Mitigable Public Bads

The economic theory of public goods is sometimes assumed to be adaptable in a straightforward manner to public bads.  Here I consider some implications of the fact that such bads are usually mitigable.

Consider a region around an airport, subjected to noise from flights.  Noise arrives at every home in the region, but any household can choose, at a cost, to moderate the noise level inside its home by installing double glazing.  Such a choice by a household has no effect on the noise level inside any other home.

Suppose the sole water supply to a poor village is polluted.  Everyone in the village will be risking their health if they drink that water as supplied.  But (for some types of pollutant) any household can treat its water to make it safer to drink, perhaps by filtering or chemical treatment.  Such a choice by a household has no effect on the safety of the water consumed by any other household.

Similarly, suppose a region’s agriculture is affected by a drought.  All farms in the region receive less rainfall than normal but some, because of the crops they have chosen to grow or their cultivation methods, are better able than others to withstand the effects of the drought. Such choices by a farm have no effect on the resilience of other farms.

These are all examples of what I call mitigable public bads.  The idea of a public bad is derived from that of a public good, commonly defined as a good which is both rival and non-excludable. By non-rival is meant that one person’s consumption of or benefit from the good does not reduce the amount available to others. Non-excludable means that neither the provider of the good nor any other agent is able to pick and choose which individuals within the scope of provision of the good can consume it (hence the free-rider problem consisting in the fact that those who decline to pay for the good cannot be excluded from benefiting from it).  Street lighting in a city, for example, is non-excludable because no one can determine that certain individuals passing through the city’s streets will have their way illuminated and others will not. The fact that the provider could turn off the lights, excluding everyone, is irrelevant, as is the fact that only those who live in or visit the city can benefit from its street lighting.

Public bads have sometimes been defined as “goods” which are non-rival and non-excludable but which tend to lower rather than raise welfare (1).  For a bad to be non-rival is usually clear enough: my disturbance by noise from flights does not render my neighbours’ disturbance any less.  But non-excludability in relation to a bad is not so clear.  No one can be expected to pay for a bad, so a “provider” such as a factory causing air pollution has no interest in excluding non-payers. And whereas it can be assumed that few would wish to exclude themselves from a public good, sufferers from a public bad will certainly so wish if they can do so at reasonable cost. 

Usage in this context seems not to be settled. But I find it convenient to use the term public bad for any welfare-lowering “good” which is both non-rival and, in the sense that no one can pick and choose who within its broad scope suffers from it and who is excluded, non-excludable. If affected individuals can take action to moderate the effects of the bad on themselves, then I shall say that the public bad is mitigable. Environmental economists sometimes refer in this context to defensive expenditure or avertive expenditure, but as an adjective qualifying public bad, my sense is that mitigable is more appropriate than defensible or avertable.  Mitigable public bads are thus a subset of public bads, but an important one, as the above examples suggest.  Indeed, it seems plausible that most public bads are mitigable, at least for some individuals and to some degree. 

Mitigation need not be limited to individuals acting alone. In the case of installing double glazing, a decision by a household, or by a group of households making a bulk purchase to obtain a discount on the installation cost, can still be regarded as mitigation.  But for most public bads a clear distinction can be made between mitigation, action by potential sufferers to protect themselves from the bad, and what I shall term reduction, consisting in action at or close to the source of the bad to limit its scale or scope.  Examples of reduction would be a ban by an airport authority on flying to and from the airport in a certain direction, limiting noise for all households in that direction, or installation of equipment by a smoke-emitting factory to capture pollutants in the smoke, improving air quality for everyone in its vicinity.   

Two more definitions will be useful. I shall refer to the state of a public bad as its condition, across the whole of the affected region, after any reduction but before consideration of possible mitigation. By an individual’s exposure I shall mean the condition of the bad as experienced by that individual after any mitigation they have undertaken.  Thus exposure can differ between individuals both because some parts of the affected region may be affected more badly than others, and because the mitigation they have undertaken may differ.

A well-known result concerning public goods can be stated informally as below:

Proposition 1 The level of provision of a public good is optimal if the marginal cost of providing the good equals the sum over individuals of their marginal benefit from the good.

