Solow Model

In the current section, as a frame of reference, the Solow model will be sketched out briefly, offering the real analysis of a growing economy. Firms manufacture a single product by making use of capital and labour. For ease of exposition, consider a Cobb-Douglas technology Y = Kα Nβ with α > 0, β > 0 and α + β = 1. Output can be devoted to consumption and investment Y = C + I. Households save a certain fraction s = const of income S = sY. Savings are invested I = S, thereby adding to the stock of capital $$ \dot{K} $$ = I. Moreover let labour grow at the natural rate $$ \dot{N} $$/N = n = const. Now it is convenient to state this in per capita terms. Output per head y = Y/N is a well-known function y = kα of capital per head k = K/N. Next take the time derivative of k = K/N and rearrange adequately $$ \dot{k}\ $$ = $$ \dot{K} $$/N − (K/N)($$ \dot{N} $$/N). Then substitute $$ \dot{K} $$ = I = S = sY and $$ \dot{N} $$/N = n, observing y = kα, to arrive at: 1$$ \dot{k}=s{{k}^{\alpha}}-nk $$