European Economic Review 17 (1982) 307-324. North-Holland Publishing Company

I N T E R G E N E R A T I O N A L E X C H A N G E M O B I L I T Y

A N D E C O N O M I C W E L F A R E

A n i l M A R K A N D Y A *

Unit'ersity College, London WC1, UK

Received February 1981, final version received July 1981

The concepts of exchange and structural mobility are introduced and their relationship with the notion of Equality of Opportunity discussed. It is then shown that exchange mobility can have an impact on social welfare when the utility of any one generation is dependent on the consumption levels of overlapping generations. Whether such consumption interrelationships imply a preference for mobility or- the reverse depends on whether the marginal utility of current consumption is raised o r lowered by an increase in the consumption level of the associated generation. The rest of this paper examines these two cases in the context of a simple two-class family model. The analysis is concerned with the description of the optimal intergen- erational behaviour within this framework. In order to be able to say anything substantial, however, it is necessary to impose some restrictions on the structure of the utility function. One restriction which has some appeal and which yields plausible patterns of optimal policy is that intergenerational links through the utility function be not too strong, in a sense that is made specific in the paper. Imposing this condition, one can show how intra-generational equity is affected both by intergenerational considerations and by the inherited past. Where intra- generational transfers involve efficiency losses, a simple tax model is examined. The cir- cumstances under which, starting with no taxes, the tax rates converge are worked out in both the cases where the utility function has a pro as well as an anti-mobility bias.

Finally a specific utility function is taken, in which the fathers utility depends not only on his own consumption but also on the average consumption of his son and himself. In this case the utility function has a pro mobility bias and one can evaluate the response of the long-run tax rate to parameters that measure mobility and equality. Using a similation model, one can also assess the speed of convergence and the sensitivity of the tax rates to the parameters.

1. Introduction

T h i s p a p e r is c o n c e r n e d w i t h a n e x a m i n a t i o n o f t h e r e a s o n s w h y , a n d t h e

i m p l i c a t i o n s of , e c o n o m i c m o b i l i t y m a t t e r i n g to i n d i v i d u a l a n d soc i a l p e r c e p -

t i o n s o f w e l f a r e , it~dependently of i ts e f fec t s o n t h e d i s t r i b u t i o n of i n c o m e o r

o n t h e e f f i c i e n c y w i t h w h i c h l a b o u r r e s o u r c e s a r e u t i l i s ed . W i t h i n t h e

s o c i o l o g i c a l l i t e r a t u r e a d i s t i n c t i o n is m a d e b e t w e e n c h a n g e s in m o b i l i t y t h a t

c a n b e a t t r i b u t e d to t h e i n c r e a s e d r e l a t i v e a v a i l a b i l i t y o f p o s i t i o n s in t h e

h i g h e r s o c i a l c l a s s e s a n d t h o s e c h a n g e s t h a t c a n b e a t t r i b u t e d to a n i n c r e a s e d

*I would like to thank Professor A.B. Atkinson and Mr. D. Ulph and a referee of this journal for their comments on an earlier draft, although I remain responsible for all the errors. An earlier version of this paper was presented to the World Congress of The Econometric Soc!ety, Aix-en-Provence, in September 1980.

0 0 1 4 - 2 9 2 1 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 �9 1 9 8 2 N o r t h - H o l l a n d

308 A. Markandya, lntergenerational exchange mobility and economic wel[are

intergenerational movement of individuals among social classes, for a giuen distribution of positions among these classes. The former is referred to as the phenomenon of structural mobility and the latter is the phenomenon of exchange mobility. It is with the latter, interpreted within the context of economic models, that this paper is mainly concerned.

Most economists t who have been interested in the question of the intergenerational links of income and status have focussed on the notion of Equality of Opportunity. From the way in which their models have been constructed, this focus has corresponded ra ther more closely to the notion of structural mobility than to that of exchange mobility. That is to say, they have identified the extent of which variables such as family background, race and sex determine income, both directly and indirectly, and have then evaluated their effects by seeing what their removal does to some measures of the inequality in the distribution of incomes. However , as Atkinson 2 has pointed out, the mechanisms by which these effects are removed are rather simplistic and essentially assume an infinite demand for labour at any given wage for a given occupation. It is probable that if family background and similar variables were not to be used in the allocation of individuals to positions, then at least part of the effect would be the reallocation of individuals among a given stock of positions. In this case the effects on the distribution of income are more complex. More importantly, however, focussing merely on the final distribution of outcomes neglects some of the social benefits that increased ' justice' in the allocation of positions should imply. This paper is concerned with examining the basis for this notion of justice and with investigating its consequences. This is not to imply of course that changes in structural mobility are unimportant . It is true, however, that as far as welfare is concerned their effects are picked up in existing measures of aggregate welfare. Consequently investigations of structural mobili ty appear to belong more properly to the realm of the study of the determina- tion of the existing distribution of income.

Tha t exchange mobility matters in some general though vaguely defined sense is fairly clear. If, for example, public opinion were solicited on this issue by asking individuals to rank two societies that were identical in every respect except that in one the economic status of every n iember was exactly the same as that of his (now deceased) father, whereas in the other there was no systematic relationship between their statuses, it is unlikely that the

ISee for example the papers by Becker and Tomes (1979), Conlisk (1974) and Taubman (1979).

2Atkinson (1980a). A similar point was made much earlier by R.H. Tawney who pointed out that an increased supply of skilled labour would probably reduce occupational differentials. Also Professor Glass in his evaluation of his own pioneering work on social mobility suggested that the distributional problem would be little affected by increased mobility. This view stands in contrast to that of economists such as Conlisk and Taubman.

