<p>The production function <span class="math-container">$f$</span> is typically taken to be concave with <span class="math-container">$f(0)=0$</span>.</p>
<p>Taking the derivative of the function you have
<span class="math-container">$$F(h) = f(h) - f'(h)(1+h)$$</span>
where <span class="math-container">$h \equiv \frac{H}{K}$</span>
yields:
<span class="math-container">$$F'(h) = f'(h)-f''(h)(1+h)-f'(h) = -f''(h)(1+h)$$</span>
which is positive by concavity. Since it is monotonically increasing and we start with <span class="math-container">$f(0)=0$</span>, if <span class="math-container">$\delta_K - \delta_H&gt;0$</span> we will only have it hit this value only once.</p>

CC BY-SA 4.0
2022-04-21T20:20:19.187Z
2022-04-21T20:20:19.187Z
Fića
<p>I have difficulty with understanding how the AK model is derived from a one-sector model with physical and human capital.</p>
<p>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
<span class="math-container">\begin{equation} 
Y=F(K,H)
\end{equation}</span>
where <span class="math-container">$F$</span> exhibits the standard neoclassical properties, including constant returns to scale in <span class="math-container">$K$</span> and <span class="math-container">$H$</span>. Using the condition of constant returns to scale, we can write the production function in an intensive form:</p>
<p><span class="math-container">\begin{equation} 
Y=K\cdot f\bigg(\frac{H}{K}\bigg)
\end{equation}</span></p>
<p>where <span class="math-container">$f'(H/K)&gt;0$</span>.
Let <span class="math-container">$R_K$</span> and <span class="math-container">$R_H$</span> 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:</p>
<p><span class="math-container">\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}</span></p>
<p>The rates of return to owners of capital are <span class="math-container">$r_K=R_K-\delta_K$</span> and <span class="math-container">$r_H=R_H-\delta_H$</span>, respectively,</p>
<p>Equalization of rates of return implies
<span class="math-container">\begin{equation}
f(H/K)-f'(H/K)\cdot (1+H/K)=\delta_K-\delta_H
\end{equation}</span></p>
<p><strong>How is it possible that this condition has a unique, constant value of <span class="math-container">$H/K$</span>? In other words, how does it follow that there exists a solution for <span class="math-container">$H/K$</span> which is also unique?</strong></p>
<p>I know that, subseqeuntly, the production function can be written as
<span class="math-container">\begin{equation}
Y=AK
\end{equation}</span>
where <span class="math-container">$A \equiv f(H/K)$</span>.</p>

<macroeconomics><endogenous-growth>
Broad capital in the AK model - derivation
freakonomist
# Broad capital in the AK model - derivation

(Asked by: username_0 on 2022-04-20)

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 $A \equiv f(H/K)$.

**Tags:** macroeconomics, endogenous-growth


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## 1 answers

### Answer by username_1 on 2022-04-21. Score: 2 | Accepted answer

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.


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