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This article provides partial mathematical analysis of Amartya Sen’s published paper “Peasants and Dualism with or without Surplus Labor”. This paper may provide useful illustrations of the applications of mathematics to economics. Here, three portions of Sen’s paper “the simplest model, production for a market response and to withdrawal of labor” are discussed in some details. Results of the study are given in mathematical formulations with physical interpretations. An attempt is taken here to make the Sen’s paper more interesting to the readers who have desire for detailed mathematical explanations with theoretical analysis.

Amartya Kumar Sen is the most important and prolific living philosopher-economist. At present, he is Thomas W. Lamont University Professor and Professor of Economics and Philosophy, Harvard University. He was born in Santiniketan (India) and studied at Calcutta and at Cambridge. He has influential contributions to economic science in the fields of social choice theory, welfare economics, feminist economics, political philosophy, feminist philosophy, identity theory and the theory of justice. He was awarded the Nobel Prize in Economic Sciences in1998 [

In this study, we have discussed peasant economies on the basis of Sen’s published paper “Peasants and Dualism with or without Surplus Labor” [

In this article, we explore elementary mathematical techniques in some details with displaying diagram where necessary. We have chosen this article of Sen for mathematical review because we have observed that we can do some work on it which will be beneficial for the modern peasants. We stress application of mathematics in the Sen’s paper so that readers can realize it easily. Although Sen’s paper was published in 1966, we thought its usefulness would remain same to some (but very few) peasant seven 50 years later in 2016. We consider here some explicit functions with the stated properties, such as the derivative being positive by Sen. In this review paper, we set two examples to examine various aspects, such as points of equilibrium clearly and in some details.

The objective of the study is to represent mathematical analysis of Sen’s paper mentioned above. Although the paper was published in 1966, we thought its importance would remain present to few farmers even in 2016. We hope detailed mathematical analysis will be helpful to the readers those who want to work on peasant family. Main objective of this review paper is to help the peasants of Bangladesh those who are in backward and may be benefited from this study.

Amartya Kumar Sen has given peasants economies in his published paper in 1966, where he discusses the economic equilibrium of a peasant family, the effect of surplus labor and withdrawal of labor, dual equilibrium between peasant and capitalist, and efficiency of resource allocation in peasant agriculture [

Michael P. Todaro and Stephen C. Smith have revealed that the agricultural progress and rural development in developing nations and expressed the progressive improvement in rural levels through increases in small-farm incomes, output and productivity, along with genuine food security [

In this study we have used the secondary data and analyze on previous published papers. This is a review paper and discusses the mathematical analysis of Sen’s paper “Peasants and Dualism with or without Surplus Labor”. In this work we introduce two examples and try to give mathematical framework which (we think) Sen has not provided in detail. We have used techniques of the optimization of differential calculus. We also discussed the geometrical interpretation of mathematical results. In addition we have displayed diagrams where appropriate.

Here we have discussed basic assumptions of Sen’s economic equilibrium of peasant model. Suppose a community of identical peasant families each with

is the marginal productivity of labor. From our common sense,

For the maximization output

On the other hand

The total income (output) of the family, Q, is shared equally among all the members of the family, but the total labor L, is shared equally among all the working members. Let q is the individual income of any member and l is the amount of labor of any working member as,

Again, every member of the family has a personal utility function

From (6) we see that the marginal utility from income is positive and non-increasing. From (7) we observe that the marginal disutility from labor is non-negative and non-decreasing [

Each person’s notion of family welfare W in a suitable sense is given by the net utility from income and effort of all members taken together attaching the same weight to everyone’s happiness. Let a subscript i represents the i^{th} individual, then the family welfare W is given by;

If it is assumed that all the functions

Each individual could equally well regard W as a function of Q and L, since,

ther, since, Q is a function of L, we can conclude that W is also a function of L;

Assume welfare is maximized by

provided that

since

From (11) and (12) we get;

Sen defined x as the “real cost of labor” which indicates that labor is applied up to the point where its marginal product equals the real cost of labor.

In the light of above discussion we consider two explicit examples as follows.

We make an ad hoc assumptions about the form of the functions

We assume

where

Also the conditions (2), (6) and (7) are all satisfied if all the constants are positive. Again, (15a) and (16a) give;

From (10) we get (15a, b, c) for the welfare function W as;

Now we get,

which is the same as the explicit form as (11). Now from (19) we get;

Now we define a new variable X in terms of L and choose the constants a and k as;

Using (21), Equation (20) becomes the quadratic equation for X as;

Solution of (22) becomes;

For the relevant solution we should consider only positive sign of (23), then we get,

where

For real solution we get from (21);

Inequality (25a) is free of

Here we made ad hoc assumptions about the form of the function, and show that these satisfy the relevant conditions, and then proceed via the corresponding welfare function, to obtain the value of L which maximizes this function at

since

We observe that if l tends to

From (26a) we get;

for

Now we can express the function

indicates that b must have dimension of inverse money. Similarly, if labor L is measured in hours, then the constant “a” has the dimension of money/hour^{2}, etc. To avoid the different form of dimension we avoid dimension in our calculations. The welfare function W is given by;

From (30) derivative of

For maximum welfare (i.e.,

We know that a cubic equation can be solved in radicals in terms of the coefficients. We observe that solution of (32) will be complicated, so that we cannot find exact and necessary information from it. In this situation we proceed in an indirect way. First, we introduce some preliminary remarks.

