In this model we will try to address the problem of the effect that the donor's lack of trust has when the process of granting social or public services is rather more structured than in the previous model. We will differentiate here between subsidy recipients and service recipients. Hence, we assume a model with three levels of agents on a hierarchical scale:

a) a donor (state) providing a subsidy for the provision of a social or public service (which it is unable to provide efficiently itself),

b) subsidy recipients providing a social or public service, who cease to exist if they are unable to do so,

c) recipients of the service.

We assume that there are multiple subsidy recipients providing the social or public service, so that the collapse of one subsidy recipient does not imply the collapse of the service.^{[1]}

We also assume that each subsidy recipient behaves in such a way that both it and the recipients of its service survive jointly (do not collapse) with maximum probability.^{[2]} In this sense, we depart from Svensson's microeconomic models of conditionality,^{[3]} in which the objective functions of the donor and the recipient are different (the principal-agent approach).

We assume that the partial probabilities of economic extinction of the service provider and recipient are independent and that the collapse of all service recipients would mean the collapse of the subsidy recipient as well, since a fatal failure in the provision of the service would disqualify the subsidy recipient in the eyes of the donor (an assumption of independence of the partial probabilities of extinction and of existential dependence of the subsidy recipient on the service recipient).

Economic extinction of the subsidy recipient (and simultaneously the service provider) can be caused by:

• a real impossibility of providing the service at the given revenue level,

• excessively low income of the subsidy recipient, for cing it to change its line of business,

• collapse of the service recipient.

The subsidy recipient will seek the optimum allocation of the subsidy between its own overheads (including subsistence) and its service recipients. A distrustful donor (state) will intervene to change this optimum. Suppose it orders the subsidy recipient to reduce its overhead spending by δ% in favour of the service recipients. In so doing it may increase the probability of the subsidy recipient walking out like an "unhappy wife” i.e. it may unwittingly create a bottleneck for the subsidy recipient which will increase its probability of collapse (extinction) in the sense of an inability to provide the given social or public service.^{[4]} We will quantify this causality in the following model.

As we Justified above, we assume a second-order Pareto distribution for the survival (non-failure) of agents. We use this probability distribution both for subsidy recipients and for service recipients.

We use the following notation in the model: a the amount of the subsidy,

d the subsidy recipient's income which is used directly by the service recipient (i.e. the value of the services provided to it),

r the overhead costs of the subsidy recipient,

bd the subsistence level (extinction zone boundary) of the subsidy recipient,

bs ditto for the service recipient,

pd the probability of survival of the subsidy recipient,

ps ditto for the service recipient.

As in the previous model we assume a second-order Pareto probability distribution for the subjective probability of survival of the two agents (the subsidy recipient and the service recipient):

for

for

for

for

For clarity we abstract from the problem of mandatory timing of expenditures used in the previous model. This corresponds to an assumption that the state leaves the time distribution of the subsidy into individual phases entirely to the subsidy recipient. In this sense, therefore, the state trusts the subsidy recipient. The lack of trust studied in this model is of a different kind: the state tries to prevent the subsidy recipient from enriching itself at the expense of the service recipients.

This can be a good thing (in areas dominated by fiercely competitive agents trying to make the maximum short-term profit], but it can also be a bad thing (in areas dominated by altruistic, long-term providers of social or public services). In our model, we will focus on the second type of subsidy recipients providing public or social services.^{[5]}

Given the above assumptions (including the assumption of independence of the partial probabilities of extinction and of existential dependence of the subsidy recipient on the service recipient; see above], the probability of nonfailure of a social service provider is given by the product of the probabilities of (economic) survival of the subsidy recipient and the service recipients:

A rational subsidy recipient decides to divide amount a between its overheads r and its costs for providing services to the service recipient d in the way described by the following constrained optimization problem:

A subsidy recipient maximizing (explicitly or implicitly] its own probability of survival solves the problem of optimal allocation of the subsidy between its overheads and the service recipient in a manner corresponding to the solution of the problem:

We can transform this constrained optimization problem into a free optimization problem using the Lagrange function:

We obtain the necessary conditions for the optimum by setting the partial derivatives of function L with respect to all its three variables r, d and λ equal to zero. In this way we derive a system of three equations:

From the first two equations we get:

If the threat to the subsidy recipient happens to be equal to the threat to the recipients of its service, the solution is, i.e. it is optimal for the subsidy recipient to spend half of the money provided by the donor on its overheads. If the threats are unequal (i.e. if), it is optimal to favour the more threatened agent.

As in the model in the previous section, we will focus on a service provider with mean income, i.e. ( for the second-order Pareto distribution) with a subsistence level that is half the income level. Assuming that:

(that is how we select the money unit),

After substituting into equation (*) we obtain:

(*)

The following figure shows the relationship between the solution to this equation (with unknown r and parameter bd) and the parameter.^{[6]} For bd = bs = 1 (equal threats to the subsidy recipient and the service recipient) the solution is r* = 2 (half of the subsidy a - 4 will go on overheads). For bd → 0+ and bs → 0+ the whole subsidy goes to the more threatened agent (i.e. r* → 0+ and r* → 4+ respectively). If bd > bs, i.e. if the subsidy recipient is the more threatened, it will choose overhead costs exceeding half of the total subsidy. If, however, bd < bs, i.e. if the service recipient is the more threatened, the subsidy recipient will reduce its overhead costs in its own interests to below half of the subsidy, to the extent illustrated in Figure 70.

