Mar 30 JDN 2460765

A lot of economists seem puzzled by the fact that co-ops are just as efficient as corporate firms, since they have this idea that profit-sharing inevitably results in lower efficiency due to perverse incentives.

I think they’ve been modeling co-ops wrong. Here I present a new model, a very simple one, with linear supply and demand curves. Of course one could make a more sophisticated model, but this should be enough to make the point (and this is just a blog post, not a research paper, after all).

Demand curve is p = a – b q

Marginal cost is f q

There are n workers, who would hold equal shares of the co-op.

Competitive market

First, let’s start with the traditional corporate firm in a competitive market.

Since the market is competitive, price would equal marginal cost would equal wage:

a – b q = d q

q = a/(b+f)

w = d (a/(b+f)) = (a d)/(b+f)

Total profit will be

(p – w)q = 0.

Monopoly firm

In a monopoly, marginal revenue would equal marginal cost:
d[pq]/dq = a – 2 b q

If they are also a monopsonist in the labor market, this marginal cost would be marginal cost of labor, not wage:

d[d q²]/dq = 2 f q

a – 2 b q = 2 f q

q = a/(2b + 2f)

p = a – b q = a (1 – b/(2b + 2f)) = (a (b + 2f))/(2b + 2f)

w = d q = (a f)/(2b + 2f)

Total profit will be

(p – w) q = ((a (b + 2f))/(2b + 2f) – (a f)/(2b + 2f))a/(2b + 2f) = a²/(4b + 2f)

Now consider the co-op.

First, suppose that instead of working for a wage, I work for profit sharing.

If our product market is competitive, we’ll be price-takers, and we will produce until price equals marginal cost:

p = f q

a – b q = f q

q = a/(a+b)

But will we, really? I only get 1/n share of the profits. So let’s see here. My marginal cost of production is still f q, but the marginal benefit I get from more sales may only be p/n.

In that case I would work until:

p/n = f q

(a – b q)/n = fq

a – b q = n f q

q = (a/(b+nf))

Thus I would under-produce. This is the usual argument against co-ops and similar shared ownership.

Co-ops with wages

But that’s not actually how co-ops work. They pay wages. Why do they do that? Well, consider what happens if I am offered a wage as a worker-owner of the co-op.

Is there any reason for the co-op to vote on a wage that is less than the competitive market? No, because owners are workers, so any additional profit from a lower wage would simply be taken from their own wages.

If there any reason for the co-op to vote on a wage that is more than the competitive market? No, because workers are owners, and any surplus lost by paying higher wages would simply be taken from their own profits.

So if the product market is competitive, the co-op will produce the same amount and charge the same price as a firm in perfect competition, even if they have market power over their own wages.

Monopoly co-ops

The argument above doesn’t assume that the co-op has no market power in the labor market. Thus if they are a monopoly in the product market and a monopsony in the labor market, they still pay a competitive wage.

Thus they would set marginal revenue equal to marginal cost:

a – 2 b q = f q

q = a/(2b + f)

The co-op will produce more than the monopoly firm..

This is the new price:

p = a – b q = a(1 – b/(2b+f)) = a(b+f)/(2b + f)

It’s not obvious that this is lower than the price charged by the monopoly firm, but it is.

(a (b + 2f))/(2b + 2f) – a(b+f)/(2b + f) = (a (2b + f)(b + 2f) – 2 a(b+f)²)/(2(b+f)(2b+f))

This is proportional to:

(2b + f)(b + 2f) – 2(b+f)²

2b² + 5bf + 2f² – (2b² + 4bf + 2f²) = bf

So it’s not a large difference, but it’s there. In the presence of market power in the labor market, the co-op is better for consumers, because they get more goods and pay a lower price.

Thus, there is actually no lost efficiency from being a co-op. There is simply much lower inequality, and potentially higher efficiency.

But that’s just in theory.

What do we see in practice?

Exactly that.

Co-ops have the same productivity and efficiency as corporate firms, but they pay higher wages, provide better benefits, and offer collateral benefits to their communities. In fact, they are sometimes more efficient than corporate firms.

Since they’re just as efficient—if not more so—and produce much lower inequality, switching more firms over to co-ops would clearly be a good thing.

Why, then, aren’t co-ops more common?

Because the people who have the money don’t like them.

The biggest barrier facing co-ops is their inability to get financing, because they don’t pay shareholders (so no IPOs) and banks don’t like to lend to them. They tend to make less profit than corporate firms, which offers investors a lower return—instead that money goes to the worker-owners. This lower return isn’t due to inefficiency; it’s just a different distribution of income, more to labor and less to capital.

We will need new financial institutions to support co-ops, such as the Cooperative Fund of New England. And general redistribution of wealth would also help, because if middle class people had more wealth they could afford to finance co-ops. (It would also be good for many other reasons, of course.)