Distributional weights, shadow wages, and the accounting rate of interest : estimates for India

World Bank Reprint Series: Number 108 - Deepak La1 I Distributional Weights, Shadow Wages, and the account in^: Rate of Interest: Estimates Tor India Reprinted with permission from Indian Economic Review, vol. 12 (New Series), no. 2 (October 19779, pp. 99-131   Distributional Weights, Shadow Wages and the Accounting Rate of Interest: Estimates for India* DEEPAK LAL University College, London Introduction Project appraisal has traditionally concentrated on the efficiency aspects of project choice. This would be acceptable if the government could deal with the relevant equity aspects by independent tax-subsidy measures. It has been argued, however, that the government's fiscal powers to redistribute income intra-tempo- rally and inter-temporally are likely to be limited in developing countries, and hence equity considerations cannot be separated from those of efficiencyin project choice. (See Little and Mirrlees [1972],Marglin [19761,UNIDO [19721.)We, thus, have to take account of the income distributional impact of the project's net benefits on social welfare in the ensuing second best world. In project appraisal, the impact on inter-temporal income distribution via the savings-consumption distribution of the net benefits of the project, and on the iritra-temporal distribution of income via income accruals from the project to different income classes amongst contemporaries must be simultaneously taken into acc0unt.l In making these income/consumption changes commensurable, we need a numeraire. The choice is between Little-Mirrlees' (LM's) "uncommit- ted social income expressed in foreign exchange" and the UNIDO Guidelines "aggregate consumption". The former is close to public savings, and if unlike LM we do not differentiate between public and private savings, the two different aumeraires can be said to correspond to "savings" on the LM and "aggregate *This is a substantially revised version of a paper "Distributional Weights and the Social Rate of Interest", Technical Paper No. 11, Project Appraisal Division, Planning Commis- sion, September 1974, Mimeo., written whilst I was working as a consultant to the Plan- ning Commission. I alone remain responsible for the opinions, errors, and omissions in this paper which should in no way be ascribed to the Indian Planning Commission. Comments by a referee are gratefully acknowledged. 1. In large countries like India it may also be desired to take account of the effects of project choice on the inter-regional distribution of income. We abstract from this aspect in this paper, but see La1 [1973, 19751,for ways in which this aspect can be incorporated.  100 DEEPAK LAL consumption" on the UNIDO methods of project appraisal. If "co~sumption" and "savings" were homogeneous "commodities", then it would be a matter of convenience which of the two numeraires we adopted (see La1 [1974]). However, consumption and savings would be homogeneous "commodities" (in the sense that the social value of one unit of the "commodity" is the same as any other unit and hence the government values each unit of consumption/savings equally no matter to whom it accrues) only if the government did not want to effect income distribution in its two dimensions through project choice. As the need for distributional weighting in project analysis arises, precisely because this assumption does not hold, the problem then is to choose an item of national income which would be relatively invariant to this distributional weight- ing and hence be relatively homogeneous as a numeraire. Given the existing in- equalities in consumption in India, we would clearly want to use project choice to affect this distribution, and hence to differentially weight consumption changes of different groups. But this implies that homogeneity cannot be ascribed to "current aggregate consumption", which would therefore not be an appropriate numeraire as its own "value" could change with the distributional weighting adopted. Similar problems arise with current savings accruing to different income classes. For the savings of each group will determine its own consumption time profile and again we may not want to value the consumption profile which accrues to one group on a par with others. This has led Little-Mirrlees to recommend the use of uncommitted social income (which is close to public savings) as the numeraire for public sector project appraisal. For of the various alternatives, this item can be ascribed (at least from the government's viewpoint) the greatest degree of homogeneity. We have, however, decided not to adopt this numeraire exactly in this paper, partly as a simplification of the actual process of project appraisal, and partly because of our belief that private savings in India are probably as valuable as public savings. We, therefore, will not distinguish between different types of savings, and consider them to be equally socially val~able.