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Famine: a simple general equilibrium model.

1. Introduction

Famine, according to Thomas Malthus (1798), 'seems to be the last, the most dreadful resource of nature', that restores demographic equilibrium after the arithmetic advance of the means of subsistence is overwhelmed by the geometric explosion of population. Such a perspective enervates relief policy, a point illustrated by the essentially negative approach to relief of the British administrators of India in the nineteenth century, well versed as they were in Malthusian theory (Ambirajan, 1976). Given that population expansion is held to be responsible for the decline in per capita food supply, the provision of relief can almost be viewed as prolonging the misery.(1)

The relationship between distress and decline in per capita food supply, referred to as the Food Availability Decline (FAD) doctrine, is far from being the sole preserve of Malthusians. The power of the approach is founded on intuition, even if the economic mechanism is not elaborated. FAD was dramatically challenged by Sen's (1981) seminal work; famines have occurred without a significant fall in the local supplies of food. The distinctive characteristic of famines has been that they represented a collapse of the entitlements, the command over food, of major sections of the population due to adverse price movements as well as endowment losses. Each agent has a bundle of endowments - seed, labour, and livestock, for instance, that by market exchange and production can be converted into a bundle of food. When this leaves the agent with insufficient food to survive he or she falls into the starvation set.

Does the adoption of the entitlement approach change the evaluation of policies to counter famine? The typical implementation of entitlement analysis divides the population into groups defined by particular endowments and then relates aggregate outcomes to average group endowments and the food price ratios that pertain. So if we are dealing with fishermen, we take the average fish output and multiply it by the fish-food price ratio. If this falls short of the subsistence level then the group is clearly at risk (see Locke and Ahmadi-Esfahani, 1993, for example). What policy recommendations flow from such an analysis?

Before attempting to answer this, several points are worth noting. The approach outlined ignores Sen's (1981) own warning that 'this drastically simple modelling of reality makes sense only in helping us to focus on some important parameters of famine analysis; it does not compete with the more general structure' we have briefly outlined above. In particular, what determines the price ratios that play such a crucial role? Secondly, 'Strategies of Entitlement Protection', the title of a chapter in the major (1989) book by Dreze and Sen, covers the principal policy instruments available together with issues relating to their implementation. However, the discussion of the issues would elicit no objection from a FAD adherent. This fits in with Osmani's (1995) position that entitlement analysis constitutes an approach, rather than an hypothesis, and so encompasses the FAD doctrine as a special case. But surely one would anticipate that an analytic advance would have more definite consequences for policy evaluation?

This paper investigates whether the entitlement approach can support a relatively general ranking of policies, in particular relief works and food distribution, to protect entitlements. In order to compare their economic impact, a general rather than a partial equilibrium approach has to be adopted. What then are the markets that are essential to include to make the model insightful while avoiding the technical distraction of generality (a rigorous general equilibrium approach to famine is developed by Coles and Hammond, 1995)?(2) At the very least the food and labour markets demand inclusion; the central price ratio is the food wage, that is, the quantity of food that the wage will purchase. If this affords subsistence, then how is there distress? If it does not, then who accepts it?

The peasant, the most common victim of famine, combines the role of producer and consumer and so we have to move beyond exchange.(3) Now the time dimension of production is fundamental since current endowments will frequently be the result of past production decisions; to allow for this we assume that the agent's decisions are influenced not only by current consumption but also the anticipated level of endowment in the next period, which is taken as a proxy for future consumption.

This does not have the sophistication of Glomm and Palumbo's (1993) study of how the threat of starvation can influence the optimal pattern of intertemporal consumption. However the relationship between time and consumption we seek to explain is a simple though vital one and is captured by the following account of the famine in Darfur: 'In Furawiya, one woman described how she had harvested just enough millet to provide seed for the next year. She had buried it, mixed with sand and gravel to prevent her children digging it up and eating it, and then gone south for the dry season. In June she returned, dug it up and planted it' (de Waal, 1989, p.149). Clearly the level of future endowment was a crucial consideration for this woman and its inclusion into the utility function employed in the model below seems entirely reasonable.

The final market we include is the one for land services. While it does not play the central role of the other two markets, any change in the food wage will have major repercussions for the optimal technique of production, which in turn will affect the demand for labour and consequently the level of distress. Land is assumed to be owned by a class of landlords who sell its services in a competitive market. The three markets examined in this paper operate within the framework of a poor and predominantly agrarian economy. It is poor in several senses: firstly, securing subsistence is a major challenge for a large proportion of the population; secondly; the lack of a capital market constrains an agent to use his own resources at each time period - he cannot borrow today and pay back tomorrow. The agrarian character justifies the two period framework adopted in this paper, reflecting the agricultural cycle.

The basic story behind our model can be briefly outlined; the cycle begins with agents endowed with the output from the previous period; this can be consumed or used to produce the next period's endowment by paying rent for land and, if the agent is wealthy, hiring labour from poorer agents; such wages provide their recipients with the income to purchase food in the current period. The model is spelt out in the following section.

The most direct trigger of famine in this simple model is loss of endowment. The crucial market is that for labour where the loss results in a wage rate below that which can support subsistence. To survive, those who supply labour must supplement earnings from their endowments; those with insufficient assets starve. The process is examined in Section 3.