A. Markandya, lntergenerational exchange mobility and economic welfare 309

rankings obtained would indicate indifference. At the same time it is true that current measures of economic welfare based on income or wealth would fail to distinguish between the two societies caricatured above. 3"4 A distinc- tion between them will arise, however, if the welfare of individuals in the current generation depends not only on their income but also on the income of their parents. Then the measure of welfare will involve the joint distribu- tion of incomes in overlapping genera t ions and in general the linkages between them will have a direct effect on welfare. In section 2 a utilitarian framework for social welfare is presented and general functional forms that incorporate intergenerational links are considered. Some plausible reasons for the existence of such links are also discussed. In section 3 some of the implications of these relationships for intergenerational welfare maximisa- tion under various constraints are elaborated. In order for the optimal policy to be able to be characterised to any significant degree, it is necessary to introduce some restrictions on the utility function. The notion introduced here is that of relatively weak intergenerational links. This concept is exp- lained and some justification for it is offered. Incorporating it in the utility function allows one to describe the optimal consumption path in the absence of efficiency losses when the government makes intragenerational transfers and, along with some other restrictions, the optimal tax path when such losses do occur. Section 4 concludes the paper by evaluating the results obtained.

2. A simple model

Consider a society in which there are a discrete number of economic status levels and in which e a c h individual has a prior classification with respect to some identifiable characteristic. Let Pii be the probability that an individual of prior classification Yi will have economic status Yi, fi the proportion of the population that ends up with status Yi, and gi the proportion of the population that starts as being of type y~. We assume that the population is exactly reproduced in each generation. Also since we are concerned with exchange mobility, we wish to restrict ourselves to changes in the transition matrix that leave the distribution of outcomes unchanged. Hence ~ = g~ and the following restrictions hold:

ZP~i=I and ~p,ig~=gi allj. (1) i i

3A referee of this journal has suggested that a lexicographic ordering may be more appropriate for this purpose-- i .e . , considering the distribution first and the linkages second. Although it would be interesting to pursue this line of reasoning, the author finds the clear dichotomy necessary for a lexicographic ordering lacking in this issue.

4 Inheritance is of course excluded in the above comparison in so far as it affects the distribution of economic status in the current generation. What looking at this distribution alone misses is the analysis of how that status was acquired.

310 A. Markandya, lntergenerational exchange mobility and economic welfare

In this m o d e l one can re fe r to equa l i ty of o p p o r t u n i t y in t e rms of the ma t r ix P whose i, j th e l e m e n t is p~j.5 C o m p l e t e e q u a l i t y could be ident i f ied by a ma t r i x in which all e l e m e n t s were equa l . C o m p l e t e inequa l i ty is no t so eas i ly ident i f ied bu t could p e r h a p s be a s soc ia t ed with a d iagona l mat r ix . I t cou ld also, however , be a s soc ia t ed with o t h e r P mat r ices which r e p r e s e n t e d comple te r ig id i ty of t rans i t ion , r a the r than no t ransi t ion between genera- t ions , as the d iagonal ma t r ix would r ep re sen t . In add i t ion it is i m p o r t a n t to no te tha t how much e q u a l i t y of o p p o r t u n i t y m a t t e r e d m a y d e p e n d u p o n how much the e c o n o m i c s ta tus levels var ied . T h e m o r e s imilar t hey are , the less r e l e v a n t in a sense is the s t ruc ture of t he t rans i t ion mat r ix p.6 E x c h a n g e m o b i l i t y is one d imens ion of the equa l i t y of o p p o r t u n i t y ques t ion , whe re the p r i o r types i a re ident i f ied accord ing to the e c o n o m i c s ta tus of the pa ren t s .

A s far as the m e a s u r e m e n t of e c o n o m i c wel fa re is conce rned , o n e m a y d is t inguish be tween we l fa re ex a n t e - - b e f o r e individuals have b e e n a l lo - ca t ed to s t a t u s e s - - a n d ex post ~ af te r t h e y have been a l loca ted . In b o t h cases, howeve r , if i nd iv idua l ut i l i ty d e p e n d s only on the ind iv idua l s e c o n o m i c s ta tus , and we t ake an add i t i ve social wel fare funct ion , the ca lcu la t ed wel fa re will be i n d e p e n d e n t of the t rans i t ion mat r ix P. L e t U(y i) be the ut i l i ty assoc ia ted with e c o n o m i c s ta tus y~ and let W be the m e a s u r e of social we l fa re then we have the e x - p o s t we l f a r e as

w = E U(yi)~, (2) i

and the e x - a n t e e x p e c t e d social wel fa re as

E ( w 3 = E E p,,fi u(~;.) = Y. f~u(y3 = w. (3) i i i

H o w e v e r , if ut i l i ty d e p e n d s not on ly on the cur ren t s ta tus e n j o y e d by an ind iv idua l bu t also the s ta tus of the g e n e r a t i o n to which it is l inked , then bo th e x - a n t e and ex -pos t wel fa re m e a s u r e s genera l ly involve the ma t r ix P. In the f o r m e r case this will a p p e a r as a p r o b a b i l i t y mat r ix in (3), whi le in the l a t t e r case, t h rough the app l i ca t ion of the law of large numbe r s the mat r ix P will r e p r e s e n t the jo in t f r equency d i s t r i bu t ion of Yi and yi.7 T h e one case

s This concept of Equality of Opportunity differs somewhat from that given in the introduc- tion and used, for example, by Taubman (1979). However both appear to be valid uses and ones which occur quite naturally in their own contexts. This suggests that this concept is in fact more subtle than would appear at first glance and that there are a number of aspects to it. For a further discussion of the various uses of this term, see Atkinson (1980a).

6 Considerable effort has been devoted to producing summary measures of this matrix P. For an interesting analysis of this question, the reader is referred to the paper by Shorrocks (1978).