~Given this assumption, current savings will provide us with a homogeneous numeraire. Moreover, as this numeraire is fairly close to the LM numeraire, it will enable us to use the methods they have suggested for estimating various "national para- meters" required for project appraisal in India. Our specific problems in this paper are (a) to derive distributional weights which will enable us to evaluate the interpersonal consumption changes in terms of their savings equivalent social value, and (b) to provide current saving equivalent weights for weighting the intertemporal consumption effects of projects. The 2. If, however, it is desired to adopt the LM numeraire, the shadow prices we derive in this paper would still remain valid. It would only be necessary to make further estimates of the shadow price of private investment (savings) in terms of uncommitted social income, if necessary, differentiated by income group.  DISTRIBUTIONAL WEIGHTS, SHADOW WAGES & ACCOUNTING RATE OF INTEREST 101 latter problem is equivalent to deciding on a discount rate, the accounting rate of interest (ARI), for social cost-benefit analysis for India. Estimating these parameters is essentially a part of the problem of delineating the optimal economic growth path for a labour surplus economy. Though we will not solve an explicit optimal growth model in this paper, our derivation of "national parameters" may be looked upon heuristically as approximations from a long-run optimal growth model for the Indian economy. It should also be noted that as the so-called em- ployment problem in developing countries - at least in what Sen 119751has termed its "income" and "output" aspects - is essentially a problem of delineating the second best "optimal" distribution of inter- and intra-temporal consumption subject to various technological and political constraints (see Marglin 119761, La1 [1974]), we will also be (c) estimating the ratios of shadow to market wage rates for the economy. 'These will determine the second best "optimal" labour intensity for the economy on the "optimal" growth path. In Part I we derive the various formulae from which the distributional weights and the ARI can be estimated, and in Part 11, we provide our estimates. As will become apparent, in the process we also have to make estimates of various other "shadow" or social prices, namely the current premium on savings/investment (So),the ratio of the shadow to the market wage rate (k), and the consumption rate of interest (CRI), as these shadow prices and the ARI and distributional weights are interdependent. Moreover, in estimating these interdependent shadow prices, judgements on some "quasi-normative" parameters is required. These are discussed in Part 111,which also provides our best estimates of all the key "national parameters". I. Methodology 1.I Distributional Weights Assuming, for simplicity, that all wages are consumed and all profits saved, project choice will affect the intra- and inter-temporal distribution of consumption, through the distribution of project benefits in the form of wages and profits. The latter, ex hypothesi, being saved are valuable at par in terms of our numeraire, which leaves the consumption changes resulting from changes in employment which have to be made commensurable in terms of our numeraire, savings expressed in foreign exchange. Assume that the government has some notional base level of income (b) at which it values czrrrent changes in consumption, socially at par. We can then postulate a social valuation function (V) which is iso-elastic in form, and which converts consumption changes into their numeraire (savings) value, as  102 DEEPAK LAL where b is consumption at the base level of income, Y is the consumption level of the relevant income group, and e is the elasticity of social marginal utility of consumption. The marginal distributional weight (w,) is then 1.2 The Premium on Savings ( S ) To determine the distributional weights, we will have to determine the value of 6, which in turn will depend upon the value of S, the current premium on savings per unit of socially weightedcurrent consumption. Ifwe define1as the utility price of investment (savings), and V' is the social marginal utility of (employment gene- rated) consumption, then by definition from which by logarthmic differentiation it follows thatS With savings (investment) as the numeraire, the accounting rate of interest (ARZ), which is the discount rate to be used in project analysis, is the pro- portionate rate of fall in the utility value of savings (investment), that is ARZ = -(dl%). Moreover, we define the consumption rate of interest (CRI) as the pro- portionate rate of fall of the social marginal utility of employment generated consumption over time; thus CRI = - (V/VO and ARI = - (ij1,) Hence, - SjS = ARZ-CRI. (6) (6) will, therefore, determine the time path of S over time. If we assume that the divergence between the ARI and CRI diminishes linearly over time, till at some date T, ARZ=CRI, and hence S,(t=T,. ., co) remains constant and equal to unity, then, the current value of S (So) will be given by (see LM [1969]) 3. Marglin [I9761 has labelled (4) the inter-temporal consistency condition, and as he shows it is valid not only for optimal growth paths but "whenever capital is consistently valued in terms of its product" (Marglin, p. 186-187).  