Price elasticity of demand

[Benjamin]

I'm confused about what represents unit elasticity in considering a 45 degree line (e.g. y = 10 - x) vs a rectangular hyperbola (e.g. y = 1 / x).

To clarify that I'm using the right formula, PED = (%change D)/(%change P)?

The CMP (Ch3, s2.3, p6) has a rectangular hyperbola as unit elasticity but if you chart P = 1 / Q and take a starting point of P = 1 where Q = 1, if you move to P = 2 and Q = 1/2, the % change in P is +100% while the % change in Q is -50%, i.e. PED = -2, not -1.

And then comparing to a -45 degree line, it appears that anywhere along the line, PED is -1 e.g. if you start at P = 5 and Q = 5 and you move along the curve to P = 7 and Q = 3, it's a +40% move in P and a -40% move in Q, PED = -1 and this seems to be true anywhere along a 45 degree line.

Now, in Ch6, s1.3, p.4, there's a graph that shows elasticity changing as you move along a straight line but as above, I'd assume it to be constant anywhere along a straight line.

Could you please clarify how all this works and regarding straight lines, if there's any difference between it being exactly 45 degrees or not (obviously more elastic if flat i.e. small change in price leads to large change in Q but I'm wondering if there's any "special case" aspects to a 45 degree line).

Thanks in advance!

 

[Graham Aylott]

The price elasticity of demand, e = % change in Q/% change in P

i.e. e = (change in Q/Q) / (change in P)/P)

Or in the limit, as the change in P becomes very small:

e = dQ/dP * P/Q

For a rectangular hyperbola, P*Q = c (constant), so Q = c/P and:

dQ/dP = -c/(P^2)

So, e = dQ/dP * P/Q = -c/(P^2) * (P / (c/P) = -c/(P^2) * (P^2)/c = -1

for all P and Q.

Consider a 45 degree straight line demand curve with P = 10 - Q. Then Q =10 - P and dQ/dP = -1.

So, e = dQ/dP * P/Q = -1*P/(10-P) = - P/(10-P)

and at P = 10 (top intercept): e = -infinity
P = 5 (midway along the line): e = -5/(10 - 5) = -1
P= (bottom intercept): e = -0/(10-0) = 0

and e varies continuously as P varies, as you move along the straight line. In addition, the the three results above hold for any downward-sloping straight line demand curve.

The results you mention re. the straight line demand curve arise because we are using % changes over discrete changes in P (and Q) and the original method. Actually, if you use the midpoint or average method, then you will get different results.

 

[Benjamin]

Hi Graham,

Thanks for the detailed response, but I'm still confused :-(

Starting at P=5 and Q=5, if you take case one per my first post of moving to P=3 and Q=7, it looks like a 40% move in both and so PED = -1.

If you move to P=2, Q will equal 8 - that's a 60% decrease in P and a 60% increase in Q, also PED = -1.

So what calculation error have I made there? As based on what you've said, |PED| should be moving away from 1 as you get further from the start point of P=Q=5.

 

[Hemant Rupani]

Hi Benjamin,
for Q=10-p
if you consider from P=7 and Q=3, suppose price changes to 5.
e will be [(3-5)/3]/[(7-5)/7]=-7/3... same as Graham says, e=- P/(10-P)=-7/(10-7)

you were considering change from 5, hence you were getting e=-1

 

[Benjamin]

Ah gotcha. So a starting point of P=Q is a specific scenario where PED = -1 and then goes to infinity/zero away from the centre (per the pic in the CMP. Funny that... ).

Thank you!