Figure 70: The optimal overhead level (from the subsidy recipient's point of view) for an agent with a threat equal to the mean of the second-order Pareto distribution

It can be seen from Figure 70 that when the general threat level is high, the preference for the more threatened agent is greater.

The most important conclusion is that a subsidy recipient acting in its own interests does not maximize its overheads. On the contrary, in certain situations it will reduce them, despite not being forced to do so. If, however, it is forced to reduce them against its will, both the subsidy recipient and the service recipient may collapse.

We will now quantify this risk for three illustrative cases:

• equal threats to both (bd = bs = 1),

• service recipient more threatened (bd = 0.5, bs = 1),

• subsidy recipient more threatened (bd = 1.5 = 2bs).

In the case of equal threats to both agents, the survival probability attains its maximum possible values for r = 2. How much does the survival probability decrease if the donor puts a cap on overhead spending?

Figure 71: The reduction in the subsidy recipient's survival probability for sub-optimal overheads (100% corresponds to the maximum survival probability for the individual cases)

It can be seen from Figure 71 that in the case of equal threats to the subsidy recipient and the service recipient (bd = bs = 1), a cap on overhead costs of less than or equal to 25% of the total subsidy will cause the certain demise of the subsidy recipient. Even for the case where the service recipient is more threatened (bd = 0.5 and bs= 1), when the maximum survival probability occurs when overheads r = 1.25 (i.e. 31% of the total subsidy), a 25% cap on overhead costs means a reduction in its probability of survival. A donor-imposed cap of one-eighth of the total subsidy will certainly ruin the subsidy recipient. Such a danger applies most of all, of course, in the case where the subsidy recipient is more threatened (bd = 1.5 = 2bs), where a cap on overhead spending of one- third of the total subsidy is enough to lead to certain extinction.

In the interests of its own survival, a rational subsidy recipient will not "rip off" service recipients, as by doing so it would be cutting its own throat (if, as we assume, it wishes to survive in the role of service provider). If the donor (state) trusts it, everything is okay. If, however, the donor (in an ef for t to ensure that service recipients get a greater share) constrains the subsidy recipient by setting a cap on overhead costs as a proportion of the total subsidy, there are two possibilities: the cap — motivated by the donor's lack of trust — is either of no significance (the subsidy recipient would satisfy it of his own free will anyway), or represents a reduction in survival probability. Such a cap can even be a lethal weapon — in all the cases described above a condition that the subsidy recipient's own consumption should not exceed one-tenth of the subsidy (see footnote 151) would itself preclude the economic survival of practically every subsidy recipient.

Wherever possible, a rational donor (state) provides subsidies with trust in the recipient's reliability, since limiting rules can fundamentally reduce the efficiency of use of the subsidy, even though their original and sole purpose was the exact opposite.

We are not trying to saying that it is possible or necessary to remove all state-imposed constraints on the use of public money If there is a low level of competition, or if subsidized public service providers have insufficiently long histories, the threat of misuse of the subsidy is greater and the donor's rules to prevent such misuse should be stricter. The same applies in reverse — gradually earned trust in a subsidy recipient allows a rational donor to increase the efficiency of the subsidy by relaxing the rules of use of the subsidy for "tried and tested" recipients.

We merely wish to point out that the standard economic trade-off applies to subsidies. The donor [state] pays for control of the use of such funding with a reduction in the efficiency of service provision. It is a prisoner's dilemma of sorts — a lack of trust between agents [in our case the donor's unwillingness to believe that the recipient will not misuse the subsidy by time-shifting part of it or by charging high overhead costs] can be a cause of economically irrational behaviour.

In this sense, too, a lack of trust can be expensive and trust can be economically rational.

[1] This assumption allows us to abstract from the moral hazard effect in the model.

[2] This does not necessarily mean that maximization of the probability of survival is an explicit criteria of public service recipients. Their criterion is some sort of subjective utility. However, the survival of a public service recipient is part of the decision-making problem of the public service provider. For the latter it is crucial that the former does not collapse, because such collapse could cause its own economic extinction.

[3] Svensson, J.: When Is Foreign Aid Policy Credible? Aid Dependence and Conditionality. Journal of Development Economics 61,1(2000): 61-84.

[4] A major Czech grant agency provides research grants with the proviso that no more than 10% of the grant is spent on wages and salaries, yet the wages of young researchers and teachers are undoubtedly a bottleneck for fundamental research work in the Czech Republic.

[5] In the future we would like to dynamize the model and investigate how the approach (allocation rule) of the donor determines the type of climate.

[6] We solved the equation numerically. For all parameters b 0 ( {0, 2} in the interval {0, 4} the equation has a unique solution. The other solutions are nonsensical.

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