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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.

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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:

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

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