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?