Law of the jungle: firm survival and price dynamics in evolutionary markets

Abstract

In this paper I develop a simple, and general model of supply and demand within which almost any theory of consumer and producer behaviour may be integrated by varying parameters. I then investigate the dynamics of this model and its implications for the theory of market evolution, and show that it unifies a number of insights from evolutionary economics. I extend upon these evolutionary theories and also characterise the distribution of prices across the market and investigate its evolution over time.

Appendices

Appendix A: An intuitive derivation of the Price equation for economics

One problem with applications of the Price equation to evolutionary economics is that the economic intuition behind it can become difficult to follow. Here I provide a re-derivation (based upon Price (1972a)) of the Price equation using only the terminology of economics. The Price equation serves to decompose the growth rate in average fitness across a population, here this means a decomposition of the growth across the population of the average size of firms. We will begin with the change over one “time period”

$$ E_{q\left( t+1\right)}^{t+1}\left( q\right)-E_{q\left( t\right)}^{t}\left( q\right)=\frac{\sum\nolimits_{i=1}^{F}q_{i} \left( t+1\right)q_{i}\left( t+1\right)}{\sum\nolimits_{j=1}^{F}q_{j}\left( t+1\right)} -\frac{\sum\nolimits_{i=1}^{F}q_{i}\left( t\right)q_{i}\left( t\right)}{\sum\nolimits_{j=1}^{F}q_{j}\left( t\right)} $$

(6.1)

it is important to note that the average here is weighted with respect to the {q _i (t)} _{i = 1}, much the same as the distribution of prices, which may seem unusual (weighting firm sizes with firm sizes), though it is less unusual once one grasps that the “second”, or “weighting” firm size included in each term of the above expression is being normalised relative to the size of the market - in a sense, but absolutely not exactly being “cancelled out”. Now, we can define what is often called a “selection coefficient” z _i for each individual firm within the market Price (1972a, p.486) which reflects the growth rate of the firms within the market²⁵

$$ z_{i}\left( t\right)=\frac{q_{i}\left( t+1\right)}{q_{i}\left( t\right)}=1+\frac{\partial q_{i}\left( t\right)}{q_{i}\left( t\right)} $$

(6.2)

Now, using this selection coefficient to expand the average size of firms within the market at time t+1, and using the fact that q _i (t+1) = q _i (t)+Δq _i (t) we obtain

$$\begin{array}{@{}rcl@{}} E_{q\left( t+1\right)}^{t+1}\left( q\right)&=&\frac{1}{\sum\nolimits_{j=1}^{F}q_{i}\left( t\right)z_{i}\left( t\right)} \sum\limits_{i=1}^{F}q_{i}\left( t\right)z_{i}\left( t\right)q_{i}\left( t+1\right)\\ &=&\frac{1}{\sum\nolimits_{j=1}^{F}q_{i}\left( t\right)z_{i}\left( t\right)}\sum\limits_{i=1}^{F}q_{i}\left( t\right)z_{i}\left( t\right)\left( q_{i}\left( t\right)+{\Delta} q_{i}\left( t\right)\right) \end{array} $$

(6.3)

Expanding this last equality and re-substituting for q(t+1) we get

$$ E_{q\left( t+1\right)}^{t+1}\left( q\right)=\frac{1}{\sum\limits_{j=1}^{F}q_{i}\left( t\right)z_{i}\left( t\right)}\sum\limits_{i=1}^{F}q_{i}\left( t\right)z_{i}\left( t\right)q_{i}\left( t\right)+\frac{1}{{\sum}_{j=1}^{F}q_{i}\left( t+1\right)}\sum\limits_{i=1}^{F}q_{i}\left( t+1\right){\Delta} q_{i}\left( t\right) $$

(6.4)

