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While studying a chapter called price elasticity of demand in my economics course, I have been presented with something called a unit elasticity curve (some sort of a hyperbola), which has supposed applications to the subject.

So being curious, I tried to find an explicit function that would fit the described properties of this curve (the book doesn't give an explicit formula and my teacher doesn't know).

The critical property of the hyperbola should be that when given a point (x,f(x)) (which is on the line), a, lets say, 10% increase in the magnitude of x, results in a 10% decrease in the magnitude of f(x), and vice versa (hence the quotient %x/%y = -1).

Clearly, if I were to plot the points resulting from the 10% increases and decreases, I would get a function (with asymptotes), but if i were to start plotting with 5% increases and decreases I would get a slightly different (though also asymptotic) function. But my book suggests that there should only be one function! God, if I were to "plot with >100% changes, it wouldn't even be a hyperbola (asymptotic) at all!

So to solve this, I treated the % change as an infinitesimal, which gives me: (dx/x)/(dy/y) = -1, next: dy/dx = -y/x. So, I guessed xy = c as the function, but this does not satisfy the previously stated property of %x/%y = -1 for non infinitesimal percentile changes.

What am I doing wrong and what is the actual explicit function? (forgive me if I am being very confusing; if there are any extra details you would like to know, just comment) Thank you very much!

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Let $Q$ be quantity and $P$ the price. Consider a demand function with the following shape:

$$Q(P)=\dfrac{a}{P},$$

where $a$ is any positive constant. Since we can change the value of $a$, in fact, you have an infinite number of such curves.

Using calculus, the formula for the elasticity of demand is: $$\varepsilon_{Q,P}=-Q^\prime(P)\times \dfrac{P}{Q(P)}.$$

For the above function we have $Q^\prime(P)=-\dfrac{a}{P^2}$ and $Q(P)=\dfrac{a}{P}$. So plugging it back in the formula for the elasticity, you get that (for this demand function): $$\varepsilon_{Q,P}=-(-\dfrac{a}{P^2})\times \dfrac{P}{\dfrac{a}{P}}=1.$$

For this reason, any Microeconomics textbook calls the above demand the unit (price) elasticity demand (despite the article "the" notice we have a family of unit elastic demands as we can change $a$).

Finally, notice as you correctly pointed that the elasticity is one only for infinitesimal (very "small") changes in the price. So we we say that a price increase of $5\%$ will bring demand down by $5\%$, we are actually saying the change in demand will be in the ballpark of $5\%$.

The reason you get an approximation is because $Q^\prime(P)=\lim_{\Delta P\rightarrow 0}\dfrac{\Delta Q}{\Delta P}$. That is, when we compute elasticity using calculus we assume small variations in price: $\Delta P\rightarrow 0$.

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Answering the second part of your question : you need to look among the so-called "Constant Elasticity of substitution" (CES) functions. You will find a lot of references online, one of them being http://www.ecpol.vwl.uni-muenchen.de/downloads/advanced/ss08/uebung/att_ces_functions.pdf. In these notes, you will find an explicit expression for this kind of functions, as well as a proof (section 3) that they display constant elasticity of substitution.

Then, to complete you quest for functions with a unit elasticity of substitution, you should look at the equality linking the constant elasticity of substitution to the parameters of the function itself, set the constant to $1$ and see under which conditions on the parameters of the function the equality holds.

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