An individual’s marginal benefit from a public good can be interpreted as their marginal rate of substitution of the good for private goods.  This is equivalent to the slope of an indifference curve connecting combinations of goods which yield the individual the same utility. Since an individual’s utility function defines a whole set of indifference curves, the question then is which curve’s slope should be included in the sum when we are considering a possible re-allocation of production between private goods and the public good. The answer I will rely on here is to take the status quo distribution of private goods between individuals and assume that, in any re-allocation of production, individuals’ quantities of private goods would all change pro rata to the total quantity of private goods.  Any quantity of the public good together with an individual’s implied quantity of private goods would then define a point identifying a single indifference curve of that individual.  An alternative approach is to assume that all the indifference curves of any one individual have the same shape and therefore the same slope at any one quantity of the public good, regardless of the individual’s quantity of private goods.  This is equivalent to assuming that individual utility functions are quasilinear in the private goods (2), and is plausible if the quantity of private goods that individuals would have to forgo in return for the public good is small in relation to the total private goods.  Under this assumption it does not matter which curve of each individual we take: the sum of the slopes of one curve per individual will be the same whichever curves we take.

A point quite properly highlighted in most discussions of public goods is the sharp contrast between Proposition 1 and the optimality condition for a private good, which requires that the marginal cost of provision equal each individual’s marginal benefit. From this it follows that a public good will be under-provided in a free market (3).  However, other features of the proposition are sometimes overlooked. Firstly, the idea of optimal provision makes no sense for natural public goods like sunshine, except to the extent that they can be manipulated by human intervention. Secondly, for those cases where optimal provision does make sense, it is often a gross oversimplification to assume that the level of provision can be adequately characterised as a number of units on a single scale.  Just consider national defence, commonly given as an example of a public good. Thirdly, the relevant marginal cost of providing the public good is marginal social cost.  In the case of street lighting, for example, marginal cost should include the marginal social cost due to any greenhouse gases emitted in generating the electricity to supply the lights.  

Having noted these points, which apply equally to public bads, we may ask what is the equivalent of Proposition 1 for a mitigable public bad. In this case there are two kinds of decision to be made: the extent of mitigation by each individual; and the extent of action at source to limit or reduce the bad. 

Given the state of the bad, a rational individual will undertake mitigation up to but not beyond the point at which their marginal cost of mitigation (MCM) equals their benefit from a marginal lessening of exposure (BMLE). This is assuming normally shaped curves, that is, marginal cost increases and marginal benefit falls with additional units of mitigation.  For the purpose of optimal mitigation by individuals, it is not important how units of mitigation and exposure are defined: the only requirement is that, for any one individual, marginal cost and marginal benefit are measured with respect to the same units.  Note that, even if mitigation is possible for an individual, they will not undertake any mitigation if their marginal cost at zero mitigation exceeds their marginal benefit at that point.  In symbols, the condition under which an individual will undertake mitigation is:

MCM0  <  BMLE0                                  (A)

where the zero subscript indicates that the marginal quantities are measured at the state of the bad.

By analogy with the case of a public good, we expect that for a state S of a bad to be optimal, the marginal cost of reducing S must equal the sum over individuals of some sort of marginal quantity.  But what exactly?  The benefit to an individual from a marginal reduction in the state (BMRS) will depend upon whether, at S, they will undertake mitigation. If inequality A is not satisfied so that they do not undertake mitigation, then their benefit from a marginal reduction in S is simply their benefit from a marginal lessening of exposure:

BMRS(MCM0 > BMLE0)  =  BMLE0                    (B)

If however inequality A is satisfied so that they undertake mitigation, then their net benefit from mitigation at the margin will be BMLE minus MCM. Hence their benefit from a marginal reduction in S is the benefit from a marginal lessening of exposure less the benefit they could instead have obtained themselves from mitigation, the difference being the marginal cost of mitigation:

BMRS(MCM0 < BMLE0)  =  BMLE0 – (BMLE0 – MCM0)  =  MCM0           (C)

Putting the above together we have:

Proposition 2 Provided individual mitigation behaviour is rational, the state of a mitigable public bad is optimal if the marginal cost of reducing the bad equals the sum over individuals of the lower of a) their benefit from a marginal lessening of exposure and b) their marginal cost of mitigation.