Given that the famine wage rate is below subsistence, one avenue for policy is to intervene in the labour market to raise the wage and for public works to soak up the excess supply. Alternatively endowments may be increased (even across the board) so that the resulting wage rate may ensure that no one starves. These policies are analysed and contrasted in Section 4 which is followed by our conclusions.

2. The model

In this section the basic model is outlined and the 'normal' case, that is when there is no acute distress, is discussed. This provides the background against which the consequences of a shock to the system can be clearly delineated.

Given the importance of subsistence in a poor economy, we assume that the only commodity produced is rice. In period 0 each agent, i, possesses an endowment of rice, [Mathematical Expression Omitted]. To be physically capable of supplying a unit of labour, at least [c.sup.*] amount of rice must be consumed.(4) The market wage w is a piece rate and so amounts to the return for completing a series of specified tasks such as cultivating a particular acreage of rice. The precise mechanism that secures this need not concern us - the point is that efficiency wage issues are not relevant.

The alternative to participating in the labour market is to work [l.sub.i]([less than or equal to] 1) on the domestic holding and thereby secure an endowment of rice in the next period. Labour, [l.sub.i], together with [k.sub.i] acres of land produces [Mathematical Expression Omitted] rice in the next period via a simple Cobb-Douglas production function

[Mathematical Expression Omitted] (1)

The output [Mathematical Expression Omitted] becomes agent i's endowment in the next period. Anticipating [Mathematical Expression Omitted] provides a sense of security in the current period and is included as an argument in the agent's utility function. The optimisation exercise is thus a static one. The optimal values of ([l.sub.i], [k.sub.i]), however, determine the agent's resource constraint in the next period (which, together with the prevailing prices determine the agent's well-being).

Agents have identical preferences

[Mathematical Expression Omitted] (2)

Consumption [c.sub.i] of rice yields utility only when [c.sub.i] [greater than] [c.sup.*]. Should consumption fall below [c.sub.*] the household falls into the starvation set - an outcome to be avoided, if at all possible. Cobb-Douglas technology in production and consumption are assumed essentially because of analytic convenience, as shown below by the form of the demand and supply schedules. How sensitive are our conclusions to the adoption of this functional form? Duffy and Outram (1994) base their criticism of Desai's (1989) rice and fish model upon the idiosyncrasies of the Stone-Geary utility function. The point carries much less force in our case for a number of reasons. Firstly, the model is primarily one of production rather than trade. All the action occurs in period 0; given that their endowment of rice and labour permits at least [c.sup.*] rice to be consumed, agents must decide between additional consumption and domestic production. The outcome of the latter is seen in period 1, but yields utility in period 0 because of the sense of security it conveys; effectively it will be the resource bound on consumption in period 1 and so the presence of future endowment in (2) may be interpreted as a proxy for future consumption.

Secondly, the boundary situation is not significant. According to (2) an agent who can just attain [c.sup.*] consumption will have zero utility, the same as one in the starvation set. This is awkward, but not critical. It is not necessary to assume that all those in the starvation set perish as in the 'hard-boiled' solution outlined by Koopmans in his (1957) survey of the standard model of competitive equilibrium.(5) Those in the starvation set will not be involved in production; other possibilities are discussed below. In terms of the economic mechanism elucidated by the model, the individual agent on the boundary does not have a crucial role in the outcome. More particularly, policy analysis is our central concern and here the restrictive nature of the functional forms adopted will apply impartially to both the policies considered.

The agent's budget constraint reflects his current resource position

[Mathematical Expression Omitted] (3)

where r is the rent paid per unit of land. We may note that (3) ignores the possibility of the agent storing rice between the two periods essentially because, given the strong assumptions of the model, either storage or production will occur but not both in general.(6) The strength of the assumptions makes the policy comparison tractable but at the cost of neglecting the potential roles of uncertainty, credit, and expectations.

Maximising (2) subject to (1), (3), and the non-negativity constraints [l.sub.i], [k.sub.i] [greater than or equal to] 0 yields demand for food, demand for land, and supply (or demand) of labour functions. For food

[c.sub.i] = [c.sup.*] + (1 - [Alpha]) [F.sub.i] (4)

where [Mathematical Expression Omitted], the 'full income' of the agent. [F.sub.i] [greater than] 0 ensures that the feasible region for the optimisation problem exceeds a point. From (4) it is clear that supply to the food market, [Mathematical Expression Omitted], increases with endowment and decreases with an increase in the food wage. Agents with endowment less than (exceeding) [c.sup.*] + 1 - [Alpha]/[Alpha]w will buy (sell) rice. Thus the poor purchase rice from the rich and finance it by labour income.

Labour supply is given by

1 - [l.sub.i] = 1 - [Alpha] [Beta] [F.sub.i]/w (5)

As might be expected, labour supply falls with an increase in endowment and with a decrease in the food wage. The labour market plays a central role in our analysis of famine so it is worthwhile highlighting several aspects of (5). In Fig. 1 we sketch the market schedules for individuals with different levels of endowment. First let us consider two poor agents with endowments [Mathematical Expression Omitted]. Labour supply will fall for both as the food wage increases - essentially the increase in full income is allocated to current consumption and future production. Since the poor need to purchase rice from the market, they will never become demanders of labour. In fact as the food wage becomes infinitely large, the individual supply schedules of all agents asymptotically approach 1 - [Alpha][Beta], the supply level of an individual with [Mathematical Expression Omitted].

