∞
∑βt Ε[u(ct)] (1)
t=0
where β is the discount rate [ 0 < β < 1 ], u is the utility function [ u '(ct ) > 0 and u ''(ct )< 0 ],
and c is consumption of a single traded good. Utility is maximized subject to a dynamic budget
constraint given by
bt+1 =(1+r)bt +qt -ct -it -gt (2)
where b is the level of foreign bonds held by the economy, r is the world rate of interest, qt is
GDP, it is the level of investment, and g t is government expenditure.
The current account balance is given by
cat = bt+1 -bt (3)
Assuming a no-Ponzi game and the first order conditions with the dynamic budget constraint,
the optimal consumption function is given by
* r , .
ct = ~ 1 bt +
t θ t
1+r
Et
∑ (1⅛ ∆(,t+j- it+j - gt+j ) [
j=0 (1 + r ) J
(4)
where Ct* is the optimal path of consumption and θ is the proportion that reflects consumption
tilting which is given by the relation between rate of interest (r) and the rate of time preference
(β).5 If θ < 1, then the country is consuming more than the national cash flow which means the
country is tilting consumption to the present. If θ > 1 then the country is consuming less than the
national cash flow which implies that the country is tilting consumption to the future. If θ = 1
then consumption equals the national cash flow. There is no consumption tilting in this case.
From optimal consumption Ct* we can compute the optimal consumption smoothing current
account Ca t* as follows
**
(5)
Cat = yt - it - gt -θCt
5 For example, if we assume a quadratic utility function, θ = βr (1 + r)∕[β (1 + r)2 -1].