# Elasticity computations and revenue

I would have appreciated some help in this issue. I have a demand schedule that looks like this

P --> Q --> Revenue

5000 --> 10000 --> 50,000,000
6500 --> 8127 --> 52,825,500
8000 --> 6218 --> 49,744,000
9500 --> 4808 --> 45.676,000
11,000 --> 2832 --> 31,141,000


Now the price elasticities at the first 4 prices calculated as (% change in Q/% change in price) each ratio calculated over the mid-points of the arc as denominators turn out to be (abs value) -0.79, 0.32, 0.37, 0.88. Each of these is less than 1, so revenue should increase with an increase in price? But the revenue increases between 5000 and 6500 and falls thereafter? Also, the percentage fall in revenue is highest for the price increase between 9500 and 11000 whereas here the elasticity is the highest. I do not have much of an issue is I just use the straightforward % change formula without using the denominator as the halfway point of the segments. Why do we use the formula with the halfway point as the denominator? For example for a price change from P1 to P2, we use ((P2-P1)/ ((P1+P2)/2)))?e

• If you use the starting point for the percentage change rather than the midpoint then you hit the problem that going from $100$ to $80$ is a $20\%$ fall but going from $80$ to $100$ is a $25\%$ increase. Better still would be to use logarithms rather than percentages for elasticity calculations Jul 6, 2023 at 8:51

$$\varepsilon=\frac{-(q_2-q_1)(p_1+p_2)}{(q_1+q_2)(p_2-p_1)}$$
Now if $$p_2>p_1$$ and $$q_2 (demand is downward sloping) then the numerator and denominator are positive, and it follows that
$$\varepsilon<1 \iff -(q_2-q_1)(p_1+p_2)<(q_1+q_2)(p_2-p_1) \iff p_2q_2>p_1q_1$$
I.e. if and only if revenue rises with the increase in price from $$p_1$$ to $$p_2$$.