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Poverty and FaminesAn Essay on Entitlement and Deprivation$

Amartya Sen

Print publication date: 1983

Print ISBN-13: 9780198284635

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198284632.001.0001

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(p.167) Appendix A Exchange Entitlement

(p.167) Appendix A Exchange Entitlement

Poverty and Famines
Oxford University Press

A.1 Fixed Price Exchanges

X is the non‐negative orthant of n‐dimensional real space, representing the amounts of n commodities; it is the set of all non‐negative vectors of all commodities. Y is the power‐set of X, i.e., the set of all subsets of X. Let x be the vector of commodities (including ‘labour power’) that the person owns, and p is the n‐vector of prices faced by him.

Given his ownership vector x, his exchange entitlement set E(x) is the set of vectors any one of which he can acquire by exchanging x.

E ( x ) = { y | y X & p y p x } .
The function E(.) from X to Y is his ‘exchange entitlement mapping’, or E‐mapping, for short.

Two explanatory points. First, clearly xE(x). Second, the exchanges covered by (A1) are not, of course, confined to selling all of x, and a part of it can be retained (since this will not affect the exchange‐possibility of the remainder, as given by (A1)).

Let the set of commodity vectors that satisfy the specified minimum food requirement be given by FX. Starvation must occur, in the absence of non‐entitlement transfers (such as looting), if E(x) ∩ F = ∅. The ‘starvation set’ S of ownership vectors consists of those vectors x in X such that the exchange entitlement set E(x) contains no vector satisfying the minimum food requirements. Obviously, S depends on F and the E‐mapping.

S = { x | x X & E ( x ) F = } .

To illustrate consider a simple two‐commodity case with commodity 1 standing for food, and let OA in Figure A1 represent the minimum food requirement. The price ratio is given by p. The starvation set S is given by the region OAB.

More generally, when food is not one commodity but many and the ‘food requirements’ can be met in many different ways, let the minimum cost of meeting the food requirements, i.e. for attaining any vector in F, be m(p, F).

m ( p , F ) = min  p x | x F . x
Appendix A Exchange Entitlement

Fig. A1 Starvation Set

The starvation set can be alternatively characterized for this case as:
S = { x | x X & px < m ( p , F ) } .

Finally, it may be noted that it is possible to specify F taking into account taste constraints (see Chapter 2). In applying these concepts to the analysis of famines as opposed to regular poverty the taste constraints may, however, play a rather limited role. It is also possible to include essential non‐food requirements in the specification of F.

A.2 Variable Price Exchanges

If the person is not a price‐taker, then the simple model outlined above will not work, in particular equations (A1), (A3), and (A4). In general, we can characterize the exchange possibilities in terms of a ‘net cost function’ f(y, z), representing the net cost of buying y and selling z:

f ( y , z ) is a real‐valued function, with f ( O , O ) = O .
The E‐mapping can now be redefined as:
E ( x ) = { ( x z + y ) | y , z X & z x & f ( y , z ) O } .
The interpretation of z is, of course, that of the vector of sales by this person, while y stands for his purchases. Obviously, xE(x).

(p.169) The starvation set S is still given by (A2), but now coupled with (A5) and (A6).

A.3 Direct Production and Trade

The person can use his ownership vector not only for trade, or for his own consumption, but also for production. The production possibilities open to him can be characterized by another mapping Q(.) from X to Y, representing, for any vector of inputs s, the set Q(s) of output vectors, any of which he can produce.

Q ( . ) mapping from X to Y with Q ( O ) = { O } , unit set consisting of the null vector .

Consider, now, the person owning x, buying r to be used as inputs, buying y to be used for consumption, selling z to meet the cost of purchases, and producing q by using a part s of x plus purchased inputs r. The exchange entitlement mapping is now given by:

E ( x ) = { ( x s + q z + y ) | r , s , y , z X & ( s + z ) ( x + q ) & q Q ( s + r ) & f ( r + y , z ) O } .

The functions f(.) and Q(.) can be defined to take note of taxes, subsidies, social security benefits, etc.

