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242 CHAPTER 9. WELFARE For proof of this, see Appendix C. So it appears that under certain circum- stances we can pick the Pareto-e¢ cient allocation that we would like to see, and then arrange that the economy is automatically moved to this allocation by the process of competition –the competitive equilibrium “supports”the allocation. But a few words of caution are appropriate here.  First, in order to manipulate the economy in this way we need to have the right property distribution ^ d as a starting point. But how do we arrange for this distribution in the …rst place? If history has thrown up some other property distribution d then perhaps it is possible to arrange transfers of entitlements of property from one group of households to another before production and trade takes place. These transfers are not base d on the activities or choices of any of the agents in the economy –in the jargon they are lump-sum transfers –and the political and administrative di¢ culties associated with them should not be taken lightly (more of this in chapter 13, page 461). A principal di¢ culty is that of identifying who is entitled to receive a transfer and who should be required to provide the resources. Some resource endowments are intrinsically non-observable; 10 some are intrinsically non transferable. 11  Second, the conditions in Theorem 9.5 are fairly stringent. Reasonably we could ask what guidance is available on e¢ cienc y grounds once we try to accommodate real-world problems and di¢ culties. These di¢ culties will involve either departures from the ideals of perfect competition or relaxation of some of the assumptions that underpin the theorem. This issues are addressed in section 9.3.2.  Third, the discussion of e¢ ciency has b ee n conducted in a world of perfect certainty. There are important issues raised by the mo de l of uncertainty that we developed in chapter 8. These are handled in section 9.3.5. 9.3.2 Departures from e¢ ciency The pair of theorems, 9.4 and 9.5, are undeniably attractive but, to be applicable they clearly impose somewhat idealistic requirements. So, two things deserve further consideration: (1) In situations where we have a private goods economy with technology and preferences that satisfy the conditions of theorems 9.4 and 9.5, how does one quantify departures from the ideal? It may be useful to have some guidance of this to have an idea of whether one imperfect state is “better”or “worse”than another in e¢ ciency terms. (2) What if the underlying assumptions about the private-goods economy were relaxed? What could we then say about the conditions for an e¢ cient allocation? We deal with each of these in turn. 10 (a) G ive an example of why this is so. (b) Because of the problems of non observability policy makers often condition transfers on individuals’ac tions, as with the income tax. Why will this giv e rise to e¢ ciency problems? 11 Again p rovide an exa mple. 9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 243 Waste Consider the problem of quantifying ine¢ ciency. Suppose that we are in a purely private-good economy. All the conditions for a competitive equilibrium are present –which would involve prices p  and incomes y 1 ; y 2 ; ::: – but we …nd that in fact one good (good 1) has its price …xed up above p  1 . Th is price- wedge might be caused by a sales tax, for example, and it will in general distort all other prices. Can we measure the loss that is induced by the price-wedge? Let u s suppos e that we actually observe the consumer prices (p 1 ; p 2 ; :::; p n ) and producer prices (~p 1 ; ~p 2 ; :::; ~p n ) such that p 1 = ep 1 [1 + ]; p 2 = ~p 2 ; p 3 = ~p 3 ::: ::: ::: p n = ~p n 9 > > > > = > > > > ; (9.16) where  is the price-wedge imposed exogenously upon good 1. To make the argument easier assume that all prices are positive and that all markets clear. To measure waste we need a reference point. Since we have argued that under the idealised conditions of a competitive equilibrium it seems natural to use as the reference point the prices p  that would have prevailed in equilibrium. Furthermore let 4p i := p i  p  i (9.17) denote the deviations from the reference prices for each good i = 1; 2; :::; n. Given that all consumers are