Chapter 5 Welfare economics and the environment



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Chapter 5

Welfare economics and the environment


Welfare economics is the branch of economic theory which has investigated the nature of the policy recommendations that the economist is entitled to make.

Baumol (1977), p. 496

Learning objectives


In this chapter you will

  • learn about the concepts of efficiency and optimality in allocation

  • derive the conditions that are necessary for the realisation of an efficient allocation

  • find out about the circumstances in which a system of markets will allocate efficiently

  • learn about market failure and the basis for government intervention to correct it

  • find out what a public good is, and how to determine how much of it the government should supply

  • learn about pollution as an external effect, and the means for dealing with pollution problems of different kinds

  • encounter the second-best problem

Introduction

When economists consider policy questions relating to the environment they draw upon the basic results of welfare economics. The purpose of this chapter is to consider those results from welfare economics that are most relevant to environmental policy problems. Efficiency and optimality are the two basic concepts of welfare economics, and this chapter explains these concepts as they relate to problems of allocation. There are two classes of allocation problem: static and intertemporal. Efficiency and optimality are central to both. In this chapter we confine attention to the static problem – the allocation of inputs across firms and of outputs across individuals at a point in time. The intertemporal problem – allocation over time – is dealt with in Chapter 11. If you have previously studied a course in welfare economics, you should be able to read through the material of this chapter rather quickly. If not, the chapter will fill that gap.

There are three parts to this chapter. The first states and explains the conditions required for an allocation to be (a) efficient and (b) optimal. These conditions are derived without regard to any particular institutional setting. In the second part of the chapter, we consider how an efficient allocation would be brought about in a market economy characterised by particular institutions. The third part of the chapter looks at the matter of ‘market failure’ – situations where the institutional conditions required for the operation of pure market forces to achieve efficiency in allocation are not met – in relation to the environment.


PART 1 EFFICIENCY AND OPTIMALITY


In this part, and the next, of this chapter we will, following the usage in the welfare economics literature, use ‘resources’ to refer generally to inputs to production rather than specifically to extractions from the natural environment for use in production. In fact, in these parts of the chapter, when we talk about resources, or ‘productive resources’ we will have in mind, as we will often make explicit, inputs of capital and labour to production.

At any point in time, an economy will have access to particular quantities of productive resources. Individuals have preferences about the various goods that it is feasible to produce using the available resources. An ‘allocation of resources’, or just an ‘allocation’, describes what goods are produced and in what quantities they are produced, which combinations of resource inputs are used in producing those goods, and how the outputs of those goods are distributed between persons.

In this section, and the next, we make two assumptions that will be relaxed in the third part of this chapter. First, that no externalities exist in either consumption or production; roughly speaking, this means that consumption and production activities do not have unintended and uncompensated effects upon others. Second, that all produced goods and services are private (not public) goods; roughly speaking, this means that all outputs have characteristics that permit of exclusive individual consumption on the part of the owner.

In the interests of simplicity, but with no loss of generality, we strip the problem down to its barest essentials. Our economy consists of two persons (A and B); two goods (X and Y) are produced; and production of each good uses two inputs (K for capital and L for labour) each of which is available in a fixed quantity.

Let U denote an individual’s total utility, which depends only on the quantities of the two goods that he or she consumes. Then we can write the utility functions for A and B in the form shown in equations 5.1:


(5.1)

The total utility enjoyed by individual A, denoted UA, depends upon the quantities, XA and YA, he or she consumes of the two goods. An equivalent statement can be made about B’s utility.

Next, we suppose that the quantity produced of good X depends only on the quantities of the two inputs K and L used in producing X, and the quantity produced of good Y depends only on the quantities of the two inputs K and L used in producing Y. Thus, we can write the two production functions in the form shown in 5.2:

(5.2)

Each production function specifies how the output level varies as the amounts of the two inputs are varied. In doing that, it assumes technical efficiency in production. The production function describes, that is, how output depends on input combinations, given that inputs are not simply wasted. Consider a particular input combination KX1 and LX1 with X1 given by the production function. Technical efficiency means that in order to produce more of X it is necessary to use more of KX and/or LX.

