10.5.1 RESULT OF NON-TRANSFERABILITY OF A SUBSIDY TO THE NEXT PERIOD-OPTIMAL SUBSIDYTIMING MODEL

Let us start by assuming that a subsidy recipient having the preferences described in the previous section can freely carry over any unused part of the subsidy to the next period and that the funding need in the two periods is the same. For simplicity, we will consider two consecutive periods with the same extinction boundary (subsistence level).

At the optimum of an economically rational donor, it must hold that the marginal transfer of funds from the first period to the second will reduce the recipient's probability of extinction in the first period to the same extent as it will increase its probability of extinction in the second period.

We will now derive the optimal subsidy timing. We use the following notation:

d the recipient's income,

b the recipient's subsistence level (extinction zone boundary),

a the total subsidy for the two consecutive periods from the donor (state),

at the subsidy from the donor (state) provided to the recipient in the t-th

period.

As we justified at the end of the previous section, we assume that the risk of extinction of the recipient as perceived by the donor is determined not by the relative income margin vis-a-vis the subsistence level, but by the change therein r'(d). We therefore assume that the agent's survival corresponds to an asymmetric second-order Pareto probability distribution^{[1]} with probability density function

for

for

and with distribution function

We assume for simplicity that the subsidy is the sole income of the subsidy recipient (a public service provider), i.e. d = a.

We will set b = 1 (that is how we select the money unit). Suppose, furthermore, that the overall subsidy amount for the two periods a is fixed and that survival in both periods is not ruled out, i.e. that

By applying the second-order Pareto distribution we obtain:

for

for

Remember that the donor's criterion is maximization of the recipient's survival probability, which is given by the product of the [let us assume mutually independent) survival probabilities in the two periods:

The donor solves the constrained optimization problem:

which can be trans formed into the free optimization problem:

Differentiating function with respect to all its three variables gives us the necessary conditions for the optimum:

The optimal solution is therefore (not surprisingly) regular granting of the subsidy:

The recipient's survival probability is given by the product of the (let us assume mutually independent^{[2]}) survival probabilities in the two periods:

Let us now examine the implications of the fact that the time distribution of the subsidy does not correspond to the recipient's needs. In the case of uniform needs, such a mismatch represents irregularity in the provision of the subsidy. Let us denote the difference between the higher and lower subsidy by δ > 0, that is:

If the subsidy is well above the threat zone boundary (d>>b = 1), a slight irregularity in the provision of the subsidy will have no effect. If, however, the subsidy is close to this boundary and the probability of survival is much less than unity, this irregularity can threaten the recipient, as its survival probability will be significantly reduced. The relative decrease as a result of an undesirable (i.e. not matching the recipient's needs) shift of a portion of the subsidy δ is:

Figure 68 and Figure 69 show the effect of the relative deviation from subsidy regularity on the relative decrease in survival probability (both in %) for an agent with a threat equal to the mean of the Pareto distribution and for an agent with a threat equal to the median of the Pareto distribution

Figure 68: The relation between the relative decrease in survival probability (in %) and the relative deviation from a regular subsidy (in %) for an agent with a threat equal to the mean of the second-order Pareto distribution

It is clear from Figure 68 that for an agent with an average subsidy (i.e. a subsidy equal to double the subsistence level) an irregularity whereby half the annual grant is moved to another period will be ruinous for the recipient Shifting 40% of the annual grant will reduce its probability of survival by roughly one half.

Figure 69 concerns an agent which is, as far as the threat is concerned, at the midpoint of the series of all agents ranked according to threat level, i.e. an agent at the median

Figure 69: Ditto for an agent with a threat equal to the median of the second-order Pareto distribution

A subsidy fluctuation of 40% of the subsidy will fatally endanger the median agent and all economically weaker agents (by comparison with the median agent) and will also significantly threaten the other agents. A 30% deviation will mean ceteris paribus an almost 50% reduction in the survival probability of the median agent (and hence also of more than half of the agents).

We arrive at the same conclusions for the case where, conversely, the subsidy is regular (equal in both periods) but the needs change. This time profile can also imply a significant bottleneck for periods when needs are greater.

The donor is reliant on information from recipients and therefore does not have objective information about their needs in individual periods.

One option is to have a regular regime. In the case of different needs in different periods (possibly ascertained by the recipient only during the second period), this fundamentally reduces the agent's probability of survival, i.e. it runs counter to the point of the subsidy. Here again, we can describe the rate of decline in the agent's survival probability on a subsidy equal to double the subsistence level by means of the chart in Figure 68.

Another option is to have an irregular but pre-agreed subsidy. However, this presupposes that the recipient knows his needs in the coming years exactly and a long way in advance and that these needs will not change over time. An unexpected change in needs — which cannot be ruled out in reality — can then cause a reduction in the probability of survival and a decrease in the efficiency of the subsidy (as measured by the effect of the social or public services per money unit issued by the state).

The restriction on carrying over public budget funding from one period (year) to another is based on numerous factors and practical problems. The main causes (explicit or implicit) of this constraint are the following:

• the technology used to draw up public budgets,

• a lack of trust of the central authority in the honesty of recipients, including a lack of trust (justified or unjustified) in the objectivity of the information provided to it by the recipients,

• a lack of trust on the part of the recipient that his information openness will not backfire on him (leading to distortion of information),

• the efforts of the central authority to reduce the "moral hazard” whereby the recipients force the donor to provide funds de facto by blackmail and the recipients of the social or public service play the role of hostages of the subsidy recipients.

Whatever the cause of the restriction on carrying over funding from one period to another, it does constitute a specific type of money wasting. It is a sui generis case of the prisoner's dilemma, since the lack of trust here prevents efficient behaviour of the system and optimal efficiency of use of the funding provided by the donor (state).

In the following model we will analyse the consequences of a different type of lack of trust of the donor in the recipients. In this case, the donor prescribes the subsidy use structure in an attempt to prevent the subsidy from being misused, for example, for excessive (in the donor's judgment) personal enrichment or for excessive overhead or investment spending.

[1] See section 1.3.2 for the properties of the second-order Pareto distribution.

[2] This is a simplifying assumption that abstracts from the fact that an increase in the subsidy in the first period can in reality cause the survival probability in the second period to either increase (owing to a smaller covered equipment maintenance debt) or to decrease (owing to higher maintenance costs of equipment purchased in the first period).

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