Now employing some further mathematical tautologies to manipulate the first term of this expression gives us the following

$$\begin{array}{@{}rcl@{}} E_{q\left( t+1\right)}^{t+1}\left( q\right)&=&\frac{1}{\sum\nolimits_{j=1}^{F}q_{i}\left( t\right)z_{i}\left( t\right)} \frac{\sum\nolimits_{j=1}^{F}q_{i}\left( t\right)}{\sum\nolimits_{j=1}^{F}q_{i}\left( t\right)}\sum\limits_{i=1}^{F}q_{i}\left( t\right)\left[z_{i}\left( t\right)-E_{q\left( t\right)}^{t}\left( z\right)\right]\left[q_{i}\left( t\right)-E_{q\left( t\right)}^{t}\left( q\right)\right]\\ &&+\frac{1}{\sum\nolimits_{j=1}^{F}q_{i}\left( t+1\right)}\sum\limits_{i=1}^{F}q_{i}\left( t+1\right){\Delta} q_{i}\left( t\right)\\ &&+\frac{1}{\sum\nolimits_{j=1}^{F}q_{i}\left( t\right)z_{i}\left( t\right)} \frac{\sum\nolimits_{j=1}^{F}q_{i}\left( t\right)}{\sum\nolimits_{j=1}^{F}q_{i}\left( t\right)} \left[\sum\limits_{i=1}^{F}q_{i}\left( t\right)z_{i}\left( t\right)E_{q\left( t\right)}^{t}\left( q\right)\right.\\ &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\left.+\sum\limits_{i=1}^{F}q_{i}\left( t\right)\left[q_{i}\left( t\right)-E_{q\left( t\right)}^{t}\left( q\right)\right]E_{q\left( t\right)}^{t}\left( z\right)\right]\\ \end{array} $$

(6.5)

Notice that when we expand this last term out carefully it collapses to $E_{q\left (t\right )}^{t}\left (q\right )$. We can now recognise that this equation is reducible using the quotidian statistical concepts of covariance and expectation

$$ E_{q\left( t\right)}^{t+1}\left( q\right)=\frac{1}{E_{q\left( t\right)}^{t}\left( z\right)}\text{Cov}_{q\left( t\right)}^{t}\left( z\quad q\right)+E_{q\left( t+1\right)}^{t}\left( {\Delta} q\left( t\right)\right)+E_{q\left( t\right)}^{t}\left( q\right) $$

(6.6)

Now subtracting $E_{q\left (t\right )}^{t}\left (q\right )$ from both sides gives us

$$ E_{q\left( t\right)}^{t+1}\left( q\right)-E_{q\left( t\right)}^{t}\left( q\right)=\frac{1}{E_{q\left( t\right)}^{t}\left( z\right)}\text{Cov}_{q\left( t\right)}^{t}\left( z\quad q\right)+E_{q\left( t+1\right)}^{t}\left( {\Delta} q\left( t\right)\right) $$

(6.7)

Which can be further reduced to the elegant Price equation by generalising to an arbitrary time period and (for the sake of it) assuming differentiability in time

$$ \frac{\partial E_{q}\left( q\right)}{\partial t}=\frac{1}{E_{q}\left( q\right)}\text{Cov}_{q}\left( z\quad q\right)+E_{q}\left( \frac{\partial q}{\partial t}\right) $$

(6.8)

Now, inputting $z_{i}=1+\frac {1}{q}\frac {\partial q}{\partial t}\partial t$ into the definition of covariance gives us

$$\frac{\partial E_{q}\left( q\right)}{\partial t}=\frac{1}{E_{q}\left( q\right)}\text{Cov}_{q}\left( \frac{\partial t}{q}\frac{\partial q}{\partial t}\quad q\right)+E_{q}\left( \frac{\partial q}{\partial t}\right) $$

Appendix B: Proofs of theorems

1.1 Proof of Theorem 1

Proof

If it is the case that p _j = 0 ∀ j then by the definition of demand elasticity $\varepsilon _{d_{i}}^{p_{j}}=\frac {p_{j}}{{q_{i}^{d}}}\frac {\partial {q_{i}^{d}}}{\partial p_{j}}=0\,\forall \,j$. This implies that $\left (\frac {\varepsilon _{d_{i}}^{p_{i}}}{F-1}\right )\frac {1}{p_{i}}\frac {\partial p_{i}}{\partial t}=0$ and $\left (\frac {\varepsilon _{d_{i}}^{p_{j}}}{\left (-1\right )}\right )\frac {1}{p_{j}}\frac {\partial p_{j}}{\partial t}=0\,\forall \,j\ne i$, since by induction $0\times \lim _{x\rightarrow \infty }x=0$. Inserting this into Eq. 3.3 yields the result. Similarly, we need only input directly the assumption ∂ p _j /∂ t = 0, ∀ j ∈ {1,…, F} into Eq. 3.3 to obtain the result. □