Of particular importance is the point at which the supply schedule commences, that is, the lowest food wage [Mathematical Expression Omitted] at which the peasant is physically capable of participating in the labour market ([l.sub.i] = 0)

[Mathematical Expression Omitted] (6)

(The agent just survives in this case by supplying his entire labour capacity to the market and consuming the resulting wage plus his endowment so that he begins the following period with no rice stock of his own). Thus labourer 2 will be able to supply the market for some levels of the food wage at which 1 will be physically incapable of matching (at [w.sup.*], for instance). Should the market wage rate be [w.sup.*], those agents with [Mathematical Expression Omitted] rice will survive by labouring while those with [Mathematical Expression Omitted] fall into the starvation set. The reservation wage falls with endowment, up to [c.sup.*].

Now consider the better off agents, for whom [Mathematical Expression Omitted]. The wage rate at which an individual switches from being a supplier in the labour market to a demander is given by [Mathematical Expression Omitted]. Thus a richer agent will require a higher food wage to become a labourer (as demonstrated on [ILLUSTRATION FOR FIGURE 1 OMITTED] for [Mathematical Expression Omitted]).

Finally we may briefly outline the implications of the above discussion for the aggregate labour supply schedule. As the wage increases from zero, those individuals with endowment just less than [c.sup.*] will enter the market, supplying one labour unit (that is, [l.sub.i] = 0). With further increases in the food wage, they will reduce their supply and start domestic production. Thus supply from this layer will be reduced, but this effect will be counteracted by another layer entering the market for the first time. The distribution of endowments will thus be the principal determinant of the elasticity of the aggregate labour supply schedule. We may note that for w [greater than] [c.sup.*] all individuals will be capable of entering the labour market and so the shape of the aggregate supply schedule will be determined by the relative magnitude of the poor reducing their supply in response to wage increases and the better-off increasing theirs. The comparative statics of a change in the food wage rate are discussed in the next section.

The remaining decision variable is the demand for land which is straightforward since in effect it may be considered simply as a current expenditure to secure future endowment

[k.sub.i] = (1 - [Beta])[Alpha] [F.sub.i]/r (7)

Given that we restrict ourselves to the situation where F [greater than] 0, an increase in the rent level will unambiguously lead to a fall in the demand for land.

In order to determine equilibrium prices we start with the labour market where the demand for labour from farmers is met within the agricultural sector. As is evident from Fig. 1, the wealthier will be demanding labour and the less well-off supplying it. For clarity we begin with the simple case where [Mathematical Expression Omitted] for all i = 1, ..., N. Summing (5) over all agents and setting the result to zero yields

[Mathematical Expression Omitted] (8)

where [Mathematical Expression Omitted] is the mean endowment level in the economy. There are thus three determinants of the wage rate: the mean endowment level, the output elasticity of labour, and the elasticity of utility with respect to consumption in excess of [c.sup.*]. Inserting (8) into (5) gives (with the e indicating equilibrium throughout)

[Mathematical Expression Omitted] (9)

Those with endowments above the mean level demand labour while those with less supply it. Agents differ only with respect to their endowment level; all agents labour but those with endowments above the mean will hire labour in addition to their own. Should endowments be allocated equally, there are no actors in the market and everyone works full time upon their domestic holding.

We assume land to be held by a class of landlords who supply its services totally inelastically - the rent level is demand determined. Repeating the above exercise with (7) and inserting (8) yields

[Mathematical Expression Omitted] (10)

where K is the total available land area. The above is the simplest way to deal with the land market, though at a substantial cost to realism. A more general treatment would be possible to take account of prevailing institutional arrangements such as common property rights and sharecropping but is not pursued here due to the possible distraction from the policy analysis.

Once the equilibrium food wage rate is determined (4) will generate the rice consumption of every agent and thus his market supply or demand. The food market consists of the poor purchasing rice from the better-off. By summing the budget constraint over all agents we see that total rice from the previous period, that is, aggregate endowments in the current period, equals the total consumption of rice (by producers) plus the total rent payments (to landlords), that is [Mathematical Expression Omitted]. Each agent divides his full income between consumption in excess of [c.sup.*] and expenditure on future production in the respective proportions 1 - [Alpha] and [Alpha]. Now by (4) we have [Mathematical Expression Omitted] so any outlay in excess of the agent's own wage comes from consuming less than the endowment and aggregating this leads to the wage payments being netted out leaving total rent payments.

Inserting (5), (7), (8), and (10) into (1) gives

[Mathematical Expression Omitted] (11)

and thus [Mathematical Expression Omitted]; the agent's share of output in the next period is in proportion to his share of aggregate full income in the current period. Aggregate output involves, in the normal case, the output associated with the utilisation of aggregate land and labour inputs (all labour is employed within the rural economy) - the size distribution of farms has no impact on total output when each agent behaves optimally.

Now the endowment in the current period is the outcome of production in the previous one, which itself reflects the result of employing total factor supplies of labour and land. Growth in the economy may thus be due to technical progress or the expansion of population. The steady state may be considered to be when endowments in aggregate are reproduced, that is, when [Mathematical Expression Omitted]. The model is in the 'wages fund' genre since that which is not consumed in the current period is allocated to production, either by paying for labour or land, and yields output in the next period.(7) Landlords are essentially rentiers and we assume that their assets are such that they can survive the famine period.