The starvation set once again is given by (A2), combined with this.

A.4 Special Cases

We can now consider some special stipulations, taking (A2), (A5), (A7), and (A8) as the general structure.

  1. Stipulation (i): r = O.

  2. Stipulation (ii): Q(s) = {s}, unit set, keeping s unaffected.

  3. Stipulation (iii): f(y, z) = p(yz), where p is a non‐negative n‐vector.

If we stipulate (i), (ii), and (iii), we are back to the case covered in Section A.1, with exchange entitlement mapping characterized by (A1) and the starvation set by (A4). If only (i) and (ii) are stipulated but not (iii), then we have the case without direct production, but also without fixed prices for exchange, essentially the same1 as the one discussed in Section A.2. If only stipulation (i) is imposed, direct production is permitted with owned resources only, without the person being able to set himself up at all as an ‘entrepreneur’, purchasing inputs for productive use. Combined with (iii) this provides an (p.170) analogue to the usual simple characterization of production and competitive trade, as in figure A2, with OAB standing for E(x), given the production frontier CD and the inter‐good exchange rate given by angle ABO.

A.5 Economic Status and Modes of Production

The landless labourer, having nothing to sell other than his ‘labour power’ and not in a position to undertake production on his own, is covered by stipulations (i) and (ii), i.e. the case discussed in Section A.2. If the wage rate is fixed and so are the commodity prices, then this reduces to the simpler case covered in Section A.1, with stipulation (iii) being imposed as well.

The small peasant farmer, undertaking production with his own resources, including his labour power, land, etc., corresponds to the case with stipulation (i). Since typically small peasants, even in the poor developing countries, buy some inputs from outside, and sometimes even labour power (especially at the time of harvesting), it is perhaps best to think of stipulation (i) as being a bit of an exaggeration, with the true situation being captured accurately only in some model within the general framework of Section A.3.

The share‐cropper also falls in this category, since he undertakes production, gets some part of the return (and Q(.) must now be seen as his return function and not the function of total production), and buys some inputs (though typically not all). If the owner provides all the resources other than the share‐cropper's labour power, then the case is one in which stipulation (i) does hold, interpreting Q(.) as a function of his own labour.

The large farmer will clearly violate all the stipulations in question. But if he is an absentee landlord, then there will be a new stipulation that s will not contain any of one's own labour. If the absentee landlord rents out his land at a fixed rent then it will again be a case as in Section A.1 or A.2, without production being directly involved in the landlord's exchange entitlement. If he leases it out to a share‐cropper, then whether the production circumstances are directly involved or not will depend on whether he plays an active part in the production decisions. If he does, then the choices introduced by Q(.) are open to him; if not, he is just selling the services of his land for a reward, which, though variable, is not within his control once contracted out.

Similar contrasts can be drawn outside agriculture as well, e.g. the industrial proletariat living on selling his labour power, the capitalist industrialist producing mainly with purchased inputs, and so on.

When a labourer fails to find employment, the entitlement question depends on what arrangements for social security there happens to be. (p.171)

Appendix A Exchange Entitlement

Fig. A2 Entitlement Set with Own Production and Competitive Trade

If there are guaranteed unemployment benefits, then the entitlements arising therefrom can be characterized as a special case of entitlement related to labour power as such. This will require a dual set of prices for labour power, viz. a wage rate w if the person finds employment and a social insurance benefit b if he does not, with w > b. The entitlement is characterized not in terms of what he expects, but in terms of whether or not he can actually find employment. The focus is not on a person's subjective assessment, but on the real possibilities. This means that even in a given market situation there may be big differences between the positions of different workers in it, depending on whether the person's entitlement gets determined by his wage rate (or wage rates) or by social security benefits. In the absence of a social security system, the contrast (p.172) is even sharper, since the entitlement of his labour power will be zero if he cannot obtain employment.2

A.6 Own‐Production Entitlement

In Sections A.1 and A.2 a person's exchange entitlement was considered in terms of trade only. Later, production was incorporated into the general structure of exchange entitlement, treating production as a form of exchange (with ‘nature’). But in some contexts, it is useful to distinguish between the entitlements arising purely from trade and those arising purely from production without any trade. The ‘pure trade entitlement relation’ T(.) can be defined in exactly the same way as the exchange entitlement relation was defined in the absence of production possibilities—the only difference being that now production possibilities can exist without being taken into account in the T‐mapping.

