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Phase Diagrams: construction and comparative statics
November 13, 2015
Alecos Papadopoulos
PhD Candidate
Department of Economics, Athens University of Economics and Business
[email protected], https://alecospapadopoulos.wordpress.com
I detail the steps for the construction of a phase diagram. Then I present different tactics
to do comparative statics, depending on how much algebraically complicated the model
is. This is not a comprehensive guide. It focuses on the most popular educational
instance of a phase diagram in economics, the representative-household and the
overlapping-generations deterministic, continuous time, saddle-path stable models of
growth.
1. Constructing the phase diagram
We will concern ourselves with the phase diagram of the overlappinggenerations model of Blanchard and Weil with government, since it essentially
nests the Ramsey representative household model, as regards the phase
diagram.
Our goal is to picture in two dimensions the long run state of an economy,
study its stability properties and get a taste of the path towards it. The "longrun" state was initially called "steady-state", but in order to accommodate the
fact that the actual levels of the aggregate variables continue to grow
indefinitely (when there is exogenous growth like that of population and
productivity), the profession has started to use the term "balanced growth
path". Sounds nice, the only problem is that in the phase diagram this "path" is
just a point, while there is another important path, the "saddle-path towards the
fixed point". To avoid confusion, we will keep the "steady-state" terminology.
The Blanchard-Weil overlapping generations model with government is
(eventually) described by the two differential equations

c(t )   r (t )    g  c(t )   n k (t )  d (t ) 
[1]

k (t )  y(t )  c(t )  cg (t )   n  g    k (t )
[2]
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the production function in intensive form
y(t )  f  k (t )  Ak (t )
[3]
and the market-clearing and profit maximizing condition
r (t )  f   k (t )   Ak (t ) 1
[4]
where y is total output(income), c is private consumption, k is physical capital
, d is public debt (held as wealth by the individual), and cg is government
consumption. These magnitudes are measured per labor efficiency unit.  is
the rate of pure time preference, r is the real interest rate, n is the birth rate of
new households that also equals the growth rate of population, g is the growth
rate of labor efficiency, and  is the rate of capital depreciation. A denotes
total factor productivity and  is the capital share in production. Preferences
have been assumed logarithmic, so the intertemporal elasticity of substitution in
consumption is equal to unity. We maintain the (not necessary but crucial, and
not unrealistic) assumption   n .
Note 1: the bar above the two government variables indicates that are treated as
exogenous to the model, functioning essentially as parameters. So the phase
diagram is for given values of these two variables.
Note 2: Consumption and capital here are average magnitudes per efficiency
unit, since in this model new households are born every instant of time
(forming a generation), and the representative household of each generation
differs from the representative households of all other generations as regards
the level of capital it owns (they are identical in every other respect).
The phase diagram here is a graph in (c, k ) space where we put consumption
on the vertical axis. It becomes informative if we draw in it
a) The zero-change loci for capital and consumption, namely the combinations


of consumption and capital that satisfy c(t )  0 and k (t )  0 . The crossings of
these loci represent fixed-points of the system, where both differential
equations are zero.
b) Arrows indicating how consumption and capital tend to change (increasedecrease), outside these zero-change loci, which provides qualitative
information on the dynamic tendencies of the system when we are not on the
fixed points. These also help establish the stability properties of the latter (a
"fixed point" is not necessarily stable).
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1.1. Zero-change loci and their shape
For the zero-change loci, it is convenient to write them both as "consumption
being a function of capital", since capital is placed on the horizontal axis and it
is customary even in economics to have the "argument of a function" on the
horizontal axis (except for price-quantity diagrams!). So

c(t )  0  c 
n k  d 
rg

n k  d 
[5]
f (k )      g

k (t )  0  c  y  cg   n  g    k  f (k )  cg   n  g    k
[6]
where I have dropped the time-index to stress that these are static relationships.
In order to draw (schematically) the graphs of these functions, we need to do
old-fashioned analysis of a function with respect to its argument: calculate
derivatives with respect to capital to see whether the function is monotonic or
not, determine if and where it crosses the axes.
We have
c
k

c (t )  0

 n  f (k )      g   f (k )  n  k  d 
 f (k )      g 
2
0
[7]
This is always positive because consumption cannot be negative (so we
examine the space where k : f (k )      g  0 ) and because we have assumed
that f (k )  0 . Therefore this zero-change loci is monotonically increasing in
capital, and moreover it has a vertical asymptote at kˆ : f (kˆ)      g .
For k  0 eq. [5] crosses the consumption axis at zero, because by the Inada
conditions, lim f (k )   (so the quotient is not an indeterminate form).
k 0
Moving to the other zero-change loci, we have
c
k

k (t )  0
 f (k )   n  g   
[8]
This is not monotonic. It has a critical point at k : f (k )   n  g    where also
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 2c
k 2

