1. Differential Equations by tarungehlots
Concepts of Differential Equation
Consider a variable k that might denote the per capita capital stock level in an economy.
When convenient, we can recognize that the level of capital depends upon time by addition the
time argument. Doing so, we would write the per capita capital stock level as k (t ) , rather than
just writing k . If we think of time as unfolding continuously, we can also think of capital as
being a continuous function of time, and we can assume that the function has derivate dk/ dt .
This derivative is the instantaneous change in per capita capital. In many presentations, dk/ dt

is written as k , but we will here use the more familiar notation dk / dt k ' (t ) . By writing the
derivate as k ' (t ) , so that we include the time argument, we are emphasizing that the value of the
derivative may change with time. Because it is typically cumbersome to repeatedly write down
the argument, we can also write the derivative as just k ' , while remembering that its value may
change with time.
A differential equation is an equation that relates the time derivative of a variable to its level.
An example is the equation
(1) k ' sf (k ) gk nk k.
The variable k is called a state variable because it gives the state of the system at any given
point in time. In our example, k gives the state or level of per capita capital stock.
2. A Dynamic System
The basic dynamic principle is the idea that “the way things are determines the way things
change.” A differential equation is one way of modeling the basic dynamic principle. The state
variable k indicates the way things are, while the time derivative k ' indicates how things change.
The differential equation itself is what relates the two. More precisely, a differential equation
typically presents a functional relationship, showing how k ' depends upon k . When we want to
emphasize the functional relationship, we can write the differential equation as k ' (k ) , where
the general functional form (k ) is the rule that tells us how the value of the time derivative k '
is determined from the level of the state variable k .
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2. A difference equation can also be used to represent the basic dynamic principle. When a
difference equation is used, time unfolds as sequential time periods, rather than unfolding
continuously. We will focus on using the differential equation here.
A solution to a differential equation is a function k (t ) that satisfies the differential equation
for all points in time t that are of interest. Some differential equations can be solved. However,
most differential equations that are of interest in economics cannot be solved. Nonetheless, we
can still learn about the path followed by the state variable over time by using a phase diagram
technique.
Assuming the function f (k ) in equation (1) is nonlinear, we can describe equation (1) as
a single state variable system, and equation (1) is a nonlinear and non-autonomous differential
equation. As a dynamic system, equation (1) is a single state variable system because there is
only one time derivative present, which is k ' . If the system were a two state variable system,
you would see two time derivatives and two different state variables. The equation is labeled
nonlinear because the functional relationship between k ' and k is not linear. Typically,
nonlinear differential equations cannot be solved. However, the presence of the general
functional form f (k ) in equation (1) assures us that we cannot solve it. Finally, equation (1) is
labeled non-autonomous because it includes variables other than state variables and time
derivatives. The variables s , g , n , and in equation (1) are exogenous variables, which are
variables that describe the environment impacting the system. Usually exogenous variables are
constants, which is the case in equation (1).
Classifying the variables of a differential equation system helps clarify how you the
model builder believe the system works. Because our system (1) is a single equation system, we
can only have one endogenous variable. The endogenous variables a differential equation
system are always the time derivatives. So, for our system, k ' is endogenous. The state
variables of a system are always classified as predetermined, so k is predetermined for our
system. The level of the variable k is predetermined at point in time t because the
instantaneous change k ' in the level of k we just determined by the model in the previous
instant of time. Non state variables are exogenous, so s , g , n , and are exogenous. The last
value that needs to be specified for the system is the initial condition k0 for the state variable k ,
which is the value of the variable k at the initial point in time t 0 . Thus, we can summarize
the classification of variables as follows:
Classification of Variables
Endogenous (1): k '
Exogenous (4): s , g , n ,
Predetermined (1): k
Initial Conditions (1): k0
It is common that differential equation models contain auxiliary equations, which are
equations that endogenized (i.e., determine) the values for other variables by relating them to the
state variable. Using standard techniques for analyzing a differential equation of this type, we
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3. can learn how the path for state variable k will evolve over time. Once we know the path
followed by the variable k , we can use the following “auxiliary” equations to determine the
paths followed by the other variables of interest.