7 In general (as economic mobility can be seen as a concern about inequality in the ex-ante distribution of economic status) it seems preferable to measure welfare in ex-ante expected value terms for this analysis. Because of the substantial degree of additivity that has been assumed in this paper, the ex-ame/ex-post distinction does not seem to matter much. Where more general forms are considered, the distinction can make a difference. For a further discussion of this issue, see Hammond (1976).

A. Markandya, Intergenerational exchange mobility and economic welfare 311

where, with intergenerational links in utility, the matrix P disappears is when the utility function UO'~,yj) is additive in Yi and Yi" Let UO,~,yi) = Vl(y,)+ V2(yj), then

E(W) = Y~ Y~ {V'(y,)+ V2(yi)}p, if~ =Y~ V ' ( y , ) f , + Y . V2(~,'~)~ = W. (4) i I i i

Hence although current wclfare depends on past status, this dependence has little impact on current policies. For example the optimal current status distribution is independent of the previous distribution. Ill the absence of such additivity, however, the elements Pii will appear explicitly, and the implications of such dependence can be seen by considering a society consisting of two families, A and B. Lct CA, the consumption by family A in the first generation, be greater than C~, consumption by family B in that generation. Furthermore, for convenience, let A Co + Co = 1. Then if status Yi is determincd by consumption C~ and if one chooses consumption in this generation, C A and C~, to maximise this generation's welfare subject to the constraining C A a < + C t = 1, it can easily be shown that, for a strictly concave utility function,

B 1 C1 <~< CA

and

c~>I>cA

o if ~oo \aC~} '

(0v < 0 if ~ o \ a C t ] "

A similar result can be obtained by altering the transition probability rather than the income levels in each generation. Let CA and C~ be held constant and let the transition probabilities vary subject to the restrictions given in (1). Then one can think of a diagonalising operation as one which increases Pu and P22 (probabilities that the father and son have identical economic statuses) by a small positive amount, e, and which decreases Pt2 and P2t (probabilities that the sons statuses are the reverse of those of the fathers) by the same amount. Now one can show that

a 0(----~ > 0 --~ ex-ante welfare increases with diagonalisation, aCo \aCl]

o aC'o < 0 ~ ex-ante welfare decreases with diagonalisation.

From this it appears that the consequences of consumption interrelation- ships for mobility depend crucially on whether the marginal utility of consumption in the current generation is raised or lowered by an increase in the consumption level of the parents generation. If it is raised, then the EER--B

312 A. Markandya, lnterger~erational exchange mobility and economic welfare

model has an anti-mobility bias; if it is lowered the reverse is true. Whether we wish to argue for the one or the o ther or be agnostic will basically depend on the source of the intergenerational link. Although a full analysis of these sources is not a t tempted in this paper , one could think of a number of factors which, on a priori grounds, would lead to the utility of one generation depending in total and on the margin on the utility of the associated generations. Firstly there is the fact that a significant part of anyone 's lifetime consumption is de termined by parental income and not personal income. However , whether this exerts a pro- or an anti-mobili ty bias is unclear. Secondly there is the unarguable fact that parents care about their children's lifetime consumption and vice-versa. Hence we observe intergenerational income transfers. Again one cannot argue entirely con- vincingly that such a linkage should result in one kind of bias or the other (although it is easier to construct examples of a pro-mobil i ty kind rather than the reverse). Finally there is little doubt that the economic status of the father has an effect on the welfare of his children through an influence on their tastes. To model this feature, one would have to regard the status of the previous generation as a pa ramete r in the utility function of the current generation. Similar considerations to the above would apply here and again it is not clear whether one should view this as a pro- or an anti-mobility factor. 8

Thus although there are strong reasons for believing in the intergenera- tional links, whether they imply a positive association between exchange mobili ty and welfare is an unanswered question. Hence it is certainly important to examine the implications of both cases for welfare optimising policies and this is done in the next section.

3. Exchange mobility and optimal distributive policies

So far we have developed a f ramework within which the issues of exchange and structural mobility can be formulated and in doing this have been able to work with a rather general utility function. If, however, the distributive implications of exchange-type mobility are to be analysed further it appears that some restriction on the structure of the utility function is necessary. We already know from the previous section that if utility is additive in the consumption of overlapping generations, then exchange mobility has no implications for economic welfare. This section shows that as long as the cross effects in the utility function are not too strong then, given certain minor regularity conditions, a number of interest- ing results about how a government ' s distributive policies are affected by the

a The concepts of pro- and anti-mobility bias developed here are of course dependent on the utilitarian approach adopted here and may not hold with other concepts of welfare.

A. Markandya, Intergenerational exchange mobility and economic wel[are 313

presence of exchange mobility follow. The sense in which the non-additivity of the utility function is restricted may be explained as follows: Consider two fathers, one poor and one rich who have sons whose income position is the reverse of that of their fathers. Assume that the utility function has a pro-mobility b i a s - - i . e . , that the marginal utility of the father with respect to his own income declines as his sons income increases. We know from the concavity of the utility function that if the sons had the same income, then the marginal utility of the rich father would be less than that of the poor father. Given the pro-mobility bias mentioned above, the reversal in the incomes of the sons, however, could make the marginal utility of the rich father greater than that of the poor father. One way in which the effect of the son's income on the father's utility could be kept small is by requiring that this reversal of marginal utilities only take place if the difference in son's incomes is greater than that of the fathers'. In other words, one could say that the main determinant of the father's marginal utility of income is his own income; the son's income has, in a sense, a smaller influence on it.

Formally, and more generally, this condition may be stated as follows:

Let U= U(C~,C2) be a concave function with the units of C1 and C2 measured commensurately. Define E as

~ U . . a U . E - ( c , , c g - ( c i , c ) .