1.2 Proof of Theorem 2

Proof

First take Eq. 3.3 and input the assumptions of normal and substitutable goods, a zero population growth rate and unchanging product attributes. Then the growth rate of the firm can be expressed as

$$ \frac{\partial q_{i}}{\partial t}={q_{i}^{d}}\sum\limits_{j\ne i}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{i}}}{F-1}\right)\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}-\left( \frac{\varepsilon_{d_{i}}^{p_{j}}}{\left( -1\right)}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}\right] $$

(6.9)

Now, it is a mere fact of logic that any output bought must have been sold, so ${q_{i}^{d}}=q_{i}$. But note that $q_{i}=0\implies \frac {\partial q_{i}}{\partial t}\ne 0\Leftrightarrow {\sum }_{j\ne i}\left [\left (\frac {\varepsilon _{d_{i}}^{p_{i}}}{F-1}\right )\frac {1}{p_{i}}\frac {\partial p_{i}}{\partial t}-\left (\frac {\varepsilon _{d_{i}}^{p_{j}}}{\left (-1\right )}\right )\frac {1}{p_{j}}\frac {\partial p_{j}}{\partial t}\right ]\ne 0$. Since the laws of physics decree that q _i ≥0 therefore, it is the case that

$$ \frac{\partial q_{i}}{\partial t}>0\Leftrightarrow\sum\limits_{j\ne i}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{i}}}{F-1}\right)\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}-\left( \frac{\varepsilon_{d_{i}}^{p_{j}}}{\left( -1\right)}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}\right]>0 $$

(6.10)

Now suppose that we can partition the set of firms {1,…, F} into two sets B and ¬B such that ∀ j ∈ B we have $\left [\left (\frac {\varepsilon _{d_{i}}^{p_{i}}}{F-1}\right )\frac {1}{p_{i}}\frac {\partial p_{i}}{\partial t}-\left (\frac {\varepsilon _{d_{i}}^{p_{j}}}{\left (-1\right )}\right )\frac {1}{p_{j}}\frac {\partial p_{j}}{\partial t}\right ]>0$. Then condition (6.10) holds if and only if

$$\begin{array}{@{}rcl@{}} \sum\limits_{j\in B}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{i}}}{F-1}\right)\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}-\left( \frac{\varepsilon_{d_{i}}^{p_{j}}}{\left( -1\right)}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}\right]>\sum\limits_{j\notin B}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{i}}}{F-1}\right)\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}-\left( \frac{\varepsilon_{d_{i}}^{p_{j}}}{\left( -1\right)}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}\right]\\ \end{array} $$

(6.11)

□

1.3 Proof of Theorem 3

Proof

This condition consists of a simple rearrangement of Eq. 3.3 when restricted to be greater than zero, so

$$ \frac{\partial {q_{i}^{d}}}{\partial N}\frac{\partial N}{\partial t}+{q_{i}^{d}}\sum\limits_{j\ne i}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{i}}}{F-1}\right)\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}-\left( \frac{\varepsilon_{d_{i}}^{p_{j}}}{\left( -1\right)}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}\right]+\sum\limits_{j=1}^{F}\sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{i}^{d}}}{{\partial\alpha_{j}^{k}}}\frac{{\partial\alpha_{j}^{k}}}{\partial t}>0 $$

(6.12)

when rearranged gives us

$$\begin{array}{@{}rcl@{}} &&\sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{i}^{d}}}{{\partial\alpha_{i}^{k}}}\frac{{\partial\alpha_{i}^{k}}}{\partial t}>{q_{i}^{d}}\sum\limits_{j\ne i}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{j}}}{\left( -1\right)}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}-\left( \frac{\varepsilon_{d_{i}}^{p_{i}}}{F-1}\right)\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}\right]\\ &&-\sum\limits_{j\ne i}^{F}\sum\limits_{k=1}^{N_{A}}\left[\frac{\partial {q_{i}^{d}}}{{\partial\alpha_{j}^{k}}}\frac{{\partial\alpha_{j}^{k}}}{\partial t}\right]-\frac{\partial {q_{i}^{d}}}{\partial N}\frac{\partial N}{\partial t} \end{array} $$

(6.13)

□

1.4 Proof of Theorem 4

Proof

The market share of firm i with respect to output is given by $s_{i}=\frac {q_{i}}{{\sum }_{j=1}^{F}q_{j}}$, and its dynamics can be readily obtained by simple application of the quotient rule

$$ \frac{\partial s_{i}}{\partial t}=\frac{\frac{\partial q_{i}}{\partial t}\sum\nolimits_{j=1}^{F}q_{j}-q_{i}\frac{\partial\sum\nolimits_{j=1}^{F}q_{j}}{\partial t}}{\left( \sum\nolimits_{j=1}^{F}q_{j}\right)^{2}} $$

