- Make our usual technical assumptions about the production function: Positive marginal products: ∂F/∂L≡FN>0, ∂F/∂K≡FK>0 and Diminishing returns: ∂2F/∂K2≡FKK ∂2F/∂N2≡FNN<0 - Also assume to “constant returns scale” i.e. raise all inputs by the same proportion and output rises by that proportion. So where z is some constant: = z∙Yt F(z∙Kt, z∙Nt) Let: z=1/Nt Then the production function can be written in per worker terms: Yt/Nt = F(Kt/Nt ,1) or: yt = F(kt,1) (y=Y/N, k=K/N) notice : = FK ∂y/∂k - Lastly assume the “Inada conditions” hold: lim FK = 0 lim FK = ∞ k→∞ k→0 - Consequences of technical assumptions: Inada conditions will push outcomes away from extremes (0 or infinite k): convenient! Are they true? No empirical evidence! (don’t see these extremes) Diminishing returns: return to k investment declines in level of k. - Plausibility? Two stories: - unit of K has little N to work with high ‘k’ then each
- Make our usual technical assumptions about the production function: Positive marginal products: ∂F/∂L≡FN>0, ∂F/∂K≡FK>0 Diminishing returns: ∂2F/∂K2≡FKK and ∂2F/∂N2≡FNN<0 - Also assume “constant returns to scale” i.e. raise all inputs by the same proportion and output rises by that proportion. So where z is some constant: z∙Yt = F(z∙Kt, z∙Nt) Let: z=1/Nt Then the production function can be written in per worker terms: Yt/Nt = F(Kt/Nt ,1) or: yt = F(kt,1) (y=Y/N, k=K/N) notice : ∂y/∂k = FK - Lastly assume the “Inada conditions” hold: lim FK = 0 lim FK = ∞ k→∞ k→0 - Consequences of technical assumptions: Inada conditions will push outcomes away from extremes (0 or infinite k): convenient! Are they true? No empirical evidence! (don’t see these extremes) Diminishing returns: return to k investment declines in level of k. - Plausibility? Two stories: - high ‘k’ then each unit of K has little N to work with