~!SCUSSIOi~ PAPER DRD83 OPTI~rr~ T~XATIOS AND SHADOW PRICING IN A DEVELOPING ECONO~ty by Christopher Heady and Pradeep K. Mitra June 1984 I I ~ve1opment Research Department E(?nomics and Research Staff World Bank f ... · - .. · . - The vieys presented here are those of the author. and they should not be interpreted as reflecting those of the World Bank FOR OFFICIAL VSE ONLY Revised draft of a chapter in Newbery,·D. -and N. Stern (eds.): The Theo~y of Taxation for Developirs Countries (forthcoming) Commer.ts ~elcome ~ I t l' Ii ." k ~. OPTIMUM TAJATION AND SHADOW.PRICINC IN A. DEVELOP INC ECONOMY' by Christopher Heady Department of Political Economy University Collpge London and Pradeep !C .. Mitra The WorH Bank" June 1984 *The authors would like to thank Christopher Harris·for formulating and· calculating the steady-states reported in Section 5 and for numerous helpful discussions. Thanks are also due to him and to Hector Sierra for exceptionally able research assistance. Useful comments from David Newbery on an earlier draft are gratefully acknowledged. However, the views expressed in this paper are the authors' alone and, in particular, do not necessarily reflect those of the World Bank. This doc::ument has a restricted distribution and may be used by recipients only in the performance . .. · . . ' W .. Optimum Taxation and Shadow Pricing in a Developing Economy by Christopher Heady and Pradeep ~itra Abstract This paper analises the determinants of optimal tax and investment polieies in a developing eountry and investigates how those polieies should respond to .'\ s·:,*tained ris"! in the price of imported intermediate inputs such as oil. Th.- ~ffeet on the terms of trade between industry and agrieulture are also repo'~ I.ed. The paper breaks new ground by combining in one model the tax ... and inves 1 .ment ?olieies considered in the literature on. slt·4w ·p.ci'=,i..n..,-.uui '':'I ineentives in tho presenee of tax restrietions with the notion of JiabeuJ:" market imperfeetion eentral to contributions on the daal- et;.q,.pom~. This general model is then subjeeted to external shoeks. A simple general equilibrium model is constructed and its parameters are chosen so that it can approximat~ly replicate the observed economic data for a particular developing country. Optimal policies are first derived analytically, providing rules for organizing production, setting taxes and offsettin~ labour market distortions. They are then computed under alternative assumptions about government objectives, its ability to tax transactions within the agricultural sector of a developing economy and the environment in which government policies must operate. While many important insights can be obtained from simple static models, an analysis of investment policy also requires ~he computation of optimal intertemporal policies. This paper presents optimal one-period polieies as well as optimal steady-state policies. The n~erical analysis demonstrates that the optimal tax structure depends crucially on the extent to which agriculture can be taxed and on the assumptions made about the distribution of land ownership among rural and urban residents. )t turns out that, whenever tax restrictions are present, increased prices ~ imported energy reduce the size of government subsidies. There is also a s~gestion that increases in the energy price sf.uuld lead to a .. reduction in the qptimal rate of investment · - · · · Table of Contents Pages 1. Introduction 1 2. The Model 4 Production and Trade 4 Consumption 5 Migration 5 Ownership 6 Tax Restrictions 6 Choice of Numeraire ~, . ' 8 Shadow Prices and Subsidies 9 Social Valuation 9 The Optimization Problem 10 Dynamic ~odel 11 Resource Flows: An Example 11 3. Analytic Results 11 "!:','/ .. Production Rules 16 Tax Rules !8" . Migration 19 4. Static Results 20 Government Revenue 29 5. Steady-State Results 31 6. Conclusions 36 Mathematical Appendix 40 Production Rules: ~anufacturing Sector 46 Setting Consumer Prices: Manufacturing Sector 47 Taxation in the Agricultural Sector 50 Migration f 55 The Government Bud'et Constraint 58 References 60 . Table 2.1, Social Accounting Matrix: Base Case with Milrants Losing Rights to Land 13 Table -4.1, Optimal Taxes in the Static Model with Land Owned · by the Rural Population (Food Untaxed) " Table '4.2, Optimal Taxes in the Static Model with Land ~wned 22 by the Entire Population (Land Untaxed) 27 Table 4.3, Optimal Taxes in the Static Model with Land Owned by the Rural Population Varying Revenue Requirement (Food Untaxed) 30 Table 5.1, Optimal Taxes in the Steady-State With Land Owned by the Rural Population (Food Untaxed) 34 ...............................................................................................~ 1. Introduction The purpose of this paper is to explore the relationships between taxation, tariff and inve~tment policies and such key prices as shadow wage rates and acc?unting rates of interest used in the evaluation of alternative policies and projects in a develo?ing economy. The exercise makes precise the crucial role of assumptions about the domain of government control, the nature of rural to urban migration, the pattern of property rights and trade , possibilities open to the economy. The effects of varying these assumptions on optimal public policies and associated shadow prices are then examined both analytically as well as numerically. Ihis ca~_.~.!!