We define the cross effect of C2 on C1 as relatively small when

E . ( C , - C i ) > 0 only if Ic=-c&l>lc,-cil. One may define, in an exactly analogous way, the cross effect of C~ on (?2. Although in general it is not true that the cross effect of C~ on C2 being small implies that the cross effect of (?2 on C~ is small, one can show that when the function is homothetic this implication holds.

It is of interest to ask what kind of concave functions will satisfy the condition stated above. One can see immediately that if U(C~,C2)- U(C~ + (?2) then the cross effects will be relatively small for all values of C1 and C2. However , this form is rather restrictive. For other concave functions that are commonly used in economic analysis the condition will generally not be globally satisfied. It is also easy to check that it will be satisfied for a second-order approximation to the concave function taken at C~, (?2 when IO2U/aC~[>Ia2U/OCt ~C2], with the derivatives being defined at that point. Again this imposes certain restrictions on the parameter values and variables of any concave function.

In this section we will make use of this concept of relatively small cross effects for the analysis of two different problems; one dealing with distribu- tive policies when there are no efficiency losses to consider, and the other

314 A. Markandya, hltergenerational exchange mobility and economic welfare

dealing with the case where such losses do occur, and a tax instrument is used for redistribution. 9

3.1. Distributive policies in the absence of efficiency losses

In order to examine how the distribution of resources is affected by exchange mobility, we consider a model where there is a fixed amount o f output to be allocated between two families or classes of families, and where the process of allocation has no effect on the amount to be allocated (i.e., there are no efficiency losses). Dealing with only two families permits us to analyse the relevant issues without having to get involved in the measure- ment of inequality and the problems related to that. Fix total output in each generation at unity and let Co be the share of that output going to one of the families in generation O. One may assume that C0>�89 Since it is also assumed that C~, the share of output going to family one in generation i can be freely controlled by the government, the constraints imposed by any transition matrix can be overridden. Hence, focussing only on forward intergenerational linkages, the welfare of generation i - 1 is given by

W ~ _ ~ = U ( C , _ , , C , ) + U ( 1 - C ~ _ ~ , I - C , ) , 1_-> C,_->O. (5)

Given Co, the welfare maximisation problem over a T generation horizon may be stated as follows:

7" max W = Y. W~ with I > = G N 0 , Cor189 (6)

C l . C 2 , . . . , C w i - - I

Concerning this problem m the following theorem may be stated:

Theorem I. Assume that:

(i) the strictly concave utility function U is such that 02U/0C1 0C2=- U!2 is always of tile sante sign - - i.e., always positive or always negative;

(ii) tlle cross effects of Ci on Ci-1 and of C H on Ci are relatively small, in the sense defined earlier; and

(iii) l i m q _ o U~(C1,C2) = % i = 1, 2.

9 The policies chosen here are of course only a few of the many alternative policies that could be undertaken. They deal with exchange mobility in somewhat different ways too. The first allows the transition matrix to he overridden but restricts total output to be constant, thus preventing changes in mobility that increase such output. The second holds the transition matrix constant and hence the proportion of the population in each ability category but allows total income to vary. Thus while both are not purely exchange mobility changes, they impose restrictions on feasible alternatives that are very much in the spirit of those that looking at exchange mobility alone would imply.

l~ additivity of W in I,V~ follows from the restrictions on welfare that generate additivity. Discounting may be introduced but adds nothing of interest to the problem. When Co = 21 the solution to (6) is of course C~ = 2 x for all i and the problem is uninteresting.

A. Markandya, b~tergenerational exchange mobility and economic wel[are 315

Then the sequence C1, Cz ..... Cr that is the solution to (6) is such that it converges towards �89 Extending the horizon T takes the terminal value closer to ~. When U12>0 the convergence is with damped oscillation around �89 whereas when U~2>0 the convergence to that value is monotonic.

The proof of this theorem is given in appendix 1. As far as the conditions are concerned, (i) and (ii) have already been discussed and there is nothing to add here. Condition (iii) guarantees an interior solution to the 0ptimisa- tion problem and removes some tiresome corner solutions that are obviously economically uninteresting.

A couple of implications of the above results are also worth pointing out. The first is that even in the absence of intergenerational transfers of income, exchange mobility leads to a situation where the inherited past influences distributive policies for many generations. When the welfare function has a promobility bias the impact of the initial equality is to keep reversing the position of families in each consecutive generation. When the function has an anti-moility bias inequality is reduced stepwise. Both statements, how- ever, do require the condition that intergenerational links be relatively weak. When such links are strong it does not appear to be possible to characterise the optimal distributive policy.

Another important implication of the above analysis is that when incomes can be freely allocated within each generation then, although the optimal allocations are effected by mobility considerations, the transition matrix itself is of no relevance. This may be contrasted with the case considered below when incomes cannot be costlessly allocated. In that case the matrix of transition probability matters directly.

3.2. Exchange mobility and taxation

In order to examine the impact of exchange mobility on distributive policy in the presence of efficiency losses, we continue to work with the two-family model, but now assume that the son's 'natural' income is related to his father's. Let the natural division of talents be such that in the absence of any intervention, two incomes would be observed, s and l - s , and let p be the probability that the son of a father who had income s will himself have income s. For stationary incomes this makes the transition matrix P,

P = 1 - p '

and the closer p is to unity the less mobility one would observe. The value of p may represent the work of either genetic or environmental factors. In order to influence the distribution of resources, a tax may be placed upon actual incomes, but the existence of such taxes will affect the level of output

316 A. Markandya, baergenerational exchange mobility and economic wel[are

of each individual. Since the utility functions do not incorporate leisure effects explicitly, it is better to think of taxes as affecting individual output through their effect on decisions involving the investment of human capital.