Although aggregate output is unchanged in normal times, this does not mean that the distribution of endowments is unaltered. In fact we have

[Mathematical Expression Omitted] (12)

so if w = [c.sup.*], that is, [Mathematical Expression Omitted], then each agent's level of endowment stays the same over time. If the wage rate is greater than [c.sup.*] then [Mathematical Expression Omitted] and vice versa. We now consider how famine intrudes upon this equilibrium.

3. Famine

Given that each agent is endowed with a unit of labour, he will fall into the starvation set only if the wage rate falls below [c.sup.*] and if his rice endowment is insufficient to cover the shortfall. Now the wage will fall by (8) if the mean endowment level drops so one obvious trigger for famine in our model will be generalised loss of endowments. Alternatively a shift in the production function (1) that reduces demand for labour will have a similar effect. We may interpret such a shift broadly; it could be due to anticipated civil unrest such that the output associated with each level of input falls as warring parties 'tax' producers. (1) would then be the net output function after such expected expropriations had occurred.

We will examine both possibilities starting with the former since it is more direct. Let us consider then the consequences of a loss of 1 - [Delta] of the crop due to disease, drought or flood. The equilibrium we have characterised above will be disturbed with the extent dependent upon [Delta]. The economic impact of the shock and the consequent adjustment process can be appreciated more clearly by examining the labour, land, and food markets in turn. We begin with the labour market because its role is central; for clarity we assume that the wage rate before the shock was [c.sup.*], so all agents were then able to survive to the next period irrespective of their level of endowment.

3.1 The labour market

The new mean level of endowments is [Mathematical Expression Omitted] and if the wage rate remains at [c.sup.*], supply of labour to the market will increase and demand will fall. We assume that the result of excess supply of labour is that the wage rate will fall. However, this will lead to a rate below [c.sup.*] and thus one which will not ensure survival unless supplemented by endowment. If the wage rate falls below [Mathematical Expression Omitted] for some i then we must take care that the summation of (5) to generate (8) does not violate (6) for any agent i.(8) Provided it is possible to strictly rank all agents by endowment, that is, [Mathematical Expression Omitted], then

[Mathematical Expression Omitted] (13)

where [Mathematical Expression Omitted] is the mean endowment of the first n agents and n is the highest integer such that [Mathematical Expression Omitted]. Since the N - n agents with the lowest resources have dropped into the starvation set, [Mathematical Expression Omitted].(9)

It is important to fully grasp the implications of (13) for the economic explanation of famine. There is no need for the market clearing wage to guarantee subsistence even though it is rational for every active worker to accept the market rate. To return to Fig. 1, we see that for [Mathematical Expression Omitted] the reservation wage falls with endowment as the poor workers use earnings from the labour market to cover the shortfall in their subsistence consumption. Should the market equilibrium wage be [w.sup.*] then the better off labourer with [Mathematical Expression Omitted] will be working while the one with [Mathematical Expression Omitted] will be destitute since [Mathematical Expression Omitted].

To appreciate the adjustment process, it helps to consider the comparative statics of the labour supply function. The endowment effect is just [Mathematical Expression Omitted] so domestic production is reduced by all agents. As the wage falls in the face of excess supply we have

[Mathematical Expression Omitted] (14)

so the (post-shock) endowment level of [c.sup.*] divides agents into those with less who reduce their domestic production and those with more who increase it. In response to a fall in the wage rate the substitution effect will lead to an increase in the labour allocated to the domestic holding. For the poor, however, this is dominated by the endowment effect. In effect, though production for the next period becomes relatively cheaper, the poor are excluded from taking advantage of this as they have to allocate more of their full income towards survival in the current time period.

Some among those who could survive on their endowments will improve their utility by supplying labour to the market despite the wage being below subsistence. If [Mathematical Expression Omitted] then by (5) we have [l.sub.i] [less than] 1.(10) In effect such agents are physically capable of labour and it is inefficient for this to be used entirely on the domestic holding. Although we have assumed that all agents experience the same proportional shock with respect to their endowments, the reduced wage rate may lead to some of the rich actually increasing their level of production.

We may summarise the effect of the loss of endowments upon the labour market as follows: A famine is characterised by a wage rate which is below that required for subsistence. Despite this the market is supplied by two sets of agents defined by their endowment levels:

(i) those for whom [Mathematical Expression Omitted] and who depend on wages to survive;

(ii) agents with [Mathematical Expression Omitted] can survive on their endowments but supply the market to maximise their utility.

3.2 The land and rice markets

The adjustment here follows directly from that in the labour market. By (7) demand for land will fall as endowments are reduced and the food wage falls - this is reinforced by the destitute dropping out of economic activity. As a consequence there will be excess supply of land services and competitive pressures will bid down the rent level. Moreover, it can be easily demonstrated that the factor prices ratio w/r will increase with the endowment loss. The intuition behind this is straightforward: the reduction in the wage rate will be achieved by agents dropping out of the labour market and becoming destitute. This means that the mean level of endowment of those that remain active will increase, providing a brake upon the decline. There is no such brake in the land market and in order to let the land which has become vacant, landlords will reduce the rent level proportionally more than the wage rate has declined - the average size of land let increases. Thus even though the wage rate had been reduced, the technique of production becomes relatively more land intensive.

