# MODEL A: UNIFORM DISTRIBUTIONS OF THE PROBABILITY OF EXTINCTION W.R.T. PRICE

In sections 8.1 and 8.2 we assume for simplicity uniform distributions of the probability of extinction with respect to price p, i.e. distribution functions in the for m

in the interval *(p*0*, p*h*),*

= 0 for p < *p*0,

= 1 for p > *p*h,

where p0 is the price at which the possibility of competitors entering vanishes entirely and *p*h is the lowest price that induces competitors to enter, and

** **in the interval *(p*p *p*2*),*

= 1 for *p < p*1,

= 0 for *p > p*2,

where *p*2 is the lowest price at which the risk of extinction due to insufficient profit vanishes entirely. The firm will certainly pay for insufficient profit with extinction for *p* < p1 < *p*h.

The following two figures show the probability distribution functions for these distributions:

**Figure 44: The probability of extinction **η1** (p) due to entry of competition at price ****p**

**Figure 45: The probability of extinction **η2**(p) due to low profit at price ****p**

The probability of survival in this model is maximized by the price p* at which the following function reaches a maximum:

for the derivative of function η*(p)* it holds that

If we set , we get the following for the argument of the maximum^{[1]}:

Here, not surprisingly, the optimal price is the average of the price levels at which the threat of extinction due to one of the two reasons under consideration materializes with 100% certainty.

# MODEL B: UNIFORM DISTRIBUTIONS OF THE PROBABILITY OF EXTINCTION W.R.T. PROFITABILITY

Now let us assume, more realistically, uniform distributions of the probability of extinction with respect to profitability . We will denote

*π*h*=π(p*h*),*

*π*j* = π(pj) for j =* 1, 2.

We assume here extinction distribution functions in the form:

** **in the interval * (p*0

*h*

**, p**

**),**= 0 for *π* < *π*0,

= 1 for *π* > *π*h,

where *π*0 is the profitability level at which the possibility of competitors entering vanishes entirely, *π*h is the lowest profitability level that induces competitors to enter with 100% probability, and similarly

** **in the interval (*π*1, *π*2**),**

= 1 for *π < π*1,

= 0 for *π* > *π*2,

where *π*2 is the lowest profitability at which the risk of extinction due to insufficient profit vanishes entirely. The firm will certainly pay for insufficient profit with extinction for *π* ≤ *π*1

The probability of survival in this model is maximized by the profitability *π** at which the following survival probability function reaches a maximum:

Plots 1 and 2 in Figure 46 represent the distribution functions of the uniform distribution of the risk of extinction *η*1*(π), η*2*(π)* and plot 3 is the probability of survival *η*B*(n).* The optimal profitability for uniform distributions of the risk of extinction is at point *π*.*

Analogously to the previous section, if we set *η'*B*(p) =* 0, we get the following for the argument of the maximum of the probability of survival:

Hence, the optimal price *p** must fulfil the condition

It therefore holds that the survival-probability-maximizing price *p** lies in the interval (min *(p*h, p1); max (*p*h, p1) and does not depend on the other parameters of the model.

**Figure 46: The probability of survival and optimal profitability for uniform distributions of the risk of extinction**

# MODEL C: NORMAL DISTRIBUTIONS OF THE PROBABILITY OF EXTINCTION W.R.T. PROFITABILITY

In this model we will work with the normal distribution of the probability of extinction due to the two reasons under consideration with respect to profitability . As it is not generally possible to determine the cumulative distribution function of the normal distribution algebraically, we will use estimates obtained by numerical methods for our subsequent conclusions.

In this model we are looking for the profitability π* at which the following survival probability function reaches a maximum

** **where

The optimal profitability π* is the root of equation *Ω(*π*) =* 0, where

The symbols used have the following meanings:

*f*j the density of the distribution of the probability of extinction due to the *j-th* reason *(j =* 1, 2),

*Ф*j the cumulative distribution function of the probability of survival (i.e. the probability of avoidance of extinction due to *j =* 1, *2),* Ω(π) the derivative [with respect to profit] of the agent's probability of survival [i.e. the probability of avoidance of both threats).

**Figure 47: The probability of survival and optimal profitability for normal distributions of the risk of extinction**

Plots 1 and 2 in Figure 47 represent the distribution functions of the probability of extinction due to *j* = 1, 2, plot 3 is the firm's probability of survival [i.e. the probability of avoidance of both threats of extinction] *η(*π), and plot 4 represents function Ω[π], The location of the optimum is π*.

