## The scarcity of cooperatives

Why do capital-managed firms predominate over worker cooperatives? Perhaps it’s because worker cooperatives have less incentive to expand.

### Introduction

The predominance of capital-managed firms (CMF) over worker cooperatives (WC) remains an open question in economics. Early explanations relied on a hypothesized comparative inefficiency of cooperatives. Subsequent empirical study has shown that cooperatives are at least as efficient as capital-managed firms (Doucouliagos 1995) (Estrin, Jones, and Svejnar 1987) (Craig et al. 1995) (Levine 1990).

A profusion of hypotheses has since arisen. (Dow and Putterman 1999) offers a good summary (though the term must be used loosely for a 126-page paper). One that I have not seen presented is: capital-managed firms predominate because capitalists have a greater incentive to expand than worker-owners in worker cooperatives. Roughly, for each market segment a capitalist expands into, their income increases by capital’s share of the new segment’s profit. For each market segment a cooperative expands into, the expanders receive no direct remuneration (supposing that the new market segment is also a cooperative). Any new profit goes to worker-owners in the new segment.

### Cellular automaton

We can make this hypothesis more tangible by representing it as a cellular automaton. In this automaton1, each cell represents a market segment requiring a fixed quantity of labor and capital. Adjacent cells represent similar market segments.

#### Initial conditions

In the beginning, the market is filled with empty market segments which have a random cost (represented in the automaton by the opacity of the red cell interiors) for a firm to expand into.

#### Step

##### Occupied market segments

In each step, an existing firm may go bankrupt (or exit the market segment in some other way) with fixed probability $$B = 0.1$$. If a firm goes bankrupt, its market segment becomes empty once again.

If it does not go bankrupt, the profits generated during that step are distributed. For worker cooperatives, all profits accrue to the worker-owners within the cooperative. For capital-managed segments, labor’s share of income (estimated at 70% (Karabarbounis and Neiman 2013) (Gomme and Rupert 2004)) accrues to the segment’s workers and capital’s share of income accrues to the capitalists of the segment’s firm.

The accumulated income of workers in a capital-managed segment is represented in the automaton by opacity of the purple cell interior. The accumulated income of the capitalists (of a given firm) is represented by the opacity of the purple cell border enclosing all segments owned by the firm. The accumulated income of worker-owners in a worker cooperative is represented in the automaton by the opacity of the green cell interior. Worker cooperatives sharing an ancestor (e.g. cooperative B and C were both founded by cooperative A) are enclosed by a single border.

##### Empty market segments

In each step, an empty market segment may be occupied by a newly formed firm with chance $$N = 0.001$$.

Also, in each step, an empty market segment may be subject to expansion from adjacent firms. Each adjacent firm has a 20% chance of attempting expansion (representing market conditions, firm conditions, &c.). The cost of expansion into the market segment must be less than the accumulated income of the expander and less than the projected value of the segment. In the event of multiple firms competing to expand into a single segment, the firm with the greatest valuation for the segment succeeds.

Valuations are determined thus:

Capital-managed firm
The total value a capital-management firm can expect to extract from a segment takes the form $$KP + KP(1 - B)(1 - D) + KP(1 - B)^2(1 - D)^2 + \ldots$$ where $$K$$ is capital’s share of income, $$P$$ is the per-step profit, $$B$$ is the bankruptcy rate, and $$D$$ is the firm’s discount rate (ranging uniformly from $$0$$ to $$0.2$$). Using the formula for geometric series, we can write this as $$\frac{KP}{1 - (1 - B)(1 - D)}$$.
Worker cooperative
Similarly, the total value a worker cooperative can expect to extract from a segment takes the form $$\frac{AS}{1 - (1 - B)(1 - D)}$$. $$S = GP - LP$$, where $$G$$ is the productivity advantage of worker cooperatives (set to 1.1 in the automaton) and $$L$$ is labor’s share of income, represents the comparative benefit for worker-owners in a worker cooperative over being laborers in a capital-managed firm. $$A$$ (“altruism”) represents the extent to which the expanding worker cooperative cares about these potential gains.

A quick consequence of this model is that worker cooperatives and capital-managed firms will value expansions equally (assuming equal discount and bankruptcy rates) when

\begin{align} \frac{AS}{1 - (1 - B)(1 - D)} &= \frac{KP}{1 - (1 - B)(1 - D)} \\ AS &= KP \\ A(GP - LP) &= KP \\ AP(G - L) &= KP \\ A(G - L) &= K \\ A &= \frac{K}{G - L} \\ A &= \frac{K}{K - 1 + G} \\ \end{align}

### Future work

• Allow manual specification of automaton initial state
• Increase fidelity of model
• Permit tweaking of other parameters
• Provide analysis in terms of evolutionarily stable strategy
• Survey of how cooperatives decide to expand
• Empirical examination of rate of worker cooperative formation and altruism (charitable giving, diversity (Putnam 2007), &c.)
• Empirical examination of rate of worker cooperative formation and labor’s share of income

Craig, Ben, John Pencavel, Henry Farber, and Alan Krueger. 1995. “Participation and Productivity: A Comparison of Worker Cooperatives and Conventional Firms in the Plywood Industry.” Brookings Papers on Economic Activity. Microeconomics.

Doucouliagos, Chris. 1995. “Worker Participation and Productivity in Labor-Managed and Participatory Capitalist Firms: A Meta-Analysis.” Industrial & Labor Relations Review. http://library.uniteddiversity.coop/Money_and_Economics/Cooperatives/Worker_Participation_and_Productivity-Meta_Analysis.pdf.

Dow, Gregory K, and Louis Putterman. 1999. “Why Capital (Usually) Hires Labor: An Assessment of Proposed Explanations,“.” Employees and Corporate Governance. http://www.econ.brown.edu/1996/pdfs/96-21.pdf.

Estrin, Saul, Derek C Jones, and Jan Svejnar. 1987. “The Productivity Effects of Worker Participation: Producer Cooperatives in Western Economies.” Journal of Comparative Economics.

Gomme, Paul, and Peter Rupert. 2004. “Measuring Labor’s Share of Income.” FRB of Cleveland Policy Discussion Paper. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.405.7400&rep=rep1&type=pdf.

Karabarbounis, Loukas, and Brent Neiman. 2013. The Global Decline of the Labor Share. National Bureau of Economic Research.

Levine, David I. 1990. “Participation, Productivity, and the Firm’s Environment.” California Management Review.

Putnam, Robert D. 2007. “E Pluribus Unum: Diversity and Community in the Twenty-First Century the 2006 Johan Skytte Prize Lecture.” Scandinavian Political Studies. http://www.aimlessgromar.com/wp-content/uploads/2013/12/j-1467-9477-2007-00176-x.pdf.

1. Sorry, Elm, Firefox, and this program don’t seem to mix well.