Our analysis of production starts with the micro-foundations of macroeconomics that the neoclassicals so greatly achieved. A main facet of such a concept is the quasi-concave short-run neoclassical production function, which is defined as, \(Y = F(N,\bar{K})\) which incorporates the rigidity of the level of capital in the economy. The firm’s maximisation problem therefore must be written as $$\max_{N} PY-WN$$ and substituting with the available function for production, we arrive at $$\max_{N} PF(N,\bar{K})-WN$$ which enables us to arrive at the first order conditions (FOCs) given by \(PF_{N}(N,\bar{K})-W=0\). This result could also be written as, \((W/P)=F_{N}(N,\bar{K})\) with the interpretation being that the households get paid the marginal product of labour as the real wage that is supplied by them. Differentiating the latest equation further with respect to \(N\) and \(\bar{K}\) yields:
$$d(W/P)=F_{NN}dN^D+F_{N\bar{K}}d\bar{K}\textrm{, or}\newline
dN^D=\frac{1}{F_{NN}}d(W/P)-\frac{F_{N\bar{K}}}{F_{NN}}d\bar{K}$$