One may formalise the model in the following fashion. Let:

t~ = (constant) marginal tax rate for generation i; #)(s,t)=output of a person who has natural ability s and faces a tax rate t; p = probability that a father of ability s has a son of ability s, and 1 - p is

the probability that he has a son of ability l - s ; g(s,t) = income received by a person of ability s under tax regime t.

The following assumptions are made about the relevant functions and controls:

(A.1) All taxes are equally redistributed: Consequently,

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

g(s,t) = (1 - t )qS(s , t )+ t/2(c~(s,t)+ck(1 - s , t)). (7)

g2(1 - s , 0 )>0 .

The abilities add up to unity and, without loss of generality, it is assumed that s >�89

The function 4, is assumed to have the following properties: c~t(s,t)> O, ~(s,O)=s, 4,(s,1)=0, ~b2(s,t)<0, d~22(s,t)<0; 0_---t~ 1.

The function g(s',t) is concave in t for all s'.

to = 0, i.e., the initial generation is untaxed, and total welfare is the sum of each generations expected welfare, starting from generation one and extending to infinity.

The maximisation problem may be stated as oo

max ~ W, {r l , t2 , . . . , t , , . . . .} f - - 1

where

W~ =p{U(g(s,t~), g(s,t~+l))+ U ( g ( 1 - s , t~), g (1 - s , t~+l))}

+(1 -p ) {U(g( l - s , t,), g(s,t,+l))+ U(g(s,t,), g(1 - s , t,+l))}

= G(ti,ti+l,s,p)

(8)

(9)

(A.7) bz fimction G(ti,ti+t,s,p) the cross effects of ti+l ol, ti a ,d of ti on ti+l are relatively small in the sense defined earlier.

The dynamic tax rule for this problem may be characterised by the following theorem:

A. Markandya, lntergenerational exchm,ge mobility an d economic welfare 317

Theorent 2. Given A.1 to A.7 the tax rates will converge in the limit to a stationary value for sufficiently large values of p. The convergence takes place with damped oscillation i[ U12<0, and monotonically if Ut2> 0.

The proof of the theorem is given in appendix 2. It is useful, however, to discuss the role of the assumptions A.1-A.7 and to give them some economic content. Assumptions A.1 and A.3 are essentially ones of con- venience, while A.2 is a regularity condition that ensures that for sufficiently large p, G2t and U2t have the same sign. It may be noted that g2(1-s , t ) > 0 for all positive values of t. A.2 merely extends that to zero tax rates. A.4 specifies the effects of taxes on output. By suitable normalisation, we can define the output with no taxes as the ability level [4)(s,0)= s] and it seems reasonable to assume that with a 100% tax rate no output will be produced [4)(s,1) =0] . It also specifies that output declines with taxes at an increasing rate [4)2(s,t)<0, 4)22(s,t)<0]. Although one can construct reasonable models which ensure these conditions it is not clear how empirically valid t h e concavity of 4, in t is. Related to the concavity of 4) is A.5, the assumption of the concavity of g(s,t) in t. This is required to ensure that a global optimum is defined by the first-order conditions, following from (8). Given A.4 , A.5 follows either if 4) is additive in s and t or if 4) is 'sufficiently' concave, n These are probably over-strong conditions for the concavity of g but it should be noted that the concavity of 4, is not enough. Finally, assumption A.7 is required for this proof in much the same way as the condition of relatively small cross effects of C2 on Ct in U(CI,C2) was required for Theorem 1. It is of course related to the latter since the function G is a linear function of the utility functions. It can be shown again that for sufficiently large values of p, G will exhibit relatively weak cross effects if U exhibits weak cross effects. It will always remain possible, however, for G to exhibit relatively strong cross effects, for low enough values of p, even when U satisfies the condition for relatively weak cross effects.

When the equilibrium tax rate t* has been attained, one may ask how that rate will be affected by changes in the parameters p and s. The use of comparative statics methods does not reveal any predictable pattern for dt*/ds, dt*/dp can, however, be shown to be unambiguously positive given relatively weak cross effects on U and U12 < 0 (the pro-mobility case) and to be negative when U t 2 > 0 (the anti-mobility case), and when both the U function and the quadratic approximation to the G function demonstrate relatively weak cross effects. 12 In order to get some idea of the sensitivity of

tt Specifically ~,,1~, >2. 4,,,/$, is of course a well known measure of concavity. t2precisely what is needed is for dt*/dp<0 is Ut2>0, [Gtzl<lGtt[ and IG~21<IG221. The

latter two conditions will hold when the quadratic approximation to G satisfies the relatively weak cross effects condition.

318 A. Markandya, lntergeneratlonal exchange mobility and economic welfare

the tax rate to these and other parameters and to find out how rapid the rate of convergence is, it is desirable to do some simulations using a specific functional form. This is done in the next section.

4. Some simulation results

In this section we consider the following specific utility function:

U ( C 1 , C 2 ) = ( a ( C a + C2)) k + B((1 - a ) C 0 k,

l > a > - 0 , B > 0 , l > k > 0 . (10)

Such a function has the property Ua2<0 and satisfies the conditions for relatively weak cross effects. One justification for this particular form would be that a proportion a of total consumption in the two generations is carried out collectively and is a separate source of utility for the parent generation from private consumption. Alternatively one could think of part of the parents ' utility as being dependent on the average consumption of both generations (a =�89 as well as on its personal consumption.

