Here we present the analysis for the case described in Section 4, of prices with a lower bound.
Suppose that
prices time 
are given by equation 39 with 
, so that prices
decrease from a maximum of 
toward a minimum of 
. In order to
make comparisons with the results of the other sections, a reasonable assumption is that the minimum price is
lower than under the constant price regime of section 2, i.e. that
, where 
is the price of section 2. Using equation 39
implies that there is no limit on the income from selling
the harvest, as there is when a=0. However,as long as a is small
compared to 1, we expect that the results given for the a=0 case
will essentially hold: namely that a nonzero equilibrium only exists
when b is above a minimum value near 
,
and that (when 
) the
equilibrium resource level is larger,
the equilibrium is more strongly attracting,
and oscillations are less likely than in the constant price case. However,
as a moves away from zero and price supports become stronger, we would
like to know how this affects the a=0 results.
When a > 0 the nonzero equilibrium of the nondimensionalized system
with 
, 
(
, 
given by equation 5)
is 
, 
, where
and we define 
,
, and
.
As we found in section 4, adding prices does not change the equilibrium
capital but significantly affects the equilibrium resource.
However, unlike the a=0 case, N-p is positive for all positive
values of b. What is different now is that if b<1 then
as 
while if b>1 then 
approaches the a=0
value as 
.
Letting
and linearizing about the nontrivial equilibrium, we find that the eigenvalues of this system are:
These eigenvalues always have negative real part. This can be seen
via the following argument. Note that in the remaining discussion of this section,
we drop the tildes.
Since 
, these eigenvalues both have negative real part when
. Using the relationship between a, b, c and 
to
replace each 
, this
condition on 
can be rewritten as
or
and applying the equality again shows that this is always true. Thus this is always a stable equilibrium point. It is a stable spiral point when
In the limit as 
, these results approach those found in section 3.
We expect that increasing a will have the effect of 0 the impact of
decreasing prices, and, in fact, increasing
the lower bound on prices, 
,
decreases the equilibrium level of the resource, 
.
However, the dependence of the eigenvalues
on a is much more subtle, and depends on the values of b and c.
For very small a, the upper bound on 
for oscillations
decreases as a increases (when 
satisfies the constraint
), implying a stabilizing effect.
When the parameters are such that 
satisfies this
inequality and the value
under the square root is negative, then the real part of 
increases in magnitude with a
when 1 < b < 3, and otherwise decreases.
These results are found by the following calculations:
which is negative for all a>0.
and