Now the aggregate level of endowment in the next period will be given by summing (11) over the n agents in the active set. We can see immediately then that the aggregate level falls but also that the mean endowment level increases, if we assume all in the starvation set perish. This provides some insight into how the rural economy recovers over time from famine since the wage rate would be expected to increase in these circumstances.

Current endowments have all been reduced by the factor [Delta], so the question obviously arises how these are distributed. The proportion of total current endowments (equivalently the output from the previous year's planting) that is accounted for by the consumption of producers increases with the loss of endowments. Landlords receive a lower proportion of the reduced aggregate endowments of those in the active set.

3.3 The evolution of the economy

How does the combination of endowment loss and a wage rate below [c.sup.*] affect the level of future output? Instead of (12) we have

[Mathematical Expression Omitted] (15)

For the agent to increase his endowment over the famine period his endowment must be greater than [c.sup.*] by the factor contained in the above equation which is determined both by [w.sub.F] and the size of the starvation set.

To appreciate their interaction consider two simple cases, those of economies where endowments follow a uniform and a triangular distribution, which are sketched in Fig. 2. The two have the same mean, [Mathematical Expression Omitted] and thus by (8) the same wage rate. For clarity we take this to be [c.sup.*] so agents reproduce their endowments over time.

Once we assume that both distributions have positive densities at zero endowment they will be completely specified. Measuring inequality by the coefficient of variation will characterise the triangular as having the higher inequality.(11) Now assume that both are hit by a loss of 1 - [Delta] of the crop. Since the previous wage rate was [c.sup.*] this loss will entail some agents falling into the starvation set.

The famine wage rates are [Mathematical Expression Omitted] with j = 3 for the triangular case

of k = t and j = 2 for the uniform k = u. Since by assumption [Mathematical Expression Omitted] we have [w.sub.t] [greater than] [w.sub.u]. The higher wage in the triangular case leads to a lower minimum endowment level, [c.sub.m], that will permit the agent to participate in the economy, since [c.sub.m] = [c.sup.*] - [w.sub.k]. However, the greater inequality in the triangular case leads to a greater proportion of the population falling below the lower [c.sub.m], and thus the number in the starvation set for the triangular, [(N - n).sub.t] is greater than [(N - n).sub.u]. Moreover, since the output level in the following year is directly proportional to [n.sub.k], the uniform case will enjoy a greater aggregate endowment level in the ensuing year. Thus the higher wage in the triangular case is combined with higher distress, making the wage rate a dubious indicator of the size of the starvation set.

Figure 2 suggests a possible outline of a dynamic adjustment process. In several papers Hoskins (1964, 1968) argued that runs of three or four bad harvests in a row were not primarily due to the weather but rather the reduced seed ratios induced by a single bad year. The mechanism has been considerably developed by Fogel (1992), within an entitlement framework, and stresses the class differences in the impact of high food prices. As might be anticipated, the landowners and large farmers are net beneficiaries while the labourers endure severe hardship. A possible further dimension is raised by our model. Consider a poor harvest that leads to the food wage falling to below [c.sup.*]. If starvation ensues then aggregate output will fall since labour input is reduced by the extent of the starvation set. This reduced output in the following year could lead to the resulting wage remaining below [c.sup.*]. In contrast to Fogel though, landlords are faced with reduced rentals.

At this point it is pertinent to consider what happens to agents that fall into the starvation set. In the less drastic circumstances of destitution Dasgupta (1993) mentions the fruits of common property rights such as hunting and gathering. A major survival mechanism in Sudan (de Waal, 1989, p.112) was the increased consumption of wild foods which at least partially met the shortfall in millet and sorghum. In the model developed above, those agents who fall into the starvation set do so because they cannot labour at the intensity demanded in the market. This does not necessarily mean that they would perish instantly; some would have resources that would permit a substantial search for alternative employment or relief (see also Corbett, 1988). However, if the former is achieved then by definition they are not in the starvation set. The hungry and ragged migrant will generally not be able to fulfil the work required by the calculating (and rational) farmer. The probability of survival to the next period of an agent in the starvation set would be an increasing function of endowment (see Ravallion, 1987, for an advanced analysis of such a situation).

Any shock that leads to a drop in labour demand can also precipitate famine. Unfortunately the form of the utility function employed above means that any 'Hicks neutral' technological regression in the production function (1) will have no change in outlays. An example of how a change in the production function can result in famine is provided by the following modification

[Mathematical Expression Omitted] (1a)

Initially we assume that d = 0 and so the first order conditions and resulting demand and supply functions are unaltered. Now consider the effect of a negative displacement in d. We have

[Alpha][l.sub.i]/[Alpha]d = [(1 - [Beta]/[Beta]).sup.[Beta]] [(r/w).sup.1-[Beta]] [l.sub.i] [greater than] 0, [Alpha][k.sub.i]/[Alpha]d = -[(1 - [Beta]/[Beta]).sup.[Beta]] [(w/r).sup.[Beta]] [l.sub.i] [less than] 0, [Alpha][c.sub.i]/[Alpha]d = 0 (16)

from which it is evident that provided factor prices are unaltered, the expenditure on future production is unchanged in the face of a small reduction in d, though less labour and more land will be employed on the domestic holding. The result will be that future output will be reduced, that is, [Mathematical Expression Omitted]. Of course the reduction in labour demand by agents will result in a change in factor prices. Since we are considering infinitesimal changes it is preferable to deal will a continuous distribution of endowments with density [Mathematical Expression Omitted]. Equilibrium in the labour market demands

[Mathematical Expression Omitted] (17)

with the result

[Mathematical Expression Omitted] (18)

The effect of a negative shock on the productivity of labour will be to lower the wage causing famine if the initial wage was at [c.sup.*]. In terms of policy, this situation will be similar to that resulting from a generalised loss of endowments though the optimal balance between consumption and production will clearly be altered. However, since famine can develop without current food availability being affected, this does provide a further illustration of the contrast between the entitlement analysis and the FAD doctrine.