Figures 48 and 49 show the plot of the probability of survival (i.e. the probability of avoidance of both threats of extinction] against profitability π and the magnitude of the optimal profitability π* for the case where the two distributions have the same standard deviation. In this case, the maximum probability of survival occurs for the profitability which corresponds to the average of the means of the two distributions and which does not depend on the other parameters of the model (the common size of the variance and the slope of the demand function].

Figure 48 assumes normal distributions of the risk of extinction *N*1*(a*1 = 0; *σ*1 = 0.1), *N*2*(a*2 *=* 0.2; *σ*2 = 0.1), Plots 1 and 2 are the probabilities of extinction due to *j = 1,2* (see Figures 44 and 45] and plot 3 is the dependence of the agent's probability of survival on profit π.

Figure 49 assumes normal distributions of the risk of extinction *N*1*(a*1 *=* 0; *σ*1 = 0.2), *N*2*(a*2 = 0,3; *σ*2 = 0.2). Plot 3 represents the probability of survival λ(π) and the optimal profitability is π*.

**Figure 48: The probability of survival and optimal profitability ****n**** for normal distributions of the risk of extinction for the same standard deviations ***σ*1* = σ*2

**= 0.1**

**Figure 49: Ditto for ***σ*1* = σ*2

**= 0.2**

Figures 50 and 51 illustrate the case where the variances of the extinction probability distributions with respect to the chosen profitability differ significantly. In both figures, plot 3 represents the agent's probability of survival and plots 1 and 2 the probabilities of extinction due to one of the reasons under consideration. The optimal profitability is again π**.* In the first case, the variance of the probability of extinction due to the entry of another competitor is the lower: *N*1*(a*1*=* 0; *σ*1 *=* 0.05), *N*2*(a*2= 0.2; *o*2 *=* 0.4), In the second case, described in Figure 45, the variance of the probability of extinction due to insufficient profit is the lower: *N*1*(a*1*=* 0; *σ*a= 0.4), *N*2*(a*2 *=* 0.2; *σ*2*-* 0.05), In both cases, the optimal profit level π* moves from the average of the means of the two distributions towards the mean of the distribution with the higher variance.

**Figure 50: The probability of survival and optimal profitability π* for normal distributions of the risk of extinction for different standard deviations ***σ*t* < σ*2

**Figure 51: Ditto for ***σ*1** > ***σ*2

It is clear that the main determinants of the size of the optimal profit are the means of the two distributions and the relationship between their variances. If we fix the means [say at 0 and 2) we can compare the size of the optimal profit as a function of the variances.

When the two variances are the same, the optimal profitability will be close to the average of the two means, i.e. to the diagonal of the square in Figure 52. For small positive variances of both distributions (*σ*1 *<* 0.25 and *σ*2 *<* 0.25) the optimal profitability will also be close to the average of the two means. The agent's behaviour changes only when the variances of the probabilities of extinction due to the two reasons differ significantly and are not very low (i.e. are greater than 0.25).

It can be shown^{[2]} that when one of the variances is fixed at a constant value and when the other one is increased, the optimal probability of survival remains constant up to a certain threshold representing a qualitative change. Beyond this threshold the optimal profitability starts to change relatively significantly.

**Figure 52: Optimal profit as a function of the variances of the random distribution of the two threats under consideration**

We have successfully investigated the problem of the firm's optimal business strategy in conditions of increasing returns to scale given high initial fixed costs of entry into the market and zero marginal costs where the firm is threatened by extinction due to both low profitability and ruinous entry of a competitor. The optimum strategy is a compromise (between the two threats) that involves choosing the price at which the rise in price is associated with an equal fall and rise, respectively, in the probabilities of extinction due to the two reasons (low profit and entry of a competitor). It turns out that this problem has a single solution (with both uniform and normal distributions of the probabilities of extinction). This allows us to construct a supply function. Where the sole threat to the firm is low profitability, this function is coincident with the standard supply function of a monopoly producer. The described approach is therefore obviously again a generalization (and not a refutation) of the standard micro- economic approach. Where both extinction threats are active, the location of the optimum depends also on the relationship between the variances of the probability of extinction. If the variances are identical, the optimal profit is at the level of the arithmetic average of the means of the two distributions. If one of the variances increases in relation to the other, the optimal profit level shifts towards the mean of the distribution having the lower variance, although only when the variance exceeds a certain level. From this we can conclude that under certain conditions (a high and non-uniform degree of uncertainty regarding the various threat factors) there is a risk that a relatively minor increase in uncertainty in a system will result in a qualitative change in the behaviour of the system and its sensitivity to changes in its parameters. This represents a potential element of instability in markets of the type under analysis.