k ( t )  0, k  k
 f (k )  0
so it is a maximum. The locus [6] takes the value cg when k  0 and also it
crosses the axis of capital at another point, since its second derivative with
respect to capital is always negative after the maximum point.
Finally we note that the vertical asymptote of the consumption locus lies to the
left of the maximum of the capital locus, since we have made the assumption
  n and so
f (kˆ)      g  n  g    f (k )  kˆ  k
With all this information gathered, we can draw representative curves for the
two zero-change loci, without the need to specify numerical values for the
parameters:
We observe that the system possess two fixed-points, the second and low one
due to the existence of government consumption (if cg  0 the capital locus
crosses at the beginning of the axis and the low fixed point disappears). We will
concentrate on the other fixed point, and leave this one as a small
application/exercise, after we have finished with the phase diagram.
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1.2. Dynamics off the zero-change loci
We want to know how consumption and capital move when not on the zerochange loci. Namely, we want

c(t )  0  c 
n k  d 
f (k )      g
[9]

k (t )  0  c  f (k )  cg   n  g    k
[10]
Inequality [9] tells us that for all points in the phase diagram "above" the
consumption zero-change locus, consumption will tend to increase (and it will
tend to decrease when below). Inequality [10] tells us that for all points in the
phase diagram below the capital zero-change locus, capital will tend to increase.
We represent these dynamic tendencies by drawing arrows. The arrows that are
parallel to an axis represent the dynamics of the variable measured on that axis.
The two zero-change loci have split the space in five sub-spaces. We draw
arrows in four of them (see the exercise at the end for the fifth), and in each we
join the arrows at their beginnings to provide a hint on the joint dynamics that
characterize the system:
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This is a classic picture of a "saddle-path stable" phase diagram: in two of the
regions the arrows representing the dynamic tendencies jointly seem to push
the system to extreme/corner solutions. In the other two, they seem to push the
system towards the fixed-point... but this does not mean that if we start
wherever in these two regions, we will end up on the fixed point. There is a
unique path that lives in these regions and leads to the steady-state: the saddlepath. Points off the saddle-path , even if inside the two regions that host it, will
lead to the other two, clearly unstable regions and then to extreme/corner
solutions.
The saddle path represents combinations of consumption and capital that will
be realized as time passes (i.e. it is a curve qualitatively different than the zerochange loci). On it, variables change in magnitude. The exact shape of the
saddle path depends on the algebraic nature of the equations involved. But the
certain thing is that it passes from the fixed point, and that it lies wholly inside
the two regions. We can add this information to the phase diagram, completing
its construction:
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1.3 A special case: the consumption zero-change locus in the Ramsey model.
In the representative-household Ramsey model the differential equation for
consumption is

c(t )   r (t )    g  c(t )
[11]
and the corresponding zero change loci is

c(t )  0  f (k )      g  0  f (k )      g
[12]
(we ignore the uninteresting case where c  0 ). This cannot be seen as
"consumption as a function of capital", but it is just an equation determining
capital, and it will be a vertical line in the consumption-capital space. But we
can still apply the standard methodology to determine the dynamic tendencies,
combined with assumptions on the production function. Specifically, we want

c(t )  0  f (k )      g
[13]
Now, by the assumption f (k )  0 , we know that as capital increases its
marginal product tends to fall. So in order for the marginal product to be higher
than the value determined by equation [12], the capital level must be lower. So
we can conclude that "to the left" of the vertical locus, consumption will tend to
increase, while to its right it will tend to decrease.
Exercise 1
Draw the dynamic arrows on the fifth region of the phase diagram, and verify
that the low fixed point is unstable.
Exercise 2
Consider the case where n  0 , (and nullify government for clarity). Verify that
the phase diagram of the Blanchard-Weil model becomes equivalent to a
representative-household Ramsey model, with the following interpretational
qualifications: if we assume that at some point in time population becomes
fixed, after having grown in overlapping-generations fashion for some time
interval, then the Blanchard-Weil model still contains heterogeneous
households. This means that the consumption and capital variables are still
proper averages here, not the magnitudes that characterize identical
households. Also, in the Ramsey model we may have growth of the size of each
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household (this is what n would represent there), while in the Blanchard-Weil
model, n is the birth rate of new households, while the size of each (old or new)
household is identical and fixed.
Still, the phase diagrams of the two for n  0 in the Blanchard-Weil model
become mathematically equivalent, as regards the general properties of the
economy, its fixed point, but also the comparative statics...
...to which we now turn.
2. Comparative statics methods
By "comparative statics" we mean studying qualitatively how the steady-state
of the economy is affected by a change in some parameter or exogenous
variable of the model. The phase diagram permits as also to say something
about the short-term response of consumption and capital, as well as the
properties of their travel towards the new steady state, especially if we assume
that we start by being on the current steady state, which we will do. Moreover,
we may want to say things not only about consumption and capital, but also
about output, the interest rate, or the wage level and the savings rate.
In general, there are three methods to approach the matter:
1) Direct method. Solve and express consumption and/or capital as an explicit
function of the exogenous parameters, and then differentiate these expressions
with respect to each parameter in turn to see what happens. This method is
rarely available -for example, in the Blanchard-Weil model it is not, as we will
see in a while, but for the Ramsey model, it is, at least for the capital variable
(which is the central variable here).
2) Implicit-function method. Express in a single equation the condition
characterizing the steady-state, and apply the implicit function theorem. This in
principle can always be done, but it does not guarantee an unambiguous
conclusion.
3) Phase-diagram method: Turn to the phase diagram, and study further the
two zero-change loci, but now as functions of the parameters, for any given level
of capital. If we can determine how each curve shifts as the parameter under
examination changes, then we are justified in picturing that shift on the phase
diagram, and conclude geometrically what will happen to the steady state. This
approach in many cases proves to be the easiest one, and more over, it is the
only one among the three that can give us directly what will the short run
response will be, and how are we going to approach the new steady-state.
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In the Blanchard-Weil model, combining the two relations [5] and [6] that
together characterize the steady-state we have, eliminating consumption