The crucial characteristic of the famine period is that the wage rate falls below subsistence and to be capable of supplying labour an agent must supplement the wage by some of his endowment. Those with insufficient endowments fall into the starvation set. The endowment loss reduces the demand for land but competitive pressure leads to the rent level falling by more than the wage rate, with the result that production becomes more land-intensive despite the presence of unemployment. The character of the food market is unaltered, though the quantities traded are obviously reduced.

4. Policy implications

The avenues through which policy can counter famine can be classified by reference to eq. (6). To remove an individual from the starvation set either endowment can be increased to lower [Mathematical Expression Omitted] below the market rate or alternatively the market rate can be increased to cover [Mathematical Expression Omitted], respectively endowment augmentation and wage fixing. In order to compare the approaches and their consequences it is clear that the policy objectives and environment will have to be the same in both cases. To normalise the objective we assume that it is to render the starvation set empty. In terms of environment we first assume that no information is available concerning the endowment level of any agent. While drastic, such a perspective allows a clear and fair comparison of the two types of policy.

4.1 Relief works (RW)

Let us first consider public works, a policy whose pedigree stretches back several centuries and whose vitality can be seen in Maharashtra Employment Guarantee Scheme today. In order to prevent starvation the wage rate offered by the government must be at least [c.sup.*] (assuming that the poorest labourer has no endowments whatsoever), which will be greater than (13). As a consequence the government accepts responsibility to employ the excess supply in the labour market. Those demanding labour will have [l.sub.i] [greater than] 1, which through (5) will in turn imply that the set of agents that demand labour, D, is defined by [Mathematical Expression Omitted]. Thus the demand for labour, [L.sub.D], is

[Mathematical Expression Omitted] (19)

where m is the number of members in D. Labour supply, [L.sub.S], can be determined in an analogous manner which will allow excess supply in the market to be found

[Mathematical Expression Omitted] (20)

where [Mathematical Expression Omitted] is the mean endowment level of the entire population. Clearly the larger [Mathematical Expression Omitted] is, the lower the excess supply. The relief works will require an outlay of rice [c.sup.*] times the right hand side of (20).

The operation of relief works increases aggregate full income, [summation of] [F.sub.i] where i = 1 to N, through

two channels: firstly, [w.sub.F]-[c.sup.*] ceases to be negative in contrast to the famine situation; secondly, since the starvation set is now empty, [Mathematical Expression Omitted] will generally be expanded. Given (7) the demand for land at the famine rent level will increase, necessitating an increase in rent to restore equilibrium; we have: [Mathematical Expression Omitted] and so landlords benefit. As a consequence, a tax on land to finance the works would appear to be reasonable.

Since the land market is assumed to be competitive, all land will be employed in agricultural production. The effects of the introduction of relief works on the level of aggregate endowments in the next period will thus depend on the extent of labour input into domestic production. It can be shown that

[Mathematical Expression Omitted] (21)

The condition on the left hand side of (21) can be expressed as

[Mathematical Expression Omitted] (22)

The left hand side of (22) is the ratio of the aggregate endowments of those in the starvation set to the aggregate endowments in excess of subsistence of those agents economically active in the famine situation. If this ratio is greater than the proportionate change in the wage rate, then relief works will increase the aggregate level of endowments in the next period. Clearly the lower the famine wage, the more likely that agricultural production will be reduced by the introduction of relief works.

When the wage prior to the loss of endowments equals [c.sup.*] then the condition in (21) is particularly straightforward: it becomes [Delta] [greater than] n/N or the proportion of endowments lost being less than the proportion of agents in the starvation set.

4.2 Food distribution (FD)

An alternative policy to relief works is to distribute food directly to agents.(12) In order that both policies are implemented under the same conditions we assume that administrators have no information on the endowment level of any agent and that all agents apply for any ration that the government distributes. In order to render the starvation set empty administrators will give [Mathematical Expression Omitted] rice to each agent where [Mathematical Expression Omitted] is the quantity that will just achieve full employment in the labour market without starvation, that is, an agent with no endowment who receives [Mathematical Expression Omitted] will just be able to survive on his earnings from the labour market. Thus we have

[Mathematical Expression Omitted] (23)

The equilibrium wage is found by following the procedure in (8) which gives

[Mathematical Expression Omitted] (24)

If the wage rate before the shock was [c.sup.*] then the new wage rate under FD is just [Delta] times this. Comparing the wage rates gives

[Mathematical Expression Omitted] (25)

so if the ration is greater than the difference between mean endowment in the active set and that overall then the wage rate will rise under FD.

Since FD gives all agents a ration this, combined with the wage rate, will be greater than [w.sub.F] and so aggregate full income, and thus rents, will be greater under FD. Given (8) and (10) the factor price ratio w/r will be less under FD because all agents will in the active set. This also implies that future production, [Mathematical Expression Omitted], will increase under FD and will regain the pre-shock level.