maximising utility we must have that MRS h ij = p j p i (9.18) for all goods i and j, and all households h, 12 and MRT 1j = p j p 1 [1 + ] MRT 2j = p j p 2 ::: ::: ::: MRT nj = p j p n 9 > > = > > ; (9.19) Now consider the net gain that person h would expe rience were one to go from the reference allocation a  to the actual allocation a: EV h = C h (p  ;  h )  C h (p;  h )  [y h  y h ] (9.20) where C h is h’s cost function. Summing (9.20) over all h expresses the total loss measured in the same units as income. Assume that all producer prices remain 12 Ske tch a diagram similar to Figure 9.5, but with a convex production possibility set, a nd superimpose a set o f indi¤erence curves; use this to illustrate the conditions (9.18) and (9.19). 244 CHAPTER 9. WELFARE constant –an implicit assumption of in…nite supply elasticities. Then (minus) the aggregation over the consumers of the loss in equation (9.20) gives the total measure of waste involved in the price distortion 4p thus: 13 (4p) := n h X h=1  C h (p;  h )  C h (p  4p;  h )   n X i=1 R i 4p i  n X i=1 q i 4p i (9.21) We have (0) = 0 and Shephard’s Lemma implies: x h i = C h i (p;  h ) (9.22) Using the materials’ balance condition and taking an approximation we then get 14 (4p)   1 2 n X i=1 n X j=1 n h X h=1 @H hi (p;  h ) @p j 4p i 4p j (9.23) where @H hi (p; h ) @p j is the substitution e¤ect of a rise in the price of commodity j on the demand for commodity i by household h – in other words the slope of the compensated or Hicksian demand curve. 15 The interpretation of this can be based on the analysis of cost changes that we developed for the …rm (page 33) and the consumer (page 91). The price increase leads to an inc ome increase for someone (because of the e¤ect on sales revenue) and the contribution to this from agent h’s consumption this is given by the lightly shaded rectangle with dimension p 1 x h 1 in Figure 9.6. However the component of cost increase to agent h represented by the change in price 4p 1 is represented by by the whole shaded area in Figure 9.6. The di¤erence between the two represents the component of the waste generated by the price distortion faced by person h directly from 4p 1 . It is illustrated in Figure 9.6 as () the area of the heavily shaded “triangle”shape, app roximated by  1 2 4p 1 4x h 1 where 4x h 1 = @H h1 (p;  h ) @p 1 4p 1 Of course, one needs to take into account the other components of waste that are generated from the induced price changes: the sum of the little triangles such as that in Figure 9.6 gives the expression for loss (9.23). 16 13 Use equations (7.8), (7.9 ) and theorem 2.7 to show how (9. 21) follows from (9.20). 14 Show how to derive (9.23) using a Taylor approximation (see pag e 494). 15 Show how the the ex pressio n for waste must b e modi…ed if supply elasticities are less than in…nite. 16 (a) Suppose there is only a single …rm producing good 1 that uses the market po wer it enjoys to force up the price of good 1. If we neglect cross-price e¤ects and use consumer’s surplus as an approximation to EV interpret the model as one of the wa ste that is attributable to monopoly. [Hint: us e the equilibrium condition given in (3. 11s)] (b) How is the waste related to the el asticity of demand for good 1? 9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 245 Figure 9.6: Component of e¢ ciency loss This idea of quantifying waste gives us the basis for developing a coherent analysis of economic policy that may be aimed at yielding welfare improvements rather than shooting just at a welfare optimum. More of this in Chapter 13. E¢ ciency and market “failures” Now let us turn to the other main thing that needs consideration. This intro- duces us to a class of economic problems that are sometimes –perhaps some what unfairly –characterised as instances of “market failure.” It is perhaps better to say that these are instances where unquali…ed reliance on the market mechanism cannot be relied upon to produce an e¢ cient outcome. This is hardly aston- ishing: the requirements for the “s upport” result in Theorem 9.5 may app ear to be unacceptably strong. Relaxing these requirements raises two key issues. 