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[Note for bibliographic reference: Melberg, Hans O. (1998), Cooperatives: A short model with surprising implications, www.oocities.org/hmelberg/papers/980309.htm]

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

 

Cooperatives
A short model with surprising implications

by Hans O. Melberg


Introduction
As promised last week, I will present a few more examples of formal reasoning in economics before I try to write a verbal argument about the value of models in social science. This week I want to present a very simple model which demonstrates what happens to the optimal number of workers in a cooperative when the price of their product increases. The original argument is from Ward (1958), but my information is from a lecture by K. O. Moene at the University of Oslo (1998).

The model
Imagine a world in which the workers divide the profit equally. In other words, the "income" to each worker is given by the equation:

y = (P F(L) - C) / L

Where P is the price of the product, F is the amount produced (which depends on L only; with a positive and decreasing marginal productivity), C is fixed non-labour costs, and L - of course - is the number of workers.

The question is now: What is the optimal number of workers in this factory if the workers want to maximize their income. To find this we simply maximize:

[(P F'(L)) L - (P F(L) - C)] / L2 = 0

P F'(L)= [(P F(L) - C)] / L

Or, in words, the cooperative will hire new workers until the marginal revenue of one more worker is equal to the money they have to give him.

What happens when the price increases?
Without the mathematical expressions it is difficult to determine what happens to the optimal number of workers when the price increases (just try!). Formally the answer can be found by taking the differential as follows:

d [P F'(L))] = d [(P F(L) - C) / L]

P dF'(L) + F'(L) dP = [L d(P F(L) - C) - (P F(L) - C) dL] / L2

P dF'(L) + F'(L) dP = [L d(P F(L) - C) - (P F(L) - C) dL] / L2

L2 P F''(L) dL + L2 F'(L) dP = L d(P F(L) - C) - (P F(L) - C) dL

L2 P F''(L) dL + L2 F'(L) dP = L P F'(L) dL + L F(L) dP - L dC - P F(L) dL + C dL

dL [L2 P F''(L) - L P F'(L) + P F(L) - C] = dP [L F(L) - L2 F'(L)]

And finally,:

dL/dP = L [F(L) - L F'(L)] / [L2 P F''(L) - L P F'(L) + P F(L) - C]

What do we know about the signs in this equation? L is positive (as long as there is production), and [F(L) - L F'(L)] is also positive; The last worker adds less to the marginal product than the second last worker. Since some people have higher productivity than the last worker, the total amount produced by L workers is higher than the marginal productivity multiplied by the number of workers, or formally: F(L) > L F'(L)

To find the signs of the other expression, we first note that from the first order condition for maximimum we have:

P F'(L) = (P F(L) - C) / L

Which means that:

C = P [F(L) - L P F'(L)]

But this implies that [L2 P F''(L) - L P F'(L) + P F(L) - C] is the same as:

[L2 P F''(L) - C - C]

[L2 P F''(L)]

And this expression is negative (assuming that the marginal productivity of labour is positive and decreasing, so F''(L) <0).

Hence we have a positive number divided by a negative number which is a negative number, or formally:

dL/dP <0

In short, when the price of their product increases, the optimal number of workers in the cooperative goes down (which in turn implies that the amount produced goes down). This is certainly surprising, and it is not easy to see without the use of formal reasoning. Usually we expect people to produce more when the price increases, not less as it seems that an optimally adjusting cooperative would do.

Criticism
In the real world, of course, there are mechanisms that prevent the perverse reaction outlined in the model above. For instance, cooperative do not fire workers (when profit is good) simply because they want even more profit (for obvious human and social reasons). Moreover, there would probably be some rules governing the exit of workers concerning compensation/payment depending on whether the cooperative was worth something (so that they could "sell" their share), or whether it had a negative net value (in which case they would have to pay to leave).

Conclusion
It seems that we have a good example of a model which is short and rich on implications. It is also counter-intuitive and to see this we need to do the math. On the other hand, being counter-intuitive is not very useful when it simply reveals that the model probably is not correct as a description of the real world (since we do not find that cooperatives fire workers easily). In that sense, being counter-intuitive simply mean being wrong compared to the facts. However, it also makes us search for mechanisms that can explain why the result described in the model is not correct (for instance, the effect of compensation to the workers that are fired), which also can be modelled.


Reference
Moene, K.O., (1998) Lecturenotes, University of Oslo


[Note for bibliographic reference: Melberg, Hans O. (1998), Cooperatives: A short model with surprising implications, www.oocities.org/hmelberg/papers/980309.htm]