The marginal utility that A derives from the consumption of good X is denoted UAX; that is, UAX = ∂UA/∂XA. The marginal product of the input L in the production of good Y is denoted as MPYL; that is, MPYL = ∂Y/∂LY. Equivalent notation applies for the other three marginal products.

The marginal rate of utility substitution for A is the rate at which X can be substituted for Y at the margin, or vice versa, while holding the level of A’s utility constant. It varies with the levels of consumption of X and Y and is given by the slope of the indifference curve. We denote A’s marginal rate of substitution as MRUSA, and similarly for B.

The marginal rate of technical substitution as between K and L in the production of X is the rate at which K can be substituted for L at the margin, or vice versa, while holding the level output of X constant. It varies with the input levels for K and L and is given by the slope of the isoquant. We denote the marginal rate of substitution in the production of X as MRTSX, and similarly for Y.

The marginal rates of transformation for the commodities X and Y are the rates at which the output of one can be transformed into the other by marginally shifting capital or labour from one line of production to the other. Thus, MRTL is the increase in the output of Y obtained by shifting a small, strictly an infinitesimally small, amount of labour from use in the production of X to use in the production of Y, or vice versa. Similarly, MRTK is the increase in the output of Y obtained by shifting a small, strictly an infinitesimally small, amount of capital from use in the production of X to use in the production of Y, or vice versa.

With this notation we can now state, and provide intuitive explanations for, the conditions that characterise efficient and optimal allocations. Appendix 5.1 uses the calculus of constrained optimisation (which was reviewed in Appendix 3.1) to derive these conditions formally.

5.1 Economic efficiency


An allocation of resources is said to be efficient if it is not possible to make one or more persons better off without making at least one other person worse off. Conversely, an allocation is inefficient if it is possible to improve someone’s position without worsening the position of anyone else. A gain by one or more persons without anyone else suffering is known as a Pareto improvement. When all such gains have been made, the resulting allocation is sometimes referred to as Pareto optimal, or Pareto efficient. A state in which there is no possibility of Pareto improvements is sometimes referred to as being allocatively efficient, rather than just efficient, so as to differentiate the question of efficiency in allocation from the matter of technical efficiency in production.

Efficiency in allocation requires that three efficiency conditions are fulfilled – efficiency in consumption, efficiency in production, and product-mix efficiency.


5.1.1 Efficiency in consumption

Consumption efficiency requires that the marginal rates of utility substitution for the two individuals are equal:

MRUSA = MRUSB (5.3)

If this condition were not satisfied, it would be possible to rearrange the allocation as between A and B of whatever is being produced so as to make one better off without making the other worse off. Figure 5.1 shows what is involved by considering possible allocations of fixed amounts of X and Y between A and B.1 The top right-hand corner, labelled A0, refers to the situation where A gets nothing of the available X or Y, and B gets all of both commodities. The bottom left-hand corner, B0, refers to the situation where B gets nothing and A gets everything. Starting from A0 moving horizontally left measures A’s consumption of X, and moving vertically downwards measures A’s consumption of Y. As A’s consumption of a commodity increases, so B’s must decrease. Starting from B0 moving horizontally right measures B’s consumption of X, and moving vertically upwards measures B’s consumption of Y. Any allocation of X and Y as between A and B is uniquely identified by a point in the box SA0TB0. At the point a, for example, A is consuming A0AXa of X and A0AYa of Y, and B is consuming B0BXa of X and B0BYa of Y.


The point a is shown as lying on IAIA, which is an indifference curve for individual A. IAIA may look odd for an indifference curve, but remember that it is drawn with reference to the origin for A which is the point A0. Also shown are two indifference curves for B, IB0IB0 and IB1IB1. Consider a reallocation as between A and B, starting from point a and moving along IAIA, such that A is giving up X and gaining Y, while B is gaining X and giving up Y. Initially, this means increasing utility for B, movement onto a higher indifference curve, and constant utility for A. However, beyond point b any further such reallocations will involve decreasing utility for B. Point b identifies a situation where it is not possible to make individual B better off while maintaining A’s utility constant – it represents an efficient allocation of the given amounts of X and Y as between A and B. At b, the slopes of IAIA and IB1IB1 are equal – A and B have equal marginal rates of utility substitution.


