THE INFINITE HORIZON MODEL
The notes in this section are based on Obstfeld and Rogoff
In a closed economy resources available for life-time consumption are constrained by the
intertemporal model, but also by the period’s output. When we allow borrowing and
lending in the international markets, in an open economy, the second constraint is
relaxed. A negative shock to output or high investment/deficit does not require a 1-to-1
fall in consumption. This is the consumption smoothing aspect of the current account.
To get the optimal consumption and the current account, we need the following
conditions:
 The optimal consumption: obtained from the Euler condition
 The infinite-horizon IBC: obtained from solving the BC forward.
 Transversality Condition (No-Ponzi games)
I. Optimal Current Account and Consumption with Single representative agent, no
cost of adjusting capital.
Assumptions:
 SOE: the interest rate r is exogenously determined in the world market.
 Producers are self-employed farmers who invest to maximize life-time
consumption.
In the SOE, the representative consumer with infinite horizon maximizes the
intertemporal utility:

(1)
U  s  stU (Cs )
Bt 1 , K t 1
Subject to the intertemporal budget constraint, capital accumulation, and the production
function:

Budget constraint: Bt 1  Bt  Kt 1  Kt  Yt  rBt  Ct  Gt
(2)
or the CA constraint: Bt 1  Bt  Yt  rBt  Ct  Gt  I t with I t  Kt 1  Kt
Note:

CAt  Bt 1  Bt  St  I t
Production function: Yt  At F ( Kt ) .
(3)
where B=net foreign assets, Y=income, C=consumption, G=government spending,
I=investment, K=capital stock, A=technology.
Procedure: Get C from (2) and Y from (3) and substitute it in (1), for unconstrained
maximization. We need to maximize with respect to Bt 1 and K t 1 , so we must find all
these variables in the objective function. For each period s, the choice of Bt 1 and
K t 1 determines Cs given the historical values of Bs and K s and other exogenous
variables.
Optimal consumption:
The Lagrange for t=1 and 2 should give us all the choice variables we need to maximize
U  U [(1  r ) Bt  Bt 1  At F ( Kt )  ( Kt 1  Kt )  Gt ]  U [(1  r ) Bt 1  Bt 2  At 1 F ( Kt 1 )  ( K t 2 Kt )  Gt 1 ]
FOC
(i) wrt Bt 1 : Euler equation U ' (C t )  (1  r )U ' (Ct 1 )
(ii) wrt K t 1 : differentiating and substituting (4): At 1F ' ( Kt 1 )  r
(4)
(5)
The infinite-horizon IBC
The optimal consumption path satisfies the FOC (i) and the budget constraint solved
forward:
From (2):
Yt  Ct  Gt  I t Bt 1

1 r
1 r
Bt 
(6)
forward it one period:
Bt 1 
Bt 
Ct 1  NOt 1 Bt 2

,
1 r
1 r
where NOt 1  Yt 1  I t 1  Gt 1 , and substitute back into (6):
Ct  NOt Ct 1  NOt 1
Bt 2


2
1 r
(1  r )
(1  r ) 2
.
(7)
Now we need to get rid of Bt 2 . For this, again forward (6) this time by two periods and
substitute it in (7).
Bt 2 
Ct 2  NOt 2 Bt 3

,
1 r
1 r
substitute in (7):
Ct  NOt Ct 1  NOt 1 Ct  2  NOt  2
Bt 3



,
2
3
1 r
(1  r )
(1  r )
(1  r ) 2
…etc.
Bt 
Thus:
  1 
(1  r ) Bt   s  t 

1 r 
s t
(Cs  NOs ) 
Bt T 1
(1  r )T

Transversality (no-Ponzi games) condition: lim

T 
T
1 
 Bt T 1  0 .
1

 r
This is an exogenously imposed condition, which rules out that debt grows at a rate faster
than r (i.e. that lim()>0).We don’t want to accumulate claims on ROW that grows faster
than r, and neither does the ROW wants to accumulate claims on us that grow faster than
r. It is the infinite-horizon counterpart of imposing B3  0 or B2   B1 in the two-period
model.
Hence the IBC becomes:
  1 
(1  r ) Bt   s  t 

1 r 
s t
(Cs  NOs )
(8)
We can rewrite the IBC in terms of the optimal consumption path:
s t
s t
1 
 1 
 
 Cs  (1  r ) Bs  st 
 NOs .
st 
 1 r 
 1 r 

Assuming   1/(1  r ) and using the Euler conditions, C1  C2  ....  Ci  ...
Using the property of infinite sums {with a<1 t1 at  1/(1  at ) }, we get:
r

Ct  (
)[(1  r ) Bt  s t (1  r ) ( s t ) NOs ]
1 r
(9)
Given the choice of Bt 1 , K t 1 and the IBC, we obtain the paths of C and NO.
Optimal CA
We define the variables in terms of their current as well as permanent levels. The
~
permanent level of a variable X (C, G, I, Y) is X and satisfies the relation:
~


(10)
st (1  r )( st ) X t  st (1  r )( st ) X s
It says that the present value (PV) of the permanent value of X is equal to the PV of
future Xs. Using the rule of infinite sum of a ratio we get the permanent level of X as:
r
~