To illustrate, let us add the endogenous variables y , r , w , and c to our system, and assume
that they are determined from the following four equations:
(2) y f (k )
(3) r f ' (k )
(4) w f (k ) f ' (k )k
(5) c 1 sy
The variables y , r , and w are directly related to the variable k , and only related to k . Thus, if
we know the path followed by k , we can use equations (2)-(4) to determine the paths followed
by y , r , and w . Equation (5) indicates that we need to know the path followed by y , along
with the value of the exogenous variable s , to determine the path followed by c . So we can
determine the path followed by c by first using equation (2) to determine the path followed by
y . This discussion indicates that we can classify the variables of the dynamic system (1)-(5) as
follows.
Classification of Variables
Endogenous (5): k ' , y , r , w , c
Exogenous (4): s , g , n ,
Predetermined (1): k
Initial Conditions (1): k0
The core of this model is the path followed by the state variable k . Once you understand the
core of a model, you can use that knowledge to then understand the auxiliary aspects of the
model. Notice that adding the auxiliary equations only changes the number of endogenous
variables. We can think of the core of the model, which is equation (1) and the first
classification of variables as a subsystem, a system upon which the auxiliary aspects of the
model depend.
3. Analyzing the Steady State
Finding the Steady States
The steady state for the state variable k is the state where the variable is not changing, or where
k ' 0 . Setting k ' 0 in (1), we know
SS1 sf (k ) g n k.
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4. There are five variables in equation SS1, the model’s four exogenous variables and the state
variable. Equation SS1 determines the steady state values of the state variable as a function of
the exogenous variables. Assuming f (0) 0 , one value of k for which equation SS1 holds is
k 0 . This steady state is not so interesting in that this result is simply telling us that if no
capital initially exists, then none will accumulate over time, (which is because capital is assumed
to be essential in the production process). If there is a non zero value for k that satisfies SS1, let
it be denoted by k .
To illustrate a situation where there is a unique value for the steady state k , consider Figure 2
below. The right side of equation SS1 is linear in k , with a zero intercept, as shown in the
figure. The assumption f (0) 0 implies the left side of equation SS1 also has a zero intercept.
The conditions f ' 0 and f ' ' 0 would ensure that the production function increases at a
decreasing rate as k increases, as shown in the figure. If it were true that f ' (0) g n , then
the only steady state would be associated with k 0 , because the two curves sf (k ) and
g n k would only intersect at the point (0,0) in Figure 2. The Inada condition
lim f ' (k ) ensures that the sf (k ) curve initial rises above the g n k curve, and the
k 0
Inada condition lim f ' (k ) 0 ensures that the sf (k ) curve eventually intersects the
k
g n k curve. Because f ' 0 and f ' ' 0 for all k , we know that this intersection at a
capital level k 0 only occurs once, which implies k is unique, as shown in Figure 2.
Figure 2: The steady state for the Solow-Swan growth model
y
g n k
f (k )
sf (k )
k k
Stability of the Steady State
The Inada conditions lim f ' (k ) and lim f ' (k ) 0 also ensure that the steady tate
k 0 k
associated with k k is stable. A steady state is locally stable if the state variable moves toward
the steady state value for the variable when it is in the “neighborhood” of the steady state.
Define the function (k ) so k ' (k ) . Notice that, if ' (k ) 0 , then the steady state k is
locally stable. This is because ' (k ) 0 implies k' 0 when k k and k' 0 when k k .
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5. For our model given by SS1, (k ) sf (k ) g n k . So, ' (k ) sf ' (k ) g n .
Because ' (k ) sf ' (k ) g n , our steady state k is locally stable if sf ' (k ) g n 0
. Notice, in Figure 2, that the slope of the sf (k ) curve, at the steady state k k , is less than the
slope of the g n k curve. That is, sf ' (k ) g n k . So, we find ' (k ) 0 , meaning
our steady state k is locally stable.
Examining Figure 2, we can also see that the steady state k 0 is locally unstable. In the
neighborhood of k 0 , with k positive, notice in Figure 2 that the slope of the sf (k ) curve is
greater than the slope of the g n k curve. This implies sf ' (k ) g n 0 , which
implies ' ( k ) 0 . If ' ( k ) 0 in the neighborhood of k 0 , then k is increasing in that
neighborhood, meaning it increasing away from the steady state value of zero.