~P\so~ate those considerations to which the results are particularly sensitive and thus poi~c to areas whic:h should receive priority in empirical work. The motivation underlying this work is the following. Cost-benefit analysis evaluates changes in the economy at shadow prices which represent social opportunity costs. Of the numerous reasons which cause shadow prices to diverge from market prices, two ~y b~ cited. First, there are numerous distortions in developing countries, especially with respect to the operation of labour markets ·. Second, the difficulty of raising resources for investment makes the latter socially more valuable than consumpt~on which accordingly · receives a lower weight in cost-benefit calculations. One of the objectives of this work therefore is to examine the relation between market prices and shadow prices in a model which includes labour market' failure as well as tax .. t ... .. · · ., · - 2 - restrictions which can prevent ne ~overnment' from mobilizing resources except in an incr~asingly distortionary .dY. 11 Another objective of this work is provided by the need to discover how optimal policies and shadow prices in a labour market-distorted, tax- restricted developing economy respond to a rise in the price of imported '.! :'(, I:},' intermediate inputs such as oil. The reasonfot" studying the impac(;of increased oil prices is clear enough. ~any developing countries have s~ffered considerably from the i'creas~d oil prices of the 1970s and early 1980s. It is then interesting to ask how taxes and shadow p.ri'Ca·~t.inat can oe~'a:!;.'.i ..f~!' decentralized evaluation of policies and projects shou,J.:j,'resp!ir..""'i.Q.$.lU;,b,a change in the price of oil. A fully satisfactory treatm~nt of these questions requires the formulation and empirical implementation of an intertemporal model of an archetypal developing economy. The intellectual antecedents of the present work are: (1) the basic model of optimum taxation and public production due to Diamond and ~irrless (1971), as extended co tax-restricted economies by Stiglitz and Dasgupta (1971), ~unk (1980) and Heady and ~itt~ (1982) and (2) . . the model of optimal development in a dual economy due to :)ixit (968), Stern . (1972) and Newbery (1972). The approach breaks new ground by combining in one model:,the tax and investment policies considered in the first area .. ith the . f notloft of dualism that is central to the second. This leads to a quancita~ive - .. · - fraInew>rk for studying the main ques:ions of cost benefit .lOalysis as · 11 This is noc to deny the p::-actical importance of PoticT-imposed distortions in trade regimes and other incencive systems. The focus of the present study is the behavior of an optimizing government in the presence of "natural" distortions. - 3 - developed in L~IDO (1972) and Little and ~irrlees (1974), and helps make clear that the context for that work is an optimizing government with limited tools at its disposal. The general model so developed is then subjected to external shocks such as a deterioration in th~ terms of trade. Empirical evidence on the magnitude of such shocks and the adjustment made by developing countries'is analyzed in Balassa (1981) and, using a macroeconomic mo'delling approach, by ~itra (1983). Numerical simulation of the effects of shocks in applied general equilibriu~ models of archetypal developi~g countries has been undertaken by Dervis, de ~elo and Robinson (1982). But in doini so within a framework which captures some of the institutional structure of a developing country, the work breaks new ground by providing a un\fied perspective on the entire range of tax, tariff and production adjustments which are desirable in response to external shocks. The methodology employed here is similar to that in Heady and Mitra (1982). A general .~quilibrium model '",hich is simple enough to capture the issues outlined above is constructed and a number of analytical results are derived. The analy~ical approach provides useful characterizations of certain ... features of a solution and serves as a valuable check on r,umerical calculations. But the resulting formulae are frtquently too complicated to indicate the ma'gnitude of the 'Jedges bet''''een maJet prices and' shadow prices · and - the sensitivity of those results to under1y~g assumptions about · · - gO'/ernment objectives, the degree of control over pol icy instruments and the en'/ironment in which those policies operate. To get a feE~l for these issues, the paper adopts a numerical approach. ?arameter ·..alues eire chosen so that the model can approximately replicate observed economic ~ata for a particular - 4 - il I country and optimal policies are then computed under a large number of different assumptions. The computation of intertemporal policies in this type of model, with several sectors and limited government control, is entirely new and has proved to be difficult and costly in computer time. To make the program of work and the presentation manageable, this paper concentrates on'the static model and on a special kind of dynamic model, viz., steady states. Many impor~ant insigh~s can be obtained from studying these cases. Results from intertemporal analysis, though available, are at a preliminary st.ag-elami-,i.'t is hoped to present that work separately elsewhere. The rest of the paper is organized as follows. Section 2 outlines the basic model. Section 3 discusses those results that can be obtained analytically. Section 4 uses the model to solve for optimal static policies. Section 5 looks at the dynamic aspects of policy by considering the model in steady-states. Section 6 summarizes the main results. The appendix gives a formal statement of the model and derivations of the analytical results discussed in Section 3. 