Efficient Resource Allocation Within the Coalition

In economics and game theory, efficiency is identified with Pareto optimality. This in turn means that no-one can be made better off, within some objective material constraints, without shifting resources in such a way as to make another person worse off. Nevertheless, typically, there are an uncountable infinity of Pareto optima. Given measures of the benefits to each individual, u, and a mathematical representation of the constraints on these benefits, we may think of an efficient joint strategy or allocation of resources as one that corresponds to the maximum of a weighted sum of the benefits, g 1iui, where g1i = 1 and the distribution weights 1i express the relative importance of the distribution to agent i. For an idealized profit-maximizing firm, for example, l = 1 for the proprietor and l = 0 for everybody else, while for an idealized consumers’ cooperative, l has the same positive value for all customers and is zero for everybody else. Each distinct vector of distributive weights 1i may give rise to a distinct Pareto optimum, and that is why there is, in general, an uncountable infinity of them.

For a model of the business firm, the objective constraints are the effective supply of labor and its productivity, along with the outside options or rationality constraints for the individual members. If we ignore the idiosyncratic productivity differences among employees, then the efficient policies of the firm are quite familiar from established economic theory. Effort is allocated among the employees so that each has the same marginal disutility of labor (this is equation A11.7j). This marginal disutility of labor, expressed in money terms, defines the marginal cost of production (equation A11.7m). Output is allocated among the customers in such a way that each has the same marginal utility of the good or service

(equation A11.7h) and output is expanded until, again in money terms, the marginal utility of the good or service is equal to marginal cost (equation A11.7m). We may then identify marginal cost with the quotient of the marginal disutility of effort (common to all employees) and the marginal productivity of labor:

This marginal cost is the efficient price p1 in equations (11.5)-(11.7) above. Thus, each customer will take a quantity qx of the coalition’s good or service such as to make the customer’s marginal utility equal to this efficient marginal cost price.

When we allow for idiosyncratic differences in productivity, the conditions are a little more complex and less familiar. Once again output will be allocated so that all customers have the same marginal utility for the product or service of the coalition. We may define the value marginal product of labor (VMP) as the marginal product of aggregate labor times the common marginal utility of the good or service. The effort commitment of agent i will then be such that i’s marginal disutility of labor is equal to the product of the idiosyncratic productivity kt and the value marginal product of aggregate labor (equation A11.7k). Thus, the more able will be assigned to work somewhat longer hours or given more difficult assignments, given the same disutility-of-effort function. This is a result of the linear aggregation of abstract labor from effort commitments by heterogeneous employees: a different aggregation scheme might give different results.

In this case the determination of marginal cost is again a little more complex. In place of (11.14) we have

°gi

dhi

(equation A11.7n). Since k is the same for all employees (equation A11.7g) expression (11.15) is well defined. Once again, the efficient price p1 is the marginal cost price.

Thus, the quantity produced, its distribution among the customers, and the labor effort supplied and its distribution among the employees are all determined by the condition of Pareto optimality, and are identical for any set of distributive weights and so for any distribution of the benefits net of side payments. Since the normalization allows us to interpret the total benefit as the sum of the producers’ and consumers’ surpluses, and this is determined by the efficient allocation of effort, efficient total production, and efficient allocation of output among the customers, we may identify this sum as the gross value produced by the coalition, V* in equations (A11.8a) and (A11.8b). The net surplus will then be that amount minus the sum of the rationality constraints of the members of the coalition, V in equation (A11.8b). This surplus will then be distributed according to bargaining power as shown in equations (A11.8f)—(A11.8j). Equations (A11.9a)-(A11.9m) demonstrate that the same conditions are derived when the two sets of decisions, allocative and distributive, are taken separately.

In summary, then, the coalition allocates its resources just as they would be allocated in a hypothetical perfectly competitive market in which the demand side comprises the customers and the supply side comprises the proprietor and employees. For such a perfectly competitive market, the equilibrium (marginal) price would correspond to equality between marginal cost and the money expression of the marginal utility to consumers, that is, to equality of supply and demand.

This is often presented as a proof that competition promotes efficiency. Further, we find that the perfectly competitive model often predicts developments in markets that divert far from the assumptions of “many small firms,” and so on. This fact is sometimes represented as proof that competition is more pervasive than the model suggests. But this is all quite backward. What the value creation model tells us is that when (ideally) rational people engage in the cooperative activity of production for exchange, they will find ways to coordinate their action efficiently. A model that describes efficient allocations of resources is thus likely to be predictive of their actions. But it is cooperation, not competition, that leads to efficiency in market outcomes, to the extent that market outcomes are efficient. This calls in particular for reconsideration of models, such as monopoly and monopolistic competition, in which market outcomes are supposed to be inefficient in the absence of externalities, taxes or subsidies.

 
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