5.1.2 Efficiency in production

Turning now to the production side of the economy, recall that we are considering an economy with two inputs, L and K, which can be used (via the production functions of equations 5.2) to produce the goods X and Y. Efficiency in production requires that the marginal rate of technical substitution be the same in the production of both commodities. That is,

MRTSX = MRTSY (5.4)

If this condition were not satisfied, it would be possible to reallocate inputs to production so as to produce more of one of the commodities without producing less of the other. Figure 5.2 shows why this condition is necessary. It is constructed in a similar manner to Figure 5.1, but points in the box refer to allocations of capital and labour to the production of the two commodities rather than to allocations of the commodities between individuals.2 At X0 no capital or labour is devoted to the production of commodity X – all of both resources is used in the production of Y. Moving horizontally to the left from X0 measures increasing use of labour in the production of X, moving vertically down from X0 measures increasing use of capital in the production of X. The corresponding variations in the use of inputs in the production of Y – any increase/decrease in use for X production must involve a decrease/increase in use for Y production – are measured in the opposite directions starting from origin Y0.


IXIX is an isoquant for the production of commodity X. Consider movements along it to the ‘southeast’ from point a, so that in the production of X capital is being substituted for labour, holding output constant. Correspondingly, given the full employment of the resources available to the economy, labour is being substituted for capital in the production of Y. IY0IY0 and IY1IY1 are isoquants for the production of Y. Moving along IXIX from a toward b means moving onto a higher isoquant for Y – more Y is being produced with the production of X constant. Movement along IXIX beyond point b will mean moving back to a lower isoquant for Y. The point b identifies the highest level of production of Y that is possible, given that the production of X is held at the level corresponding to IXIX and that there are fixed amounts of capital and labour to be allocated as between production of the two commodities. At point b the slopes of the isoquants in each line of production are equal – the marginal rates of technical substitution are equal. If these rates are not equal, then clearly it would be possible to reallocate inputs as between the two lines of production so as to produce more of one commodity without producing any less of the other.


5.1.3 Product-mix efficiency

The final condition necessary for economic efficiency is product-mix efficiency. This requires that

MRTL = MRTK = MRUSA = MRUSB (5.5)

This condition can be understood using Figure 5.3. Given that equation 5.3 holds, so that the two individuals have equal marginal rates of utility substitution and MRUSA = MRUSB, we can proceed as if they had the same utility functions, for which II in Figure 5.1 is an indifference curve with slope MRUS. The individuals do not, of course, actually have the same utility functions. But, given the equality of the MRUS, their indifference curves have the same slope at an allocation that satisfies the consumption efficiency condition, so we can simplify, without any real loss, by assuming the same utility functions and drawing a single indifference curve that refers to all consumers. Given that Equation 5.4 holds, when we think about the rate at which the economy can trade off production of X for Y and vice versa, it does not matter whether the changed composition of consumption is realised by switching labour or capital between the two lines of production. Consequently, in Figure 5.3 we show a single production possibility frontier, YMXM, showing the output combinations that the economy could produce using all of its available resources. The slope of YMXM is MRT.

In Figure 5.3 the point a must be on a lower indifference curve than II. Moving along YMXM from point a toward b must mean shifting to a point on a higher indifference curve. The same goes for movement along YMXM from c toward b. On the other hand, moving away from b, in the direction of either a or c, must mean moving to a point on a lower indifference curve. We conclude that a point like b, where the slopes of the indifference curve and the production possibility frontier are equal, corresponds to a product mix – output levels for X and Y – such that the utility of the representative individual is maximised, given the resources available to the economy and the terms on which they can be used to produce commodities. We conclude, that is, that the equality of MRUS and MRT is necessary for efficiency in allocation. At a combination of X and Y where this condition does not hold, some adjustment in the levels of X and Y is possible which would make the representative individual better off.


An economy attains a fully efficient static allocation of resources if the conditions given by equations 5.3, 5.4 and 5.5 are satisfied simultaneously. Moreover, it does not matter that we have been dealing with an economy with just two persons and two goods. The results readily generalise to economies with many inputs, many goods and many individuals. The only difference will be that the three efficiency conditions will have to hold for each possible pairwise comparison that one could make, and so would be far more tedious to write out.