2. The !'fodel Production and Trade The economy is divided into a rural sector and an urban sector. The rural sector produces food, which is a traded good, using land and labour, .. f - .. both of which are factors spec~ic to the sector, and fertilizer, which is · · purchased from the urban sector and the rest of the wl)rld. The urban sector produces three goods: a traded consumer good (clothing). a nontraded consumer good (services) and an intermediate good (fertilizer). using capital, urban labour and i~ported energy. These factors are assumed to be fully mobile - 5 - among the three urban-based industries. As there is no domestic capitaL goods industry, any investment takes the form of buying capital goods from abroad. The country is assumed to be unable to borrov or lend internationally. Thus, investment is paid for by a balance of trade surplus on all other goods. It is assumed that food, clothing, fertilizer and energy are internationally traded at fixed vorld prices. With the eXC'l!ption of energy, vhich must btf imported, the direction of trade is determined endogenously in the model. Consumption There is a constant population, and people in the economy are assumed to be identical, except that some live and "Jork in the rural:. sector and the ~ rest live and work in the urban area. They are assumed to choose thelr hours of work and consumption of food, clothing and services by maximizing their utility subject to the going vrices and other incomes that they have. Migration Individuals have the cho:ce of migrating from one sector to the other. This raises the issue of what assumption is appropriate regarding a rural to urban migrant's rights to income from land. The paper presents t~o polar alternatives. !iodel 1 assumes that people who miyrate from the rural I sector give up their rights to land, uhile migrants into the rural sector · acquire rights to land. Thus, land is divided equally among all rural . ... re!idents. Model 2 assumes that the ownership of land and the income accruing frim it are divided equally among the entire population. . It is recognijed · - · that these alternatives embody different assumptions about the property rights enjoyed by potential migrants. However, empirical evidence on this maccer is regrettably scarce in developing countries, 50 that the present approach to - 6 - this question must be taxonomic. Sensitivity of the results to these alternati~e specifications will be illustrated through numerical analysis. It is assumed that everyone has the same utility function in terms of leisure and consumption goods. However, it is also assumed that people dislike living in urban areas so that their utility is reduced by a given prop~rtion. Thus, if a particular consumption and leisure choice would result in a utility of V~ for a rural resident, it would produce a reduced',utility of ev~ (where e < 1) for an urban resident. A migration equilibrium occurs when the utility of rural residents, VA'~. is equal to the reduced utility of urban residents, ev~. Ownership Rural and urban labour are owned by the inhabitants of the respective sectors. The ownership of l&nd is also private but its division across sectors depends on which of tne two ~odels of migration is considered. All capital stock is owned by the government, although nothing would be altered if it were privately owned but subject to a 100% profits tax. Tax Restrictions All private sector agents behave competitively in as much as they · take market prices as given. It is in principle possible to allow for differences in prices between producers and consumers as well as between rural and urban prices. In that event;!there are tour different sets of prices: urban consumer prices (q), urban ~oducer prices (m), rural consumer prices (r) and rural producer prices (p).· However, given the institutional structure of the economy, it may be virtually impossible to vary those price~ independently. It is reasonable to impose the restriction that the government cannot drive a wedge between the price at which a rural producer sells food - 7 - and that at vhich a rural ~onsumer buys it. Nor, similarly, can the government be interposed betveen a rural agent in its manifestation as a supplier of lab~ur and land and its manifestation as a demander of those inputs into food production. To assume the contrary vould be to allov taxation or. intra-household trades. For this reason, it is supposed that r · p, i.e., that it is not possible to tax or subsidize internal cramsactions vithin the rural sector. Taxes or subsidies can and vill hovever be levied on the sector's transactions vith the rest of the economy and the outside vorld, viz., on the marketed surplus Jf food and the purchase of clothing, utilities and fertilizer. This limitation to taxing only net trades'means chat, .. iJ::ol.'ll 4 public finance point of viev, there is no longer any significance r:(J-tlte~· distinction betveen production and consumptio~ in rural areas. There is no loss in simply regarding a farm as a consumer thac purchases clothing, services and fertilizer .hich it finances by