Intertemporal approach to current account: small open economy

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1 Intertemporal approach to current account: small open economy Ester Faia Johann Wolfgang Goethe Universität Frankfurt a.m. March 2009 ster Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economy March 2009 / 7

2 Basic Assumptions Multi-period models and micro-founded: allow to account for dynamic e ects (growth, business cycles, etc.) Use start with model nite horizon and in absence of uncertainty Final models: in nite horizon and uncertainty (economy is subject to shocks) Small open economy with representative agent World interest rates are taken as exogenous Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economy March / 7

3 Preferences and technology Economy starts at any time t and ends at T. In the in nite horizon case: T!. Population is normalized to Time separable utility function (bounded and satis es Inada conditions) is: U t = u(c t ) + βu(c t+ ) + β 2 u(c t+2 ) β T u(c ) = β s t u(c s ) () For the time being the foreign interest rate is constant over time and equal to r Technology: Y = AF (K ), with F 0 (K ) > 0,F "(K ) < 0 At s = t, K s and B s are taken as given. B is the stock of accumulated foreign asset positions Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economy March / 7

4 Budget constraint (T periods) De nition of the current account: CA t = B t+ B t = Y t + rb t C t G t I t Investment is given by: I t = K t+ K t Rearranging we have the intra-temporal budget constraint: ( + r)b t = C t + G t + I t Y t + B t+ (2) Forward this identity by one period and divide by ( + r): B t+ = C t+ + G t+ + I t+ Y t+ ( + r) + B t+2 ( + r) (3) This can be used to substitute B t+ from equation 2 to obtain: ( + r)b t = C t + G t + I t Y t + C t+ + G t+ + I t+ Y t+ + r + B t+2 + r (4) Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economy March / 7

5 Intertemporal budget constraint Do forward substitution of B t+2 +r, B t+3 +r,... Repeat until we get intertemporal budget constraint: ( + r )s t (C s + I s ) + ( + r )T B + (5) = ( + r)b t + ( + r )s t (Y s G s ) Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economy March / 7

6 Dynamic optimization and Euler condition B s+ : Derive consumption C s from intra-temporal budget constraint: B s+ B s = rb s + A s F (K s ) C s (K s+ K s ) G ss (6) And substitute into U(C s ) to obtain: U t = β s t u[( + r)b s B s+ + A s F (K s ) (K s+ K s ) G s ] (7) Take rst order conditions with respect to B s+ and K s+ : u0(c s ) = ( + r)βu0(c s+ ) (8) K s+ : u0(c s ) = βu0(c s+ )(A s+ F (K s+ ) + ) (9) Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economy March / 7

7 No-Ponzi condition Plug 9 into 8 to obtain: A s+ F 0(K s+ ) = r (0) No-Ponzi condition: foreign lender will not allow domestic residents to die with unpaid debts Also domestic resident will not die with unused assets. Hence in the nite horizon model the following No-Ponzi condition holds: B + = 0 () Optimality satis es 9, 8, and budget constraint (on which we impose B + = 0 ): ) ( + r )s t (C s + I s ) = ( + r)b t + ( + r )s t (Y s G s ) (2) Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economy March / 7

8 Consumption function Using the above expression for the budget constraint we can obtain an expression for the consumption function: Use the following limit of a sequence: ( +r )s t = + ( +r ) + ( +r ) ( +r )T = [ ( +r )T + ] ( +r ) Also notice that: +r +r = r +r Rearranging equation 2 leads to the following consumption function: C T = " ( r) ( + r)b t + (T +) r + r # ( + r )s t (Y s G s I s ) (3) Which states that consumption is given by the total future discounted wealth (permanent income hypothesis) Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economy March / 7

9 Consumption function (continued) As T! : = ( r ) (T +) 0 = Hence consumption function becomes: r C t = ( + r)b t + + r ( + r )s t (Y s G s I s ) (4) C t depends on future discounted value of wealth ) Permanent income hypothesis Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economy March / 7

10 In nite horizon model Preferences: at in nite horizon we need to make sure utility is a bounded above. Assumption: U T = lim u(ct ) + βu(c t+ ) + β 2 u(c t+2 ) +... = T! β s t u[c s ] (5) Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economymarch / 7

11 Dynamic optimization: in nite horizon As before substitute C s from the intra-temporal budget constraint and substitute it into the in nite-horizon utility function to obtain: U t = β s t u [( + r)b s B s+ + A s F (K s ) (K s+ K s ) G s ] First order conditions are as before (see equations 9 into 8) (6) Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economymarch 2009 / 7

12 No-Ponzi condition: in nite horizon Intuitively and in parallel with the nite horizon case, we could guess that now the No-Ponzi condition reads as follows: lim B + = 0 T! The guess is incorrect. To prove it assume that we have constant output, Y, and no government expenditure and investment, G = 0, I = 0 and that β = +r Using the Euler: u0(c s ) = ( + r) +r u0(cs+ ), u0(c s ) = u0(c s+ ) We get constant consumption: C s = C s+ Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economymarch / 7

13 No-Ponzi condition: in nite horizon (continued) With constant consumption the intra-temporal budget constraint now looks as follows: B + = ( + r)b + Y C = ( + r) ( + r)b + Y C + +Y C = ( + r) T + B t + T ( + r) s (Y C ) = ( + r) = ( + r) T + T + B t + (Y C ) = r ( + r) T + = B t + (rb t + Y C ) r h i But if T! (+r ) T +, r!, so unless C = rb t + Y, lim B + = + T! Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economymarch / 7

14 No-Ponzi condition: in nite horizon (continued) What is the right No-Ponzi condition? By inspecting the inter-temporal budget constraint: lim T! ( + r )s t (C s + I s ) + ( + r )T B + (7) = ( + r)b t + ( + r )s t (Y s G s ) The conditions that guarantees that nal wealth is zero is: h i T B (+r ) + = 0 This transversality condition guarantees that the net present value of wealth converges to zero, which impliesh (+r ) i T B + = 0 Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economymarch / 7

15 Sustainability of the current account Some countries have run persistent current account de cits. This raised the question of conditions for current account sustainability (under which countries are foreseen to be able to repay debts) Persistent large current account de cits are seen as a indicator of future borrowing problems. However persistent current account de cits can be sustainable Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economymarch / 7

16 Sustainability of the current account (continued) To obtain an indicator of current account sustainability start to rearrange the inter-temporal budget constraint: ( + r)b t = ( + r) s t (Y s C s G s I s ) {z } Equation 8 states that economy resource transfers to foreigners must equal to the economy initial debt Assume that output grows at a rate g, Y s+ = ( + g)y s, and assume a steady state B s Y s so that B s+ = ( + g)b s From the current account (intra-temporal budget constraint) this implies: B s+ B s = gb s = rb s + TB s where: TB s = Y s C s G s I s This implies that: TB s Y s = (r g )B s Y s = B s Ys (r g ) (8) Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economymarch / 7

17 Sustainability of current account (continued) To maintain a steady TB s Y s the country needs to repay the excess of the interest rate over the growth rate of output Notice however that also political and institutional factors determine sustainability as they could induce foreign lenders to " r. Ester Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt approach a.m.) to current account: small open economymarch / 7

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