C 2 = F (K 2 ) + K 2 = F (K 1 + F (K 1 ) C 1 ) + K 1 + F (K 1 ) C 1 Horizontal intercept sets C 2 = K 2 = 0 to maximize C 1 C 1 = K 1 + F (K 1 ) Vertical intercept sets C 1 = 0 to maximize C 2 C 2 = F (K 1 + F (K 1 )) + K 1 + F (K 1 )

21 Slope of PPF dc 2 dc 1 = F 0 (K 2 ) 1 < 0 Capital s diminishing marginal product makes the PPF concave d 2 C 2 dc 2 1 = F 00 (K 2 1 < 0 Consider point where C 1 is high and C 2 and K 2 are low F 0 (K 2 ) is high and falls as C 1 falls therefore, as C 1 falls, slope increases (becomes atter since slope is negative) with high C 1 and low C 2 ; diminishing marginal productivity implies that give up little C 1 to get a lot of C 2

22 as continue to substitute get less and less C 2 for each unit of C 1

23 2.5 Current Account as Intertemporal Trade Graph with social optimum in autarky Indi erence curve tangent to PPF at point A determines C 1 ; C 2; and I 1 Slope of indi erence curve and PPF at optimum is 1 + F 0 (K 2 ) = 1 + r A Let the world interest rate be lower r < r A ; such that future goods are relatively expensive abroad and cheap at home r > r A

24 Production occurs at point B where world relative price of future to current goods is tangent to PPF since r < r A, slope is atter Consumption occurs at point C where relative price of future to current goods is tangent to indi erence curve Domestic country has a comparative advantage in future goods, so it imports current goods and exports future goods Higher indi erence curve with trade illustrates gains from trade

25 Horizontal distances Between A and B is extra investment generated by opening trade Between A and C is extra consumption of C 1 Between B and C is period 1 current account de cit Government consumption shifts PPF inward

26 3 Two-Country Endowment Economy World goods market equilibrium Y t + Y t = C t + C t World savings must equal zero S t = Y t C t = S t = C t Y t Assume savings is increasing in the interest rate and that r A < r A home country has a comparative advantage in current goods

27 with opening of trade, home interest rate rises, increasing savings and CA surplus while foreign interest rate falls, reducing foreign savings and foreign CA Welfare is positively related to the term of trade (TOT) T OT = price of exports price of imports Home country exports current goods whose price relative to future goods is 1 + r Therefore, home T OT = 1 + r Let Y 2 increase, reducing savings and increasing r A Equilibrium r increases to return world savings to zero

28 Domestic TOT increase and foreign TOT fall, increasing home welfare at the expense of foreign welfare Immiserizing Growth: An increase in current output (growth) increases savings and reduces the equilibrium interest rate. When home country exports current goods, the increase in current income can make it worse o Justify savings as an increasing function of the interest rate Assume CRRA utility u (C) = C1 1= 1 1=

29 Use Euler equation C 1 = [ (1 + r)] C2 and budget constraint to solve for C 1 C 2 = (Y 1 C 1 ) (1 + r) + Y 2

30 C 1 = Y 1 + Y 2 1+r 1 + (1 + r) 1 The e ect of an increase in interest on savings is the negative of its e ect on C 1 Substitution e ect [(1 + r) ] Higher interest makes current goods relatively more expensive causing agents to substitute out of current goods into future goods, raising savings Income e ect h (1 + r) 1i Given PVLR, higher interest allows more future consumption, reducing savings. If > 1; substitution effect dominates and higher interest reduces consumption and raises savings.

31 Wealth e ect: Reduces PVLR, reducing current and future consumption and raising savings

32 Alternative expression C 1 = [ (1 + r)] [(Y1 C 1 ) (1 + r) + Y 2 ] Totally di erentiate with respect to C 1 and r C2 1+r dc 1 dr = (Y 1 C 1 ) 1 + r + C 2 =C 1 Second term is negative due to substitution e ect First term is ambiguous TOT e ect and depends on whether agent is a borrower (negative e ect) or a lender (positive e ect) If lender (Y 1 C 1 > 0), comparative advantage in current goods, T OT = 1 + r; and increase in interest increases TOT and real income

33 4 Two-Country Production Economy with Investment Savings dc 1 dr = Y 1 C 1 I r + C 2 =C 1 C2 1+r Investment where output is AF 0 (K) = r A F 0 (K ) = r Y = AF (K) Y = A F (K )

34 World equilibrium Y 1 + Y 1 = C 1 + C 1 + I 1 + I 1 S 1 + S 1 = I 1 + I 1 Metzler diagrams and shocks increase in home productivity in period 1 Increase in home productivity in period 2

35 Capital market restrictions A country with a current account de cit has a comparative advantage in future goods and its T OT = 1 1+r Can improve home welfare (at expense of foreign welfare) by increasing TOT (lowering r) Lower world interest rate if tax home consumption, raising home saving Both countries have the incentive to manipulate the TOT to their advantage, and the result can be reduced intertemporal trade Beggar thy neighbor policy

36 Oil prices and world interest rates in 1970 s, severe increase in world oil prices and redistribution of income toward oil exporters world savings increased and world interest rates fell in 1980 s world savings fell with very large US government budget de cits and larger OPEC spending world interest rates increased currently very high savings rates in emerging market countries Bernanke s savings glut low interest rates

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