# How to derive the a = rB identity from the CES production function?

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.

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

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

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:

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

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

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

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:

$$$$MP_K = \frac{\partial Y}{\partial K} = \alpha \beta^{\frac{-1}{\sigma}}.$$$$ Assuming perfect competition, the rental rate of capital, $r$, is equal to the marginal product of capital, $MP_K$. Therefore, we have:

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

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

$$$$r = \frac{rK}{K} = \frac{r}{\beta Y} K$$$$

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

Simplifying this expression, we get:

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