How to represent the elasticity function with demand on the X axis? How to transform the elasticity formula to this effect?

Question

The formula for price elasticity of demand uses price as independent variable.

But I wonder what the elasticity graph looks like on an ordinary microeconomics system of coordinates, where demand is on the X axis.

After some reflection, it seems to me that my problem amounts to : how to convert the elasticity formula ( in which the input is price) into an equivalent formula in which the input ( indep. variable) is demand. For it seems to me that only the transformed formula would be suitable for ordinary microeconomics representation ( with demand on the X axis).

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My goal is to visualize the graph of the elasticity function for a linear demand curve .

The problem I face is that the elasticity function graph I came up with looks unfamiliar.

I suppose my formula for the elasticity function contains a mistake, but I can't locate it.

Here is what I've done ( and I add a Desmos image below).

(1) I start with a demand function ( with price as independent variable) :

$$D(x)= a-bx$$.

(2) I transform this function into a price function ( with demand as independent variable), in order to obtain the traditonal demand curve ( with demand on the X axis and price on the Y axis) :

$$P(x)= - \frac 1b x +\frac ab$$.

(3) I use the calculus version of the elasticity function, namely :

$$ \epsilon_{\small P}= \frac {\mathit dD(P)} {\mathit dP} \times \frac {\mathit P} {\mathit D}$$

wich ( so it seems) should yield

$$ \Large\epsilon(x) = D'(x) \frac {P(x)}{Q(x)}$$

and finally ( since $D'(x)=-b$ here)

$$ \Large\epsilon(x) = -b \frac {P(x)}{Q(x)}$$.

But, as I said above, the graph of my alledged elasticity function looks unfamiliar.

In particular, it seems to me that the elasticity should be equal to $1$ for the X-value of the middle point on the demand curve.

Desmos (https://www.desmos.com/calculator/fp7elscgtq) :

Answer 1

The graph is not correct for your parameters $a=9.3$ and $b=1.5$, elasticity will be equal to 1 (in absolute value) at price:

$-1 = -1.5 \frac{P}{Q(P)} \implies -1 = -1.5 \frac{P}{9.3−1.5P} \implies P \approx 3.1$. At the price of 3.1 the quantity demanded is supposed to be $Q=9.3-1.5P=4.65$ If the green curves are the elasticity they clearly cross the point $(4.65,3.1)$ nowhere.

To plot this correctly you also have to solve the elasticity formula for $P$ as a function of $Q$ so you need to plot:

$$P = -\frac{1}{b} \frac{Q}{\epsilon} \implies P= -\frac{1}{−1.5} \frac{Q}{-1.5 \left(\frac{P}{9.3−1.5P}\right)} $$

and then plot this, however you would get a nonsensical graph (or to be more specific the graph would be a shadow of elasticity projected onto (Q,P) space, but such shadow has no interpretation and is not informative as you cannot know what the elasticity is from it). Elasticity is neither price nor quantity so it is impossible to plot it on a same 2D graph as quantity and price, you need to add at least one extra dimension.

If you want to plot it properly you can use R:

library("plotly")
p<-(1:10)
demand = =9.3-1.5*p
elasticity = -1.5 *(p/(9.3−1.5*p))
  
fig<-plot_ly(x=demand, y=p, z=0, type="scatter3d", mode="lines") %>% add_trace() 
fig %>% add_trace(x=0, y=p, z=elasticity, type="scatter3d", mode="lines")

I forgot to label axes but $x$ is quantity, $y$ price and $z$ elasticity. Demand is orange elasticity is green.

We can verify that the picture above is correct by plotting all combinations of $P,Q, \epsilon$. The graph does not have high granularity (I generated $P$ at increment of 1) but you can see that at $P\approx 3$, $Q\approx4.65$ and $\epsilon \approx -1$ exactly as we calculated above. Hence these graphs are correct.

PS: Also price is not function of $x$ in the elasticity formula. Moreover, $x$ cannot be at the same time price and quantity, you use it for all variables.