@jpfeifer basically I have three firms that produce from capital and labor inputs from a standard CES function.

Let say that I have an agriculture sector, industry sector and other industries sector. Each has a different amount of labor and capital inputs. My issue is with aggregation in total labor and total capital for market clearing. I wondered if I could sum the amount of inputs across sectors to have the total amount of labor and capital in the economy.

Wages and rental capital price are the same across sectors. There is no differentiation of labor or capital quality, so labor and capital across sectors are of same function and quality. Their prices are the same.

That means aggregate capital and labor are just the sums of the ones in the respective sectors. So the above is correct. The only complication may be the timing for capital: Timing of capital in two sector economy

I have a hard time finding the steady state equation that determines this l1.

My households supply labor inelastically, so they do not choose endogenously labor. The three firms choose the sectoral split of labor, but I somehow cannot manage to find the SS equation for this split l1, l2 and l3 that are the labor share of the sectors in total labor. Is this split exogenous rather than endogenous ? My first hint would be that the split in endogenous as one sector can become bigger than the others and demand for a higher share of labor than the others that are declining.

@jpfeifer thanks. How do you make sectoral shares endogenous if labor is supplied inelastically but households, such that labor does not show up in the householdâ€™s utility function? When it does, it is pretty straightforward.

There is one equation that Iâ€™m missing here in my model. For now, I fixed the sectoral share l_r and l_f. If I want to make l_r and l_f endogenous, I need to add two equations, of which one would be given by the labor market equilibrium condition: 1 = l_f + l_r. The other equation to had, I have no clue (maybe a problem with p_r the price of the second firm. Price of the final good firm is normalized to 1 and is the numeraire).

I am not sure I understand the problem. You know that households will supply 1 unit of total labor. Demand needs to adjust so that that one unit is demanded (unless there is unemployment). If there is competition, wages in all sectors typically need to be equal. Then the marginal products of labor implicitly define labor demand of each sector, which together with the adding up constraint should allow to obtain the solution.

Iâ€™ve been trying to find the steady state value of labor sector shares, but I always hit the wall. I somewhat end up with two unknowns in one equation. Thereâ€™s something that Iâ€™m not seeing, even though this looks very basic.

I am not saying you need to start from there, but that these equations should help determine the equilibrium. If you had endogenous labor supply, you still would not be able to do without them.

I am jumping back on this thread. You said â€śYou know that households will supply 1 unit of total labor. Demand needs to adjust so that that one unit is demanded (unless there is unemployment). If there is competition, wages in all sectors typically need to be equal.â€ť. Does that mean that the sectoral labor shares will be constant through time as wages equates in all sector at each time, so that there is no incentive to move from on sector to another ?