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Analysis of Variable Prices

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

 



next up previous
Next: References Up: Appendix Previous: Conditions for oscillations



weisbuch
Fri Feb 7 13:18:37 GMT+0100 1997