c(t )  0 , k (t )  0 
n k  d 
f (k )      g
 f ( k )  cg   n  g    k
 G   f (k )  cg   n  g    k   f (k )      g    n  k  d   0
[14]
As long as the production function is non-linear in capital (as is our case), the
above equation determines only implicitly the level of steady-state capital
(while in the Ramsey model one can obtain from [12] a direct expression for
capital). So the Direct method cannot be applied here.
Let's attempt the Implicit-function method, say, as regards a change in total
factor productivity A which is present in the production function f  k   Ak (t ) .
By the implicit function theorem we have
dk
G A
 

dA
G k

k   f (k )      g    f (k )  cg   n  g    k  ak  1
 f (k )   n  g      f (k )      g   f (k )  f (k )  cg   n  g    k    n
Assume that we start at the current steady state c0 , k0 . Then we have that
f (k0 )      g  0 because as an equation, it defines the vertical asymptote to
the right of the zero-change locus for consumption. Also,
f (k0 )  cg   n  g    k0  c0 .
For the denominator we also have that f (k0 )   n  g     0 (you should
remember why). So we can compact the expression a bit into
dk
dA
k  k0


k0  f (k0 )      g   c0 ak0 1
 f (k0 )   n  g      f (k0 )      g   f (k0 )c0   n
We know that the numerator is positive, so the sign of the derivative will be the
opposite of the sign of the denominator. But what is the sign of the denominator?
The first term is positive while the second and third terms are negative since
f (k0 )  0 . At a first glance, it is ambiguous. Sometimes, other assumptions of
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the model may help determine the sign after all, but this is not guaranteed.
Moreover, note that intuition does not help here: it is not clear that a higher
total factor productivity will lead to a higher level of steady-state capital: capital
has just become more productive, so maybe we can get away only with higher
consumption while steady state capital will decrease? (After all, we care about
capital only because we care about consumption).
Note that the sign ambiguity is not related to G A but to G k , so it will be
present to all such calculations we may attempt, with respect to any other
parameter of the model.
So let's try the Phase-diagram method.
We are looking at the two zero-change loci
n k  d 
c 
f (k )      g
,
c  f ( k )  cg   n  g    k
How each of these shifts as A changes? Note that, as in the Implicit-function
method, the variables themselves are not differentiated with respect to the
parameter we examine (unlike the Direct method) because here they act as an
argument of a function, they do not represent a specific value (the steady-state).
We have
c
A

c (t )  0

 k  1 n  k  d 
 f (k )      g 
2
0,
c
A

k (t )  0
 k  0
These results provide clear guidance as to how the two zero-change loci will
shift if we increase A :
The first tells us that for any given level of capital the corresponding level of
consumption must be lower on the consumption locus. The second tells us that
for any given level of capital, the corresponding consumption level must be
higher on the capital locus. Schematically then, the phase diagram will change
as follows (eliminating government for clarity):
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C
S2

k (t )  0
S1

k (t )  0

c(t )  0

c(t )  0
K
The economy is initially in steady state S1 , characterized by the red loci. The
change in total factor productivity A shifts the two loci which now become blue
(the black dashed arrows indicate this shift, they are not dynamic arrows). The
new steady state will be at S 2 . We get a clear long-run result: both consumption
and capital will be higher at the new steady-state. We can also conclude that in
the mid-term we will see both consumption and capital rising to attain their new
steady state values.
But the short-term reaction of the decision variable consumption remains
ambiguous: whether the new saddle-path will pass "above" or "below" the old
steady state (the two straight blue lines represent possible saddle paths)
depends further on the values of the various parameters. But this is what
determines whether we will see consumption jump up or down immediately
after the increase in A . So we don't get a definite result on that. Still, by this
third method we were able to finally obtain some answers to our questions.
Exercise 1
Practice on other comparative statics results on consumption and capital, using
the Phase-diagram method. Try also to answer questions about the effects on
output, the savings rate, the real interest rate and the real wage.
Exercise 2
Turn to the Ramsey model, where the Direct method is also applicable, and try
to apply also the Implicit-function method as well as the Phase-diagram
method for some of the parameters of the model. Which one is more easy and
tractable? Which one provides more definite results among the three? --