In addition to the specific utility function, we take the following forms for the functions d~(s,t) and q~(1-s, t):

r = (s](e c - 1))(e c - e c') (11)

4>(1 - s, t) = ((1 - s ) l ( e c - 1))(e c - e c') (12)

The functions ~b represented above are concave with a constant degree of 'risk aversion' and result in the function g(.) being concave for values of C greater than two. In addition they have the properties that 4 , (x ,0)=x, 4 , (x ,1)=0, and 4,~,(x,t)<0, for all x_->0. Consequently they and the utility function imply that all the conditions for Theorem 2 are satisfied as long as p is large enough. Given these functions, the functions G(.) can be defined explicitly and numerical solutions to eqs. (A.7) in appendix 2 obtained explicitly for given values of the parameters or, B, k, p, s, and C. . I t is possible, however, that a long-run positive equilibrium tax rate may not always exist, and even when it exists, the tax path may not converge to it. However , rather interestingly, for the range of parameter values given at the bottom of table 1, it was found that the existence of a positive solution was only sensitive to the values of C, k, and s. Furthermore when a positive solution existed, convergence to it was virtually complete in two periods. Typically the first period tax rate would be between 2 and 10 percent above the long-run tax rate, and the second period would be between 2 percent of the 'equilibrium' rate. In very few cases was the deviation from the long-run rate much greater than that. Thus the long-run tax rate appears to be a very good guide to the appropriate tax policy. A number of other interesting findings can be summarised as follows:

(i) 'p ' can be very small and still the tax path converges. For values of p

A. l~larkandya, b~tergenerational exchange nzobility and economic wel[are

Table 1

Long-run tax rates (percentages) for a = 0.2, B = 1 (NPS = no positive solution exists))

319

C=3 C=4

p=0.8 p=0.6 p=0.5 p=0 .8 p=0.6 p=0.5

k = 0.4

k =0.8

s = 0.7 5.9 4.3 3.6 NPS NPS NPS s = 0.8 22.5 21.1 20.3 11.5 9.5 8.3 s = 0.9 33.9 32.6 31.9 25.9 24.2 23.2

s = 0.7 NPS NPS NPS NPS NPS NPS s = 0.8 2.4 1.3 0.9 NPS NPS NPS s =0.9 15.2 14.4 13.9 3.6 2.8 2.0

a Range of parameter values tried: 0.8_>- k >_-_ 0.2, 0.8 ->_ a ~ 0.2, 1.0 ~ s _>- 0.6, 0.8>-p>=0.2, 4>_-C~2, 4w_- B >_-0.5.

up tO 0.4 c o n v e r g e n c e was o b t a i n e d . F o r l ower va lues of p tax ra tes d id no t converge . I t a p p e a r s then that T h e o r e m 2 will ho ld for all p laus ib le va lues of p.

(ii) T h e m a g n i t u d e of the l ong - run op t ima l tax r a t e is ve ry sensi t ive tO s, C and k bu t n o t to et, h and p. I t does inc rease wi th p and dec l ine wi th a as the m o d e l w o u l d p red ic t , but an increase in p f rom 0.5 to 0.8 resul ts in an increase in the tax ra te of b e t w e e n 2 and 7 p e r c e n t , d e p e n d i n g on the va lue of a . A s one wou ld expec t , the g r ea t e r is the va lue of a , the m o r e sens i t ive is the tax r a t e to p. S imi la r ly va ry ing a ove r the r a n g e 0 . 2 - 0 . 9 and B ove r the range 0 . 5 0 - 4 . 0 affects tax ra tes by on ly a coup le of pe r c e n t a ge poin ts . O n the o t h e r h a n d , an increase i n the va lue of s f rom 0.7 to 0.9 resul ts in the tax r a t e inc reas ing by b e t w e e n 12 and 30 pe rcen t . T h e magn i tude of the r ise in the tax r a t e wi th s increases with k, the d e g r e e of concavi ty of the u t i l i ty funct ion. T h e l a t t e r also has, ceteris paribus, a c o n s i d e r a b l e d i rec t effect on the tax ra te , wi th a r ise in k resu l t ing in a subs tan t i a l fall in the ra te as can be seen f rom tab le 1. F ina l ly the effects of C o n the tax ra te a re as one would expec t . A n inc rease in C, which is a m e a s u r e of the r e spons iveness of the w o r k effor t to the tax ra te resul ts , o t h e r things be ing equal , in no t i c e a b l e r educ t ions in t h e tax ra te .

5. Condusions

This p a p e r s t a r t ed by m a k i n g a d is t inc t ion b e t w e e n exchange m o b i l i t y and s t ruc tura l mobi l i ty . I t was a rgued that the f o r m e r on ly affects we l fa re th rough the p r e s e n c e of i n t e r g e n e r a t i o n a l l inks in ut i l i ty . Since a pe r suas ive case can be m a d e for the p re sence of such l inks, it fol lows that exchange mob i l i t y is a m a t t e r of some conce rn for e c o n o m i c welfare . A crucial d is t inc t ion a r o s e b e t w e e n the case w h e r e the m a r g i n a l ut i l i ty of c u r r e n t c o n s u m p t i o n is l owered by an inc rease in the c o n s u m p t i o n of the assoc ia ted

320 A. Markandya, bltergenerational exchange mobility and econotnic welfare

generation and the case where it is raised. The former imparts a pro- mobility bias to economic welfare whereas the latter does the reverse. It is difficult to argue convincingly on a priori grounds for either the pro- or the anti-mobility case. Arguments in favour of both have some merits, although on balance the author is more inclined to accept the pro-mobility scheme as being more realistic.

