Lecture 2
The Static Model : An Infinite Period Case
Macroeconomics
Ph.D. Program, Chulalongkorn University
1/2008
By Kornkarun Kungpanidchakul, Ph.D.
Reading list: Manuelli’s Note, chapter 2
Consider the representative household who lives for an infinite number of periods.
The utility function of the household can be written as:
u (c1 , c 2 ,..)  u (c1 )  u (c 2 )   2 u (c3 )  ...

   t u (c t )
t 0
In each period, given the initial amount of capital k t and xt id investment with  is
the depreciation rate. Then we can write down the maximization problem as:

max   t u (ct )
ct , xt
t 0
s.t. ct  xt  f (k t )
k t 1  (1   )k t  xt
Truncate time to T periods and take limit when T goes to infinity to find the first
order condtion. Also, Impose assumptions to assure interior solution:
lim u ' (c)   (Inada condition)
c 0
f (k 0 )  c and f is increasing, k t 1 , xt  0
Then we get the first oder conditions as follows:
u' (ct )
 f ' (k t 1 )  1  
u' (ct )
(1)
(Euler Equation)
lim  T 1T kT 1  0
(2)
(TVC)
ct  xt  f ( k t )
(3)
(4)
T 
k t 1  (1   )k t  xt
Steady State Analysis
Steady state is the environment in which the allocations are equal at all time. It is
the situation that a planner chooses not to grow. We can regard the steady state as the
long run equilibrium.
Definition: A steady state is a feasible allocation such that ct  c , xt  x and k t  k for
all t.
Consider when there is a government spending, then the budget constraint
becomes:
c t  xt  g t  f ( k t )
With (c*,x*,k*) is steady state allocation, the condition can be simplified as:
1   [ f ' (k *)  1   ]
(5)
c *  x *  g*  f (k *)
(6)
*  u ' (c*)
(7)
(6) implies that the steady state investment is just enough to cover depreciation of the
capital stock.
Existence of the steady state
From (5)-(7), in order to find the existence of the steady state, it suffices to show
that there is a capital stock k* that satisfies (5) and that at the level of capital stock k=k*,
the level of consumption c* is non negative.
1. The existence of k*
Consider the right hand side of (5), since f is twice differentiable and f’ is
continuous, then to show the existence of k*, we need to show that for small k,
1   [ f ' (k )  1   ] and for large k, 1   [ f ' (k )  1   ] . If this is the case, then there is at
least one k s.t. 1   [ f ' (k )  1   ] . Therefore, to have the existence of k*, it is enough to
impose the condition saying that MPK is high when the capital stock is low and vice
versa.
2. The uniqueness of k*
From the assumption that f is strictly concave, then there is only one k* s.t.
1   [ f ' (k )  1   ] .
3. c*  0
From (6), c*  f (k *)  k *  g * . First consider when g=0, then
c*  f (k *)  k * . To show that c*  0 , we need to show that f (k *)  k * . Since f(.) is
concave function, it is sufficient to show that f ' (k )  k at k*.
How about if g>0? The condition (5)-(7) we derive is still true when g is small.
Therefore, if g is not too large, a steady state exists. However, if g is sufficiently high,
there is no steady state. For example, if f (k *)  g , then c*  0 .
Remarks:
1. The stock of capital per worker (k) is independent of the level of government
spending and the form of utility function.
2. The permanent changes in government spending plays no role. If will only result
in a decrease in consumption. So the model in the steady state does not have
crowding out effect. The investment decision does not depend on g in the steady
state. So an increase in g has pure income effect in the way that it reduces
consumption in the same amount.
3. The more patient economies will have higher steady state capital stocks.
4. To consider the impact of better technology, let’s modify the production function
to allow for different technological levels. For example, let production function be
given by zf(k), then an increase in technology (higher z) increases the steady state
capital stock.
Example 1: Show that higher z brings about higher k* when production function become
zf(k).
Example 2: Suppose instead that there is a labor needed in the production function. The
household is endowed with one unit of labor. Then the utility function is represented by:

max   t u (ct ,1  nt )
ct , xt
i)
ii)
iii)
iv)
t 0
s.t. ct  xt  g t  f (k t , nt )
k t 1  (1   )k t  xt
Construct the steady state of the economy.
Prove for the existence and the uniqueness of the steady state.
Find the impact of z on n.
Find the impact of an increase in government spending on a capital stock per
labor and the output.