Short-Run Equilibrium

First of all have a look at the goods market. Output is determined by consumption plus investment Y = C + I, hence the goods market is in equilibrium. Firms produce a homogenous commodity with the help of capital and labour. Assume a Cobb-Douglas technology Y = Kα Nβ with α > 0, β > 0 and α+ β = 1. Firms maximize profits II = pY − rpK − wN under perfect competition. For that reason, the interest rate corresponds to the marginal product of capital r = ∂Y/∂K = αY/K. From this one can deduce the desired stock of capital K* = αY/r. Likewise real wages agree with the marginal product of labour w/p = ∂Y/∂N = βY/N. Properly speaking N denotes labour demand which adjusts endogenously. On the other hand $$\overline{N}$$ symbolizes labour supply which is given exogenously. Let labour supply $$\overline{N}=\overline{{{N}_{0}}}{{e}^{n\Gamma }}$$ expand at the natural rate n = const, where τ stands for time.