Analysing the implications of both schemes for distributive policies leads to the need for another important concept in this field, and that is the idea of relatively weak cross effects. This idea is made precise and its implications considered. When it is possible to assume relatively weak cross effects, intragenerational distributive policies are systematically affected by in- tergenerational links. In the pro-mobility cases such policies imply a con- vergence to some long-run equilibrium with damped oscillation; there is a tendency to want to encourage the reversal of positions in successive generations. In the anti-mobility case such a tendency does not exist. For this scheme, however, there is the rather worrying implication that when redistribution is done by taxation then a rise in p, the probability that father and son will end up with identical earning capacities, reduces the optimal tax rate t*.

A simulation model was tried for the pro-mobility case. The results of this, however, were rather suggestive of mobility as rather less important a phenomenon than intragenerational equity in determining tax rates. Both p and a did affect the tax rates but the magnitude was rather small.

The model should be tested for robustness and developed further. One important question that is not tackled here is the one of intergenerational transfers. Such transfers could be built into a model of this kind but a number of difficult issues are raised and further analytical work is required.

Appendix 1

The proof of Theorem 1 may be sketched out as follows: The first-order conditions following from (8) may be written as

{U2(Co,C,)- U2(1 - Co, 1 - C,)}

+{U~(C, ,C2)- U , ( 1 - C , , 1-(:72)}=0, (A.1,)

{U2(C,,C2) - U2(1 - C,, 1 - (72)} +{U,(C2,C3)- U , ( 1 - C 2 , 1=C3)}= 0 , (A.12)

{ U2(Gr_2,Gr_~)- U2(1 - Cr-2, 1 - Gr_~)}

+ { U , ( C r _ ~ , C r ) - U ~ ( 1 - C r _ , , 1 - C r ) } = O , (A.1T-,)

{U2CCr-,,Cr) - U2tL - C r - , , 1 - Cr)} = 0. (A.1T)

We may assume w.i.g, that Co>�89

A. Markandya. lntergenerational exchange mobility and economic wel[are 321

If Cr = ~ and U2t is of one sign [assumption (i)], then Ca--i = ~ is required to satisfy (A.lr) . This applies, however, to all equations and hence implies Co--21 which is a contradiction. Therefore C-r:/:�89 and assume w.l.g, that CT>~. Consider first the case where U12<0. Then from (A.1T), Cr - l< �89 and by assumption (ii),

1 - C r - I - C r - , > C T - I + C - r or - ~ - C r - , > C r - � 8 9

Now consider eq. (A.1T-1). Given Cr- t <~ and relatively weak cross effects of Cr on Co--t, we know that the second term in { } is positive. Hence the first term in { } must be negative and applying the same reasoning we get

CF_2-1+CT_2>I -CT_I - -CT_ 1 or C r - 2 - ~ > ~ - C r - t .

The same argument applies to the earlier equations and consequently we have shown that the sequence Ct, (:72 . . . . . CT is converging with damped oscillation towards �89

When U12>0 from (A.1T) we obtain by concavity and assumption (ii),

c~_1-�89 ~-�89

and from (A.lr_l),

c~_:-~> c~_,-�89

and so on. Hence the sequence converges to �89 from above. In order to show that extending tile horizon takes the terminal value

closer to I, one may proceed as follows: When the horizon is extended from T - 1 to T, the new welfare function may be written as

W ( C o . . . . . C-r) = W(Co,C~ . . . . . CT- t ) + W ( C r - I , C r ) , (A.2)

where

W(CT_,,CT) = U(C-r-,,CT) + U(1 - C-r_t, 1 - C-r). (A.3)

Let the optimal solution for the T - 1 period horizon be given by a star and the solution for a T period horizon be given by a prime superscript; w.l.g, let

I 1 C r < : and consider the case where U~2<0. Then if the proposition is not true we would have

C . !~_1 (~,.~c', -�89 (A.4) T - - I - - 2 ~ 2 - - ~"" T ~ I ' - " T - - I

Hence Cr - t > C'*T-1 >�89 Choosing CT tO maximise W(CT-1, (ST) given C r - i , denote the optimal'value of W by I~. Differentiating W w.r.t. Cr-t we get

c31}V = ( c30_~_ t _ _ 0 U ( 1 - C r _ t , 1 - C r ) ) 0Cr- t (Cr-t 'CT) 0(1-- Cr)

+ (Cr_,,C-~) O(I _ CT--------~ (I -- CT_,, i ,

(A.5)

322 A. Markandya, lntergenerational exchange mobility and economic wel[are

a n d

~ ,~ ' 0(0___~_~ 0 U - C r ) ) ( A . 6 ) 0 ~ _ _ , = ( G r - ~ , C r ) 0(1 - C r ) (1 - Ca-- t , 1 .

T h e s e c o n d t e r m in ( A . 5 ) is z e r o b y the f i r s t - o r d e r m a x i m i s a t i o n c o n d i t i o n s t h a t d e f i n e W ( C r - 1 ) . E x p r e s s i o n ( A . 6 ) , h o w e v e r , is n e g a t i v e as C r - i >21, a n d the max imisa t ion has been carried out wi th w e a k cross effects on U. H e n c e it f o l l ow s tha t

r162 > 9r

S i n c e * * W( Co,Ct ,C2 . . . . . CT.- 1)> W( Co,C'~,c~ . . . . . C~--t), it f o l l ows t ha t

W(Co, c* . . . . . c * _ 0 § 1 6 2 1 6 2 W(Co,C; . . . . . c~-_O+ ~(c~-_1)

W(Co,Ci . . . . . c~-)

B u t th is c o n t r a d i c t s t h e d e f i n i t i o n W(Co,C'~ . . . . . C~-). The p r o o f fo r the c a s e w h e r e U 1 2 > 0 p r o c e e d s a l o n g s i m i l a r l i nes .