4.3 A comparison of RW and FD

The total cost of RW will be the right hand side of (20) times [c.sup.*]. But this is [Mathematical Expression Omitted] and so the total cost of the two policies is the same (if the pre-shock wage was [c.sup.*] then the cost equals replacing the endowment loss). This is despite the apparent profligacy of FD, which gives the same ration to the rich as to the poor. By construction the wage under FD is lower than RW. Aggregate full income is the same under the two policies, as (23) shows that the wage plus ration under FD equals the wage under RW, and so too will be the rent level.

The crucial difference between the two policies relates to the level of production. Under RW the wage is above the market clearing rate and so excess supply exists and the government has to become an employer. With FD all labour is employed in domestic production and aggregate endowments are regained in the next period. Clearly this is the major advantage of FD, given that both policies are constrained to render the starvation set empty. By contrast, the lower level of aggregate endowment in the next period under RW makes it possible that famine will persist. No matter what possible infrastructural improvements are brought about by RW, these will not influence the output of the next period. Indeed, it highly unlikely under the circumstances of possible famine that these will be executed efficiently (even the title of Basu's, 1981, paper does not inspire confidence in this matter). The diversion of agents from labour on domestic holdings is the critical weakness of RW in comparison to FD. Only if the opportunity cost of labour for the last agent taken onto the works is zero will this objection cease to hold.(13) However, the ability of FD to achieve the ex ante endowment level fails to link relief and development, as desired by most relief agencies today, since the level of vulnerability is unchanged. RW though has the potential to accomplish this longer term objective.

These points can be seen by comparing the level of endowment in the next period of the individual agent under the two policies. The full income of each agent is the same under both policies since [c.sup.*] will be met by either the wage under RW or the wage plus ration under FD and so we have [Mathematical Expression Omitted].

Then

[Mathematical Expression Omitted] (26)

and thus under RW the agent in the next period will have [Theta][c.sup.*-[Beta]] of his current endowment while under FD it will be greater than this by a factor of [[Delta].sup.-[Beta]] if we assume that the wage before the shock was [c.sup.*]. Under FD each agent regains his preshock endowment in the following period while under RW it will be less by a factor of [[Delta].sup.[Beta]]. Relative inequality remains the same under both policies.

4.4 Targeting relief

Both RW and FD assume that administrators of relief have no information concerning the endowments of agents. What is the impact on the assessment of the policies if this assumption is relaxed? An objective comparison demands that the available information is the same in both cases; let us say that administrators know whether an agent is currently in the starvation set, though are not able to rank agents beyond this. With relief works then the wage remains at [c.sup.*] but now admission is restricted to those in the starvation set. The wage for those in the active set thus stays at [w.sub.F]. Clarity in this discussion is furthered by assuming that the economy consists of a unit of labour continuously distributed according to the cumulative distribution function G. We determine employment on the targeted relief works (ETRW) as before which yields

[Mathematical Expression Omitted] (27)

where Gs = G(S), the proportion of agents in the starvation set and [Mathematical Expression Omitted] is their mean endowment level. It is obvious from (20) that the excess labour supply falls and thus that the level of endowment in the next period is increased by targeting. The second term on the right hand side of (27) accounts for the domestic labour of those agents who were in S. For those not on the public works the level of domestic labour is the same as in the famine situation, so aggregate endowments are greater in the next period for TRW than in famine.

The total cost of TRW is (27) multiplied by [c.sup.*] from which we derive

[Mathematical Expression Omitted] (28)

so targeting reduces costs since the famine wage is below [c.sup.*].

Targeting with food distribution is less straightforward than with RW. The injection of rice into S can lead to an increase in labour supply with a consequent fall in the wage rate. Thus the proportion of agents receiving rations, [Mathematical Expression Omitted] in general. [Mathematical Expression Omitted] is such that when given to [Mathematical Expression Omitted] agents the resulting wage rate just ensures that no-one starves. We obtain

[Mathematical Expression Omitted] (29)

so [Mathematical Expression Omitted]. Since

[Mathematical Expression Omitted] (30)

targeting reduces the cost of the relief program. As with FD all labour is employed in production so that the pre-shock level of endowment is regained in the next period. Moreover

[Mathematical Expression Omitted] (31)

where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] otherwise. The second equality in (31) follows from the assumption of a pre-shock wage of [c.sup.*] Thus the targeted food distribution policy has the potential of reducing inequality in the next period, by allowing the wage to fall below [c.sup.*] so reducing the price of future endowment and by selectively augmenting the endowment of the poor.

Comparing TFD and TRW produces little in addition to the comparison of their unrestricted alternatives. The crucial weakness of relief works is that the level of current production and hence future endowments is less than potential; the advantage of targeting is that it reduces the extent of this. The advantage with regard to cost cannot be established analytically.(14) However, it must be pointed out that the merit of public works is frequently maintained to be that willingness to participate is an effective self-targeting device (see Besley and Coate, 1992). The notion of targeting the works relinquishes the strongest argument in their favour. Two observations are germane at this point: firstly, as pointed out in Section 2, the reservation wage falls with endowment among the poor, so lowering the wage on the works will not sort the poor from the not-so-poor; secondly, even the EGS, a highly successful public works scheme, has been forced to ration employment (Ravallion et al., 1993).