(6.14)

It is fairly straightforward then to confirm that market share only grows if the firm grows at a faster rate than output across the market

$$ \frac{\partial s_{i}}{\partial t}>0\Leftrightarrow\frac{1}{q_{i}}\frac{\partial q_{i}}{\partial t}>\frac{1}{\sum\nolimits_{j=1}^{F}q_{j}}\frac{\partial\sum\nolimits_{j=1}^{F}q_{j}}{\partial t} $$

(6.15)

using the additive law of differential calculus we can expand this into a slightly more tractable (in view of the model above) expression

$$ \frac{\partial q_{i}}{\partial t}>\frac{s_{i}}{1-s_{i}}\sum\limits_{j\ne i}\frac{\partial q_{j}}{\partial t} $$

(6.16)

Now substituting in Eq. 3.3 for the growth rate of firm size we obtain

$$\begin{array}{@{}rcl@{}} &&\frac{\partial {q_{i}^{d}}}{\partial N}\frac{\partial N}{\partial t}+{q_{i}^{d}}\sum\limits_{j\ne i}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{i}}}{F-1}\right)\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}-\left( \frac{\varepsilon_{d_{i}}^{p_{j}}}{\left( -1\right)}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}\right]+\sum\limits_{j=1}^{F}\sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{i}^{d}}}{{\partial\alpha_{j}^{k}}}\frac{{\partial\alpha_{j}^{k}}}{\partial t}\\ &&\quad>\frac{s_{i}}{1-s_{i}}\sum\limits_{j\ne i}\left\{ \frac{\partial {q_{j}^{d}}}{\partial N}\frac{\partial N}{\partial t}+{q_{j}^{d}}\sum\limits_{n\ne j}\left[\left( \frac{\varepsilon_{d_{j}}^{p_{j}}}{F-1}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}-\left( \frac{\varepsilon_{d_{j}}^{p_{n}}}{\left( -1\right)}\right)\frac{1}{p_{n}}\frac{\partial p_{n}}{\partial t}\right]\right.\\ &&\quad\left.+\sum\limits_{n=1}^{F}\sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{j}^{d}}}{{\partial\alpha_{n}^{k}}}\frac{{\partial\alpha_{n}^{k}}}{\partial t}\right\} \end{array} $$

(6.17)

We can reduce this inequation by isolating behavioural and firm strategy terms which refer to firm i as follows

$$\begin{array}{@{}rcl@{}} \frac{\partial {q_{i}^{d}}}{\partial N}\frac{\partial N}{\partial t}&+&\frac{s_{i}}{1-s_{i}}\sum\limits_{j\ne i}\frac{\partial {q_{j}^{d}}}{\partial N}\frac{\partial N}{\partial t}\\ &+&{q_{i}^{d}}\sum\limits_{j\ne i}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{i}}}{F-1}\right)\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}-\left( \frac{\varepsilon_{d_{i}}^{p_{j}}}{\left( -1\right)}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}\right]\\ &-&\sum\limits_{j\ne i}\frac{s_{i}}{1-s_{i}}{q_{j}^{d}}\left[\left( \frac{\varepsilon_{d_{j}}^{p_{j}}}{F-1}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}-\left( \frac{\varepsilon_{d_{j}}^{p_{i}}}{\left( -1\right)}\right)\frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}\right]\\ &-&\sum\limits_{j\ne i}\frac{s_{i}}{1-s_{i}}\left\{ {q_{j}^{d}}\sum\limits_{n\ne j,i}\left[\left( \frac{\varepsilon_{d_{j}}^{p_{j}}}{F-1}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}-\left( \frac{\varepsilon_{d_{j}}^{p_{n}}}{\left( -1\right)}\right)\frac{1}{p_{n}}\frac{\partial p_{n}}{\partial t}\right]\right\} \\ &+&\sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{i}^{d}}}{{\partial\alpha_{i}^{k}}}\frac{{\partial\alpha_{i}^{k}}}{\partial t}+\sum\limits_{j\ne i}\sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{i}^{d}}}{{\partial\alpha_{j}^{k}}}\frac{{\partial\alpha_{j}^{k}}}{\partial t}-\frac{s_{i}}{1-s_{i}}\sum\limits_{j\ne i}\sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{j}^{d}}}{{\partial\alpha_{i}^{k}}}\frac{{\partial\alpha_{i}^{k}}}{\partial t}\\ &-&\frac{s_{i}}{1-s_{i}}\sum\limits_{j\ne i}\sum\limits_{n\ne i}\sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{j}^{d}}}{{\partial\alpha_{n}^{k}}}\frac{{\partial\alpha_{n}^{k}}}{\partial t}>0 \end{array} $$

