1 .**General Framework**

The lack of car parking space is one of the main problems of the largest cities in the world, with increasing expression in medium-sized cities or towns. Therefore, the implementation of the most efficient models for the use of parking space is a priority issue in modern Urban Economics, generally taking the form of tariff (or pricing) mechanisms for the parking space being used during each period of time[1].

Effectively, one of the most common solutions to this problem has been the application of time-based user tariffs, especially by municipalities or other types of public entities (such as municipally or locally based public companies). However, the use of these mechanisms is predominantly based on a standard model for charging the use of parking space during a certain period of time (Δ*t*), applying a predetermined value for each unit of time (*t*), which is usually charged in minutes-currency[2], without any differentiation between “good” and “bad” parking, with good parking being understood as the most efficient use of parking space (*E*) during Δ*t*, in a logic of 1 parking space – 1 car, and bad parking being understood as the use of, at least, 2 parking spaces – 1 parked car during Δ*t*.

Assuming a world of rational agents[3], zero informational costs[4] in which parking spaces all have the correct delimitations (typically marked on the ground) allowing for good parking (1 parking space – 1 car), this standard model ends up not providing an incentive for good parking, since users who pay the tariff for Δ*t* do not support any additional charge if they have incorrectly parked, for example, in the middle of a delimited space corresponding to two parking spaces, in such a way that 2 parking spaces – 1 car, instead of 2 parking spaces – 2 cars.

Assuming that each user is a utility maximiser[5], and therefore has a utility function for parking space, it can be assumed that there is a utility function per parking space in the following terms:

Where, respectively:

- U (P
) is the utility of the parking space at each moment in time (_{n}*t*); and *E*is the available space for parking at each moment in time (_{n}*t*).

Assuming the existence of this general utility function, it can be proved how social utility[6] is detrimentally impacted in a bad parking scenario, thus reducing not only the allocative efficiency of a scarce resource (parking space) but also the associated social welfare (including additional parking search time, opportunity costs, among others).

2. **Mathematical Assumptions**

For the following purposes, we will assume the existence of a standard model for the use of parking space, as used in most cities and towns around the world.

In this last standard model, users pay tariffs based on their use, during a certain period of time (Δ*t*), of a parking space, supporting some type of tariff surcharge depending on factors such as the geographical location of each zone, the supply of public transport, the characteristics of the aggregate demand function for parking, and also the spaces exclusively or partially intended for residents – but not, at least as a rule, bad parking, as defined above.

Considering the formulation of the general utility function defined in (1), we can generically assume that the function describing *E** _{n}* is:

Where, respectively:

A* _{S}* is the available parking space, during a certain period of time (Δ

*t*), where 0 ≤A

*<*

_{S}*Φ*, with

*Φ*being the maximum value of A

*during Δ*

_{S}*t*(including not only free parking spaces, but also spaces unavailable due to works or any other temporary impediment); and

*N* is the number of users searching for parking during the same period of time (Δ*t*), where 1≤* N* < ∞.

Therefore, considering (2), (1) can be written as:

Since the standard model is not based on the optimal use of a parking space (1 car – 1 parking space), an individual (I_{1}) who parks a car in both space E_{1 }(for 1 car) and E_{2} (for 1 car) will pay a zero amount for the use of E_{2}, thus not providing an incentive for good parking and resulting in maximum inefficiency in E_{2} allocation during Δ*t*.

By also parking in E_{2}, I_{1} is essentially depriving individuals I_{2}, I_{3} and I_{n}, where 2 ≤ n < ∞, of the opportunity to park during Δ*t* and, as such, I_{1} is incurring a utility loss (or cost) for those who want to park, during Δ*t*, in E_{2}.

Since I_{2}, I_{3} and I_{n} could not all park during Δ*t* in the same E_{2} parking space, the value of the utility loss caused by I_{1} should correspond to the average of the utilities lost by all those unable to park in E_{2}[7], as opposed to the sum of the utility losses of I_{2}, I_{3} and I_{n}.

**3. The Model**

The purpose of the following model is to review the standard model used in most European (and world-wide) cities and towns to charge the available parking space, that is, based on the period of parking time (Δ*t*) and on factors such as the geographical location of each zone, the supply of public transport, the characteristics of the aggregate demand function for parking, and also the spaces exclusively or partially intended for residents.