Figure 3 is a phase diagram. The traditional phase diagram for a single state variable
differential equation is constructed by plotting the state variable along the horizontal axis and
plotting the time derivative of the state variable along the vertical axis. So, for our model k is
plotted horizontally and dk / dt k ' is plotted vertically. The function (k ) shows how k '
depends upon k . The Inada conditions tell us that the function (k ) is positive and diverging
to infinity as k approaches zero, and they tell us (k ) decreases as k increases, eventually
becoming negative, as shown in Figure 3. The steady state value for k is where (k ) equals
zero. The arrows along the horizontal axis are presented to emphasize k is increasing when
k k but decreasing when k k .
Figure 3: Phase diagram for the Solow-Swan growth model
(k )
k
k
(k )
To summarize, we know that the steady state k 0 is locally unstable and the steady state k k
is locally stable. The Inada conditions indicate that the path taken over time by the state
variable k is as shown in Figure 3. If the initial capital stock level k0 is between zero and k ,
then k increases over time and approaches k . Alternatively, if k0 is greater than k , then k
decreases over time and approaches k . Figure 4 shows these two potential time paths for k by
plotting k as it depends upon time.
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6. Figure 4: Two possible time paths for the state variable k in the Solow-Swan growth model
y
k0 k
k
k0 k
t
4. A System of Two Differential Equations
Suppose a dynamic system is represented by the following system of equations and the
following classification of variables
(A) k ' (t ) f (k (t )) c(t ) [ g n]k (t )
U ' ' (c(t ))
(B) c' (t ) g n f ' (k (t ))
U ' (c(t ))
Endogenous (2): k ' (t ) , c ' (t )
Exogenous (2): , g, n
Predetermined (2): k (t ) , c (t )
Initial Conditions (2): k (0) , c(0)
How can we characterize the paths followed by the two state variables?
5. Using a Phase Diagram to Characterize the Two Dimensional Differential Equation
System
We now characterize the path followed by c (t ) and k (t ) diagrammatically in a “phase
diagram.” The first step in constructing the phase diagram is to plot the “nullclines” associated
with the steady state conditions c' (t ) 0 and k ' (t ) 0 . From (B), we can see that c' (t ) 0
implies [ g n] f ' . The usual restrictions on the production function f imply there is a
unique value k such that [ g n] f ' (k ) . (Notice that we need the Inada conditions
lim f ' (k ) and lim f ' (k ) 0 to conclude this. The restrictions f ' ( k ) 0 and f ' ' (k ) 0
k 0 k
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7. alone are not enough.) When k k , the assumption f ' ' 0 implies f ' (k ) f ' (k ) , so
f ' (k ) [ g n] which implies the right side of (B) is negative. Because U ' ' 0 and U ' 0
are typically assumed for the utility function, we know c ' (t ) must be positive for the left side of
(B) to be negative. Thus, when k k , c' (t ) 0 . Using analogous reasoning, when k k ,
c' (t ) 0 . This is shown by the arrows in Figure 1.
We obtain the k ' (t ) 0 nullcline by setting k ' (t ) 0 in condition (A), which implies
c(t ) f (k (t )) [ g n]k (t ) . Because f (0) 0 , we know c(t ) 0 when k (t ) 0 . The Inada
conditions lim f ' (k ) and lim f ' (k ) 0 allow us to draw conclusions about the shape of
k 0 k
this nullcline consumption level c (t ) . The assumption lim f ' (k ) ensures that
k 0
f (k (t )) [ g n]k (t ) when k (t ) is near zero, which implies c(t ) 0 when k (t ) is near zero.
