# Stormwater Ponds in Coastal South Carolina Appendix A6-2 – Theoretical Economic Models

## A Theoretical Model of Maintenance Preferences

Here we present a relatively simple, theoretical model of pond maintenance to demonstrate several economic factors that influence a pond owner’s selection of annual maintenance activities. The presented model borrows heavily from Becker et al. (2006) but is adapted to illustrate stormwater management, specifically. Individual landowners and HOAs are responsible for pond maintenance activities on their land and often hire private pond management companies to maintain their ponds. Here, we refer to property owners as “pond managers” because they designate funds to be spent on pond maintenance activities. We assume that the objective of a pond manager is to select the cost-minimizing level of maintenance per acre of pond. We also assume that the manager provides funding to service and maintain the pond(s) through their useful lifecycle.

If ponds are regulated, then the consumer price, *P*, for stormwater management is given by:

where *c*(*E*) denotes the individual manager’s constant unit costs, which is assumed to be increasing along with the government’s resources (or enforcement activities) that are dedicated to ensuring that ponds are operating effectively (per the regulator’s code or standards).

The term *T* denotes the additional costs imposed on a second-party builder or developer through reduced convenience or criminal punishment. The variable *T* can be interpreted as the government’s efforts to thwart the improper, initial construction of a pond – for example, a developer decided to build an improperly constructed pond in order to reduce costs. Because we are only considering the private pond owner in this model, we will effectively set T equal to 0 for the remainder of the discussion.

One could interpret *c*(E) as the government’s efforts to prevent the mismanagement of the ponds, such as a manager deciding to forego regular maintenance in order to reduce costs. In the absence of any government oversight of ponds, *E* = 0, which implies that *P* = *c*(0), we interpret *c*(0) as the legal or regulatory (absolute) minimum level of maintenance activities required. **Figure A6.1** illustrates a linear demand curve for the competitive market of pond maintenance. We depict the benefits of management activities in terms of the services the pond offers (e.g., flood control, contaminant mitigation), for ease of exposition. However, **Figure A6.1** can easily be generalized to other types of services offered as well.

**Figure A6.1** Demand curve for pond maintenance that includes the costs (c) of both government enforcement (E) and free market (0) expenditures. We see that as the costs of management rise due to pricey government enforcement practices, the quantity of runoff approaches the welfare-enhancing “optimal level” (Q_{w}) while under the free market the quantity of runoff is high (Q* _{f}*) due to low levels of management.

Hence, one can interpret movement along the horizontal from left-to-right as decreasing pond maintenance results in an increasing quantity of un-managed runoff, with the potential for increased flooding and poor downstream water quality. Without government enforcement, so *E* = 0, the equilibrium outcome is represented by point *f* (the free-market, competitive outcome) and the equilibrium quantity of pollution control is represented by the competitive quantity, *Q _{f}*.

On the other hand, if the government engages in monitoring and enforcement, then the equilibrium outcome becomes point *w* (the welfare-enhancing level) in the diagram. In this case, the price rises to the cost level *c*(*E*), and the equilibrium quantity of pollution becomes *Q*_{w}. Note that the quantity of flooding, or pollution associated with increased runoff, is decreasing from right-to-left. We define the pollutant and flood water as stock variables, as opposed to flow variables, without any loss in generality.

We use the following notation throughout this section: *Q* is the quantity of runoff; *P* is the price of stormwater maintenance (or the lack thereof); demand is defined as D = P/Q; *F* is a monetary fine or penalty equivalent to a punishment for non-compliance; pond maintenance is assumed to be characterized by constant returns to scale; *c = c*(0) is the competitive costs of stormwater maintenance with or without enforcement; *A* is the manager’s private expenditures on avoidance of enforcement per unit of maintenance (more on avoidance expenditures below); *E* is the level of government enforcement per unit of maintenance; and *p*(*E*,*A*) is the probability that a manager is caught with a pond in non-compliance.

As in Becker et al. (2006), we make the following assumptions regarding the relationship between enforcement, *E*, avoidance, *A*, and the probability of being caught:

The first assumption, equation (2), implies that government enforcement, or the perception of government enforcement of the regulations, increases the probability of being caught for a violation. Whereas the second assumption, equation (3), implies that increasing avoidance expenditures decreases the probability of being caught.

We further assume that if a manager is caught – that is, his or her pond is not in compliance – then he or she is penalized *F* (per unit of maintenance). Assuming that the market of ponds is competitive and maintenance activities are characterized by constant returns to scale, then the price (of pollution or pollution mitigation) will be determined by the expected, minimum unit costs, *u*, of maintenance. A unit cost can be interpreted as the maintenance costs per pond-acre. The expected unit cost is equal to a weighted sum of the (odds ratio) probabilities of occurrence. Following Becker et al. (2006), we present this as an odds-ratio by dividing the expected cost of neglected maintenance by one minus the probability of being caught in violation, as follows:

We assume that *F* = 0 if the manager is not caught in non-compliance – that is, a penalty fee is not assessed against the manager if he or she is not caught. After some algebra, the weighted sum of probabilities can be reduced to:

Equation (5) implies that the expected unit cost is equal to the sum of competitive costs of pond maintenance, the avoidance costs, and the expected probability of paying a penalty fee if in non-compliance – all relative to the probability of not being caught. Further, we can define the odds ratio (The odds ratio represents the odds that an outcome will occur, relative to the odds of the outcome not occurring. We implicitly assume here that the odds of an outcome are based on discrete probabilities) as follows:

Given our definition of the odd ratio, we can rewrite the expected costs per unit as:

With this definition, the expected unit costs are linear in the odds ratio, since it gives the probability of being caught per unit of maintenance, and the expected costs are linear in the probability of paying the penalty fee, *F*, if caught. Provided that the penalty fee is exogenously determined, the competitive price of stormwater maintenance is given by the minimum level of unit costs:

Given a pre-determined penalty fee and level of enforcement, the first-order condition for cost minimization is given by:

Equation (9) implies that the manager’s expected costs are increasing in the odds of being caught in non-compliance. To see this, assume the manager’s total unit costs remain fixed. The partial derivative on the left-hand side of (9) is positive by assumption (3) above. Thus, as the odds of being caught in non-compliance increase, the manager’s expected unit costs are rising (i.e., the entire term on the left-hand side of equation (9)). This intuitively implies that if there is more government oversight and monitoring, then the manager has an incentive to increase avoidance expenditures in order to reduce the odds of being caught with a non-compliant pond. In other words, the manager does not necessarily expend efforts to ensure the intended functions of the pond (flood and nutrient mitigation), but instead expends efforts on potentially non-critical functions of the pond.

We interpret expenditures on avoidance, *A*, as including the entire increase in direct costs of maintaining non-critical functions within a pond. Non-critical functions include maintenance activities such as ensuring the pond and surrounding area is landscaped and well manicured or the manager only treats the pond for mosquito control. Expenditures on landscaping increase the aesthetical appeal of the appearance of a pond but generally do not improve the primary functions – flood and nutrient mitigation. HOA covenants generally have specific rules and regulations regarding the appearance of retention ponds, which improve curb appeal, but the covenants often do not have explicit policies regarding the actual function or efficacy of the ponds. Therefore, even under increasing government oversight, the pond manager still does not necessarily have the correct incentives in place in order to maintain the critical functions offered by the pond.

Based on the solution to equation (9), the minimum of the expected maintenance cost per unit is then given by:

where *A*^{*} is the cost-minimizing level of expenditures. The competitive equilibrium price in equation (10) exceeds the legally-binding minimum level of maintenance, *c*, by *A*. The term is the expected costs of pond maintenance when caught in non-compliance; and the term is expected costs of punishment if caught in non-compliance. From a regulatory standpoint, equation (10) implies the price of pond maintenance is increasing in government enforcement and the assessed penalty fee for non-compliance:

From equations (11) and (12) above, it can be seen that increasing the enforcement of government regulations, *E*, and increasing the fine level, *F, *raises the expected cost of stormwater maintenance.

## Optimal Enforcement Model

In order to determine the optimal level of enforcement for stormwater management, we must first determine the government’s enforcement expenditures. Following Becker et al. (2006), we assume the government has a cost function of enforcement defined by:

We assume that the government’s costs of enforcement are increasing in the level of enforcement:

However, we also assume, without a loss of generality, that the government’s costs are convex in enforcement. Equation (14) implies that enforcement costs depend not only on the level of enforcement, but also the number of ponds (*Q*) monitored and the fraction of non-compliant managers punished (through ).

The equilibrium level of enforcement depends on the government’s objectives. We assume that government wants to reduce the amount of runoff relative to what would be found in a competitive market (*Q*_{c} in **Fig. A6.1**). We define *V*(*Q*) as the social value function, and assume that the marginal social value is no larger than the marginal cost of providing stormwater management:

With these preferences the government chooses a level of enforcement to maximize the value of consumption, less the sum of production and enforcement costs:

With the assumption of constant returns to scale and perfect competition within the production of stormwater retention ponds, then , and we substitute the cost function from (15) into (16). Hence, the social planner’s problem becomes:

The first-order condition is given by:

where denotes marginal revenue. The left-hand side of equation (18) represents the marginal social benefits associated with a reduction in runoff, whereas the right-hand side represents the marginal social costs of enforcement. If we assume that the marginal social costs are equal to zero, then equation (18) simplifies to:

where denotes the price elasticity of demand for runoff (Becker et al. 2006). The term denotes the ratio of the social willingness to pay relative to the private marginal willingness to pay for stormwater maintenance (or flood and nutrient mitigation). Equation (19) implies that if *V _{q}* is positive, then stormwater management has a non-negative marginal social value, and if the price elasticity of demand is inelastic,

*MR*< 0, then the optimal level of enforcement would be zero.

The key to disentangling the marginal social benefits and marginal social costs is to examine elasticity of demand for runoff. Becker et al. (2006; p. 50) show that the marginal costs are also based on the price elasticity of demand (and the elasticity of the odds ratio with respect to government enforcement). The derivation is beyond the scope of the current manuscript. However, the interested reader is referred to the original work for a formal poof. In words, if demand is elastic, then total maintenance costs fall as a result of a reduction in runoff, and enforcement costs increase slowly. An elastic demand implies that any percentage increase in the costs of stormwater management would lead to a proportionally larger decrease in the quantity of runoff. We argue that an elastic demand for runoff is highly unlikely, although this is difficult to assert with absolute confidence until more research is conducted to assess the costs and benefits of stormwater management. However, several past studies have found that the residential demand for drinking water has a tendency to be highly inelastic (Espey et al. 1997). Residential drinking water offers different services than does stormwater control; nevertheless, the demand elasticities for drinking water and stormwater control are likely to be highly correlated.

In contrast, if the demand for runoff is inelastic (a more likely scenario), then total management costs rise as the amount of runoff falls, and enforcement costs rise more rapidly (Becker et al. 2006). However, the results imply that in order to reduce the amount of runoff to the socially optimal level (*Q _{w}* in

**Fig A6.1**), then the marginal social value should be very negative. Given the general public’s lack of education in regards to stormwater management (Roy et al. 2008), it is not very likely that the marginal social value would be characterized as “very negative” – that is a large value in magnitude in an absolute sense. Consistent with the claims by Becket et al. (2006), reducing runoff to the socially optimal level would arguably absorb a lot of society’s resources – implying that such an option is not necessarily cost effective.