# Lecture 1: The intertemporal approach to the current account

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1 Lecture 1: The intertemporal approach to the current account Open economy macroeconomics, Fall 2006 Ida Wolden Bache August 22, 2006

2 Intertemporal trade and the current account What determines when countries lend and when countries borrow in international capital markets? The current account (driftsbalansen) Let B t+1 be the value of an economy s net foreign assets at the end of period t. De nition of current account: net increase in foreign asset holdings or where NX t denotes net exports CA t B t+1 B t (1) CA t = NX t + rb t (2)

3 Table 1 Global current account balances Current Account Balance, 2004 Country or Aggregate US Dollars (billions) % of GDP (a) Largest Deficits United States Spain United Kingdom Australia Turkey Italy Portugal Hungary France Greece (b) Largest Surpluses Japan Germany China Russia Saudi Arabia Switzerland Norway Sweden Singapore Korea (c) Country Aggregates United States Other advanced economies Emerging economies Middle East Developing Asia World (discrepancy) 86.3 Source: International Monetary Fund, World Economic Outlook Database, September 2005.

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5 The simplest possible model: small open economy two periods, labeled 1 and 2 one good at each date endowment economy: output in each period is given: Y 1 and Y 2 all individuals are identical, population size normalised to one. perfect foresight (no uncertainty)

6 The representative consumer s problem: Lifetime utility U = u(c 1 ) + u(c 2 ); 0 < < 1 (3) where is the subjective discount factor and u 0 (C) > 0; u 00 (C) < 0 and lim C!0 u0 (C) = 1 Period budget constraints (B 1 = B 3 = 0) C 1 = Y 1 B 2 (4) C 2 = Y 2 + ()B 2 (5) where r is the (exogenous) world real interest rate Current account CA 2 = B 2 = (Y 1 C 1 ) = CA 1 (6)

7 Intertemporal budget constraint C 1 + C 2 {z } Present value of consumption = Y 1 + Y 2 {z } Present value of output (7) Consumer s optimisation problem: Maximise (3) subject to (7) From (7) C 2 = ()(Y 1 C 1 ) + Y 2 (8) Substitute into (3) U = u(c 1 ) + u(()(y 1 C 1 ) + Y 2 ) (9) First-order condition with respect to C 1 u 0 (C 1 ) u 0 (C 2 )() = 0 u 0 (C 1 ) = ()u 0 (C 2 ) (10)

8 This is the consumption Euler equation: at an optimum the consumer cannot increase utility by shifting consumption between periods. A one unit increase in consumption in period 1 increases utility by u 0 (C 1 ) Alternatively the consumer can save in period 1 and get () extra units of consumption in period 2 which increases utility by (1+r)u 0 (C 2 ) Consumers have incentive to smooth consumption over time Rearrangement yields u 0 (C 2 ) u 0 (C 1 ) {z } Marginal rate of substitution of present for future consumption = 1 {z } Price of future consumption in terms of current consumption Optimal consumption plan found by combining (10) and (7)

9 What determines whether a country runs a current account de cit or a current account surplus? Autarky real interest rate r A : interest rate that would prevail in economy which could not borrow or lend internationally In autarky: C 1 = Y 1 and C 2 = Y 2 u 0 (Y 2 ) u 0 (Y 1 ) = 1 A (11) An increase in Y 1 or a fall in Y 2 causes the autarky interest rate to increase If Y 1 = Y 2 =) = 1 1+r A Gains from intertemporal trade as long as r 6= r A

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11 Special case: = 1 1+r Euler equation implies C 1 = C 2 = C (perfect consumption smoothing) Budget constraint yields C + C = Y 1 + Y r C = Y 1 + Y 2 C = Y 1 () + Y r (12) (13) (14) Autarky interest rate u 0 (Y 2 ) u 0 (Y 1 ) = A (15)

12 Assume economy initially expects Y 1 = Y 2 =) r = r A and CA 1 = 0. Permanent changes in output (dy 1 = dy 2 = dy ): no e ect on r A or CA 1 Temporary increase in output in period 1 (dy 1 > 0; dy 2 = 0): r A falls (r A < r), CA 1 > 0 Temporary increase in output in period 2 (dy 1 = 0; dy 2 > 0): r A increases (r A > r), CA 1 < 0.

