Broad capital in the AK model - derivation

Question

I have difficulty with understanding how the AK model is derived from a one-sector model with physical and human capital.

The interpretation of the AK model is that capital should be viewed broadly to include both physical and human capital. Based on the notation of Barro and Sala-i-Martin (2004, p. 211-212), the production function can be written as \begin{equation} Y=F(K,H) \end{equation} where $F$ exhibits the standard neoclassical properties, including constant returns to scale in $K$ and $H$. Using the condition of constant returns to scale, we can write the production function in an intensive form:

\begin{equation} Y=K\cdot f\bigg(\frac{H}{K}\bigg) \end{equation}

where $f'(H/K)>0$. Let $R_K$ and $R_H$ be the rental prices paid by competitive firms for the use of the two types of capital. Profit maximization then implies that the marginal product of each input equals its rental price:

\begin{equation} \partial Y/\partial K=f(H/K)-(H/K) \cdot f'(H/K)=R_K \\ \partial Y/\partial H=f'(H/K)=R_H \end{equation}

The rates of return to owners of capital are $r_K=R_K-\delta_K$ and $r_H=R_H-\delta_H$, respectively,

Equalization of rates of return implies \begin{equation} f(H/K)-f'(H/K)\cdot (1+H/K)=\delta_K-\delta_H \end{equation}

How is it possible that this condition has a unique, constant value of $H/K$? In other words, how does it follow that there exists a solution for $H/K$ which is also unique?

I know that, subseqeuntly, the production function can be written as \begin{equation} Y=AK \end{equation} where $A \equiv f(H/K)$.

Answer 1

The production function $f$ is typically taken to be concave with $f(0)=0$.

Taking the derivative of the function you have $$F(h) = f(h) - f'(h)(1+h)$$ where $h \equiv \frac{H}{K}$ yields: $$F'(h) = f'(h)-f''(h)(1+h)-f'(h) = -f''(h)(1+h)$$ which is positive by concavity. Since it is monotonically increasing and we start with $f(0)=0$, if $\delta_K - \delta_H>0$ we will only have it hit this value only once.