X 
st (1  r ) ( st ) X s
1 r
It is the weighted average of the discounted value of future Xs.
We can express the optimal consumption path in (9) by applying the permanent values of
variables:
Ct  rBt  (
~
r

)st (1  r ) ( st ) NOs  rBt  NO
1 r
(11)
Substituting this expression into the CA equation in (2), we get the optimal CA in terms
of deviations of variables from the long-run values:
~
CAt Bt 1Bt Y trBt  (rBt  NO)  It  Gt
~
~
~
~
CAt  NOt  NO  Yt  Y  ( It  I )  (Gt  G)
(12)
Fundamental equation of optimal intertemporal CA theory. Predictions:

~
Y  Y : deviation of current output from its permanent level leads to a CA+
because of consumption smoothing. C does not rise 1-to-1, accumulation of net
foreign assets B through an increase saving.
When Y rises, at time t,
Ct / Yt  r (1  r ) 1 ,
but CAt / Yt  1  r (1  r ) 1  (1  r ) 1  S t / Yt  Ct / Yt .

~
~
I  I or G  G : deviation of I or G from its permanent level leads to a CA-
because of consumption smoothing again. C falls but less than 1-to-1, the country
borrows from abroad to finance the departure of I or G from its LR value.

~
When A rises (permanent technological progress): both Y (future output) and I
rise and lead to a CA-.
Thus, CA- is caused by contemporaneous departures of Y below or those of I/G above
their permanent levels.
Exemples:
1. Define TB=Y-C-I-G=NO-C and look at the implications for national solvency.
2. NO grows at a constant rate: NOs1  (1  g ) NOs . Implications for debt growth.
1. National Solvency:
If countries go bankrupt then the IBC doesn’t hold, they cannot get loans because
potential creditors perceive them to be high risk.
Define TB (primary deficit/surplus) as Y-C-I-G. The infinite horizon BC can be written
i.t.o. TB:
s t
s t
  1 
 1 
(1  r ) Bt   s  t 
 ( NOs  Cs )   s  t 
 TBs
1 r 
1 r 
IBC holds if and only if the country can pay its initial debt through a stream of future
primary surpluses. If the country is net debtor (B is net debt), for it to decline, either
future Y must rise or a combination of present and future G/I/C must fall. Conversely if
the country is net creditor (B is net foreign assets), the country can run future deficits and
allow a combination of present and future C/I/G to be higher than the stream of income.

Often the IBC is defined as a ratio to GDP. Then the IBC will be modified as:
s t

1 
 TBs / Ys
(1  r  g ) Bt / Yt  st 
1 r  g 
where g is growth rate of real output and 1+r-g is the growth adjusted real gross rate. We
have to have or assume r>g otherwise transversality condition implies that we can run
infinite Ponzi games since the no bubbles condition will tend to zero as long as change in
debt < change in Y (because B/Y will decline).

2. Implication on CA of constant growth in NO
Assume constant growth of NO: NOs1  (1  g ) NOs and a stable debt-to-GDP ratio B/Y.
Thus B also grows at the rate g. This means that the country is solvent (or external debt
is sustainable) even if the optimal CA is negative at all t.
s t
 1 r 
1 g
 1 
1 g 
s t  1  r  NOs  NOt  1  r NOt   1  r  NOt  ...  NOt  r  g  if g<r, i.e if


debt growth is less than interest paid on it.

2
Replace in optimal consumption:
Ct  (
r

)[(1  r ) Bt  st (1  r ) ( st ) NOs ]
1 r
 r 
 NOt .
 rBt  
rg
Substitute it in CA:
 r 
 g 
 NOt  
 NOt <0 if r>g.
CAt  rBt  NOt  rBt  
rg
rg
Extensions:
1. When   1/(1  r ) , how does the optimal CA change?
2. Time varying interest rates:
If r was time varying such as rs 1 is the rate on loans offered between s and s+1.
Then the discount ratio can be defined as
1
Rt , s 
s
v t 1 (1  rv )
and after imposing the transversality condition
lim Rt ,t T Bt T 1  0
T 
the IBC can be rewritten as:


s t
Rt , sCs  (1  r ) Bs  s t Rt , s NOs

Now the optimal CA will be slightly different.
II. Optimal CA with capital markets
Until now we assumed producers invest to maximize their lifetime consumption. Now
introduce two separate markets (capital and labor). Results change little.
Assumptions:
 Y=AF(K,L) where F is HD(1) --CRTS—
 Labor supply constant, no international mobility of labor.
 Y produced by a domestic firm owned domestically, which hires labor at wage w,
invests and sells shares of future profits in the stock market.
The consumer’s problem becomes:
Maximize U  s  stU (Cs )
(13)
Bt 1 , xt 1
Subject to:
Bt 1  Bt  vt xt 1  vt 1 xt  rBt  dt xt  (vt  vt 1 ) xt  wt L  Ct  Gt
(14)
where v=price of a claim to firm’s future profits, x=share of firm owned by consumer, d=
dividend the firm issues.
 LHS=Savings=increase in NFA + increase in value of shares.
 RHS= interest income from NFA + Dividends on shares + cpt gains/losses on
shares + human capital – private and government consumption(taxes).
 There is no I in the BC because capital stock decisions are taken by the
~
representative firm, hence there will be no I in the CA expression. Consumption
decisions are separated from investment decisions.
Substitute for C in the objective function, and maximize with respect to the choice
variables.
Optimal consumption
First order conditions:
Bs 1 : U ' (Ct )  U ' (Ct 1 )(1  r )  0  U ' (Ct )   (1  r )U ' (Ct 1 )
xs1 : U ' (Ct )vt  U ' (Ct 1 )(vt 1  dt 1 )  0  U ' (Ct )vt   (vt 1  dt 1 )U ' (Ct 1 )
(15)
(16)
Substituting (15) into (16) we get:
1 r 
d t 1  vt 1
vt
or r 
dt 1  vt 1  vt
vt
(17)
The FOCs say that on the margin, consumers are indifferent between investing in FA and
shares since both returns are equal (return on FA=return on domestic shares=div+cpt
gain/loss).
Intertemporal budget constraint
We now derive the consumption path from the BC:
Bt 1  Bt  vt xt 1  rBt  dt xt  vt xt  wt L  Ct  Gt
thus Bt 1  (1  r ) Bt  vt xt 1  (dt  vt ) xt  wt L  Ct  Gt
Using from FOC (17) that (dt  vt ) xt  (1  r )vt 1 , we can rewrite the BC as:
Bt 1  (1  r ) Bt  vt xt 1  (1  r )vt 1 xt  NI t  Ct
where NI=wL-G (net income).
Define Qt 1  Bt 1  vt xt 1 financial wealth.
Thus, we can rewrite the BC in terms of Q:
(1  r )Qt   NI t  Ct  Qt 1 and apply the forward iteration as before to get:
s t
Q
 1 
Qt  st 
 (Cs  NI s )  t T 1T .
(1  r )
 1 r 

T
 1 
Transversality condition lim

 Qt T 1  0
T 
 1 r 
Imposing this condition on the IBC, we get the optimal consumption path as before:
s t
s t
1 
 1 
 
 Cs  (1  r )Qs  st 
 NI s
 1 r 
 1 r 
st 

or expanding:
s t
s t
1 
 1 
 
 Cs  (1  r ) Bt  (dt  vt ) xt  st 
 NI s
 1 r 
 1 r 
st 

This equation says that the PV of consumption is equal to the initial stock of financial
wealth (consisting of FA and stocks) + PV of after-tax labor income (human wealth).
 Optimal Consumption path:
For this, we define again consumption in terms of permanent levels of variables and
assume that   1/(1  r ) .
Ct  rQt  (
r
~

)st (1  r ) ( st ) NI s  rQt  NI
1 r

Optimal Current Account:
CAt  St  I t  rQt  NI t  Ct  I t . Substitute in C:
~
~
~
~L
CAt  NIt  NIt  It  (wt Lt  w
)  (Gt  Gt )  It .
Firm’s behavior
(i). No adjustment cost for capital stock
The firm maximizes the PDV of the future stream of profits (=dividends)
Max Firms market value at t= st (1  r)( st ) [ As F (Ks , Ls )  ws Ls  (Ks1  Ks )]
Kt 1 , Lt , I t
FOC:
K t 1 : At FK ( Kt , Lt )  r
Lt : At FL ( Kt , Lt )  w
Firms can adjust the capital optimally instantaneously because there is no cost of
investment, thus no well-defined investment function. Hence, the CA remains as before.
(ii). Adjustment costs for capital stock: Tobin’s q
The firm incurs higher costs the faster is the installation of new capital.
Max Market value at t= s t (1  r ) ( st ) [ As F ( K s , Ls ) 

Kt 1, Lt
 Is
2
2 Ks
 ws Ls  I s ]
Subject to: Kt 1  Kt  I t
In this case we don’t substitute for C, instead we use the Lagrange multiplier explicitly
because this multiplier is interpreted as Tobin’s q, where q is the shadow price of capital
at the end of period s. Maximizing with respect to I, L, and K, assuming marginal
q=average q, we get

The optimal consumption path:
s t
s t
1 
 1 
 
 Cs  (1  r ) Bt  (1  r )qt 1 Kt  st 
 NI s
st 
1

r
1



 r


The optimal CA:
~
~
~
CAt  Yt A  Y A  ( I t  I )  (Gt  G)
where output adjusted for investment cost Yt A  At F ( K t , Lt ) 
 I t2
2 Kt
.
Productivity shocks:
1. In a two-period model: a rise in A makes future capital K 2 more productive, hence
next period output Y2 rises, thus C1 and C2 increase, CA1 <0 while CA2 >0. Thus, K
increases to its new level in one period. There are no adjustment costs.
2. In an infinite-horizon with adjustment costs, the CA deficit lasts long time
because the capital stock adjusts over several periods. K rises slower and slower,
I falls and S rises gradually because Y increases slowly, CA gets back into
balance (tends to 0) only in the long run.