5.2 An efficient allocation of resources is not unique

For an economy with given quantities of available resources, production functions and utility functions, there will be many efficient allocations of resources. The criterion of efficiency in allocation does not, that is, serve to identify a particular allocation.

To see this, suppose first that the quantities of X and Y to be produced are somehow given and fixed. We are then interested in how the given quantities of X and Y are allocated as between A and B, and the criterion of allocative efficiency says that this should be such that A/B cannot be made better off except by making B/A worse off. This was what we considered in Figure 5.1 to derive equation 5.3, which says that an efficient allocation of fixed quantities of X and Y will be such that the slopes of the indifference curves for A and B will be the same. In Figure 5.1 we showed just one indifference curve for A and two for B. But, these are just a small subset of the indifference curves for each individual that fill the box SA0TB0. In Figure 5.4 we show a larger subset for each individual. Clearly, there will be a whole family of points, like b in Figure 5.1, at which the slopes of the indifference curves for A and B are equal, at which they have equal marginal rates of utility substitution. At any point along CC in Figure 5.4, the consumption efficiency condition is satisfied. In fact, for given available quantities of X and Y there are an indefinitely large number of allocations as between A and B that satisfy MRUSA = MRUSB.




Box 5.1 Productive inefficiency in ocean fisheries


The total world marine fish catch increased steadily from the 1950s through to the late 1980s, rising by 32% between the periods 1976–1978 and 1986–1988 (UNEP, 1991). However, the rate of increase was slowing toward the end of this period, and the early 1990s witnessed downturns in global harvests. The harvest size increased again in the mid-1990s, was at a new peak in 1996, and then levelled off again in the late 1990s. It is estimated that the global maximum sustainable harvest is about 10% larger than harvest size in the late 1990s.

The steady increase in total catch until 1989 masked significant changes in the composition of that catch; as larger, higher-valued stocks became depleted, effort was redirected to smaller-sized and lower-valued species. This does sometimes allow depleted stocks to recover, as happened with North Atlantic herring, which recovered in the mid-1980s after being overfished in the late 1970s. However, many fishery scientists believe that these cycles of recovery have been modified, and that species dominance has shifted permanently towards smaller species.

Rising catch levels have put great pressure on some fisheries, particularly those in coastal areas, but also including some pelagic fisheries. Among the species whose catch declined over the period 1976–1988 are Atlantic cod and herring, haddock, South African pilchard and Peruvian anchovy. Falls in catches of these species have been compensated for by much increased harvests of other species, including Japanese pilchard in the north-west Pacific.

Where do inefficiencies enter into this picture? We can answer this question in two ways. First, a strong argument can be made to the effect that the total amount of resources devoted to marine fishing is excessive, probably massively so. We shall defer giving evidence to support this claim until Chapter 17 (on renewable resources), but you will see there that a smaller total fishing fleet would be able to catch at least as many fish as the present fleet does. Furthermore, if fishing effort were temporarily reduced so that stocks were allowed to recover, a greater steady-state harvest would be possible, even with a far smaller world fleet of fishing vessels. There is clearly an inefficiency here.

A second insight into inefficiency in marine fishing can be gained by recognising that two important forms of negative external effect operate in marine fisheries, both largely attributable to the fact that marine fisheries are predominantly open-access resources. One type is a so-called crowding externality, arising from the fact that each boat’s harvesting effort increases the fishing costs that others must bear. The second type may be called an ‘intertemporal externality’: as fisheries are often subject to very weak (or even zero) access restrictions, no individual fisherman has an incentive to conserve stocks for the future, even if all would benefit if the decision were taken jointly.

As the concepts of externalities and open access will be explained and analysed in the third part of this chapter, and applied to fisheries in Chapter 17, we shall not explain these ideas any further now. Suffice it to say that production in market economies will, in general, be inefficient in the presence of external effects.