selling food. In addition to the above restriction , vhich is less an assumption than a virtual necessity dictated by the institutional structure of the problem, it is assumed that there are certain commodity specific tax r~strictions. This applies to agricultural output, A, and clothing, C. It is supposed that the ratio of the consumer prices of goods A and C in the rural And urban sectors are equal. The restrictions may be vritten: r A qA = l+~ rC - - QC= 1+1/1 · · · - Thus, a consumer from one sector cannot get more favourable terms on these commodities by pretending to be a consumer from the other sector. The assumption is intended to reflect the practical difficulties of implementing - 8 - dual pricing policies in situations where possibilities of arbitrage are present. It does not apply to services which can oft~n be provided in a discriminatory manner. For example, rural electricity can be sold a~ a different price from urban electricity. It is worth noting here and elsewhere that a subsidy is ~roduced by a negative tax on a good an agent buys ,or a positive tax Qn & lood an agent sells. Choica of Numeraire The calculation of taxes and subsidies requires a numera.it'e to be chosen and it is tnerefore worth noting that the. ~rcieal_ ~t.a~ c.a1'tes presented depend critically on that chotee';"The specirf<:;!I~i ... ndDigrdnts' rights to land income has implications for the choice of numeraire. In both modei~, decisions by rural consumers depend solely on rural consumer prices, while decisi~ns by r~ral (resp. urban) producers depend solely on rural (resp. urban) producer prices. The situation is however different for urban consumers. With all connections to the rural sector severed in Modell, their ecol'Jtt:iction. This result has not been derived before 1n the published literature. However, it does not hold for Mocel 2 where land 1S owned by all agents in the economy, so thp.t urban consideration~ inter alia influence rural tax rates. Finally, it should be noted that the tax rule described in the preceding paragraph will generally imply non-zero taxes for land and labour despite the fact that neither can be taxed directly. This apparent paradox can be resolved by realizing that the taxes used in the tax rule refer to divergences " bet~een con~umer prices and shadow pric~s, . . while the tax restriction refers to divergences between consumer prices and producer , prices ·. Thus, the taxes reported below cpr land and labour in agriculture are . ... . 'edges between consumer pr1ces and shadow pr1ces that are caused by - .. - distortions elsewhere in the economy, not'by taxation applied directly to them. Migration A marginal rural, to urban migrant has no direct effect on social welfare. This is because the extra urban utility gained, 1 (ev l~ exactly u M' - 20 - balances the rural utility lost ~ VA~ , by the migration condition. In !iodet 2, where the entire po~ulati~n owns land, indirect net benefit of the move is the cost of the rural consumption bundle he give up, less the cost of the urban consumption bundle to whfch he lays claim, where both are evaluated at shadow prices. Since this must be zero at an optimum, the shadow cost of a consumption bundle in the two sectors must .be equal. This h easily:shown to imply an equal tax burden per capita on rural and urban consumers at'an optimum. Thus, this condition states that the tax system should be such as to neither encourage nor discourage migration~ It Js_similar to the production efficiency result in that it forbids government intervention in all aLlocatiY~ proeesses despite the existenee of distortiona~y :axation. The situation is different in !"Iodel 1 where rural and urban migrants lose all rights to land. This creates an externality in the form of an increase in land income for those left behind in the rural sector. The eorresponding policy intervention takes the form of the tax burden per capi:a on rural consumers exceeding that on urban consumers by the net social value of the ensuing change in land incomes. These results on optimal migration in the presence of distortionary taxation have not been derived beforu i~ the published literature. 4. Static Results The results of solving the !"Iodels 1 and 2 with specific functio~l period~ptimal policies are reported in Tables 4.1 and ;'2 .. forms for single - - 21 - respectively. 11 The case that is of central concern is characterized by untaxability of trades internal to agriculture and by the commodity-specific tax restriction. The figures appear in column 1. As mentioned before, the untaxed numeraire is food for ~odel 1 and land for ~odel 2. In interpreting the results, it should be noted that a positive tax on a good supplied by consumers constitutes a transfer from the government to those con$umers. In colu~~ l of Table 4.1" urban consumers gain both from the negative taxes on clothing and services and from the p09itive tax on labour. The only good that is supplied by the rural sector is food, which is the untaxed numeraire. However, it benef:ts from the negative taxes Oft all its. purchases: clothing, utilities and fertilizer. Land and. labour in agriculture are assumed to be untaxed and so the taxes reported on them do not represent transfers of resources, simply differences between private prices and shadow prices that result from the other distortions 1n the economy. To remind the reader of this fact, these taxes are enclosed 1n parentheses