The critical point, which underpins this revision, continues to be the fact that the standard model does not include any type of incentive to discourage bad parking.

Therefore, if we consider a tariff for the use of parking space (*T*), we can easily assume that it is a function of *E _{n}*, among other variables, as follows:

However, following (2), *E _{n}* depends on the available parking space (P), which is detrimentally influenced by bad parking, without the utility value of I

_{1}being necessarily affected (because bad parking is not included in the tariff policy), contrary to what happens with the utility of I

_{2}, I

_{3}and I

_{n}, where 2 ≤ n < ∞, during the Δ

*t*when bad parking is in evidence.

This means that in a scenario where, for the use of E_{1 }and E_{2}, U_{I1} = U_{I2}, and another scenario where, for the same use of E_{1 }and E_{2}, U_{I1} > U_{I2}, the amount paid by I_{1} is always constant (*k*).

Therefore, the value of T must include a tariff penalty for bad parking determined based on several criteria, depending on the option of the public decision maker – although, for the benefit of simplification, it seems more favourable to us to establish a derivation from the value of the hourly tariff itself, for example, by charging those who park badly 50% more for the same use of E_{1} or, if budget constraints allow, to offer a discount coupon for the next parking space to all I_{n} who park correctly.

Only empirical evidence associated with the effects of a fare tariff (or pricing) policy change such as the one we propose will allow public decision-makers to conclude, simultaneously, whether bad parking is efficiently discouraged and whether an increase in parking space allocation is achieved.

4. **Preliminary conclusions**

In the standard model presented, the imposition of space usage tariffs exclusively based on time of use (although with variations) does not provide an incentive for good parking.

As we have seen, if the payment of these tariffs is totally unrelated to optimal use of the space itself (1 car – 1 parking space), the sum of the utility of the individual who parked the car incorrectly during Δ*t* is always constant, as opposed to what happens with the utilities of the I_{n} individuals, where 2 ≤ n < ∞, who want to park in spaces E_{2}, E_{3} and E_{n}.

As a result of the above, an optimal tariff policy should include (tariff) surcharges on drivers who park badly, thus promoting efficiency in the allocation of resources and social well-being. As a substitute or complement, and to the extent that budget constraints allow, discount vouchers (to be used for the next parking space) could be awarded to users who park correctly.

**BIBLIOGRAPHY**

Arrow, K.J., *Social Choice and Individual Values* (New York: Wiley 1951)

“Economic theory and the hypothesis of rationality”, in J. Eatwell, M. Milgate, and P. Newman, Eds., *Utility and Probability* (New York: Norton 1990), pp. 25–37

Herden, Gerhard, “On The Existence of Utility Functions”, *Mathematical Social Sciences*, 17, (1989), pp. 297-313

Stigler, George J., *Theory of Price* (4^{th} Edition, MacMillan Publishing Company, New York 1966)

[1] Which we ultimately assume derives from Law, regardless of whether it is a national law (Law, Decree-Law) or a municipally based legal normative (such as a Regulation).

[2] In this case, for example, Euro.

[3] Therefore assuming that the preferences of these agents reflect, at least, the following assumptions: (i) continuity; (ii) completeness (often mathematically known as the axiom of order); and (iii) transitivity.

[4] For these purposes, we assume as (only) relevant informational costs those arising from the demand for a good parking space delimitation (1 parking space – 1 car).

[5] Since we will disregard, in the simple mathematical model presented, the risk component, we will not need to resort to the von Neumann-Morgenstern axiomatic and, as such, to utility maximisation in the same sense. This type of axiomatics can be fruitfully used for the “lottery” (i.e. a probabilistic distribution) concerning the finding of a parking space, at each time interval, but no longer for the specific resource allocation problem arising from bad parking.

[6] That is,, where *n* is the number of users that potentially want to park in a badly occupied space, excluding the user who is affected by the bad parking (i.e. in relation to its own good parking).

[7] Or still E_{n}, where 2 ≤ n < ∞, if I_{1 }parks in a way that implies the use of n spaces beyond E_{1}.