Sources: WRI (2000), WRI web site www.wri.org, FAO web site www.fao.org

Now consider the efficiency in production condition, and Figure 5.2. Here we are looking at variations in the amounts of X and Y that are produced. Clearly, in the same way as for Figures 5.1 and 5.4, we could introduce larger subsets of all the possible isoquants for the production of X and Y to show that there are many X and Y combinations that satisfy equation 5.4, combinations representing uses of capital and labour in each line of production such that the slopes of the isoquants are equal, MRTSX = MRTSY.

So, there are many combinations of X and Y output levels that are consistent with allocative efficiency, and for any particular combination there are many allocations as between A and B that are consistent with allocative efficiency. These two considerations can be brought together in a single diagram, as in Figure 5.5, where the vertical axis measures A’s utility and the horizontal B’s. Consider a particular allocation of capital and labour as between X and Y production which implies particular output levels for X and Y, and take a particular allocation of these output levels as between A and B – there will correspond a particular level of utility for A and for B, which can be represented as a point in UA/UB space, such as R in Figure 5.5. Given fixed amounts of capital and labour, not all points in UA/UB space are feasible. Suppose that all available resources were used to produce commodities solely for consumption by A, and that the combination of X and Y then produced was such as to maximise A’s utility. Then, the corresponding point in utility space would be UAmax in Figure 5.5. With all production serving the interests of B, the corresponding point would be UBmax. The area bounded by UAmax0UBmax is the utility possibility set – given its resources, production technologies and preferences, the economy can deliver all combinations of UA and UB lying in that area. The line UAmaxUBmax is the utility possibility frontier – the economy cannot deliver combinations of UA and UB lying outside that line. The shape of the utility possibility frontier depends on the particular forms of the utility and production functions, so the way in which it is represented in Figure 5.5 is merely one possibility. However, for the usual assumptions about utility and production functions, it would be generally bowed outwards in the manner shown in Figure 5.5.


The utility possibility frontier is the locus of all possible combinations of UA and UB that correspond to efficiency in allocation. Consider the point R in Figure 5.5, which is inside the utility possibility frontier. At such a point, there are possible reallocations that could mean higher utility for both A and B. By securing allocative efficiency, the economy could, for example, move to a point on the frontier, such as Z. But, given its endowments of capital and labour, and the production and utility functions, it could not continue northeast beyond the frontier. Only UA/UB combinations lying along the frontier are feasible. The move from R to Z would be a Pareto improvement. So would be a move from R to T, or to S, or to any point along the frontier between T and S.

The utility possibility frontier shows the UA/UB combinations that correspond to efficiency in allocation – situations where there is no scope for a Pareto improvement. There are many such combinations. Is it possible, using the information available, to say which of the points on the frontier is best from the point of view of society? It is not possible, for the simple reason that the criterion of economic efficiency does not provide any basis for making interpersonal comparisons. Put another way, efficiency does not give us a criterion for judging which allocation is best from a social point of view. Choosing a point along the utility possibility frontier is about making moves that must involve making one individual worse off in order to make the other better off. Efficiency criteria do not cover such choices.

5.3 The social welfare function and optimality

In order to consider such choices we need the concept of a social welfare function, SWF, which was introduced in Chapter 3. A SWF can be used to rank alternative allocations. For the two-person economy that we are examining, a SWF will be of the general form:

W = W(UA, UB) (5.6)

The only assumption that we make here regarding the form of the SWF is that welfare is non-decreasing in UA and UB. That is, for any given level of UA welfare cannot decrease if UB were to rise and for any given level of UB welfare cannot decrease if UA were to rise. In other words, we assume that WA = ∂W/∂UA and WB = ∂W/∂UB are both positive. Given this, the SWF is formally of the same nature as a utility function. Whereas the latter associates numbers for utility with combinations of consumption levels X and Y, a SWF associates numbers for social welfare with combinations of utility levels UA and UB. Just as we can depict a utility function in terms of indifference curves, so we can depict a SWF in terms of social welfare indifference curves. Figure 5.6 shows a social welfare indifference curve WW that has the same slope as the utility possibility frontier at b, which point identifies the combination of UA and UB that maximises the SWF.