5. Conclusions

According to Baird Smith, writing in 1861, famines in India were 'rather famines of work than of food' (see Dreze, 1990, p.17); essentially food was available but the effective demand by the poor was absent. This paper has sought to present this proposition within a simple general equilibrium model. To be brutally direct, a starving man is an unemployed man and the cause of the latter is asserted to be in the functioning of the labour market. Nutritional considerations set a floor to the food wage rate that any individual can accept and be physically capable of performing the required labour. Moreover, among the poor this floor declines with endowment. Consequently those in direst need are underbid for work by those slightly better-off. The market clears, but with starvation.

The obvious response to Smith's analysis is the provision of relief works, as was frequently done with some success in India during the colonial period and with greater success under independence. Such a policy can be shown, however, to be inferior to a general distribution of food to all agents, when the level of the ration is taken as that which just permits the labour market to clear with no starvation. This permits all rural labour to be employed on agricultural production so that, unlike the situation with public works, the level of aggregate endowments returns to normal in the following period. Targeting lessens the disadvantages of public works but at the cost of abandoning its supposed central advantage - that it is self-targeting. These implications may temper some of the enthusiasm with which public works are advanced as a policy to counter famine.

Acknowledgements

The author is grateful to Parimal Bag, Siddiq Osmani, and an anonymous referee for their comments upon an earlier draft that led to many improvements. The remaining deficiencies are the author's sole responsibility.

1 However, Watkins and Menken (1985) downplay the demographic role of famines.

2 Coles and Hammond (1995), demonstrate that, once each agent's consumption set is modified to deal with the inherent non-convexities introduced by the dichotomy between survival and death, all the classical existence and efficiency theorems apply.

3 The food market has been the subject of a theoretical model developed by Desai (1989) where two agents, a farmer and a fisherman (specialisation in production is complete), produce an equilibrium solution in which relative prices fail to play an allocative role. While the plausibility of Desai's model has been attacked by Duffy and Outram (1994), his attempt to model famine-prone economies formally was considered by Outram (1994) to constitute 'a great service' by indicating the feasibility of the approach.

4 The agent may be considered a household and so [c.sub.*] (just) covers the consumption required to ensure survival, that is, it secures the energy required for the resting metabolism of household members, together with the energy necessary for physical labour by the worker and household duties, such as cooking and cleaning, for those members not participating in the labour market.

5 Following Coles and Hammond (1995) survival itself could be considered as a dimension of consumption space. But given that our concern is principally to appreciate the nature of market adjustment during a famine such complexity can be distracting.

6 This reflects the production framework employed: storage can be easily included by modifying (1) to

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the amount of rice saved in the current year. Defining [Mathematical Expression Omitted] means that the Kuhn-Tucker conditions imply [Gamma] [less than or equal to] [z.sub.i] and [Beta][f.sub.i]/w[l.sub.i] [less than or equal to] [z.sub.i]. Constant returns to scale imply that the term on the left hand side of the latter equation will be the same irrespective of the scale of production and its value is determined independently of [Gamma]. For utility to be positive at least one of these equations will have to be zero and in general both will not be. Thus either storage or production is undertaken and since the former would imply no production whatsoever in the rural economy only the latter case is considered.

7 I am grateful to an anonymous referee for pointing this out.

8 The equilibrium wage rate may be determined as follows: rank all agents by endowment so that [Mathematical Expression Omitted]. For n = 2, 3 . . . N, let [m.sub.n] be the mean of the [Mathematical Expression Omitted] and define [mw.sub.n] = [Alpha][Beta]/1 - [Alpha][Beta]([m.sub.n] - [c.sup.*]) where it is assume that [mw.sub.n] will be non-negative for some n. Next define [Mathematical Expression Omitted]. Then [w.sub.e] is the highest value of n for which [mw.sub.n] [greater than or equal to] [S.sub.n]. There is, however, a difficulty, here. Around equilibrium suppose there may be a number of agents with the same level of [mw.sub.n]. As these are taken on, one by one, into employment, [mw.sub.n] will be falling but [S.sub.n] will be constant. Thus equilibrium may occur with some of these labourers employed and others not, despite having identical endowments. A sufficient condition to avoid this would be that it is possible to rank endowments strictly, i.e. [Mathematical Expression Omitted] which would mean [S.sub.n] was strictly increasing.

9 Dasgupta (1993) has used the nutrition-productivity trade-off to explain general but not immediately fatal destitution. Those that cannot find employment are forced to live upon common-property rights. The reasoning in this paper may be considered a cruder and more extreme version.

10 [Mathematical Expression Omitted].

11 The variance of the triangular is [Mathematical Expression Omitted] while that of the uniform is [Mathematical Expression Omitted].

12 The policies are being compared at the level of some generality; the issue of whether the government should provide cash or food, analysed by Coate (1989), requires that market structures other than the perfectly competitive one be developed.

13 Datt and Ravallion (1994) find that in one village in Maharashtra about 80% of EGS employment came out of male unemployment (days when work was sought but not found). The rest came largely from own farm labour which will obviously reduce endowment in the following year. Unfortunately their data start 1975/6 and so just miss the drought period where the relief works achieved a notable success but when the relative labour displacement would be more relevant to our analysis.

14 [Mathematical Expression Omitted].

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