T ( x ) = { ( x z + y ) | y , z X & z x & f ( y , z ) O } .

The other pure case is production without any trade, and this leads to the ‘own‐production entitlement relation’ P(.), as defined below. In Chapter 5 P(.) was also called the ‘direct entitlement relation’.

P ( x ) = { ( x s + q ) | s X & s x & q Q ( s ) }

It is easily checked that T(x), P(x) ⊆ E(x), but E(x) is not in general ⊆ T(x) ∪ P(x). Note also that x belongs to both T(x) and P(x).

The own‐production entitlement relation gives an idea of what the person can secure independently of the working of the rest of the economy. If P(x) ∩ F is non‐empty, then the person can see to it that he does not starve, no matter how the rest of the economy operates. This consideration is of some importance when the trade relations are subject to sharp fluctuations owing to forces operating on the economy as a whole, as is frequently the case in times of famine. The case of P(x) ∩ F ≠ ∅ will be called ‘trade‐independent security’.

In the literature of ‘general equilibrium’, it is typically assumed that every one has trade‐independent security. As Tjalling Koopmans (1957) puts it, ‘they assume that each consumer can, if necessary, survive on the basis of the resources he holds and the direct use of his own labor, without engaging in exchange, and still have something to spare of some type of labor which is sure to meet with a positive price in (p.173) any equilibrium’ (p. 59).3 But this is a very exacting assumption, and is violated by most of humanity in modern societies. While a peasant with his own land and other resources needed to grow food may indeed have trade‐independent security, an industrial worker with only his labour power to sell clearly does not. Nor would even the industrial capitalist, unless he happens to keep a large stock of food, since the strength of his position in terms of command over food arises from exchange and not from direct holding or of own‐production entitlement.

Even within the rural economy, landless agricultural labourers have little chance of survival except through selling their labour power, and the position contrasts sharply with that of peasants. Indeed, the growth of a labouring class with nothing but labour power to sell (i.e., the emergence of labour power as a ‘commodity’ in the Marxian sense) has led to a very widespread absence of trade‐independent security, and—as discussed in Chapter 5—the problem of vulnerability to famine situation has much to do with this development. The phase of economic development after the emergence of a large class of wage labourers but before the development of social security arrangements is potentially a deeply vulnerable one.4

Finally, even for landless rural population, the exchange entitlement can vary a great deal depending precisely on tenancy arrangements. Security of tenure gives an entitlement of a kind that, while formally involving trade, can be seen as something very like own‐production entitlement. Even a share‐cropper with security of tenure is, in this respect, in a much less vulnerable position than an agricultural labourer, who can be fired quite easily. Another advantage that the share‐cropper has over agricultural labourer relates to the fact that his returns typically take the form of a part of the actual output. If the output happens to be foodgrains, this makes him a good deal less vulnerable to the vagaries of the market than the agricultural labourer employed at a monetary wage. This lower vulnerability can, of course, co‐exist with vicious ‘exploitation’ of the share‐cropper, viewed from a different perspective.


(1) The only difference is a purely formal one, viz., that production being ‘undertaken’ with s yielding just s is not really a ‘production’ at all, though Q(s) = {s} makes it look like that, formally.

(2) This is one reason why the concept of ‘exchange entitlement’ cannot be reduced to a derivative of ‘terms of trade’, since the possibility of trade is itself a part of the picture captured by exchange entitlement, including non‐trade (e.g. unemployment). Another reason is, of course, the fact that exchange entitlements include production possibilities as well.

(3) See also Arrow and Hahn (1971), pp. 116–22.

(4) Some problems of this “pure exchange system transition”—PEST for short—are discussed in Sen (1980d).