The reasoning which establishes that b corresponds to the maximum of social welfare that is attainable should be familiar by now – points to the left or the right of b on the utility possibility frontier, such as a and c, must be on a lower social welfare indifference curve, and points outside of the utility possibility frontier are not attainable. The fact that the optimum lies on the utility possibility frontier means that all of the necessary conditions for efficiency must hold at the optimum. Conditions 5.3, 5.4 and 5.5 must be satisfied for the maximisation of welfare. Also, an additional condition, the equality of the slopes of a social indifference curve and the utility possibility frontier, must be satisfied. This condition can be stated, as established in Appendix 5.1, as


(5.7)

The left-hand side here is the slope of the social welfare indifference curve. The two other terms are alternative expressions for the slope of the utility possibility frontier. At a social welfare maximum, the slopes of the indifference curve and the frontier must be equal, so that it is not possible to increase social welfare by transferring goods, and hence utility, between persons.


While allocative efficiency is a necessary condition for optimality, it is not generally true that moving from an allocation that is not efficient to one that is efficient must represent a welfare improvement. Such a move might result in a lower level of social welfare. This possibility is illustrated in Figure 5.7. At C the allocation is not efficient, at D it is. However, the allocation at C gives a higher level of social welfare than does that at D. Having made this point, it should also be said that whenever there is an inefficient allocation, there is always some other allocation which is both efficient and superior in welfare terms. For example, compare points C and E. The latter is allocatively efficient while C is not, and E is on a higher social welfare indifference curve. The move from C to E is a Pareto improvement where both A and B gain, and hence involves higher social welfare. On the other hand, going from C to D replaces an inefficient allocation with an efficient one, but the change is not a Pareto improvement – B gains, but A suffers – and involves a reduction in social welfare. Clearly, any change which is a Pareto improvement must increase social welfare as defined here. Given that the SWF is non-decreasing in UA and UB, increasing UA/UB without reducing UB/UA must increase social welfare. For the kind of SWF employed here, a Pareto improvement is an unambiguously good thing (subject to the possible objections to preference-based utilitarianism noted in Chapter 3, of course). It is also clear that allocative efficiency is a good thing (subject to the same qualification) if it involves an allocation of commodities as between individuals that can be regarded as fair. Judgements about fairness, or equity, are embodied in the SWF in the analysis here. If these are acceptable, then optimality is an unambiguously good thing. In Part 2 of this chapter we look at the way markets allocate resources and commodities. To anticipate, we shall see that what can be claimed for markets is that, given ideal institutional arrangements and certain modes of behaviour, they achieve allocative efficiency. It cannot be claimed that, alone, markets, even given ideal institutional arrangements, achieve what might generally or reasonably be regarded as fair allocations. Before looking at the way markets allocate resources, we shall look at economists’ attempts to devise criteria for evaluating alternative allocations that do not involve explicit reference to a social welfare function.


5.4 Compensation tests


If there were a generally agreed SWF, there would be no problem, in principle, in ranking alternative allocations. One would simply compute the value taken by the SWF for the allocations of interest, and rank by the computed values. An allocation with a higher SWF value would be ranked above one with a lower value. There is not, however, an agreed SWF. The relative weights to be assigned to the utilities of different individuals are an ethical matter. Economists prefer to avoid specifying the SWF if they can. Precisely the appeal of the Pareto improvement criterion – a reallocation is desirable if it increases somebody’s utility without reducing anybody else’s utility – is that it avoids the need to refer to the SWF to decide on whether or not to recommend that reallocation. However, there are two problems, at the level of principle, with this criterion. First, as we have seen, the recommendation that all reallocations satisfying this condition be undertaken does not fix a unique allocation. Second, in considering policy issues there will be very few proposed reallocations that do not involve some individuals gaining and some losing. It is only rarely, that is, that the welfare economist will be asked for advice about a reallocation that improves somebody’s lot without damaging somebody else’s. Most reallocations that require analysis involve winners and losers and are, therefore, outside of the terms of the Pareto improvement criterion.