(6.18)

if we multiply and divide every term in ${\sum }_{j\ne i}^{F}{\sum }_{k=1}^{N_{A}}\frac {\partial {q_{i}^{d}}}{{\partial \alpha _{i}^{k}}}\frac {{\partial \alpha _{i}^{k}}}{\partial t}$ by $\frac {{\partial \alpha _{i}^{k}}}{{\partial \alpha _{i}^{k}}}$ we can by grouping like terms obtain

$$\begin{array}{@{}rcl@{}} &&\frac{\partial N}{\partial t}\left[\frac{\partial {q_{i}^{d}}}{\partial N}-\frac{s_{i}}{1-s_{i}}\sum\limits_{j\ne i}\frac{\partial {q_{j}^{d}}}{\partial N}\right]\\ &&+{q_{i}^{d}}\sum\limits_{j\ne i}\left\{ \frac{1}{p_{i}}\frac{\partial p_{i}}{\partial t}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{i}}}{F-1}\right)+\frac{s_{i}}{1-s_{i}}\frac{{q_{j}^{d}}}{{q_{i}^{d}}}\left( \frac{\varepsilon_{d_{j}}^{p_{i}}}{\left( -1\right)}\right)\right]-\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}\left[\left( \frac{\varepsilon_{d_{i}}^{p_{j}}}{\left( -1\right)}\right)-\frac{s_{i}}{1-s_{i}}\frac{{q_{j}^{d}}}{{q_{i}^{d}}}\left( \frac{\varepsilon_{d_{j}}^{p_{j}}}{F-1}\right)\right]\right\} \\ &&-\frac{s_{i}}{1-s_{i}}\sum\limits_{j\ne i}\left\{ {q_{j}^{d}}{\sum}_{n\ne j,i}\left[\left( \frac{\varepsilon_{d_{j}}^{p_{j}}}{F-1}\right)\frac{1}{p_{j}}\frac{\partial p_{j}}{\partial t}-\left( \frac{\varepsilon_{d_{j}}^{p_{n}}}{\left( -1\right)}\right)\frac{1}{p_{n}}\frac{\partial p_{n}}{\partial t}\right]\right\} \\ &&+\sum\limits_{k=1}^{N_{A}}\frac{{\partial\alpha_{i}^{k}}}{\partial t}\left[\frac{\partial {q_{i}^{d}}}{{\partial\alpha_{i}^{k}}}-\frac{s_{i}}{1-s_{i}}\sum\limits_{j\ne i}\left( \sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{j}^{d}}}{{\partial\alpha_{i}^{k}}}-\frac{\partial {q_{i}^{d}}}{{\partial\alpha_{j}^{k}}}\frac{{\partial\alpha_{j}^{k}}}{{\partial\alpha_{i}^{k}}}\right)\right]-\frac{s_{i}}{1-s_{i}}\sum\limits_{j\ne i}\left\{ \sum\limits_{n\ne i}\sum\limits_{k=1}^{N_{A}}\frac{\partial {q_{j}^{d}}}{{\partial\alpha_{n}^{k}}}\frac{{\partial\alpha_{n}^{k}}}{\partial t}\right\} >0\\ \end{array} $$

(6.19)

Which when we rearrange to isolate on the left hand side any market forces over which firm i has influence gives us our result. □

1.5 Proof of Lemma 1

Proof

This proof is rather trivial and well known to mathematicians and statisticians, but will be included for the sake of rigour. Firstly, if the price distribution is degenerate then

$$ E_{q}\left( p\right)=\frac{\sum\nolimits_{i=1}^{F}q_{i}p_{i}}{\sum\nolimits_{j=1}^{F}q_{j}}= \frac{\sum\nolimits_{i\in{\Xi}\left( p\right)}q_{i}p_{i}}{\sum\nolimits_{i\in{\Xi}\left( p\right)}q_{j}} $$

