<p>To derive the accounting identity <span class="math-container">$\alpha = r \beta$</span> using the production function</p>
<p><span class="math-container">$$
\begin{equation}
Y = F(K,L) = \left(\alpha K^{\frac{\sigma -1}{\sigma}} + (1 - \alpha)L^{\frac{\sigma -1}{\sigma}}\right)^{\frac{\sigma}{\sigma -1}}
\end{equation}
$$</span></p>
<p>We can use the following steps:</p>
<p>Rewrite the production function as a function of capital-output ratio, <span class="math-container">$\beta$</span>:
Since <span class="math-container">$\beta = K/Y$</span>
We can rearrange the production function to get <span class="math-container">$K$</span> in terms of <span class="math-container">$Y$</span> and <span class="math-container">$\beta$</span>: <span class="math-container">$K = \beta Y$</span>. Substituting this expression into the production function yields:</p>
<p><span class="math-container">$$
\begin{equation}
Y = F(\beta Y, L) = \left(\alpha (\beta Y)^{\frac{\sigma -1}{\sigma}} + (1 - \alpha)L^{\frac{\sigma -1}{\sigma}}\right)^{\frac{\sigma}{\sigma -1}}.
\end{equation}
$$</span></p>
<p>Take the partial derivative of the production function with respect to <span class="math-container">$K$</span> and holding <span class="math-container">$L$</span> constant, we get:</p>
<p><span class="math-container">$$
\begin{equation}
\frac{\partial Y}{\partial K} = \alpha \left(\frac{Y}{K}\right)^{\frac{1}{\sigma}} = \alpha \beta^{\frac{-1}{\sigma}}.
\end{equation}
$$</span></p>
<p>Rewrite the partial derivative in terms of the marginal product of capital:
Since the partial derivative of output with respect to capital represents the marginal product of capital, we can write:</p>
<p>$$
\begin{equation}
MP_K = \frac{\partial Y}{\partial K} = \alpha \beta^{\frac{-1}{\sigma}}.
\end{equation}</p>
<p>$$
Assuming perfect competition, the rental rate of capital, $r$, is equal to the marginal product of capital, $MP_K$. Therefore, we have:</p>
<p><span class="math-container">$$
\begin{equation}
r = MP_K = \alpha \beta^{\frac{-1}{\sigma}}.
\end{equation}
$$</span></p>
<p>Since the rental rate is the return on capital per unit of capital, we can write:</p>
<p><span class="math-container">$$
\begin{equation}
r = \frac{rK}{K} = \frac{r}{\beta Y} K
\end{equation}
$$</span></p>
<p>combining the last 2 expressions of <span class="math-container">$r$</span>, we get:
<span class="math-container">$$\frac{r}{\beta Y} K = \alpha \beta^{\frac{-1}{\sigma}}$$</span></p>
<p>Simplifying this expression, we get:</p>
<p><span class="math-container">$$\alpha = r \beta^{\frac{1}{\sigma}}$$</span></p>

CC BY-SA 4.0
2023-04-12T13:22:01.667Z
2023-04-12T13:56:32.407Z
2023-04-12T13:56:32.407Z
Macosso
<p>I am wondering how using this particular production function</p>
<p><span class="math-container">$$ Y = F(K,L) = \left(\alpha K^{\frac{\sigma -1}{\sigma}} + (1 - \alpha)L^{\frac{\sigma -1}{\sigma}}\right)^{\frac{\sigma}{\sigma -1}}$$</span></p>
<p>where <span class="math-container">${\sigma}$</span> is the elasticity of substitution between labour and capital, we can derive the accounting identity <span class="math-container">$\alpha = r \beta$</span> (where <span class="math-container">$\beta$</span> is wealth-income ratio = <span class="math-container">$K/Y$</span>). I know how it can be done using an ordinary Cobb-Douglas production function, but I doubt that taking partial derivatives for <span class="math-container">$K$</span> would work here. Am I supposed to take partial derivatives for both part? derivatives for labour and capital and then find it out? I tried doing that so that I got <span class="math-container">$$w/r = (1-\alpha L^{\frac{-1}{s}})/(\alpha K^{\frac{-1}{s}})$$</span> but I don't know where I am supposed to go from there.</p>

<macroeconomics><mathematical-economics>
How to derive the a = rB identity from the CES production function?
Notthedarkweb
# How to derive the a = rB identity from the CES production function?

