While the IS-LM model already tells us something about inflation - via the quantity of money - there is also the labour market where wages drive up costs and prices. The IS-LM sectors of the economy and the labour market are linked via Value Added Y.
The production function
For our purposes we can use a Cobb-Douglas function with employment LE and capital KE:
YR = Y0 LE a KE 1 - a
Y
P YR = W LE + i PK KE,
We assume that firms maximise profits - and since we assume constant returns to scale, there is no surplus. If firms accept wage W, then the marginal productivity of labour equals the real wage W / P, and then this determines LE which must be at most labour supply LS. Unemployment then follows as u = 1 - LE / LS. If companies also accept the rental price of capital, then the marginal productivity of capital must equal i PK / P, and this determines the employed real capital stock KE, which must be at most total stock KS.
The additional equations from these marginal conditions are (and we assume expectational equilibrium on these too):
LE =