# Dynamics of current account in a small open economy

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1 Dynamics of current account in a small open economy Ester Faia, Johann Wolfgang Goethe Universität Frankfurt a.m. March 2009 ster Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March 2009 / 5

2 Recursivity property of the optimal solution In t consumer chooses the consumption plan fc t g t=0 But if he re-maximizes in t + the utility function: U t+ = will he choose the same consumption plan? β s (t+) u(c s ) () + Ester Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

3 Dynamic consistency Since optimization in any subsequent period is undertaken facing the same preferences and the same budget constraint, the optimal consumption plan will be the same In any subsequent period t + x consumption must satisfy the same Euler equation and the same intertemporal b.c.! This is a dynamic consistency property Ester Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

4 Time-inconsistency, Strotz (956) Assume the following utility function: U t = ( + γ)u(c t ) + + This utility places a high weight on current consumption β s t u(c s ) (2) Let s choose the optimal consumption plan at time t. The Euler condition between time t + and all subsequent periods s, for any s > t + reads as following: β s t u0(c s ) u0(c t+ ) = () (s t ) (3) Ester Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

5 Time-inconsistency, Strotz (956) (continued) Let s now re-optimize at t +.The utility function now reads as follows: U t+ = ( + γ)u(c t+ ) + +2 β s (t+) u(c s ) (4) The Euler condition between C t+ and C s,for any s > t +, is: β s t u0(c s ) ( + γ)u0(c t+ ) = () (s t ) (5) Since equation 4 is di erent than equation 5 the new optimal consumption plan will be di erent In this case the optimization problem is time-incosistent Ester Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

6 Variables under permanent levels De ne the permanent level of a variable x as: Recall that: s t s t ex t = x s (6) lim s! s t = +r = r +r = r This implies that ex is the annuity value of the variable x s : ex t = r s t x s Ester Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

7 Dynamic of the current account, Sachs (982) We start with the case in which consumption ís at the steady state In the steady state: β = +r. This implies that consumption is constant and reads as follows: " C t = C = r s t ()B t + (Y s G s I s )# (7) Let s now substitute the above consumption function into the intra-temporal budget constraint: We obtain: CA t = B t+ B t = Y t + rb t G t C t I t CA t = Y t + rb t G t I t (8) " r s t ()B t + (Y t G t I t )# ster Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

8 E ects of changes in current income or investment Let s now use of the de nition of a variable under permanent level (ex t = r s t xs ) into equation 8 +r +r From equation 8 we obtain: " r s t CA t = Y t (Y s G s I s )# = Y t ey t G t eg t I t ei t When current output is larger than the permanent level, Y t > ey t, agents will lend to the rest of the world,ca t > 0, since agents aim at consumption smoothing. If current output, Y t, grows,consumption, C t, stays constant and foreign asset accumulation grows If instead investment, I t, grows, consumption, C t, stays constant and agents have to borrow from abroad ster Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5 G t I t (9)

9 Dynamic of the current account outside the steady state Let s now consider the dynamic of the current account when consumption is obtained outside of the steady state If β 6= +r (outside of steady state) consumption is not constant and is given by: C t = r + θ " where θ = () σ β σ. ()B t + Substitute it into the current account equation: s t (Y s G s I s )# CA t = Y t + rb t G t I t (0) " r + θ s t ()B t + (Y s G s I s )# Ester Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

10 Dynamic of the current account outside the steady state Let s de ne: ω t = ()B t + s t (Y s G s I s ) () After substituting the de nition of permanent level equivalent of each variable and ω t we get: CA t = Y t ey t G t eg t I t ei t {z } θ ω t {z } Due to the consumption smoothing motive agents want consumption to be constant along the balanced growth path Changes in consumption outside of the steady state, due to di erence between β and, induce changes in the current account (+r ) If β >, θ > 0 agents borrow from the rest of the world (+r ) ster Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

11 Variable interest rates Suppose now that, r t, the real interest rate changes over time The new compound discount factor, R t,s, between date t and a generic date s, is given by: R t,s = Π s v =t+ ( v ) (2) For instance: R t,t =, R t,t+ = ( t+ ) R t,t+2 = ( t+ )( t+2 ) The new current account equation (intra-temporal budget constraint) is: CA t = B s+ B s = Y s + r s B s C s G s I s (3) As before one can substitute recursively, B t+, and obtain the intertemporal budget constraint ster Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March 2009 / 5

12 Optimality conditions under variable interest rates The intertemporal budget constraint now reads as: R t,s (C s + I s ) = ( t )B t + The new transversality condition reads as: lim R t,t+b t+t + = 0 T! R t,s (Y s G s ) (4) To obtain the optimal consumption plan, as before, obtain consumption from equation 3 and substitute it into the utility function Given the production function, Y t = AF (K t ), the rst order conditions with respect to B t and K t now read as follows: u0(c s ) = ( s+ )βu0(c s+ ) (5) A s+ F 0(K s+ ) = r s+ (6) Ester Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

13 Dynamic of current account with variable interest rates With iso-elastic preferences, u0(c ) = C σ σ. Euler condition between date t and date s implies Ct = Rt,s β (s t) σ C s. Raise both sides to the power of σ and invert to obtain C s = Rt,s σ β σ(s t) C t. Substitute C s into 4 and obtain: ( t )B t + R t,s (Y s I s G s ) C t = R t,s h R σ t,s β σ(s t)i (7) With variable interest rates the permanent level of a variable is given by: R t,s ex t = R t,s x s ; ex t = R t,s x s R t,s ster Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

14 Dynamic of current account with variable interest rates (continued) De ne eγ t as the discount-rate-weighted average of consumption growth rates between date t and any date s : = R t,s h R σ t,s β σ(s R t,s Write the consumption function, as obtained from equation 4, as: t)i C t = ( t )B t g Γ t R t,s + g Γ t(y s I s G s ) Ester Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

15 Dynamic of current account with variable interest rates After substituting the above consumtpion function in the CA we obtain: ~ CA t = (r t r t )B t + Y t ey t G t eg t I t ei t + +( e Γ t eγ t )( ~ r t B t + ey t ei t eg t ) Ester Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current a.m.) account in a small open economy March / 5

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