A p p e n d i x 2

T h e o r e m 2 m a y b e p r o v e d as fo l l ows :

T h e f i r s t - o r d e r c o n d i t i o n s d e f i n i n g the o p t i m a l s o l u t i o n m a y b e w r i t t e n as

G2(t~_~,t~)+ G~(t,t~§ i = 1, 2, 3 . . . . (A.7)

W e h a v e s u p p r e s s e d the a r g u m e n t s p a n d S in the G f u n c t i o n as t h e y a r e i n v a r i a n t . C o n s i d e r f i rs t t he case w h e r e U 2 t < 0 , ~3 a n d a s s u m e w.l .g , t h a t t~-i > tv I t can b e s h o w n t h a t a d j a c e n t t ax r a t e s c a n n o t b e e q u a l , ~4 a n d so t h e r e a r e t h r e e pos s ib i l i t i e s to c o n s i d e r r e g a r d i n g t~+x,

(i) t~_t>t~>t~+t>O, (ii) t~+l>t~_~>t~>=O, (iii) t~_~ > t~§ > t~ >_- 0.

C o n s i d e r (i) a n d (ii) in t u rn :

Case (i). F r o m eqs . ( A . 7 ) , t a k i n g the f i r s t - o r d e r c o n d i t i o n s fo r t~ and t~+t

t3 Recall that, given A.1 to A.7, U21<0---* G21<0. 14 t~_t = ti = i" may be excluded by considering the three cases: (a) t i+l> t, (b) ti+ I < ~, (c)

ti§ =/'. In case (a), from eqs. (A.7) one obtains by an argument identical to that used in the paper, the condition ti. I < / > t~+ 2, and by repeated consideration of pairs of equations, Its. I - t j [ increases without limit. A similar argument applies in case (b) and yields the same result. If t,+ t = ~, then consider cases (a)-(c) applied to t~+2. Again either one has equality or taxes increasing without limit. However, if t,_ t = i" for i t ->- i - 1, then one may easily show that all tax rates must be equal, i.e., also for i t<i-1 . In that event, however, ~=to=0 .

A. Markandya, lntergenerational exchange mobility and economic welfare 3 2 3

and subtracting, we have

[G2(ti-l,t~) -G2( t i , ti+l)]+[G1(ti,ti+l)-Gl(ti+l,ti+2)]>=O. (A.8)

Given G21 < 0 and G22 < 0, the first term in [ ] is negative. Hence the second such term must be positive, and given t~ > t~+l and the weak cross effects conditions this will be so only if

t i + 2 - t i + t > h - t i + l or ti+2>ti.

Repeating the argument on first-order conditions (A.7) for t~ and ti§ yields

t i + i - t i + 3 > t i + 2 - t i > O or t i + l > t i + 3 .

Consequently the tax sequence can be shown to go

�9 . . t i + 3 > ti+2> ti > t i + l > t i + 3 > t i + 5 . . . ,

which implies that ti+ ~ - t j is increasing with j and so the tax rate must eventually become one, which is impossible.

Case (ii). t i+ l> t i - 1> h >0. Again taking conditions (A.7) for ti and ti+ 1 and subtracting yields

[G2(t~_~,t~)-G2(ti,t~+~)]+[G~(t,t~+~)-G2(t~+~,h+2)]<O. (A.9)

The first, term is positive since t~<t~+~ and t~_~-t~<t~+~-t~. Hence the second term must be negative and since t~<t~+~ we must have t~+l-t~+2> t~+~- ti or t i > ti+ 2.

Now rcpcat the abovc procedurc on the equation for ti+t and the inequality for t~+2. This yields

[G2(t,,t,+,)-G2(t,+,,t,+2)]+[Gl(t,+~,t,+2)-G~(t,+2,t,+3)]>-_O. (A.10)

We know that h+~ > h+2 and t~+l- h+2 > t~+l- t~. Consequently the first term is negative. For the second term to be positive we require, by the weak cross effects condition that

t~+3--ti+2> ti+l--ti+2 or ti+3> ti+l.

Consequently the tax sequence goes

. . . ti+5 > ti+3> ti+l > ti_1> ti > ti+2> fi+4 . . . .

and so again the amplitude of adjacent tax rates [tj+~- tjl is increasing with ]. Having eliminated (i) and (ii) this means that case (iii) must hold. Since it

must hold for all i, t~ = 0 is excluded. From the definition of (iii) one can define two infinite sequences,

ti-l>ti+l>ti+3>ti+5 . . . . t i>ti+2>ti+4>ti+6 . . . .

Since these are two strictly monotonic sequences defined on the unit interval

324 A. Markandya, Intergenerational exchange mobility and economic welfare

they each must have a unique limit point. Let the limit of the first sequence be to* and that of the second by t~*. In accordance with the assumptions of the paper to * > =re * > 0 . Given that equations G2 and G1 are continuous in t, taking the limits of eqs. (A.7) gives

Gdto,t, ) = O. Gz(to,fe)+Gt(t, , to)-O, G2(t~,to)+ (A.11)

Gl(to,t,) GI(t, , to)- to > t, would imply that both G2(to,tr Gz(t, ,to) and are negative. Since both inequalities cannot hold we have to* = t* and so the tax rates do indeed converge to a unique equilibrium value.

In the case where U~z>O one can show by a similar argument to that employed above that with ti-l>ti all the three cases listed above are excluded by the need to keep the tax rates within the unit interval. Consequently ti > ti_, and considering the three cases

(iv) ti+, > ti > t~_x,

(v) t~>ti+,>t~_,,

(vi) t~>ti-x>ti+,,

it can be shown that (v) and (vi) lead to the same contradiction. (iv) however is only possible if t~+x-t~ < t i - t~-1. Consequently we have a sequence of tax rates 11, 12 . . . . . t,~ such that each is greater than the one before and the difference between them is declining. Hence they must converge to a value t~'< 1.

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