1. The characterisation problem. Where the conditions for Theorem 9.4 are violated the FOCs (9.12)– (9.13) are no longer valid. Furthermore, in the presence of nonconvexities the FOCs are no longer su¢ cient to pin down a unique allocation –see the two parts of the …gure where points on di¤erent parts of on contour have the same MRS or MRT. So in all these 246 CHAPTER 9. WELFARE cases the FOCs for the Pareto e¢ cient allocation need to be replaced or supplemented in order to characterise an e¢ cient allocation. 2. The implementation problem. If the market mechanism cannot do the job of supporting a particular allocation in this case, then what else might work? We shall discuss nonconvexities and the di¢ cult implementation issue further in chapters 12 and 13. The characterisation issue where the conditions for Theorem 9.4 are violated can be handled by a series of tweaks as follows in sections 9.3.3 and 9.3.4. 9.3.3 Externalities We have already seen the mechanics of externalities in a simple example of in- teractions amongst …rms, discussed in chapter 3 (pages 55¤). Here we also need to take into account a similar ph enome non of interactions amongst consumers. We will handle each in turn under the labe ls production and consumption ex- ternalities. Production externalities Unfortunately there are all too many practical examples of “negative”produc- tion externalities–emissions into rivers, acid rain, tra¢ c congestion –where the unregulated actions by one …rm signi…cantly a¤ects the cost function of other …rms. So we shall focus on such detrimental interactions although virtually all of the results can be easily reworked to deal with positive externalities too. We can see the essential nature of the problem by considering a two-…rm example. Suppose that q 1 1 the output of good-1 by …rm 1 a¤ects the technological possi- bilities of other …rms: …rm 1 produces glue. Consider the position of …rm 2, a restaurant. In the no-externality case we would normally write  2 (q 2 )  0 to characterise the net-output vectors q 2 that are technologically feasible. How- ever, in view of the externality, …rm 1’s output (q 1 1 ) will shift …rm 2’s production function. If the externality is detrimental (the smell of glue does not enhance enjoyment of the restaurant’s meals) then we have: @ 2 @q 1 1 > 0 (9.24) Why? Consider a net output vector ^q 2 that was just feasible for …rm 2, before …rm 1 increased its output; this means that – in terms of the …gure – the relevant point lies on the boundary, so that  2 (q 2 ) = 0. Now suppose that …rm 1 increases its output q 1 1 : if the externality is strictly detrimental 17 , then this must mean that ^q 2 –which had hitherto been just in the feasible set –must now be infeasible (you have to use more electricity to run air conditioning). This in 17 Suppose …rms 1 and 2 experience diminishing returns to scale and generate negative externa li ties: will production overall exhibit dimini shing returns to scale? 9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 247 Figure 9.7: The e¤ect of pollution on a victim’s production set turn means that we now …nd  2 (^q 2 ) > 0 and that q 1 1 has shifted  2 inwards: in other words condition (9.24) holds –see Figure 9.7. 18 We could then appropriately de…ne the value, at the margin, of the damage in‡icted upon …rm 2 by the externality generated by …rm 2. We could measure this in terms of …rm 2’s output:  1  2 2 @ 2 @q 1 1 (9.25) where  2 2 is the conventional di¤erential of …rm 2’s production function with respect to its own output. More generally, in the multi…rm case, we can represent an externality by writing the production function for …rm g as:  g  q g ; q 1 1 ; q 2 1 ; :::; q g 1 1 ; q g +1 1 ; :::  (9.26) and if the externality generated by any of these …rms is potentially detrimental we would have: @ g @q f i > 0 (9.27) Once again this means that if the detrimental externality (noxious emissions) by other …rms were to increase, then …rm gs production p ossibilities are reduced –see Figure 9.7. 18 Rework the analysis in equations (9.24) to (9.29) for a favourable externality. 