Given this, welfare economists have tried to devise ways, which do not require the use of a SWF, of comparing allocations where there are winners and losers. These are compensation tests. The basic idea is simple. Suppose there are two allocations, denoted 1 and 2, to be compared. As previously, the essential ideas are covered if we consider a two-person, two-commodity world. Moving from allocation 1 to allocation 2 involves one individual gaining and the other losing. The Kaldor compensation test, named after its originator, Nicholas Kaldor, says that allocation 2 is superior to allocation 1 if the winner could compensate the loser and still be better off. Table 5.1 provides a numerical illustration of a situation where the Kaldor test has 2 superior to 1. In this, constructed, example, both individuals have utility functions that are U = XY, and A is the winner for a move from 1 to 2, while B loses from such a move. According to the Kaldor test, 2 is superior because at 2 A could restore B to the level of utility that he enjoyed at 1 and still be better off than at 1. Starting from allocation 2, suppose that 5 units of X were shifted from A to B. This would increase B’s utility to 100 (10  10), and reduce A’s utility to 75 (15  5) – B would be as well off as at 1 and A would still be better off than at 1. Hence, the argument is: allocation 2 must be superior to 1, as, if such a reallocation were undertaken, the benefits as assessed by the winner would exceed the losses as assessed by the loser. Note carefully that this test does not require that the winner actually does compensate the loser. It requires only that the winner could compensate the loser, and still be better off. For this reason, the Kaldor test, and the others to be discussed below, are sometimes referred to as ‘potential compensation tests’. If the loser was actually fully compensated by the winner, and the winner was still better off, then we would be looking at a situation where there was a Pareto improvement.

The numbers in Table 5.1 have been constructed so as to illustrate a problem with the Kaldor test. This is that it may sanction a move from one allocation to another, but that it may also sanction a move from the new allocation back to the original allocation. Put another way, the problem is that if we use the Kaldor test to ask whether 2 is superior to 1 we may get a ‘yes’, and we may also get a ‘yes’ if we ask if 1 is superior to 2. Starting from 2 and considering a move to 1, B is the winner and A is the loser. Looking at 1 in this way, we see that if 5 units of Y were transferred from B to A, B would have U equal to 75, higher than in 2, and A would have U equal to 100, the same as in 2. So, according to the Kaldor test done this way, 1 is superior to 2.

This problem with the Kaldor test was noted by J.R. Hicks, who actually put things in a slightly different way. He proposed a different (potential) compensation test for considering whether the move from 1 to 2 could be sanctioned. The question in the Hicks test is: could the loser compensate the winner for forgoing the move and be no worse off than if the move took place. If the answer is ‘yes’, the reallocation is not sanctioned, otherwise it is on this test. In Table 5.1, suppose at allocation 1 that 5 units of Y are transferred from B, the loser from a move to 2, to A. Now A’s utility would then go up to 100 (10  10), the same as in allocation 2, while B’s would go down to 75 (5  15), higher than in allocation 2. The loser in a reallocation from 1 to 2 could, that is, compensate the individual who would benefit from such a move for its not actually taking place, and still be better off than if the move had taken place. On this test, allocation 1 is superior to allocation 2.

Table 5.1 Two tests, two answers





Allocation 1




Allocation 2




X

Y

U




X

Y

U

A

10

5

50




20

5

100

B

5

20

100




5

10

50

Table 5.2 Two tests, one answer



Allocation 1





Allocation 2




X

Y

U




X

Y

U

A

10

5

50




20

10

200

B

5

20

100




5

10

50

In the example of Table 5.1, the Kaldor and Hicks (potential) compensation tests give different answers about the rankings of the two allocations under consideration. This will not be the case for all reallocations that might be considered. Table 5.2 is a, constructed, example where both tests give the same answer. For the Kaldor test, looking at 2, the winner A could give the loser B 5 units of X and still be better off than at 1 (U150), while B would then be fully compensated for the loss involved in going from 1 to 2 (U = 10  10 100). On this test, 2 is superior to 1. For the Hicks test, looking at 1, the most that the loser B could transfer to the winner A so as not to be worse off than in allocation 2 is 10 units of Y. But, with 10 of X and 15 of Y, A would have U = 150, which is less than A’s utility at 2, namely 200. The loser could not compensate the winner for forgoing the move and be no worse off than if the move took place, so again 2 is superior to 1.