However, extreme care is needed to ensure consistency in the units of measurement of the various marginal quantities.  Suppose we have a defined scale on which to measure the state of a bad.  For each individual, units of exposure must be such that the harm suffered from u units of state together with sufficient mitigation to limit exposure to u – 1 units must be the same as the harm suffered from u – 1 units of state with no mitigation.  A unit of mitigation is then simply that quantity of mitigation which will reduce exposure by one unit.

This has the important implication that both the physical requirements for and the cost of a unit of mitigation may vary greatly between individuals according to their circumstances.  Suppose the state of noise in the region around an airport is measured by average loudness at a defined location near the airport.  A given number of units will then be associated with more noise at some locations than others.  Suppose, as may be the case, that noise falls off with distance from the airport.  A reduction of one unit in the state of the noise may be quite significant for someone living close to an airport, but barely noticeable for someone on the edge of the affected region. Consequently the latter would need to do less than the latter, in physical terms, to achieve one unit of mitigation, and their marginal cost of mitigation will be lower.

Notwithstanding the above, a person living further from the airport will be less likely than someone nearer to it to undertake mitigation if, as is likely, benefit from a marginal lessening of exposure falls even faster with distance than marginal cost of mitigation, eventually reaching a point at which further lessening is imperceptible and offers no benefit.

The public policy implications of Proposition 2 are that those mitigable public bads which are due to human activity tend to be over-provided in a free market, but also that the extent of government action to correct that market failure should have regard to the availability and cost of mitigation by individuals.  If for example a tax is the preferred policy instrument, then it would be sub-optimal if the chosen tax rate per unit of state exceeds the sum over individuals as defined in Proposition 2.

An important question now is how the optimal state of a mitigable public bad compares with what it would be if no mitigation were possible.  This is illustrated in Chart 1 below.

The horizontal axis shows the state of the bad, reducing (ie improving) from left to right.  The vertical axis shows the quantity of a composite good representing all private goods.  It is assumed that there are no other public bads and no public goods.  The production possibility frontier PPF is the outer boundary of the possible combinations of the public bad and private goods, its concave shape reflecting the standard assumption of a diminishing marginal rate of transformation between goods.  The status quo is assumed to lie somewhere on the PPF.

S is the optimal state on the assumption that mitigation is impossible, and line I is the sum of individual indifference curves on the same assumption, chosen as explained above.  The slope of I at S will be the sum over individuals of their BMLE’s at S, and given the status quo assumption, I will be tangent to the PPF.

The red line IM is a sum of indifference curves passing through the intersection of PPF and I, on the assumption that mitigation is available. If inequality A above were not satisfied for any individual so that no mitigation would be undertaken, the slope of IM at S would also be the sum over individuals of their BMLE’s at S, and IM would be coincident with I in the vicinity of S, which would still be optimal despite the availability of mitigation.  If however A is satisfied for at least some individuals so that mitigation is undertaken, then the slope of IM will be a sum which includes MCM for at least some individuals, and therefore less than the slope of I.  Hence there is a region to the left of S within which PPF meets curves higher than IM, and the optimal state, given mitigation, will lie within this region, perhaps at SM.  So we have:

Proposition 3 Provided individual mitigation behaviour is rational, the optimal state of a public bad for which mitigation is available is greater (ie worse) than or equal to what the optimal state would have been if mitigation were not available, and is strictly greater if at least some individuals would have undertaken mitigation at the latter state.

In developing here a theory of mitigable public bads, I have passed over several points suggesting that Propositions 2 and 3 are in need of qualification.  Without following through the implications of each, I will just mention that an individual’s private cost of mitigation may not equal the social cost (for example the manufacture of double glazing may involve external costs not reflected in its price), that mitigation expenditure may be taxed or subsidised, and that rational mitigation behaviour may involve an element of gamesmanship if individuals believe that mitigation may lead to less action by government to reduce a bad.  It also needs to be borne in mind that optimality does not imply an acceptable income distribution: sometimes considerations of equity can properly take precedence over optimality.

Notes and References

1. Wikipedia  Public bad https://en.wikipedia.org/wiki/Public_bad

2. Wikipedia Quasilinear utility  https://en.wikipedia.org/wiki/Quasilinear_utility

3. Wikipedia  Public good – Free rider problem https://en.wikipedia.org/wiki/Public_good#Free_rider_problem

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