# What is the difference between constant elasticity of substitution and elasticity of substitution?

What is the difference between constant elasticity of substitution and elasticity of substitution? Are these formulas related? How are they different concepts? I just did a whole problem set involving these and got the right answers. But my thought process was all math. I don't feel like I understand them economically. And that's what I want explained.

By CES, I mean $$U(x,y) = A \left(\alpha x^{-\rho} + (1-\alpha) y^{-\rho}\right)^{-\frac{1}{\rho}}$$

By ES, I mean

$$\sigma = \frac{\mathrm{d}\log \left(\frac{x}{y}\right)}{\mathrm{d}\log \left(\frac{U_y}{U_x}\right)}$$

Well, you didn't get the math all right, otherwise you might have seen it yourself!

By Elasticity of substitution, you should refer to

$$\sigma(x,y) = \frac{\mathrm{d}\log \left(\frac{x}{y}\right)}{\mathrm{d}\log \left(\frac{U_y}{U_x}\right)}$$

Then, constant elasticity of substitution is a property of a function $U$ such that for some constant $\bar \sigma$,

$$\sigma(x,y) = \bar\sigma \, \forall y, x$$

That is, the elasticity doesn't depend on $x,y$. What you showed as $U$ is a function that has exactly that feature. If you compute $\sigma(x,y)$ (do it!), you will see that

$$\sigma(x,y) = \bar\sigma = \frac{1}{1+\rho}$$

That is, the elasticity of your $U$ function is constant for any allocation. Hence, we refer to that class of function (for different $\rho$) to Constant-Elasticity-of-Substitution-Functions (with share parameters $\alpha$). Cobb-Douglas preferences would be a special (limiting) case of that, with $\sigma(x,y) = 1$

### Example

Say, we produce happiness ($U$) with apples ($x$) and oranges ($y$). Lets look at the allocation $\{5,1\}$. We only have one orange, but we would really want more. Hence, we are willing to give up 2 apples to get one orange, and we would be indifferent:

$$U(5,1) = U(3,2)$$

It is very easy to substitute apples with oranges, we have a high elasticity of substitution.

Now look at the allocation $\{1,6\}$. Do you think that you would again be willing to lose $40\%$ of $x$ to get an increase of $50\%$ in $y$? Is it true that

$$U(1,6) = U(0.6, 12)$$

If yes, that means that the elasticity of substitution of $U$ (or my rather handwaving approximation to it) is the same at the two allocations $\{1,6\}$ and $\{5,1\}$. If that was true for all possible allocations, $U$ was CES.

There are some reasons why you would think some preferences are perhaps not CES: For example, if $x$ and $y$ are good complements. Then, when you have little of $x$, you're willing to give up a lot for the other, but that elasticity might decrease the closer you are to equality between the two inputs.

• But what does that mean? I got that answer. Like I don't understand why the $U$ function has to be so ugly looking. – Stan Shunpike Apr 20 '15 at 13:56
• @StanShunpike I've added an example, hope that helps. – FooBar Apr 20 '15 at 14:08
• @StanShunpike it doesn't have to be so ugly looking, that's just a very general form. You can set $A = 1$, remove the $\alpha$, $1-\alpha$ terms and still have CES. – FooBar Apr 20 '15 at 15:18
• Thanks for your replies. I still am confused, so let me try to be more specific: What does $A$ do? What is $\alpha$ do? Is $\alpha$ related to strict convexity? Why do we the the $-\rho$ power of the goods? – Stan Shunpike Apr 20 '15 at 16:19
• @StanShunpike I suggest you open up a new question about this, as this one was about the difference between ES and CES and I feel like we're diverging. – FooBar Apr 20 '15 at 16:59

Elasticity of substitution is a property of a function, like slope or curvature. Constant elasticity of substitution means that this property has the same value at all arguments of the function.

There are many explanations of the elasticity of substitution online, e.g. http://en.wikipedia.org/wiki/Elasticity_of_substitution If after reading these, you are still confused, you can ask a more specific question.