DISTRIBUTIONAI, WEIGHTS, SHADOW WAGES & ACCOUNTING RATE OF INTEREST 103 But the value of So can also be estimated in an alternative way. Supposethat as a result of a marginal investment project in the industrial sector, there is a margi- nal increase in employment, which leads to workers being drawn out of various sectors in the economy. Say that the proportion of labour drawn out of sector ,j (j=1 .. .n), is nj, with En,=1. Suppose that the output foregone by withdrawing a worker from sector j is wj, at market prices. Moreover, define acc~unting ratios for commodities (Ai), where Ai = P,"/ Pi , where P,"is the market and P," m the shadow price of commodity i. Then the output foregone by creating one more industrial job will be Moreover, if the wage paid to the worker in his new occupation is wf, and is greater than that (w, . . . . .w,) received by the n,. . . n,workers withdrawn, in their previous occupations, then in addition the economy will be committed, ceteris paribus, to providing extra resources to meet the ensuing increase in con- sumption. Given the expenditure weights (q,j) from the pattern of consumption of different types of workers, we can define consumption conversion factors Cj for each type of labour, which convert Re.1 of consumer expenditure at market prices into values at shadow prices as then the social cost of this incremental consumption in terms of the numeraire, savings will be However, to the extent that this increase in consumption accrues to relatively poor workers (or equivalently that employment creation is considered socially valuable), the increase in consumption will also have some social value, which given the social valuation function (1) will be where nj are the average number of adult equivalent consumption units per household, and c* = cln and = aj/n,that isthe "per capita" consumption at shadow prices of industrial sector workers and workers in sector j's households respectively. Thus the social value of the extra consumption generated per unit of savings  104 DEEPAK LAL foregone will be (11)/(10), and this must by definition be equal to l/S,, that is and clearly, for consistency (12) = (7) (13) 1.3 The Accounting Rate of Interesf (AIR) It can be shown that on the "optimal" growth path -():/I") which is the ARI must equal the social marginal product of capital (see LM [1974], Marglin [19761, Newberry [I9721, Stern 119721).Discounting the time stream of inputs and outputs of a marginal investment project (evaluated at shadow prices), along the optimum path by the relevant ARTSwill, thus, yield a zero net present value. Assume that current investment of Re.1 yields a perpetuity of Rs.(r+w) of which r is saved and reinvested, and w is paid in wages and consumed in the period in which it accrues. If, moreover, the ratio of the shadow to the market industrial wage rate is k, then the ARI will be given by ARI = p = r + (1 -k)w. (14) 1.4 The Ratio of the Shadow to the Market Wage Rate (k) Along the"optimum"path, the industrial shadow wage rate (SWR) will be given by the cost of the output foregone at shadow prices (8), plus the social cost of increased consumption (10) less the social value of this increased consumption (ll), that a marginal increase in employment (in the industrial sector) entails (see LM, Stern [1972], Newberry [1972]). As the market industrial wage is Mi, this implies that k is given by 1.5 The Consumption Rate of Znterest (CRI) The CRI from (5) has been defined as the proportionate rate of fall of the social marginal utility of employment generated consumption over time. Thus, the CRI at time t, (it),given our general val~ationfunction (I), will be given by (1+ it) = (dV/dCt) (dV/dC,+,) l (16) that is 1 plus the CRZ is equal to the marginal rate of substitution between con-  DISTRIBUTIONAL WEIGHTS, SHADOW WAGES & ACCOUNTING RATE OF INTEREST 105 sumption in period t (C,) and t +1 (C,, ,). In deriving the CRI we need to take account of both the differing per capita consumption levels in rural and urban areas and the possibility that they will grow at different rates along the "optimum" path. Then denoting rural per capita consumption in the base period as a, and urban per capita consumption as c, and if the current proportions of the rural and urban population in total population are n, and na(n, +na= l), and given the expected