Determination of L in the cross model

As firms will produce less than Y_OPT, they require less labor than L_OPT. We can determine exactly how much L they need in order to produce Y* and this level of L is denoted by L*.

Figure: Determination of L in the cross model

1. Start at bottom left. Here equilibrium level of GDP (denoted by Y*) is determined. We can add Y* on the y-axis as well because YD = Y* in equilibrium. 

2. Extend Y* to bottom-right graph. This is aggregate supply. 

3. From the production function we can determine exactly how much labor we need to produce Y*. This amount is signified by L*. 

4. Extend L* up to upper right-hand graph. As real wage is fixed, we should be on horizontal line and we find the equilibrium for the labor market. 

5. In the same figure you would also find L_OPT, quantity of labor firms would choose if aggregate demand was enough. 

Note a crucial difference amid classical and Keynesian model: in classical model we first figure out L and go from L to Y whereas in cross model we go from Y to L.