(Asked by: username_0 on 2023-04-11)

I am wondering how using this particular production function

$$ Y = F(K,L) = \left(\alpha K^{\frac{\sigma -1}{\sigma}} + (1 - \alpha)L^{\frac{\sigma -1}{\sigma}}\right)^{\frac{\sigma}{\sigma -1}}$$

where ${\sigma}$ is the elasticity of substitution between labour and capital, we can derive the accounting identity $\alpha = r \beta$ (where $\beta$ is wealth-income ratio = $K/Y$). I know how it can be done using an ordinary Cobb-Douglas production function, but I doubt that taking partial derivatives for $K$ would work here. Am I supposed to take partial derivatives for both part? derivatives for labour and capital and then find it out? I tried doing that so that I got $$w/r = (1-\alpha L^{\frac{-1}{s}})/(\alpha K^{\frac{-1}{s}})$$ but I don't know where I am supposed to go from there.

**Tags:** macroeconomics, mathematical-economics

---

## 1 answers

### Answer by username_1 on 2023-04-12. Score: 1

To derive the accounting identity $\alpha = r \beta$ using the production function

$$
\begin{equation}
Y = F(K,L) = \left(\alpha K^{\frac{\sigma -1}{\sigma}} + (1 - \alpha)L^{\frac{\sigma -1}{\sigma}}\right)^{\frac{\sigma}{\sigma -1}}
\end{equation}
$$

We can use the following steps:

Rewrite the production function as a function of capital-output ratio, $\beta$:
Since $\beta = K/Y$
We can rearrange the production function to get $K$ in terms of $Y$ and $\beta$: $K = \beta Y$. Substituting this expression into the production function yields:

$$
\begin{equation}
Y = F(\beta Y, L) = \left(\alpha (\beta Y)^{\frac{\sigma -1}{\sigma}} + (1 - \alpha)L^{\frac{\sigma -1}{\sigma}}\right)^{\frac{\sigma}{\sigma -1}}.
\end{equation}
$$

Take the partial derivative of the production function with respect to $K$ and holding $L$ constant, we get:

$$
\begin{equation}
\frac{\partial Y}{\partial K} = \alpha \left(\frac{Y}{K}\right)^{\frac{1}{\sigma}} = \alpha \beta^{\frac{-1}{\sigma}}.
\end{equation}
$$

Rewrite the partial derivative in terms of the marginal product of capital:
Since the partial derivative of output with respect to capital represents the marginal product of capital, we can write:

$$
\begin{equation}
MP\_K = \frac{\partial Y}{\partial K} = \alpha \beta^{\frac{-1}{\sigma}}.
\end{equation}

$$
Assuming perfect competition, the rental rate of capital, $r$, is equal to the marginal product of capital, $MP\_K$. Therefore, we have:

$$
\begin{equation}
r = MP\_K = \alpha \beta^{\frac{-1}{\sigma}}.
\end{equation}
$$

Since the rental rate is the return on capital per unit of capital, we can write:

$$
\begin{equation}
r = \frac{rK}{K} = \frac{r}{\beta Y} K
\end{equation}
$$

combining the last 2 expressions of $r$, we get:
$$\frac{r}{\beta Y} K = \alpha \beta^{\frac{-1}{\sigma}}$$

Simplifying this expression, we get:

$$\alpha = r \beta^{\frac{1}{\sigma}}$$


---