growth rates of per capita con- sumption as ga for rural and g, for urban areas, the social value of consumption in period 0, and 1, using valuation function (1) is therefore4 4. It should, however, be noted that once inter-personal consumption differences are valued differentially, consumption ceaxs to be homogeneous, and there is no unique consumption rate of interest (CRI) nor shadow price of savings (investment) in terms of consumption (S). The accounting rate of interest (ART) and the shadow wage rate (SWR) are, however, clearly defined in terms of our nurneraire savings (=investment). Thus, consider asimple optimal growth model for a dual economy, in which the advanced sector draws upon an elastic supply oflabour at a wage c, from the traditional sector where wages are also assumed constant, at a. Wages in both the sec:ors are consumed, and profi's (in the advanced sector) are saved. There is a given labour force N, of which L can be employed in the industrial ~ector,and this is the government's control variable. With the usual neo-classical production function in industry, the change in the capital stock K (=dk/dt) will be K =.f (K,L) - cL (i) with K, K, and L time dependent and c, ex hypothesi, constant. Total utility using the J\ 6d-&tt- instantaneous social valuation function (1)Lwould then be [L V(c) + (N-L). V(a)], or a [L{V(c) - V(a)) + CLl L3 N . V(n)l. The upper bound of social utility being N. V(c), the objective y t w boi,\h-Q, function would be 00 or I (L-N) [V(c)- V(n)] dt. (ii) 0 The Hamiltonian of the problem: Maximise (ii) subject to 0 <L < N, and with K 2 0 given K(O), is given by ff = (L-N) [V(c) - V(d)] +A[ f (K, L) - d l (iii) The necessary conditions for an optimal path are given by  106 DEEPAK LAL where Nc,is the total national population at the base date and g, is its growth r From (16) and (17), therefore, For the special case where the rural and urban per capita consumption growth rates are the same, say g, the above expression reduces to 1.6 The Inter-relationships of the Variables From (7), (12), (14), (15), (18a) we can derive the followingequation which relates all the various variables which will determine a consistent set of distributional HL = 0; Hk = and AA = K Thcse im?ly f k = j . / ~ and fL = c-{ V(c) --V(a))/A The former, (v), states that the ARI equals the marginal product of capital at all times on the optimum path. The latter (vi) provides the expression for the SWR. There is clearly no unique S, but rather two, corresponding to Sc = A/V'c and Su = A/V'n. Nor is there a unique CRI, but rather two consumption rates of interest for the two income groups with incomes c and a. However, the inter-temporal consistency condition, in terms of the con- sumption of those at consumption level c, or a, still holds; for as can be checked, s / ~ ~ = ( i / ~ )(v,/v/,)and sa/sa 6 (vra~ ' 2 . - = 1~)- In a fully specified optimal growth model, there is no need to determine the S's and the CRl's explicitly, as for project analysis (with savings as the numeraire), we merely require the values of the ARI, and the shadow wage rate. For numerical simulations of these "na- tional parameters" within such a framework, see Newberry and Stern. In our more heuristic approach, however, the introduction of the CRI and S is useful. Following Little-Mirrlees (Chps. 13 and 14), we define the CRI "as the rate of fall of the utility gain from using a unit of consumption to make possible the transfer of people from (the) occupation (with income level a to that with income level c)" (p. 302). Then defining [V(c)- V(a)]/(c-u)=q, as this utility gain per unit of conlmitted consumption, we have the CRI = --qlq,and a natural definition of S, is then A/q. Again this implies that -$s h / ~ - q /= = ~ ARZ-CRI. Moreover, the SWR in (vi) then becomes Note that m does nct appcar in this formulatioil as we have implicitly assumed that a = rn.  DISTRIBUTIONAL WEIGHTS, SHADOW WAGES & ACCOUNTING RATE OF INTEREST 107 weights, the accounting rate of interest and the ratio of the shadow to the market wage rate. Note that we have for notational ease not transcribed the full equation ((18)for i, (the CRI), but its special version (18a) -fie parameters in this messy expression are: c = the consumption level of the industrial worker at shadow prices a = the previous consumption level of the industrial worker at shadow prices r = the portion of social value added of a marginal industrial project which is saved and invested t v = the portion of social value added that is paid in wages on the marginal industrial project g = the expected rate of growth of aggregate per capita consumption g, = the expected rate of growth of total population w f = the market industrial wage nz = the output foregone by a marginal increase in industr'al employment, valued at shadow prices nj = the proportion of workers drawn from sector j, when one extra indus- trial job is