1. Notes on Current Account and Investment
G6904

2. Current Account and Investment with Adjustment Costs
Small open economy selecting consumption and investment path subject to adjustment costs. Households with inelastic labor supply wish to maximize value of firm by selecting a sequence of Is
Subject to
Note that value of firm plus human capital, wage income drops out and just want to maximize present value of net output (net of adjustment cost).

3. Current Account and Investment with Adjustment Costs
Set up the Lagrangian
First order condition for Is

4. Current Account and Investment with Adjustment Costs
The Lagrangian
First order condition for Ks+1

5. Dynamics
Dynamics of the system evaluated at steady state
Totally differentiate and use fact that square term drops out and that Fkk dK + Fkk dq term remains evaluated at steady state.

6. Model is Saddle Path Stableq 1
dk /dt = 0
dq /dt = 0 k

7. Current Account Implications
With r equal to time preference, the a version of the Sachs equation holds
What happens in response to permanent rise in productivity? We see that dq /dt = 0 must shift up and to the right

8. A Rise in Productivity q
Q rises in anticipation of higher future MPK and adjustment costs. GDP, consumption, and investment boom 1
dk /dt = 0
dq /dt = 0 k

9. Dynamics Y C A I t0

10. Current Account in Deficit Along Adjustment Path
With Y rising and I falling along adjustment path, CA must be in deficit until new steady state with CA balance. In new steady state, a trade surplus to finance the interest bill on debt incurred during adjustment. Y A CA

11. Derivation
Totally differentiate at dq/dt = 0 and use fact that square term drops out at q = 1.