# Equilibrium with Externalities: Solving without FOC

I have a general equilibrium problem, where households given objective function $U(c,n)$ solve for working hours and consumption $n, c$, and firms use labor to produce a consumption good - their objective function is $V(n)$. There are potentially profits, which is paid out to the workers through wages $w$. Households spend all their labor income on the production good.

There is the externality that working more leads to higher income on the household side, and spending higher income onto the consumption good yields higher profits for the firm - in turn leads to higher wages for the household. These things just happen in my environment, take my word for it - I'm trying to simplify as much as possible.

Usual Equilibrium Household optimality, firm-side optimality and market clearing means that all FOC have to hold. Usually, I would solve this taking the first-order conditions for $U(c,n)$, taking firm's demand $Y$ as exogenous, and replace - after taking the FOC - $Y = nw$.

Here, instead I want to rather find my equilibrium without FOC. It is $w, n, c$ such that

• Firms behave optimally: $V(n) \geq V(n')$ for all $n'$
• Households behave optimally: $U(c,n) \geq U(c',n')$ for all $c', n'$
• $wn = Y$

This makes much more sense for me since it is in this environment much easier to find the stationary state in the objective functions rather than the roots of the first-order conditions.

However, I'm worried that without two steps, I cannot "first solve FOC, then replace $Y=wn$. How do I ensure that household's are neglecting the externality of their labor/consumption choice?

• I'm aware that the abstraction might make my question hard to understand - I'm ready to clarify anything that needs so. – FooBar May 26 '15 at 19:30
• What $V(n)$ is, seems important to your question. – Alecos Papadopoulos May 26 '15 at 23:06
• @AlecosPapadopoulos In the standard Firm's problem with only labor, we would have that $V(n) = An^\alpha - wn$ – FooBar May 26 '15 at 23:09
• I'm a little rusty on solving general equilibria, but it sounds to me like you want to find a pareto optimal solution, where the externality makes participants more active in the pareto optimal state than the general equilibrium? If so, don't you just solve for ideal values of C and N, rather than solving for either parties FoC? – RegressForward May 28 '15 at 14:57
• I didn't understand what is the difference between usual equilibrium and your statement. In both cases agents optimize the behaviour and prices are determined by market clearance? – AnilB Jul 26 '15 at 19:43

If firms want to maximize $V(n)$ which is some "surplus revenue over costs" function, then surplus they will produce. Meaning, that this surplus/profits cannot be given to workers, in the form of wages, because wages are cost. It may be given as dividends or, say, end-year bonus, but in any case it won't be related to $n$.
So we do not have $Y=wn$ but rather $Y = wn + d$, and the externality appears to break down.
The $Y=wn$ relation is consistent with a "non-profit" organization, which would demand as much $n$ as needed in order to have zero profits. In such a case, the wage would be equal to the average product of labor, not the marginal one.
• If firms maximize profits, then it does not hold that $Y=wn$, this is all I am saying. The profits are given to workers as some lump-sum payment, not through an ex post adjustment of the profit maximizing level of wage. – Alecos Papadopoulos May 27 '15 at 0:37