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title: CES Function

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# CES Function

## Definition 
**Constant Elasticity of Substitution Function**

$$Q (x,y)= (a x^\rho + b y^\rho )^\frac{1}{\rho}$$, where $a,b,\rho$ are constant in $\mathbb{R}$.

## Properties of CES Functions

### Constant elasticity of substitution
$$\frac{dy}{dx} = -\frac{a}{b}(\frac{x}{y})^{\rho -1}\\\frac{y}{x} = (\frac{b}{a} TRS)^{\frac{1}{1-\rho}}\\\log(\frac{y}{x})=\frac{1}{1-\rho}\log(TRS) + \frac{1}{1-\rho} \log(\frac{b}{a})\\ \text{Elasticity of substitution}= \frac{1}{1-\rho}$$

## Linear Case
CES becomes linear function when $\rho = 1$

$$Q=ax+by$$

## Cobb-Dougalas Case
CES becomes Cobb-Dougalas function when $\rho$ closes to $0$

$$\log Q = \frac{1}{\rho} \log (a x^\rho + b y^\rho)\\\lim_{x \to 0}\log Q= \lim_{x \to 0}\frac{a x^\rho \log x+ b y^\rho \log y}{a x^\rho + b y^\rho}=\frac{a \log x + b \log y}{a+b}\\\lim_{x \to 0}Q=x^{\frac{a}{a+b}}y^{\frac{b}{a+b}}$$, where the second quation is derived by [L'Hôpital's rule](https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule).

## Leontif Case
CES becomes Leontief function when $\rho$ closes to $-\infty$

If $x > y$,

$$\\lim_{x \to -\infty}\log Q= \lim_{x \to -\infty}\frac{a x^\rho \log x+ b y^\rho \log y}{a x^\rho + b y^\rho}= \\ \lim_{x \to -\infty}\frac{a \frac{x}{y}^\rho  \log x+ b \frac{y}{y}^\rho \log y}{a \frac{x}{y}^\rho + b \frac{y}{y}^\rho} = \frac{b \log y}{b} = \log y$$

If $x=y$,

$$\lim_{x \to -\infty}\log Q=  \lim_{x \to -\infty}\frac{a \frac{x}{x}^\rho  \log x+ b \frac{y}{x}^\rho \log y}{a \frac{x}{x}^\rho + b \frac{y}{x}^\rho} = \frac{a \log x + b \log x}{a+b} = \log x$$

If $x<y$,

$$\lim_{x \to -\infty}\log Q=  \lim_{x \to -\infty}\frac{a \frac{x}{x}^\rho  \log x+ b \frac{y}{x}^\rho \log y}{a \frac{x}{x}^\rho + b \frac{y}{x}^\rho} = \frac{a \log x}{a} = \log x$$


Hence, $$Q=\min\{x,y\}$$