created ra = the number of adult equivalent consumption units per household. We + are also assuming that all n, = n, in (19) c = defined as cln, that is the "per capita" consumption at shadow prices of + worker households in the industrial sector a = defined as uj/n, that is the "per capita" consumption at shadow prices of worker households in "sector" j This leaves b = the base level of income at which marginal changes in con~umptionare socially valued at par in terms of the numeraire (savings) e = the elasticity of social marginal utility of consumption T = the date at which consumption and savingsare expected to be of equal social value We can choose the values of any twoof these latter variablesand that willdetermine  108 DEEPAK LAL the value of the third from (19). Once the values of b and e are thus determined within this consistent framework, then the distributional weights, the ARI, and the ratio of the shadow to the market wage (k) can be determined from equations (2), (14), and (15) respectively. This is the procedure we shall follow in Part 11. It should be noted that for our purposes "unemployment" will also be consi- dered to be a sector, and the nj terms will include the effects of rural-urban mig- ration (see Harris-Todaro [1970], Lal [1973a]).We turn to the estimates in the nexf part. 11. Estimates ,2.1 The Proportion of Workers Withdrawn from Sectors (nj) For purposes of estimation we have divided up the Indian economy into five "employment" sectors, namely, organized urban (industrial), unorganized urban, unemployed urban, agricultural rural, and unemployed rural. We need to estimate Sector II wages Sector I and wager Supply Price - - W' W EII E EI Labour Force and Employment Fig. 1  DISTRIBUTIONAL WEIGHTS, SHADOW WAGES & ACCOUNTING RATE OF INTEREST 109 the proportion of workers withdrawn from these sectors (on average) as a result of a marginal increase in industrial employment at the all-India level. We use a model due to Scott (see Scott et al. [1976]),depicted by Fig. 1 to deter- mine these proportions. We consider labour allocation between any two sectors I and 11, but the argument holds, mutatis mutandis, for many sectors. For simpli- city, initially we ignore the possibility of unemployment in the two sectors, and also assume that the wage rate is the same in both the sectors. Then if D,Dj is the demand curve for labour in sector j=I, Il, and SI, is the supply curve of labour from sector11to sector I,5the initial labour allocation betweenthe two sectorswill be E b in sector j=1, 11, at the wage rate W, and with a given total labour force of LIZ.,, in the two sectors. Now suppose the demand for labour rises in sector Iby E,E,,. Thiswillraisethewageratesinboththesectors.Inthenewequilibrium, the sector I wage will have risen by WW";at this wage given S,, supply curve of labour, EE,, workers will move from sector IIto sector I, and the wage in sector 11will only rise by WW.(There will now be a wage differentialbetweenthe two sectors.) Of the increaseof EIEr,in sector Iemployment, EEI will comefrom within ithe sector and EEIIfrom sector II. Hence, x1 = EEI/EIEIIand nII= EEII/EIEI,. If the elasticity of the supply curve of labour from sector II is e,,, and of the demand curve of labour in sector I is e,,, then the slopes of the two curves S,, and D, (assuming they are straight lines) will be given by where NI and NII are the labour forces in the two sectors I and II. Clearly then, it follows that Of our five "employment" sectors, three are urban and two rural. We thus first need the proportions of workersdrawn from the rural and urban sectors when one extra industrial job is created. From (20), we require estimates of the two elas- 5. As drawn, we have implicitly assum~dthat the effective supply price of sector I1 labour to sector I is greater than the wage rate (equ4 to the marginal product cf labour) in sector I. This is meant to take account of various imperfectionsin labour markets (like imperfections in information flows) as well as psychic disutilities and various real resource costs of moving from one szctor to the other. If we consider sector I to be rural and sector I1 to be urban, then the difference between the supply price and wage rate of rural labour will also depend upon theinstitutional features posited in the rural sector, like the relative proportion of landless and landed peasants, the extent of pure familylabour operated family farms,etc. (see La1[1976al).  110 DEEPAK LAL ticities es,l and e,, (where 1I is the rural and I is the urban sector). No estimates of theseare available.We have, therefore, estimated the proportio~lsn,,/n, directly from a simple rural-urban migration model of the Harris-Todaro type. In the simplest of these models, the flow of rural-urban migration resulting from a marginal increase in industrial employment depends upon the rural-urban wage differentialand the probability of