The property of a cubic equation that it has three roots, all real, or one real and two complex. In this example we are confined to find a root in the interval

From (30) we see that welfare function

i.e.,

, (by (29),). (34)

Here the constant

Now the second derivative of

We observe that this function is negative for all values of L in our expected interval

which

examine if reasonable parameter values can be found that will achieve this circumstance. Now we write (32) using (25) and (29) as follows:

Since

From (34) we get;

Since

But we need a more consistent value and we choose (39) for our convenience way as follows:

Using (37) to (40) we can write (36) in a more convenience way as solvable form as follows:

Since

Solution of 2nd equation of (42) is;

Hence,

fare function (30) as follows:

where,

Since

For

From (44) and (45) we have two properties as;

i.e.,

We represent (47) in

Again

approached from below. Since

this property of

Let us fix the values of

and set

For

Let,

The quadratic

with

mum at

Now consider the mild pathological situation. For this we consider (47) the in equation as equation,

, (by (29),). (54)

Using (54) in (32) we get;

The solutions of (55) are;

As we have seen earlier that two roots of (56) are complex, let us now choose

A. K. Sen has considered the circumstance when the product Q is not directly useable by the peasants, so it is exchanged for goods directly enjoyable by the peasants. Also it may happen that part of the product Q is used while the rest is exchanged for other goods. If the whole amount C of the new product, the individual share be-

ing

The price of output Q in terms of C is p per unit;

So that the maximum of the family welfare is given by;

Let us now consider a situation in which a part of the product Q is sold and a part is consumed. Individually, C amount of the purchased commodity and q of the self product one is enjoyed per member. Let y be the properties of output that is marketed. Sen defines a utility function with the following properties;

Again we have;

with allocation rules;

We have also used the same form of the utility function for both of the examples, A and B. Now we consider the utility function,

Taking derivatives of (63) with respect to q we get;

Hence from (64) we have;

We observe that (65) agrees with (60) and also agrees with examples A and B.

Sen also discusses the problem of surplus labor and response of peasant output to withdrawal of labor. The surplus labor is defined as that part of the labor force in this peasant economy that can be removed without reducing the total amount of output produced, even when the amount of other factors is not changed [

where (66a) is an equation but not identity. Here maximization of welfare function occurs at

sume (66a) to be valid for all

to l we get;

Now if

So that when one working member leaves, he provides support for K members (including himself) and so the peasant family is left with one less working member and K less consuming ones.

Taking derivatives of (14) with respect to

We have,

Differentiating (14),

Using (70) to (72) in (69) we get;

Simplifying (73) we get;

Using (14),

This is Sen’s Equation (31) but we have derived the equation more detailed than Sen has. Sen introduces some elasticities as follows [

E is the elasticity of output with respect to the number of working members, m is the absolute value of the elasticity of the marginal utility of income with respect to individual income, n is the elasticity of marginal disutility from work with respect to individual hours of work, G is the elasticity of output with respect to hours of labor, g is the absolute value of the elasticity of the marginal product of labor with respect to hours of labor. These quantities are defined by the following relations:

Also we have,

Using (5), (76) and (77) in (75) we get the response equation;

Now we consider the example A. From (15a) we get;

From (76), using (15a) we get;

From (76) and (15a), (16a, b, c) we get;

Using (79) to (81) in (78) we get;

If

Using (83) and (21), (82) becomes;

In this case L be very large to satisfy (84) and marginal disutility schedule approach to the vertical position, which of course will tend to toward constancy of the change in labor hours proportional to the change in the number of working people [

Now we consider a special case for

Using (76) and (21), (85) becomes;

Equation (86) implies

people are withdrawn from the peasant economy, with an unchanged number of hours of work per person, the marginal physical return work will increase [

In this study, we have analyzed some parts of Sen’s paper “Peasants and Dualism with or without Surplus Labor” with detail mathematical calculations. We have tried to give the physical interpretations of the mathematical results clearly (as far as possible). We hope the readers will feel comport when they study this article. We have not discussed all the portions of the paper of Sen. So that readers can take the opportunity to discuss the parts which we have not tried. In their study, they can set new examples to discuss the paper of Sen.

Haradhan Kumar Mohajan, (2016) Amartya Sen’s Peasant Economies: A Review with Examples. Open Access Library Journal,03,1-15. doi: 10.4236/oalib.1102337