Melberg, Hans O. (1998), Pay according to need or effort? About models in economics

_{[Note for bibliographic reference: Melberg, Hans O. (1998), Pay according to need
or effort? A starting point for thinking about the value of models in economics , www.oocities.org/hmelberg/papers/980303.htm]

[Note: This paper is not proof-read, and HTML makes it a bit difficult to write
"nice" equations]

Pay according to need or effort?

A starting point for thinking about the value of models in
economics

by Hans O. Melberg

Introduction

For some time I have intended to examine the value of models in economics and the best way
of doing so - I think - is to start by presenting some models. This is what I will do in
this review. Based on this model, and a few more, I hope to present a more verbal argument
on the general value of models and formal reasoning in a future paper.

In this review I want to examine a model by Amarthya Sen which tries to clarify the
problem of pay according to need or effort. The model I will use is a simplified version
of Sen's which was developed by K. O. Moene (University of Oslo). Since I want the review
to be useful to people with less than perfect mathematical backgrounds, I have included
not only the final equations, but also the calculations (and some explanation). Hopefully
this is useful for some people.

In order to avoid making this a pure mathematical exercise, I shall try to criticise the
model, and more generally reflect on why we build models at all. For instance, in the
model I shall use it emerges that - contrary to our immediate intuition - pay according to
effort is not socially optimal. This is one criterion for a good model; that it
brings out a something we did not know.

The Model

Imagine that your utility depends positively on your income (y) and negatively on your
effort (e), so that we have the following utility function for every person (i):

U_i = y - c(e)

Now, assume that some people also care about the utility of other people. Hence:

V_i = U_i + sigma h_ij U_j

This equations says that person (i) evaluates the "goodness of the world"
according to how much utility he himself has (U_i) and the sum of utilities of
other people (U_j). The purpose of hij is to allow the person (i) to give more
or less weight to different persons (j), specifically that he attaches greater importance
to his own utility than another person's utility if h is less than one. To make things
simple we shall assume that there are N identical people and that each person gives the
same importance to everybody else (the same h for everybody). This implies that:

V = U + h (N -1)^U

Or, in words: a person's evaluation of the situation is given by the size of his own
utility and the total sum of the utility of all the other people multiplied by his degree
of concern for these people (h). The number of other people is (N-1) and all the other
people have the same utility ^U.

Using the information we have so far we are able to construct a measure of a person's
degree of "sympathy" for other people. Consider the following equation:

S = [1 + h (N - 1)] / n

If h=0 you do not care about other people, and S=1/n (a small number, no sympathy). If h=1
you think another person's utility is as important as your own utility S=1 (maximum
sympathy).

The welfare of society is given by the following utilitarian welfare function:

W = N U, or since we know U (which is the same for all people):

W = N [y - c(e)]

In order to derive some results, we also need a production function:

Q = Q(L, A, v₁...v_n)

Total production (Q) is a function of the number of efficient labour units used, a fixed
factor (like land) and variable capital goods (₁...v_n).There are
three things to note here. First, that L does not simply measure the number of workers,
but the number of people multiplied by their effort (e), i.e. the number of efficient
labour unit inputs:

L = e N

Second, the existence of A - a factor except for labour and normal capital goods - is
important, as we shall see later. Third, we are assuming that the production function has
constant returns to scale (CRS), which means that a doubling of all inputs will double the
output.

Income per capita is given by total production (at price = 1), minus non-labour costs (C):

y = [Q - C] / N

And costs are the amount of capital goods used multiplied by their price:

C = sigma P_k v_k

We have now defined the relationships necessary to compare pay by need and pay by effort.
Before we examine this, however, we should find the socially optimal result.

The Social Optimum

To find the social optimum we simply maximize the welfare function. We have:

W = N [ y - c(e)]

We know that y = Q-C / N, so substituting this and rearranging we have:

W = Q - C - N c(e)

To maximize this, we derive with respect to e (we can control effort) and v (since we also
control the amount of capital input):

dW/de = NQ'_L - N c'(e)

dW/dv = v_k - P_k

[For those who do not get this immediately, here is an explanation: Q is a function of L,
A and v, and L defined as Ne. This means that we have to derive Q with respect to e
implicitly. First we derive L with respect to e (and gets N), second we multiply this by
the derivative of Q with respect to L (Q'_L). As for the derivative of W with
respect to v we note that both Q and C are functions of v.]

