Figure 1 represents three neighbours located along a street. Each derives certain benefits from certain consumption-production activities such as using their property for a business, playing music, repairing cars or extending the property. Due to diminishing returns, the marginal benefits decline as the level of activity increases and these are represented by the demand curves . Unrestrained, households would produce/consume q2 of the activity -- the point at which marginal benefit is zero. However, each household generates nuisance (externality) costs for its neighbours, represented by e and these are assumed to rise with the level of activity. The question arises, what will happen if neighbours negotiate with full information? The classic analysis is presented in Coase (1960) but in his account, negotiation involves real compensatory payments (in cash or kind). The case depicted in Figure 1 is a special and interesting case of the general problem since compensation is possible in terms of restraint. Neighbour 1 has the incentive to cut back on an activity that lowers neighbour 2's welfare if the latter offers compensation in the form of restraint on an activity that lowers neighbour 1's welfare. To simplify exposition, neighbours are assumed to be identical in preferences. The bargaining problem is Pareto relevant in that there are mutual gains from trade. Between q2 and q1, the welfare loss of a neighbour due to externalities is greater than the welfare gain due to consumption/production (E+F>F). A household will be willing to lose F by voluntary restraint in order to gain E+F from his neighbour's restraint. Since this is true of both parties, both gain from voluntary restraint and an informal contract can be assumed to emerge, which sets the acceptable level of consumption-production at the social optimum q1. By striking a good-neighbour agreement each party gains E and the neighbourhood of two gains 2E.
FIGURE 1: Consumption/production decisions by three neighbouring households
The simulation described in the next section models this behaviour by making the assumption that voluntary restraining contracts will emerge if at least three adjacent households make simultaneous bids to negotiate for mutual gain. In the one-dimensional case of houses on a street, this assumes that a household requires both neighbours to opt-in to the contract as a condition for opting-in itself. If one neighbour is not offering to restrain (because it's other neighbour is not offering) then the contract fails to emerge -- or dissolves if it had already emerged. Contracts only emerge, therefore where households stand to gain. If a neighbour on one side does not comply, the gains from the compliant neighbour are assumed to be wiped out (it only takes one bad neighbour to reduce welfare and once reduced, an additional source of nuisance adds only a small marginal cost). In a two-dimensional neighbourhood in which contracts can be made with households across a street or adjoining to the rear, the simulation models a situation in which a threshold of two others in the immediate joint-consumption sphere is required in order for a household to offer voluntary restraint.