248 CHAPTER 9. WELFARE Figure 9.8: Production boundary and e¢ ciency with externalities The general form of the marginal externality caused by …rm f when it pro- duces good 1 (again evaluated in terms of good 2) may thus be written: e f 2l := n f X g =1 1  g 2 @ g @q f i (9.28) We can then plug the production function with externalities into the problem de…ning an e¢ cient allocation. We then …nd: 19  f 1  f 2  e f 2l =  1  2 (9.29) which can be expressed as: ratio of MRT – externality = shadow prices One implication of this is that market prices, that the …rm would use, do not correspond to the “scarcity prices”of commodities in an e¢ cient allocation: there is a “wedge” between them corresponding to the value of the marginal externality. 20 This is illustrated in Figure 9.8. If the MRT were to equal just the 19 Substitute (9.26) into equation (9.8) and di¤erentiate to get this result. 20 Discuss how e quation (9.29) m ight be interpreted as a simple rule for setting a “polluter pays” levy on output. 9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 249 ratio of scarcity prices the …rm would produce at point ^q f . If the scarcity prices are adjusted by the marginal externality then we …nd the e¢ cient allocation at point ~q f . Consumption externalities Consumption externalities can be handled in a similar manner, and the main idea conveyed by means of a simple example. Alf is an asthmatic non-smoker who is a¤ected by the actions of Bill a boorish smoker. To simplify the example we use the device of bundling together all goods in the economy except one. Let good 1 be tobacco, and good 2 the composite of everything else. Then, we can write the utility function for Bill as U b (x b 1 ; x b 2 ) and for Alf as U a (x a 1 ; x a 2 ; x b 1 ). The signs of the partial d erivatives of these functions are f airly obvious; in par- ticular we may assume that @U a =@x b 1 < 0, since Alf su¤ers when Bill consumes commodity 1. But how awful is it for asthmatic Alf to be in boorish Bill’s company? One way of capturing this is to try to measure Alf’s willingness to pay to have the nuisance reduced –to get Bill to cut down on the tobacco. We can do this by computing the amount that Alf would be prepared to sacri…ce in order to get Bill to have one less cigarette; this is: e b 2l := 1 U a 2 @U a @x b 1  0 (9.30) where U a 2 is Alf’s marginal utility derived from other goods de rived in the usual way. From Alf’s point of view expression (9.30) is the marginal externality –or the marginal damage –in‡icted through the consumption of good 1 by Bill the boor. Translating this into our more general model of e¢ ciency with n goods and n h households let us suppose the consumption of good 1 by any household h potentially a¤ects the utility of some other household `, possibly as the result of some side e¤ect. We would then write:  ` = U ` (x ` ; x 1 1 ; x 2 1 ; :::; x `1 1 ; x `+1 1 ; :::) (9.31) If the externality is detrimental by nature then we have @U ` @x h 1  0 (9.32) for any two distinct households h and `. Analogous to (9.30) we may de…ne the marginal externality imposed on others by household h as: e h 21 := n h X `=1 1 U ` 2 @U ` @x h 1 (9.33) Notice that the summation is required because we want to know the marginal damage in‡icted on all parties, evaluated appropriately at the su¤erers’ mar- ginal utility of other goods. When we take this relationship into account in the 250 CHAPTER 9. WELFARE FOCs for e¢ ciency, we …nd the following: 21 U h 1 U h 2 + e h 21 =  1  2 : (9.34) In other words we again have a simple relationship: ratio of MRS + externality = shadow prices Clearly, if there is a negative externality, then the marginal rate of substitution of good 2 into good 1 will be greater than the price ratio in an e¢ cient allocation. 22 The interaction between …rms or between consumers leads to fairly straight- forward extensions of the rules covering the characterisation of e¢ cient alloca- tions. However, although the characterisation problem is relatively simple in this case, the implementation problem may p rove to be intractable –even for production externalities –in the absence of external intervention. 