in the table. The reason for the government's ability to transfer resources to both urban consumers and the rural sector is that it receives all the profits from the urban sector, where it owns the capital stock. In distributing these profits 'to' the two groups, and in de'ciding the tax rates on part icular goods, . . it is guided by considerations of efficiency. Equity is not an issue because r the migration me~hanism completely determines the relative utilities of the , · 1/ Demsnd functi~~s are obtained from a linear expenditure syst~m that · applies to both sectors. The choice of technique is made from a nested constant elasticity of substitution production function. In the rural sector, land and fertilizer are combined to form ~ subaggregate, which is then combined with labour to produce output. The manufacturing industries are treated similarly but capital and energy form the subaggregate, instead of land and fertilizer. Table 4.1 Opll.al Taul In th. Static Hodel .., With Land OVII ... by tbe lural PopulaUoo (food oota.~x~e~d~)___________. (I) (2) (l) (4) U) (b) ('. (6) (9) ·· , "SOl C... "SOl C. . . "1M eas. No Tax No Tn Balle eale Iii:-·-- "Ie CaBe I, · ~ No C-S-T II!. · I N. restriction i, dropped. A comparison of columnLl-.§.n~L3 shows that'"tas extlected, the tax rates on clothing change ·Jhe'!..,t"he_.restriction is eliminated. However, the effects are not limited to that. In the urban area there is a shift away f.n'm commodity subsidies and an increase in the (utility raising) t~x on labour. Also, in the rural sector, the increased subsidy on clothing allows the goverrJment to reduce the subsidies to services and fertilizer. Thus, the impact of the commodity-specific tax restriction on the pattern of taxes is considerable. However, the reported values for VA and eVM sho~ that its impact on utility levels is small. The significance of point.(b), in addition to point (a), can be eval ,dted by looking at column 9 of Table 4.1, which reports the optimal taxes in the (unlikely) even~ of the government's being able to tax trades internal .. f to the agricultural sestor. The most dramatic effect can be seen in agriculture where the 1rossibility of taxing land means that the government can transfer resources to agriculture in a non-distortionary manner. Thus, no other taxes are levied on that sector. The pattern of taxation in the urban sector is not affecced significantly in COMparison to column 3, but ail the tax rates are reduced, so that each ~rban consumer receives less from the government. This is con3istent with the f~ct that rural employment (LA) - 24 - increases, in response to the less distortior.ary form of rural subsidy, thus leaving fewer urban workers and a ~m~l& ; urban profit for the government to distribute. The figures for VA and av~ indicate a more substantial utility gain to the relaxation of the untaxability of agriculture ~ban the commodity-specific ta~ restriction. .' , Finally, the significance of migration CQ~ ~q considered by looking at columns 5, 6 and 1 of Table 4.1, which report the optimal tax rates with all the tax restrict ions in place, but with nQ aU-Ct'.... i,Q.~ ·. F9C 4!~'6!"~.. comparison, the distribution of the labour i~'be~~een urban .and. o:-.w..sa.l"., areas has been fixed at the optimal level in column 1. Two sets of results are reported because the absence of migration means that the urban-rural utility diiferential is no longer fixed. Thus, the government will take distributional considerations in(o account in setting taxes. Column 5 gives the result3 for u = 1, where the government is not concerned about :nequality. Column 1 gives the results for u = -5, where the government is greatly concerned to reduce inequality. 1/ A comparison ef columns 1 and 5 shows that the eli~ination·or migration unambiguously increases the subsidies to rural areas. There is a significant reduct:ion .in the (utility-raisi'1.g) tax on ulban labour, and some ... increase in the subsidy to urban services. The increase in the suosidy to urb~n~lothing is required by the commodity-specific ta~ restriction when the subsidy to rural clothing is increased. In fact, rural utility increases while urban utility falls. The reason for this shift in government support is the structure of rights to land income in ~odel 1. Recall that migration 1/ The results for column 6 are discussed in the footnote on page 29. - 25 - creates an externality in the form of an increase in land income for those left in the rural sector. This tends to lead to too little migration, ~hich the government attempts to correct by gi~ing larger sUDsidies t~ urban consumers. However, when migration is prohibited and the distribution of labour fixed at its optimal base case level, as in col~ 5, this reason 'for differential subsidy disappears and resources are swit~hed back into agric~lture. This p~int will be reinforced in the discussion of Table 4.2 for Model 2 where no such externality is present. This inequality in outcome in column 5 is clearly not ~hat is required by a government that desired equality. Thus, column 7 shows a pattern of taxation much closer to that in column 1, which produced the same value of LA when equality ~as imposed through the migration mechanism. The next set of experiments consider the impact of increases in energy prices on optimal government policy. Clearly, much of that impact will relate to such issues as the rate of in~estment. However, there