(6.20)

and by the definition of Ξ(p), p _i = p ∀ i ∈ Ξ(p) and so ∀ i ∈ Ξ(p), E _q (p) = p _i , while by the definition of degeneracy this must be true for any i:q _i > 0. The zero variance of the distribution follows immediately by inputting p _i = E _q (p) ∀ i:q _i > 0 into the definition of Var _q (p). Conversely if it is to be the case that p _i = E _q (p) ∀ i:q _i > 0, and Var _q (p) = 0 then by the definition of the first and second moments it must be the case that ∃{i}⊂{1,…, F}:p _i = E _q (p) ∀ i ∈ {i} and for any i not in this set q _i = 0, which is the definition of degeneracy. □

1.6 Proof of Theorem 5

Proof

In order for a uniform price to prevail across the market the price distribution must be degenerate by Lemma 1. But for the price distribution to be degenerate it can seen that we will need the dynamics of the market to lead to convergence to a situation in which ${\sum }_{i\in {\Xi }\left (p\right )}s_{i}=1$, since

$$\sum\limits_{i\in{\Xi}\left( E_{q}\left( p\right)\right)}s_{i}=1\iff\sum\limits_{i\in{\Xi}\left( E_{q}\left( p\right)\right)} \frac{q_{i}}{\sum\limits_{j=1}^{F}q_{j}}=\frac{1}{\sum\limits_{j=1}^{F}q_{j}}\sum\limits_{i\in{\Xi}\left( E_{q}\left( p\right)\right)}q_{i}=1\iff{\sum}_{i\in{\Xi}\left( E_{q}\left( p\right)\right)}q_{i}={\sum}_{j=1}^{F}q_{j} $$

But given that Ξ(E _q (p))⊂{1,…, F}, for each q _i on the left hand side there is a corresponding q _i on the right hand side, and so for this to be the case requires that q _i = 0 ∀ i∉Ξ(E _q (p)), which is the key to the distribution being degenerate by the definition referred to in Lemma 1. So saying that ${\sum }_{i\in {\Xi }\left (p\right )}s_{i}=1$ is identical to saying that the price distribution is degenerate. Hence if it is to be the case that in the market there is convergence to a situation in which ${\sum }_{i\in {\Xi }\left (E_{q}\left (p\right )\right )}s_{i}=1$, it must also be the case by definition of market share that there is a convergence to a situation in which s _i = 0 ∀ i∉Ξ(E _q (p)). But then for it to be the case that limt→∞ s _i = 0 ∀ i∉Ξ(E _q (p)), for each and every i∉Ξ(E _q (p)) from an initial market share of s _i we must have

$$s_{i}+{\lim}_{T\rightarrow\infty}{{\int}_{t}^{T}}\frac{\partial s_{i}}{\partial t}\partial t=0 $$

$$\implies s_{i}=-{\lim}_{T\rightarrow\infty}{{\int}_{t}^{T}}\frac{\partial s_{i}}{\partial t}\partial t $$

It should be fairly clear that this will hold if and only if there are a sufficient number of time periods for which $\frac {\partial s_{i}}{\partial t}<0$ so that $\lim _{T\rightarrow \infty }{{\int }_{t}^{T}}\frac {\partial s_{i}}{\partial t}\partial t<0$. That is, there must exist a set of time periods $T_{i\notin {\Xi }\left (p\right )}\subset \mathbb {R}_{+}$ for each and every i∉Ξ(E _q (p)) such that

$$\frac{\partial s_{i}}{\partial t}<0\,\forall\,t\in T_{i\notin{\Xi}\left( E_{q}\left( p\right)\right)} $$

that is, for these periods Theorem 4 are violated for each firm i, and

$$s_{i}=-{\lim}_{T\rightarrow\infty}\left\{ {\int}_{t\in T_{i\notin{\Xi}\left( E_{q}\left( p\right)\right)}}\frac{\partial s_{i}}{\partial t}\partial t-{\int}_{t\notin T_{i\notin{\Xi}\left( E_{q}\left( p\right)\right)}}\frac{\partial s_{i}}{\partial t}\partial t\right\} $$

□

1.7 Proof of Corollary 1

Proof

This is simply the reverse argument of Theorem 5 for the emergence of a dominant firm rather than the non-survival of those which do not conform the uniform price. □