I assume a demand function ( of price) : $Q(x)= 10-2x$ , and therefore, a price function (of demand ): $P(x)= -\frac 12 x +5$.

The demand curve ( $y =-\frac 12 x +5$) represents the price function, with demand on the X axis and price on the Y axis.

My aim is to obtain, for each point $( a, P(a))$ of the demand curve , the price elasticity of demand at that point; that is , the price elasticity of demand for a $P(a)$ price.

More precsely, I want to have a point $ (a, P(a))$ moving on the demand curve and to know the value of the $PED$ for each of its positions.

I apply the general formula :

$$ E_P = \frac {\mathit dQ}{\mathit dP} . \frac P Q= Q' .\frac P Q $$

which, for a price equal to $P(a)$ corresponding to a demand equal to $a$ seems to yield :

$$E_P= Q'(P(a)) . \frac {P(a)}{a}$$.

My question is to know whether this last formula is a correct application of the general (price) elasticity of demand formula.

Below the Desmos graph : https://www.desmos.com/calculator/8bbbef98iu

enter image description here

  • $\begingroup$ you should correct your Q you say "The demand curve ( y=−1/2x+5) " but thats not a demand curve. Demand curve is Q = 10-2x, the first equation is the inverse demand function. Also you use x for both price and quantity and it can't be both at the same time $\endgroup$
    – 1muflon1
    Apr 28, 2022 at 22:20

1 Answer 1


The formulas are not correct. You do not use constant variable labels which is a problem that can easily lead to mistakes.

In your case $a=Q$, you also sometimes use $Q=x$, I will ignore that part as it is inconsistent. You also use $P=x=y$ I will only use $P$ as its unnecessarily convoluted to have 2 labels for 1 variable and again you are being inconsistent by calming that $P=x$ and $Q=x$, which is absurd as it implies $P=x=Q$. So you have following demand function:

$$ Q(P)=10−2P$$

and following inverse demand function:

$$P(Q) = - \frac{1}{2} Q +5 $$

If you want to substitute $P(Q)$ into $Q$ you will get:

$$ Q = 10 - 2(- \frac{1}{2} Q +5) \implies 0=0. $$

You basically get rid of $Q$ because by substituting inverse demand function into demand function you are essentially asking what quantity demanded will give you the demanded and answer is simply any quantity demanded since quantity demanded is just quantity demanded. That has infinitely many solutions because any number you say for demand will be correct answer to the question what is demand.

Next the derivative of $Q$ wrt $Q$ does not make sense. If you would implicitly differentiate the equation above you would end up with $dQ = dQ \implies 0=0$.

It also does not make sense to ask what is the percentage change in quantity when quantity changes by some percent. If quantity increases by 10% then clearly quantity increased by 10%.

If you want to have the $Q$ coordinate corresponding to $P$ coordinate for some elasticity, just first derive the elasticity as you would normally do:

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

which in your case will be:

$$\epsilon = \frac{-2P}{10-2P}$$

Then you can substitute the inverse demand function that tells you what $Q$ is associated with the $P$ in the elasticity:

$$\epsilon = \frac{Q-10}{Q}$$

Then you can plot the above to get graph like this:

enter image description here

Intuitively the graph is correct, we can disregard the negative region since $Q\geq 0$. It is well known fact that for linear demand elasticity is equal to $\infty$ at point where $Q=0$ and then monotonously declines (both in absolute value) until it reaches 0 at the maximum quantity (in your case 10). Clearly the graph above shows such relationship.

Your graph suggest there is linear relationship between $\epsilon$ and $Q$ which should not hold for linear demand function. Moreover, your graph does not have the well know property that PED should be infinite at point $Q=0$ and then decline from there until it reaches $0$ at $Q_{max}$ (in your case 10) which is property that all linear demand functions have, except for the constant function, but your graph clearly does not show such relationship.

  • $\begingroup$ Thanks for all these valuable and usefull comments. I do not know how I managed to make so many mistakes in this calculation. $\endgroup$ Apr 29, 2022 at 19:52
  • 1
    $\begingroup$ @VinceVickler no worries you are still learning my advice is next time take it easy don’t rush make sure you use single letter for each variable go through the calculations step by step until you get more comfortable. Also sometimes even smart PhDs make mistakes that get discovered during conferences or peer review don’t let that discourage u from the pursuit of econ $\endgroup$
    – 1muflon1
    Apr 29, 2022 at 21:21

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