9.3.4 Public goods The precise meaning of a public good is given in de…nition 9.4. So, if good 1 is a pure public good it must be non-rival which requires that x h 1 = x 1 for all non-satiated households. It must also be non-excludable, which can be interpreted as an extreme case of consumption externality: once provided there is no means of charging for it. Let us explore the e¢ ciency implications of non-rivalness. In fact we only require a di¤erent form of aggregation in the e¢ ciency condition. Notice that in this case if, for some household h we have x h 1 < x 1 and yet U h 1 > 0; then a Pareto- superior allocation can be attained by allowing household h’s consumption of good 1 to increase (as long as x h 1 is strictly less than x 1 no additional resources have to be used up to increase h’s consumption of this non-rival good, so we might as well let household h increase its own utility since it will not thereby reduce any one else’s utility). Therefore at the Pareto-e¢ cient allocation for each household h, either x h 1 = x 1 so that the household is consuming the non-rival good to its maximum capacity, or x h 1 < x 1 and U h 1 = 0 so that the household is consuming less than it could, but is satiated with the public good 1. Let us assume that everyone is non-satiated; 23 each person must consume exactly the same amount at a Pareto e¢ cient allocation. Thus we put x h 1 = x 1 ; h = 1; :::; n h 21 Substitute (9.31) into equation (9.8) and di¤erentiate to get this result. 22 In a two-good model, show ho w condition (9.34) might be used to sugges t an an ap- propriate tax on the good causing the exter nality, or an appropriate subs idy on th e other good. 23 Derive the same condition assuming that the …rst h  ho useholds are non-satiated, and the remaining n h  h  ho useholds are satiated. 9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 251 in the Lagrangean (9.8) as our new aggregation condition. Di¤erentiate the Lagrangean with respect to x 1 and set it equal to zero: n h X h=1  h U h 1 (x h ) =  1 (9.35) Now pick any other pure private good i that is being consumed in positive amounts by everyone: from equation (9.35) we get n h X h=1 U h 1 (x h ) U h i (x h ) =  1  i (9.36) So we have established the result Theorem 9.6 (E¢ ciency with public goods) In a Pareto e¢ cient state without externalities for any pure private good i consumed by everyone and a non-rival good 1 we have: MRS 1 i1 + MRS 2 i1 + ::: + MRS n h i1 =  1  i = MRT f i1 ; f = 1; :::; n f : 9 = ; (9.37) Figure 9.9 illustrates the two-good, two-person case in the case where pro- duction is carried out by a single …rm. The top part of the diagram plots Alf’s marginal rate of substitution of the private good (good 2) for the pu blic good (good 1) as a function of the total supply of good 1. It is a graph of his willing- ness to pay for additional units of the public good and it is downward sloping on the assumption that Alf’s utility function is quasiconcave. The second part of the diagram does the same job for Bill. At any level of provision of the public good x 1 we can imagine asking ourselves “what is the total willingness to pay for an extra unit of the public goo d ”(remember: because it is nonrival both parties will bene…t from the extra unit). The graph of this total willingness to pay is the downward sloping line in the b ottom part of the …gure (MRS a 21 +MRS b 21 ); the marginal cost of providing the public good is given by the graph of MRT 21 against x 1 ; the intersection of these two curves gives the e¢ cient supply of public goods x  1 . 9.3.5 Uncertainty It is reasonably straightforward to apply the e¢ ciency concept in de…nition 9.2 to the case where an economy is characterised by uncertainty, following on the analysis of section 8.6. The key issue is whether e¢ ciency is to be viewed before or after the uncertain state-of-the-world is revealed (be careful not to confuse the concept of a state-of-the -world ! 2  with that of a social state  2 ). A standard approach is as follows. Consider a situation in which social states are completely described by allocations. Take an allocation ^a in which the consumption of household h under state-of-the-world ! is ^x h ! and the resulting utility for household h is ^ h , h = 1; 2; :::; n h . [...]... winners found for the left-handers and the right-handers separately! 5 Would the above paradox occur if one used de Borda voting?