will be some impact on tax rates and patterns of production in the short run. Accordingly, columns 2, 4 and 9 report the opti~~l taxes that corLespond to the models of column 1, 3 and a, but, in each case, it is assumed that the world price of - imported energy has increased by a factor of five. A comparison of columns t and 2 shows the effect on optimal taxes in the base case of an increase in the energy prict- This ~ho~s that the government reduces (the absolute size of) all taxes, thus Leducing transfers to both sectors of the economy. This is partly the result of the reduced profitability of urban production and partly the result of a slight shift of the labour force out of the relatively lightly subsidized rural sector into the urban sector, thus increasing the cost of any subsidy programme. This - 26 - shift in the labour force is mainly accounted for by import substitution in clothing in response to increased energy prices. The reduction in energy imports is accompanied by a fall in production of services -- the most energy- intensive sector in the economy. Another variable of interest, apart: from the taxes, ds the terms of trade between agricultural products and manufactured goods. In this model the effect of the energy price rise is to increase the prices of both clothing and services relat i ve to the price of food in Doth the urban .&ild.--~h.~~.... r4~ sectors. Thus, the relat ive price of clothin"S"~~C)'S'eht,aQ.-~'"'tQo.9.8l for both urban and rural residents. The relative price of services rose from 0.99 to · 1.12 for urban residents and from 0.73 to 0.85 for rural residents. As well as looking at the impact of energy price increases in the base case, it is of interest to see whether the extent of tax restrictions on the government affects its ability to deal with those increases. Thus, a comparison of columns 3 and 4 shows the effect of energy price increases when there is no commodity-specific tax restriction, while a comparison of columns 8 and 9 shows the effect when there are no tax restrictions at all. It turns out that the effects are very similar to those 1n the main case: all tax rates are ~duced, with the exception of a slight increase in the tax on urban clothing . f i~colu~ 9. Furthermore, the magnitude of the utility reductions are all ve~ similar (either 0.20 or 0.19). Thus, there appears to be little · connection between the extent of tax restrictions and the ability of the government to deal with energy price changes, at least over the range of restrictions analyzed here. Table 4.2 reports the corresponding figures for ~odel 2 where the entire popUlation owns land. In looking at these results, it is worth T.bl. 4.2 ·· Optl.al TaK.. 1o tb. St.tlc Hod.l With Land Owaed bl thl Iot1r. 'opulatloo (Land lD,alL.d) (1) (2) (3) (4) (S) h) (7) (I) (II) · la.. c.·· .... c.·· .... c.·· NG TaK Nu Tn I ··· Ca ·· " · I " . .... c. ·· ~ Nu C-S-T " · I NG C-S-T " . ~ NG ttl,utton ·1 · I .. · 1 ,!o ttllUtloo " - ' ·· - I " Nu "'Iutlon - '. If - -) ... trlcllon. " - I ...trlclhm. 'at · ) .!!!l!!!. '004 .... 7.11 -6·· 21 -74.U -13.11 -78.41 -79.01 ·.9.71 -12.U; -7u.1I Clo,hlo, -19.01 -71.71 -74.61 -13.11 -79.01 -79.01 -19.UI -12.11 -70.B% N hnlc·· -70.71 -7).1l -74.61 -73.11 -78.81 -79.01 -77.01 -12.11 -7u.ul ..... Labour -76.71 -n.3% -74.61 -73.11 -78.81 -79.01 -77.01 -12.11 -70.11 ~ .ood (3.01) ·· ~l·O\Li ,I (-1.01) (-1.11) (O.ll) (01) (l.2I) -12.11- -10."'* Cloth10, -32.41 -32.01 -U.U -44.21 -2.41 01 -28 ··1 -12.11 -lO.til hnlce. -42.91 -41.91 -42.71 -41.41 -4.41 01 -39.21 ~.72.11 -lO.U Labour (7.11%) (6.)1) (-1.41) (-1.71) (0.61) (01) \0.;1) -72.11* -10.11* 'ertlUaar -7.01 -6.91 -6.11 -~.91 -0.)1 01 -6.11 01 'A 3.96] 3.769 3.t77 3.711 3.0n 2.9~ 3.829 4.011 3.1121 I'M 3.963 3.169 3.971 3.781 4.340 4·.,4 4.024 4.011 3.1211 LA 0.247 o.na 0.269 0.271 0.247 O. ~"7 0.247 . 0.401 U.394 - nl··· '.K r.t ·· appl, '0 rural conlu.. rl. Thl corr ·· pondlD, tea r.t.. for rural producer. ar. a.ro. "1.--';:~-+;-1' . v·.,.. .te-t"",...,~;';zr'1 "_..O-I)OA S - 4I AA (p) .p ) s A P T (p 0) (A.13') + rAAA(p)YA - \1YA - Qw ~ 0 ~ p - 45 - 3A T f T f f = AS ~(m) + r~ AH(m) ~ 0 aY M CIA T T = AS AA(P) + rAP A(p) ~ 0 aY A ClY 1 _ V ] + ((l_t)y~-l - 'J] A - CIA at. ::I 1J (aY )1I m lI A at. (1 ~ 0) (A.16) (A.J£a') ::I GT s ::r 0 (A.17) where subscripts denote derivatives and each inequality bears the relation of complementary slackness with the corresponding variable appearing in brackets on the right. Equations (A.Il) to (A.17) are necessary conditions Eor a restricted tax optimum, provided certain regularity conditions are satisfied. Equation (A.17) is the standard condition that government sector projects (in this case, trade) should not make profits at shadow prices. Equations (A.14) and (A. IS) imply that any chosen private activity that breaks even at producer .. f .. pric.s must do so at shadow prices as well (cE. Diamond and ~irrlees (19nr». A widely known special case of this result is the desirability ~E shadow pricing a small country's tradeables at their world prices [Little and Mirrlees (1974)]. - 46 - Production rules: manufacturing sector From (A.14), Premuitiplying (A.12) by sT and using the above, one has T f By complementary slackness, 41 f > 0 m on 1y l'f mAH=O. From (A.14), this implies that Q f m > 0 only if sT~ SO. Thus, from (A.18) f f L T Since is positive semidefinite (by homogeneity), s T (t~~ y ) , 0 · ~ a f -l'!m m By homogeneity, .. f Hence, it is possible to choose - .. where m ::II ks k is a constant. - · Furthermore, if ~ is of maximal possible (A.19) rank, l.e., one fewer than the number of commodities in the manufacturing sector, (A.19) is the only possible solution. This is stated as . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .~!'I - 47 - Proposition 1: The manufacturing sector of the dual economy with tax restrictions 1n the agricultural sector should be characteri;ed by production efficiency. Setting Consumer Prices: ~anufacturir.g Sector Consider the ith equation in (A.ll). Using Roy's identity, '.10 1 - [ 1 + - - - - - - B~D~i - \t t s.D"IJ 1 + u " 6.~. = o (A.20) ~ ~-l j J 1 1 lG V!1 where = = private marginal utility of urban income (so that is the gross social marginal utility of urban income). D~. · 1 is the ith element "f D!1(q) D!1" is the derivative of D >Jith respect to the ith element of q· · J1 Mj 6. 1 = 1 for goods subject to (A.7) = 0 , othervise. Define consumer taxes as deviations of consumer prices from shadow P rices, i.e., t . !1J = q. J - s. J (A.21) t The budget constraint facing a consumer in the manufacturing sector- - .. is t q.D~. j J. J = 0 · .. ·- Differentiating with respect to qi = -D u · '-11 (A.22) ~""""""""""""""""""""""""""""""""~.J - 48 - Substitution of (A.20) and (A.21) in (A.19) yields - [1 + va 1]8 D . + At t tu.Du " + ~lDu' + a.~. · 0 · talJvlJ- !1 Ml j nJ nJ 1 .'11 1 1 !1 The Slutsky equation gives where the superscript c denotes compensated demands and IM: lump sum income of an urban consumer. Substitution into (A.22) and symmetry of compensate~ demand derivatives c Ou" · nJl .. °!11J leads to c = _ [1 + va J8K D · + lOu. laJ,lvlJ-1 A:11 I'll . . K · . ' - ... · .. .... · · Dividing by 10 Mi ' · · c t. tu.IOn1 u' nJ __ J _ ...._ _ ::: lOu· n1 (A.23) r-------------------------------------------------------------------------------------------------~.~ - 49 - Define = ~. is the net social value of an extra unit of income accruing to an u~ban -:'I 8 M consumer. It has three cor.ponents. The first, tA' is the direct social valuation, divided by , the shadow price of government revenue, and thu'l A SM va a '.10 M expressed in terms of revenue. The second term, 2alJ vlJ - l equals u. u .M which is the effect on social welfare (again expresse d tn terms of revenue) of the change in average urban utility compared to rural utility. The third, 30Mj ~ t Mj aIM' is the extra tax revenue accruing to the goy.e~~t. as~a'I!"I!Surt of increased urban consumption. The tax rule then becomes: c I t~c 10:-1' · M 5.t!!. j · J ,1J - lOW,1 = (l-b') + 1 UO 1 Mi (A.24) where 5. 1 =1 for goods subject to (A.7), =0 otherwise. Equation (A.24) may be stated as: Proposition 2: The proportionate reduction in the ccmpensated demand for good · i that would result from a small equiproportionate intensification of urban consumer taxation if shadow prices remained constant (the lef~ hand side f of A.24) is commodity-independent fo. those goods not subject to a commodity specific tax restriction. The rule ireds modification in other cases to · account for the fact that ratios of consumer prices must be equal in ~ural and urban sectors. - 50 - Taxation in the Agricultural Sector Consider the ith equation in (A.ll). U5ing ROY's identity. 3a'A -[1 - v _11sADA.-~(1-t) t s.DA,. + ~ts. ~ Y (!-t)V~ 1 j J Jl j J gPi A (A.2S> ..,her:-e 6. = 1 1 for goods subject to (A.1), =0 othervise, akA is the kth element of the vector AA(P) ,and SA' DAi and DAji are defined as the rural e~uivalents of SM' DMi' DMji · Define taxes as deviations from shadow prices t Aj = Pj - Sj (A.26) Define S. J = ajAY A &s the net suppLy of good j in the agriculturaL sector, as . ~S, 1 whence .-...l ap. by symmetry of the net supply function. :11- ap.1 j · Al3~, by homogeneity: as.t t Pj ap. = o. J J · · Thus (A.2S) becomes: - 51 - The budget constraint facing a rural consumer is Differentiating with respect to Pi' The Slutsky equar.ion yields c aDA' c aDA' D = D -D --!l=D -D --!l Aji Aji Ai aI A Aij Ai ~IA' where I A: lump sum income of a rural consumer. Substitute in the tax equation to get , . . f . - ~ - which may be rewritten c .. S, . t J . t AJ. [ (l-I. )DA1J - , ,1J ) :: _ ....._ _-.-:_""""""'_ _ _ !l-I.)D (A.2n Ai - 51. - wher.l b ::I A the total effect of an extra unit of i.1.I~' n1 .1ceruing to a rural consumer. ~quation (A.27) mar be simplified considerably when it is noted that land is in fixed supply. Thus if k is the in~ex for land: .. 0 for all j · Also, the agricultural techniques can all be normalized so that they use the same quantity of land at unit level of operation. Thus: ::I 0 for aU j · Finally, land is not affected by the commOdity specific tax and supply must equal demand for land as ,. is not it tradec' . (A.18) This enables (A.27) for 1 =k to be w~tten as: o = = - 53 - Therefore: (A.29) Substitution back into (A.27) gives: 6. c.J. c - u:tA,[U-t.)OA ,.