In sum, the first order conditions for social optimum are:

Q'_L = c'(e)

P_k = v_k

The first says that we should choose effort so that the last unit of effort equals the
marginal increase in production of using more effort. The second condition simply means
that we use less of the more expensive factors of production.

The interesting issue is now to compare the first order conditions for social optimum with
the results that emerge if we pay according to effort or need.

Paying according to need

Assume we let each person choose his own level of effort, and that we pay according to
need. For the sake of simplicity, I shall simply use the following equation to define
"pay by need":

y = [Q - C] / N

In other words, that everybody has the same need and gets paid the same. One could, of
course, imagine alternative ways of doing this. For instance, one might define need in
terms of number of children, degree of physical handicap or some other characteristic.
However, the equation above is by far the easiest to handle mathematically.

Which level of effort would a person choose if he was paid according to need defined in
the equation above? To answer this we assume that each person maximize V (satisfaction of
extended preferences). Thus:

max V = U + h (N - 1) ^U

max V = [Q - C] / N - c(e) + h (N - 1) [(Q - C) / N - c(ê)]

We take the derivative with respect to e (the person's level of effort):

Q'_L / N - c'(e) + h (N - 1) Q'_L / N

Setting Q'_L outside we get:

Q'_L [ (1 + h (N-1) / N ] = c'(e)

Recall the expression for S, we have the following first order condition for optimum when
pay is distributed according to need:

Q'_L S = c'(e)

If we compare to the first order condition for social optimum, we understand that if there
is perfect sympathy, we still have the optimal level of effort. However, if there is less
than perfect sympathy (S is less than one) people will supply less effort than is socially
optimal. The intuition is simple: The extra income coming from an additional unit of
labour is shared with everybody so you do not work as hard as you would if you had
received all the extra income from your work. This is a relatively well know conclusion:
paying according to need (and not effort) means that people will work less hard. What is
not obvious, however, is that the "opposite" - pay according to effort - need
not be better.

Pay according to effort

One way of formalizing pay according to effort, is the following:

y = [ (Q - C) / L] e

The equation says that people are paid a share of the profit (revenue - costs) and that
the share depends on the level of effort you supply; the more effort, the higher pay.

If we substitute this income equation into the utility function, and the utility function
into the extended preferences function, we get the following equation which we want to
maximize:

V = [ (Q - C) / L] e - c(e) + h (N - 1) [[ (Q - C) / L] ê - c(ê)]

Derive with respect to e to get:

(Q - C) / L + e [ [Q'_L L - (Q - C)] / L²] - c'(e) + h (N - 1) [ [Q'_L
L - (Q - C)] / L²] ê

Which is the same as

(Q - C) / L + Q'_L e/L - (Q - C) / L² - c'(e) + h (N - 1) [Q'_L
ê/L - ê (Q - C) / L²]

In equilibrium ê=e and L=Nê so the above can be rearranged as:

Q'_L [ 1/N + h (N - 1) / N ] + (Q - C) / L [1 - 1/N - h (N - 1) / N] = c'(e)

If we now define:

b = (Q - C) / Q (average profit per unit produced)

and c = (Q'_L/Q) L (elasticity of production with respect to labour)

The we can write:

Q'_L [1/N + h (N-1) / N + b/c (1 - S)] = c'(e)

[If you are having problems understanding this, plug in for b and c, and you will find
that you get the original equation]. Or, and this is the final result:

Q'_L [S + b/c (1 - S)] = c'(e)

What does this equation tell us about the nature of optimum when people are paid according
to effort? If S=1 (perfect sympathy) we get Q'_L=c'(e) and we are in the social
optimum. But, if b>c and S<1 (and A>0) we can see from the equation that people will supply more
effort than is socially optimal. In short, under the stated conditions S + b/c (1 - S) is
larger than one and we need to supply more effort to make Q'_L go down in order
to make the left hand side equal to the right hand side.

How do we know that b>c? Remember that we assumed a production function that has
constant returns to scale. Mathematically:

Q = Q(L, A, v₁...v_n)

CRS implies that:

c + e_A + sigma e_k = 1 (sum of elasticities of factors of production
is one)

e_k is sigma (Q_k v_k/Q), but we know that in equilibrium Q_k
= P_k. Moreover, we know that sigma P_k v_k/q is the same as
C/Q (since C=sigma P_k v_k). Thus, we have:

c + e_A + C/Q = 1

Or, to make it more obvious:

1 - C/Q = c + e_A

Which is the same as saying:

(Q - C) / Q = c + e_A

Remember than b = (Q - C) / Q, we now have:

b = c + e_A

This proves that as long as A is not zero then b > c.