(Moulin 2003) Left-handers # 10 6 0 12 0 0 00 Right-Handers 6 18 00 17 00 00 00 00 0 0 0 Table 9.4: Left-handed and right-handed voters 9 .5 Suppose social welfare is related to individual incomes y h thus: W = nh X (y h ) h=1 where ( ) has the form (x) = and x1 1 1 is a non-negative... (1948) The standard references on e¢ ciency with public goods are Samuelson (1 954 , 1 955 ) 29 Prove this using the results from Chapter 8 9.8 EXERCISES 2 65 Keenan and Snow (1999) summarise a variety of criteria for potential superiority and the relationship between them; the literature on the “reversals” problems associated with potential superiority was initiated by Kaldor (1939), Hicks (1946) and Scitovsky... basis for a social welfare function s is attributable to Vickrey (19 45) and Harsanyi (1 955 ) On the social-welfare interpretation of inequality and income distribution and its relationship to risk aversion see Atkinson (1970) The developments of social-welfare criteria for use in applied economics are reviewed in Harberger (1971) and Slesnick (1998) 9.8 Exercises 9.1 In a two-commodity exchange economy... problems The principles of economic analysis that we will develop will provide a basis for the discussion of chapters 11 and 12 and provide essential tools for the wider study of microeconomics Why a change in the direction of analysis? Our analysis of strategic behaviour in economics focuses on the theory of games Game theory is an important subject in its own right and it is impossible to do it justice... h=1 dy h (9 .50 ) 9 .5 THE SOCIAL-WELFARE FUNCTION 261 The right-hand side of (9 .50 ) is proportional to the change in national income y 1 + y 2 + ::: + y nh Now consider a change in the prices p leaving incomes y h unchanged Differentiating (9.48) we …nd that the e¤ect on social welfare is # " n nh X X Wh Vih dpi (9 .51 ) h=1 i=1 But, since each household is assumed to be maximising utility, (9 .51 ) becomes26... two-stage decision process To …x ideas, consider the example of a government which has to decide whether or not to build an airport, and assume that the airport is a “one-o¤” 9.3 PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 255 project – either one has an airport of given size and quality or one does not There is in fact a huge range of possible social states associated with this decision even though there... through the di¢ culties where general principles appear indecisive But where is it supposed to come from? On what basis can we compare the utility levels or utility scales of one person with another? 9.7 Reading notes A good overview of the main issues in welfare economics is provided by Boadway and Bruce (1984) On the “constitution” approach see Arrow (1 951 ), Black (1 958 ) and the excellent paper by Vickrey... function W in general and in the particular cases cited in part 3 (Atkinson 1970) 9.6 In a two-commodity exchange economy there are two groups of people: type a have the utility function 2 log(xa ) + log(xa ) and an endowment of 30 units 1 2 of commodity 1 and k units of commodity 2; type b have the utility function log(xb ) + 2 log(xb ) and an endowment of 60 units of commodity 1 and 210 k 2 1 units... possible projects of di¤erent types and sizes – the idea in step 1 is that the alternatives are mutually exclusive and irreversible, and that in step 2 all the states can be reached from one another by steps that are in principle reversible Clearly the distinction between the two may be somewhat arbitrary and is reminiscent of the distinction between the “short” and “long run” Nevertheless one can... sections 10.2 and 10.3 we review some of the ideas that were taken for granted in the case of perfect markets (chapters 2– and rethink the notion of equilibrium Section 10.4 applies 7) these concepts to industrial organisation Time In section 10 .5 we examine how the sequencing of decisions in strategic interactions will a¤ect notions of rationality and equilibrium Section 10.6 examines these principles . not to build an airport, and assume that the airport is a “one-o¤” 9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 255 project – either one has an airport of given size and quality or one does. commodity j on the demand for commodity i by household h – in other words the slope of the compensated or Hicksian demand curve. 15 The interpretation of this can be based on the analysis of cost. demand for good 1? 9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 2 45 Figure 9.6: Component of e¢ ciency loss This idea of quantifying waste gives us the basis for developing a coherent analysis