13 Adding government consumption Period utility u(c) + (G) (16) Assume balanced budget each period (no government de cits or surpluses) Representative agent s budget constraint C 1 + C 2 = Y 1 G 1 + Y 2 G 2 (17) Euler equation same as above

14 What is the e ect on the current account? Assume = 1 1+r ; Y 1 = Y 2 = Y, G 1 > 0 and G 2 = 0: Closed form solution for consumption Current account C + 1 C = Y G 1 + Y (18) 2 + r C = 2 + r Y G 1 (19) C = Y 2 + r G 1 (20) CA 1 = Y C G 1 (21) = 2 + r G 1 G 1 (22) G = 1 (23) 2 + r

15 Adding investment Production function Y = F (K); F 0 (K) > 0; F 00 (K) < 0; F (0) = 0; lim F 0 (K) = 1 K!0 (24) Investment (ignoring depreciation) Current account I t = K t+1 K t (25) CA t = B t+1 B t = Y t + rb t C t G t {z } National saving S t I t = S t I t (26) Current account balance = national saving minus investment

16 Returning to the two-period model B 1 = B 3 = 0, K 3 = 0 =) I 2 = K 2 Period budget constraints Y 1 = C 1 + G 1 + I 1 + B 2 (27) Y 2 + ()B 2 = C 2 + G 2 + I 2 (28) Intertemporal budget constraint C 1 + I 1 + C 2 + I 2 {z } Present value of consumption and investment = Y 1 G 1 + Y 2 G 2 {z } Present value of output (29) Solve for C 2 and substitute in for Y = F (K), I 2 = K 2 and K 2 = I 1 + K 1 C 2 = ()(Y 1 G 1 C 1 I 1 ) + Y 2 G 2 I 2 (30) = ()(F (K 1 ) G 1 C 1 I 1 ) + F (K 2 ) G 2 + K 2 = ()(F (K 1 ) G 1 C 1 I 1 ) + F (K 1 + I 1 ) G 2 + K 1 + I 1

17 Optimisation problem max C 1 ;I 1 u(c 1 ) + u ()(F (K 1 ) G 1 C 1 I 1 ) +F (K 1 + I 1 ) G 2 + K 1 + I 1! (31) First-order condition with respect to C 1 u 0 (C 1 ) = ()u 0 (C 2 ) (32) First-order condition with respect to I 1 (note that K 1 is given at date 1) () + F 0 (K 2 ) + 1 = 0 F 0 (K 2 ) = r (33) Note! Desired capital stock is independent of preferences Note! Government consumption does not crowd out investment

18 Equilibrium Assume for now G 1 = G 2 = 0 Intertemporal production possibilities frontier (PPF): the technological possibilities for transforming period 1 consumption into period 2 consumption (in autarky) C 2 = Y 2 + K 2 (34) = Y 2 + I 1 + K 1 = F (I 1 + K 1 ) + K 1 + I 1 = F (Y 1 C 1 + K 1 ) + K 1 + Y 1 C 1 = F (F (K 1 ) C 1 + K 1 ) + K 1 + F (K 1 ) C 1 Maximum consumption in period 1 (horizontal intercept) C max 1 = F (K 1 ) + K 1 (35) Maximum consumption in period 2 (vertical intercept) C max 2 = F (F (K 1 ) + K 1 ) + K 1 + F (K 1 ) (36)

19 Shape of PPF dc 2 dc 1 = F 0 (K 2 ) 1 < 0; Autarky equilibrium (point A) d 2 C 2 dc 2 1 = F 00 (K 2 ) < 0 (37) Production in open economy (point B) Consumption in open economy (point C) u 0 (C 2 ) u 0 (C 1 ) = F 0 (K 2 ) = 1 A (38) F 0 (K 2 ) = (39) u 0 (C 2 ) u 0 (C 1 ) = 1 (40)

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