Robust mix

So far I have showed how pay according to need results in not enough effort, while pay
according to effort gives too much effort (compared to social optimum). An interesting
question is how we can find a balance between paying according to need and effort which
produces the socially optimum level of effort. It turns out that the answer to this
question is not too difficult to find.

Assume that each person receives some of his income according to need (specifically: a),
and some of his income in proportion to his effort (1 - a). We have:

y = a (Q - C) / N + (1 - a) [(Q - C) / L] e

Before we start substituting this and maximizing, we should note that this work has
essentially been done in the first section of the paper, the only difference being that we
now have to include (a) and (1- a). To reduce our workload we simply take the first order
conditions we already have and combine them as follows:

a S Q'_L + (1 - a) Q'_L [ S + b/c (1 - S)] = c'(e)

Rearranging we have the following first order condition:

Q'_L [S + (1 - a) (1 - S) b/c] = c'(e)

For this to produce the socially optimal result, we must have:

1 - a = c/b

[Since the condition then becomes Q'_L = c'(e)]

So, the problem of whether to pay according to need or effort is reduced to a problem of
choosing the size of "a" (how much weight to place on payment according to need
relative to effort). Moreover, from the model we understand that the sice of "a"
depends on b (the size of value added) and c (the elasticity of production with respect to
labour). This sounds intuitively correct, the more sensitive the production is to changes
in effort, the more you pay according to effort.

The intuition

Even without the model we might understand that paying according to effort may make people
work too much. Imagine that you are one of ten people who live on an island and that the
only thing people desire is coconuts. Unfortunately, there are no coconut trees on the
island. However, luckily for you a helicopter arrives with 100 000 free coconuts each
year. The question then arises: How should these coconuts be distributed? Assume you
decide to give most coconuts to the person who digs the deepest whole (roughly equal to
pay according to effort). The digging does not serve any productive purposes, it only
functions as a way of determining who gets most coconuts. It should be obvious that this
is not a very good way of distributing coconuts.

Of course, the real world is more complex: Work usually makes the cake larger and it is
difficult to find a good criterion for need (which raise the possibility of manipulation,
political problems, rent seeking and so on). Yet, the general intuition still holds: As
long as some part of the cake is independent of effort, it would create too strong
incentives to work if we distribute the whole cake according to effort.

Criticism

I find it difficult to criticise models of this sort. To argue that it is not realistic,
would be besides the point. It does not attempt to explain an empirical phenomenon, so it
cannot fail in this respect. Of course, one might criticise it for making unrealistic
assumption (for instance CRS - Constant Returns to Scale), but this is a relatively
standard assumption. One might argue that Sen's way of modelling prefernces is a bit
curious; Why not include other people's welfare directly in your utility function if you
really care about other people (and not through the extended preference function as Sen
has done)? However, doing so would not change the conclusion (Sen says), so it is not a
serious objection here. One might also criticise the model for leaving out some variables.
For instance, the payment method does not only determine the size of the current cake, but
also size of the future cake (rate of growth, innovation). This (may) give greater force
to the "pay according to effort" argument, but it is a question of degrees.

I guess the point of the model is really to show something that we previously had not
given close thought. When faced with an incentive system that does not pay according to
effort, most people (including me) immediately responds that it must be more efficient to
change it so effort is given a greater reward. Sen's model shows that this need not be
true, and he shows the variables that determine the relationship. In this sense the model
is valuable; it produces a rigorous, policy relevant and surprising result.

Conclusion

This review is supposed to form the starting point for a larger reflection on the use of
models and formal reasoning in economics. The point of this review was simply to present a
model since I believe a methodological discussion on the use of models should be based on
concrete examples. the next essay will also present a model (a somewhat easier and shorter
model), while the last essay will be a more verbal discussion of some of the arguments
(including those presented by Paul Krugman in the essay "Two cheers for
formalism" and his book about economic theory, development economics and economic
geography.

_{Reference

Sen, A. (1966), Labour allocation in a cooperative enterprise, Review of Economic
Studies, vol. 33, no. 96, pp. 363-371

Moene, K.O., Lecturenotes}

_{[Note for bibliographic reference: Melberg, Hans O. (1998), Pay according to need or
effort? A starting point for thinking about the value of models in economics , www.oocities.org/hmelberg/papers/980303.htm]}}