- ,J 1J s,.J 1J 1 l+~ 1 (A.28) J Now define marketed surplus as the excess of agricultural supply over agricultural demand for any good: s.1 -f.l-t.)OA' 1 c M, . A 1J .. s .. - 1J c (l-t.)OA' , 1J This notation enables (A.28) to be written as c t tA,M ,. _ . J A L-:-:-__1J 5. c.J, 1 1 (A.29) HAi . . Equation (A.29) has an appealing form and is stated as: - - Proposition 3: .. At. restricted tax optimum, for an economy with land ownership confined:to the rural sector, an equiproportionate intensification · of agricultural taxation at constant shadow prices will produce a commodity- · · independent proportionate reduction in the compensated marketed surplus of all goods not subject to the commodity specific tax restriction. There are divergences from tbis rule for those goods that are subject to the commodity specific tax restriction. - S4 - The results require some modification for Model 2 since the price of land enters the demand and indirect utility functions of urban consumers. Since land is owned by the entire population. its supply-demand equality reads = tO~k + (l-t) OAk (A.2S t ) ~anipulation of the land tax equation in a way analogous to that following (A.27) leads to ).(l-b) (A.28 t ) tOMkbM + (l-t) 0AkbA where b = 10Mk + (1-1) OAk Th4J, b is a weighted average of net marginal social values of income in the urban and rural sectors, with the weights being the proportions of land owned ~y each sector. The examples worked out in the paper assumed that all land ~as divided equally, in which case · OAk · and the above reduces to b = tb + {l-t) b , M A It where the weights are the proportion of the pgpulation in each sector. Sub'stitute back into (A.27) for comaildities other than land to get - ... · · · - ). r tAJo [(1-1) j 0A~Jo - So oj lJ = The fact that ;. bA ' for income distributional reasons, maO·-es clear that b an equal proportionate reduction rule in com~ensated marketed surplus is not available in Model 2. - 55 - !1igration The allocation of labour across rural and urban areas is governed by (A.l6) and (A.l6'). It is easier to consider (A.16') first. An extra urban worker improves social welfare both directly, 1 u [AV]u M' and indirectly by giving up a rural consumption bundle which lS worth AsTOA at shado~ prices. The marginal cost of the move is the direct welfare loss, ~ v~, and indirectly, the urban consumption bundle to which he lays claim, AsTD · (A.t6') equates marginal costs and benefits at the M optimum. Not ice that the migration constraine· ", .. g'~,~Qq;a~~s .,the. .~~~."'; marginal costs and benefits, so that the.c~ho~ is simply in termS,...9f· the indirect effects. With the migration constraint holding as an equality, (A.16') becomes T s DM (1 ~ 0) Using the definitions tA = . P - s tM = q - s - .. f and'" the budget constraints - .. · · ~ p.DA· - J J J = 0 ~ q.D M· J . J = a J yields - 56 - T (I. ~ 0) (A.30') t D~ So at a restricted tax optimum. This establishes Proposition 4: When income from land ownership is unaffected by migration, an optimum with migration is characteriz.ed by an equal tax' burden per capita on both rural and urban consumers, where taxes are measured relative to shadow prices. In Modell, by contrast, land ownership is affected by migration which therefore confers an externality on rural agents. In the absence of intervention, this can be expected to lead to the extent of migration being suboptimal. It would therefore be optimal to tax the agricultural sector more heavily. The ensuing discussion establishes this formally as well as deriving the magnitude of the difference in the tax burden between the two sectors. Equation (A.16) can be given an interpretation analogous to that given (A.16'). The extra terms arise from the fact that the movement of a worker to the urban area has three additional effects. First, it improves - avA · we If are 1n rura I areas by 1.1-1 (1-8) VA ~ ~ Second, any increase in rural utility tightens the migration cons~raintf imposing a welfare cost equalling .avA f v~ · Third, it raises net dema~ds in the rural areas, the welfare cost of T aDA : which is h (1-1.) at. _ - Applying equation (A.9) to (A.16) and considering interior solutions, - 57 - Since where, as before, aA = private marginal utility of rural income y ry 1-1 (lump sum income of a rural consumer) Differentiating the rural consumer's budget constraint, ,. 0 Since = p. - s. l l the above may be written · , ... .-- .. - 58 - The term 1n curly brackets is the net marginal social value of rural income, Thus, T T = s OM - s 0A which, on using the rural and urban budget constraints, becomes T T t 0A - t O~ = This result is intuitively appealing and is stated as Proposition 5: When income from land ownership changes owing to migration, the tax burden per capita on rural consumers exceeds that on urban consumers by the net social value of the ensuing change in land income, where taxes are measured relative to shadow prices. Finally, equations (A.ll) and (A.l3) show that, as stated in Section 3, the introduction of migration simply changes the weights on the derivatives · of urban and rural utility functions, by intrOducing the terms v6V and .q M . ... \JV AP · . t - .. The government budget constraint · · · It remains to show that the government's budget constraint may be derived from the equations of the model. Its revenue is - S9 - + (r c - s c ) 0c A + (q c - s c ) 0c M + (s c - m ) Yc c The budget constraints for rural and urban consumers and the zero profit conditions in production allov this to be written as where R's denote government demands. This in turn may be written, using the market clearance conditions as .. This last, from the balance-of-payments constraint, eq~ls f ... - .. · sARA ·· sFRF + + scRc + muRu which 1S government expenditure. - 61) - References Ba1assa, 8.1981, "Adjustmer. to Extern~! Shoc~s in Developing Economies," World Bank Staff Working Paper, No. 449. 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