Perfect Competition, Zero profit rule and General Equilibrium

Question

I'm reading a book where the definition of an equilibrium for a competitive economy is given as in

Kenneth J. Arrow; Gerard Debreu (1954) Existence of an Equilibrium for a Competitive Economy Econometrica, Vol. 22, No. 3. , pp. 265-290.

where the authors define an equilibrium roughly as a price vector $p$ where households maximize utility, firms maximize profit, and consumption of households satisfies the income restraint.

However, it seems to me that the zero profit rule which is often associated with equilibrium in a competitive economy is not assumed (nor implied).

So my question is how does the zero profit rule enter into general equilibrium theory?

Is it imposed as a further assumption and if so are there any seminal papers discussing the role of the zero-profit rule and perhaps like the above article giving conditions for the existence of competitive equilibrium?

Handbook chapters could also be interesting but I am not sure which handbooks one should consult when looking for microeconomic theory and general equilibrium theory.

Answer 1

Parallel to Arrow and Debreu, there is the approach of Lionel McKenzie, in which no ownership is specified and all technology has constant returns to scale. In such a model, firms can make no profit.

If all firms in an Arrow-Debreu economy have constant returns to scale, equilibrium profits must necessarily be zero. A firm then always has the option to produce nothing without any input, so profits cannot be negative. If a firm were to make positive profits, it could double the profit by doubling the production plan, which would contradict profits being maximal. So the Arrow-Debreu approach is seemingly more general than the McKenzie approach.

However, there is a way to represent the Arrow-Debreu approach within McKenzie's approach. Let the commodity space be $\mathbb{R}^l$. If $Y\subseteq\mathbb{R}^l$ is a convex production set, then there exists a convex production set with constant returns to scale and an additional factor, $Y'\subseteq\mathbb{R}^{l+1}$ such that $$Y=\big\{y\in\mathbb{R}^l:(y_1,\ldots,y_l,-1)\in Y'\big\}.$$ Indeed, the set $Y'$ must be given by $$Y'=\big\{\alpha(y_1,\ldots,y_l,-1):y\in Y, \alpha\geq 0\big\}.$$ So one can always think of decreasing returns to scale as constant returns to scale with an unmodeled factor whose net supply is $1$. Now, in terms of profits, let $p=(p_1,\ldots,p_{l+1})$ be a price system for the extended commodity space. The firm with the production set $Y'$ must make zero profits in equilibrium. So if $(y_1,y_2,\ldots,y_l,-1)$ is a profit-maximizing production plan we must have, by the mentioned zero profit condition, that $$p_1 y_1+p_2 y_2+\cdots p_l y_l + p_{l+1}(-1)=0$$ $$p_1 y_1+p_2 y_2+\cdots p_l y_l=p_{l+1}.$$ So profits for the firm with the original production set $Y$ can be interpreted as returns to the unmodeled production factor.

The economic relevance of these results is that one can always view profits as returns to unspecified factors in an economy that actually satisfies a zero-profit condition.