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We will assume that the firm is hit by a large negative demand shock. Besides a reduction in demand, it faces a large deterioration in the solvency of some of its customers. It therefore decides whether to supply to its "unsound" customers (non-payers) and, if yes, at what price. For simplicity, we assume that these non-payers are solvent in the sense that they will pay in the long run, but that they have liquidity problems and are not able to pay immediately. This allows us to abstract from customer credit risk problems. We also assume that the volume of production is already given by the firm's past decisions (for example, its investment or recruitment decisions), hence it is exogenous from the firm's perspective. The costs associated with this production are sunk and cannot be influenced by the firm.

We will use the following notation in the model: y volume of production (in natural units), yp production sold to payers, yn production “sold” to non-payers, w payroll and other immediate costs, c total costs (immediate and non-immediate), q the price for payers (the price for non-payers is equal to unity; that is how we define the money unit).

We assume that demand from payers is linear[1]

i.e. at q = 1 (the price charged to non-payers) the firm will sell nothing to payers. If the firm wants to sell its entire output, it must sell part of it to non-payers. We assume that demand from non-payers is unlimited.

The decision-taker knows that it is at risk on the one hand of recording a book loss, i.e. book revenues less than c (immediate and non-immediate costs), and on the other hand of having a shortage of cash, i.e. disposable funds less than w (payroll and other immediate costs).

The decision-taker's problem, therefore, is to split production y between payers and non-payers in such a way as to maximize the probability of avoiding both a for ementioned threats:

Given the higher (unit) price charged to non-payers, it is optimal purely in terms of book revenues to sell all output to non-payers (i.e. q = 1; maximum profit is equal toy - c). On the other hand, maximum cash is generated by price . We will establish this by solving

A reduction of the price below leads both to lower book revenues and to lower disposable funds. The firm therefore necessarily chooses a price in the range of

If , the firm will gain (relative to q = 1) more disposable funds at the cost of lower book revenues. The total reduction in book revenues (and, since output y and costs c are given, the total reduction in book profit as well] resulting from lowering the price for "payers” from q = 1 to q < 1 is

Book revenues are given by the difference yz. This is not (as it may seem) a difference between variables in different units, because y is the volume of output in natural units multiplied by the unit price.

We consider the situation where book revenues equal costs to be the boundary of extinction due to low book revenues:

According to our assumptions the probability of avoiding the risk associated with low book revenues is proportional to the margin of salesy - z relative to the extinction boundary c:

The amount of output sold to payers (with the a for ementioned payer demand function and at price q) is. The amount of disposable funds is there fore

We consider the situation where disposable funds equal payroll and other immediate costs to be the boundary of extinction due to insufficient disposable funds.

The probability of avoiding the risk of insufficient disposable funds is again proportional to the relative margin:

In the following calculations we will assume that the decision-taker has two possible strategies. In the first (so-called "minimax") strategy he identifies the larger of the two risks under consideration and tries to minimize that risk, even at the cost of an increase in the other risk. The second strategy maximizes the probability of simultaneous avoidance of both risks under consideration.[2]

  • [1] The assumption of linear demand from payers implicitly entails unrealistic behaviour of payers at very low prices q << 1/2. However, as we will see shortly, the seller will never choose q < 1/2, as such a large discount would lead counterproductively to a decrease in revenues.
  • [2] The standard microeconomic approach in the homo economicus paradigm would involve choosing one of the two variables considered here (book revenues and disposable funds) as the variable to be maximized and the other as the constraint defining the set of feasible solutions. The result of this con-strained maximization problem would be a situation right on the threat boundary. In the extremely opaque situation of a firm under “banking socialism”, this is not realistic—no decision-taker would re-gard such a risky strategy as optimal.
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