For an unambiguous result from a (potential) compensation test, it is necessary to use both the Kaldor and the Hicks criteria. The Kaldor– Hicks–Scitovsky test – known as such because Tibor Scitovsky pointed out that both criteria are required – says that a reallocation is desirable if:

Table 5.3 Compensation may not produce fairness





Allocation 1




Allocation 2




X

Y

U




X

Y

U

A

10

5

50




10

4

40

B

5

20

100




15

16

240

    (i) the winners could compensate the losers and still be better off

    and


    (ii) the losers could not compensate the winners for the reallocation not occurring and still be as well off as they would have been if it did occur.

In the example of Table 5.2 the move from 1 to 2 passes this test; in that of Table 5.1 it does not.

As we shall see, especially in Chapters 11 and 12 on cost–benefit analysis and environmental valuation respectively, compensation tests inform much of the application of welfare economics to environmental problems. Given that utility functions are not observable, the practical use of compensation tests does not take the form worked through here, of course. Rather, as we shall see, welfare economists work with monetary measures which are intended to measure utility changes. As noted above, the attraction of compensation tests is that they do not require reference to a SWF. However, while they do not require reference to a SWF, it is not the case that they solve the problem that the use of a SWF addresses. Rather, compensation tests simply ignore the problem. As indicated in the examples above, compensation tests treat winners and losers equally. No account is taken of the fairness of the distribution of well-being.

Consider the example in Table 5.3. Considering a move from 1 to 2, A is the loser and B is the winner. As regards (i), at 2 moving one unit of Y from B to A would make A as well off as she was at 1, and would leave B better off (U = 225) than at 1. As regards (ii), at 1 moving either two of X or one of Y from A to B would leave A as well off as at 2, but in neither case would this be sufficient to compensate B for being at 1 rather than 2 (for B after such transfers U =140 or U = 105). According to both (i) and (ii) 2 is superior to 1, and such a reallocation passes the Kaldor–Hicks–Scitovsky test. Note, however, that A is the poorer of the two individuals, and that the reallocation sanctioned by the compensation test makes A worse off, and makes B better off. In sanctioning such a reallocation, the compensation test is either saying that fairness is irrelevant or there is an implicit SWF such that the reallocation is consistent with the notion of fairness that it embodies. If, for example, the SWF was


W = 0.5UA + 0.5UB

then at 1 welfare would be 75 and at 2 it would be 140. Weighting A’s losses equally with B’s gains means that 2 is superior to 1 in welfare terms. If it were thought appropriate to weight A’s losses much more heavily than B’s gains, given that A is relatively poor, then using, say



W = 0.95UA + 0.05UB

gives welfare at 1 as 52.5 and at 2 as 50, so that 1 is superior to 2 in welfare terms, notwithstanding that the move from 1 to 2 is sanctioned by the (potential) compensation test.


In the practical use of compensation tests in applied welfare economics, welfare, or distributional, issues are usually ignored. The monetary measures of winners’ gains (benefits) and losers’ losses (costs) are usually given equal weights irrespective of the income and wealth levels of those to whom they accrue. In part, this is because it is often difficult to identify winners and losers sufficiently closely to be able to say what their relative income and wealth levels are. But, even in those cases where it is clear that, say, costs fall mainly on the relatively poor and benefits mainly on the better off, economists are reluctant to apply welfare weights when applying a compensation test by comparing total gains and total losses – they simply report on whether or not £s of gain exceed £s of loss. Various justifications are offered for this practice. First, at the level of principle, that there is no generally agreed SWF for them to use, and it would be inappropriate for economists to themselves specify a SWF. Second, that, as a practical matter, it aids clear thinking to separate matters of efficiency from matters of equity, with the question of the relative sizes of gains and losses being treated as an efficiency issue, while the question of their incidence across poor and rich is an equity issue. On this view, when considering some policy intended to effect a reallocation the job of the economic analyst is to ascertain whether the gains exceed the losses. If they do, the policy can be re-commended on efficiency grounds, and it is known that the beneficiaries could compensate the losers. It is a separate matter, for government, to decide whether compensation should actually occur, and to arrange for it to occur if it is thought desirable. These matters are usually considered in the context of a market economy, and we shall return to them in that context at the end of Part 2 of the chapter.


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