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# MODEL A: MINIMAX STRATEGY

Let us start by assuming that the decision-taker tries to avoid the larger of these risks, i.e. that his (subjective) feeling of threat is due exclusively to the greater of the two risks under consideration:

Given that a change in price q increases one of these risks and decreases the other, his smallest "minimax" feeling of threat is minimal when the relative margins for the two threats are equal:

Suppose that the volume of production y is sufficient to avoid extinction due to both reasons under consideration. If this condition is met, and under the a for ementioned assumption c - 2w < 0, we can rearrange the previous relation to obtain the non-negative optimal price q' for payers:

The firm then splits the sale of its output as follows:

The share of production "sold” to non-payers in total production is characteristic of the degree of secondary insolvency. Under the assumptions adopted, the following holds for this share:

The proportion of production sold to payers is therefore given by the share of immediate costs in total costs:

In other words, decision-takers maximizing the probability of their own survival will set a reduced price for payers q* in such a way that the division of their production between payers and non-payers "copies” the ratio of their immediate costs to their non-immediate costs. Consequently, all agents — both strongly threatened and less strongly threatened — will adopt the practice of declaring secondary insolvency. This does not mean, however, that all agents will behave identically. Higher secondary insolvency will (in the logic of this model) be recorded by firms with a higher proportion of immediate (especially payroll) costs.

# MODEL B: MINIMUM EXTINCTION RISK STRATEGY

According to the assumptions made above for the probability of avoidance of the two risks under consideration, it holds that

At the optimum q = q (for the optimal price reduction for "payers"), for the value of distribution function F and the value of probability density function/ it holds that

The solution of this non-trivial equation depends, of course, on the relationship between parameters y, c and w. If the values of distribution functions F1(q) and F2(q) are close, the optimal strategy will be similar to the "minimax” strategy described above. If F1(q) >> F2(q), the "bottleneck” is the constraint relating to disposable funds and the optimal price q* will be lower (i.e. the discount for payers will be bigger). If F1(q) << F2(q), the "bottleneck" is accounting profit and price q* will be close to unity (corresponding to a zero discount).

We will denote by k the ratio of total costs to revenues at the maximum unit price:

Furthermore, we will denote by m the ratio of immediate costs to total costs:

By substituting into the above equations we obtain:

We obtain the optimal price for payers q (i.e. the price maximizing the decision- taker's probability of survival) by solving the equation:

(*)

Let us calculate the derivative and substitute into (*):

so the optimum condition is

After rearranging we obtain:

We have solved this equation numerically for various values of k and m. The results are summarized in the following table:

Table 7: Optimal price for payers in relation to the ratio of total and immediate costs to production

 m = 0.9 m = 0.8 m = 0.7 m = 0.6 m = 0.5 m = 0.4 m = 0.3 m = 0.2 m = 0.1 k = 0.95 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.77 0.85 k = 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.73 0.81 k = 0.8 0.5 0.5 0.5 0.5 0.5 0.5 0.58 0.7 0.78 k = 0.7 0.5 0.5 0.5 0.5 0.5 0.5 0.6 0.68 0.75 k = 0.6 0.5 0.5 0.5 0.5 0.5 0.53 0.6 0.66 0.74 k = 0.5 0.5 0.5 0.5 0.5 0.5 0.56 0.61 0.66 0.73 k = 0.4 0.5 0.5 0.5 0.51 0.54 0.57 0.61 0.65 0.71 k = 0.3 0.5 0.51 0.52 0.54 0.56 0.58 0.61 0.65 0.71 k = 0.2 0.53 0.53 0.54 0.55 0.57 0.59 0.61 0.64 0.7 k = 0.1 0.54 0.55 0.56 0.56 0.58 0.59 0.61 0.64 0.69

Source: Authors' calculations

The greater is the risk stemming from insufficient disposable funds, the more sellers will reduce the price for payers. However, they will not reduce it below the level that maximizes the amount of disposable funds acquired, i.e. below q = 0.5. For practically all combinations of k and m our model producer also appeals to non-payers. If the ratio of immediate costs is low (as is the case in the last four columns of the table), the producer is not forced to offer the maximum discount.

* * *

Economically non-standard environments do not rule out rational (albeit nonstandard from the perspective of the homo economicus paradigm) economic decision-making. The methodology of generalized microeconomics allows us to model and microeconomically analyse such decision-making.

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