Moreover, lim f (k ) ensures f (k (t )) increases faster than [ g n]k (t ) as k (t ) increases
k 0
from a number near zero, which implies c (t ) is increasing as k (t ) increases from zero. The
assumption lim f ' (k ) 0 ensures that, when k (t ) is large enough, [ g n]k (t ) increases faster
k
than f (k (t )) as k (t ) increases. This implies that, as k (t ) increases, c (t ) must reach a peak and
then decline thereafter. This k ' (t ) 0 nullcline separates the region where k ' (t ) is increasing
from where it is decreasing. Condition (A) indicates that, when c (t ) is greater than its nullcline
value for a given level k (t ) , then k ' (t ) 0 . Conversely, when c (t ) is less than its nullcline
value for a given level k (t ) , then k ' (t ) 0 . This is shown by the arrows in Figure 1.
Figure 1: Dynamics implied by the Euler Equation and Capital Accumulation Constraint
for the Neoclassical Model of Optimal Growth
c (t )
c' (t ) 0
c(t ) f (k (t )) [ g n]k (t )
k (t )
k
k ' (t ) 0
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8. Starting at any point in the ( k , c ) of Figure 1, the arrows tell us the direction of
movement. This allows us to trace the path from any starting point (k0 , c0 ) . The starting point
for k 0 is given to us in the problem. The starting point for c0 is a choice. Without additional
restrictions, we cannot determine which value of c0 will occur. However, proceeding with some
analysis can provide an indication of what those restrictions might be.
Figure 2 is drawn under the assumption that k 0 k. (After understanding this analysis,
it should be apparent how to present the case for k 0 k .) Path 1 is associated with a relatively
high initial consumption choice c0 . Because (k0 , c0 ) is in the upper left quadrant of the phase
diagram, the direction of movement must be up and to the left ( c increasing and k decreasing).
k 0 k . Path 2 is still associated with a relatively high initial consumption choice c0 , but one
where (k0 , c0 ) falls in the lower left quadrant. Initially, the direction of movement must be up
and to the right ( c increasing and k increasing). However, the k ' (t ) 0 nullcline is eventually
reached, and the direction of movement of k is reversed. Thus, the paths of type 2 eventually
become like those of type 1. For all paths of type 1 and type 2, k (t ) 0 must eventually occur.
To rule out such paths as equilibrium paths, we need only add the reasonable condition k (t ) 0
to the problem.
Figure 2: Phase Diagram for the Neoclassical Model of Optimal Growth
c (t )
c' (t ) 0
1
2
c
4
*
c0 3
k (t )
k0 k
k ' (t ) 0
Path 3 is associated with a relatively low initial consumption choice c0 . Because (k0 , c0 )
is in the lower left quadrant of the phase diagram, the direction of movement must initially be up
and to the right ( c increasing and k increasing). However, on this path, the c' (t ) 0 nullcline
is reached, and the direction of movement of c is reversed. Eventually, such a path must
involve a negative consumption level. One way to rule out such a path is to recognize that, at
any given point in time t , when the capital-consumption combination is ( k (t ), c(t )) , the
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9. consumer can discontinuously increase or decrease the consumption level, for such an action is
equivalent to the choosing c0 when t 0 . On path 3, by choosing to jump paths by
discontinuously increase c (t ) , (by choosing to reduce the saving level k ' (t ) ), the consumer can
move to a new path that will provide a higher consumption level in every period, for all time,
which is clearly preferable. Thus, it is not optimal to choose a consumption level c0 that
generates a path of type 3.
Having ruled out paths of types 1 and 2 because they are not feasible, and paths of type 3
because they are clearly not optimal, we can conclude that the unique optimal choice for c0 ,
*
given the initial capital level k 0 is the level for c0 that generates path 4, or the level c 0 shown in
Figure 2. On this path, consumption and capital each increase en route to the steady state (k , c ) ,
never crossing a nullcline.
It is worth noting that, if the consumer believes that the economy would end at a
particular point in time T , rather than assuming the economy will never end, then it would be
rational to choose a consumption path such that k (T ) 0 . That is, the consumer should choose
the consumption path where capital is just used up when the economy terminates. (The Euler
equation and capital accumulation equation would be the same; only the “boundary condition”
condition would change to k (T ) 0 .) This implies, rather than choosing path 4 in Figure 2, the
consumer would choose a path either of Type 1 or Type 2, so that vertical axis where k (t ) 0 is
eventually reached. Which path is chosen would depend upon the length of time T. The optimal
initial consumption level becomes lower as the length of time T becomes longer.
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