Question

3. For each of the following demand curves i) Find the price-elasticity of demand in terms of P ii) Determine the range of P values for which the de- mand curve is perfectly elastic, elastic, unitary elas tic, inelastic and perfectly inelastic (your answer will look like, the demand is inelastic for 0< P < 10, unitary elastic at P 10, etc) iii) Calculate the price-elasticity of demand at P-3 and give an interpretation in words of what that means in terms of percentages ery (a) Q(P) 77 -2P (b) Q(P) = 5p-1/4

Answer 1

Solution:

Formula for price elasticity of demand, ed = $(\partial Q/ \partial P)*(P/Q)$ . Furthermore, price elasticity is elastic if |ed| >1, inelastic if |ed|< 1, and unitary elastic if |ed| = 1. Then, using this formula we can find the following:

(a) Q(P) = 77 - 2P

i) So, $\partial Q/ \partial P$ = -2

ed = -2*(P/Q) = -2*(P/(77-2P))

Thus, ed = -2P/(77 - 2P)

ii) This elasticity is

Elastic if |ed| >1 implying |-2P/(77 - 2P)| >1. This gives P > 77/4 = 19.25

Unitary elastic if |ed| =1 implying |-2P/(77 - 2P)| =1. This gives P = 77/4 = 19.25

Inelastic if |ed| <1 implying |-2P/(77 - 2P)| <1. This gives P < 77/4 = 19.25

Thus, we have the following range for P: Inelastic if 0 < P < 19.25, unitary elastic if P = 19.25, Elastic if 19.25 < P < 38.5 (since demand, Q cannot be negative)

iii) At P = 3, ed = (-2*3)/(77 - 2*3) = -6/71 = -0.085

This means that as price of good increase from 3 by 1%, the demand for the good falls by 0.085%, holding other determinants of demand constant.

(b) Q(P) = 5P^-1/4

So, $\partial Q/ \partial P$ = (-1/4)*(5P^-5/4)

ed = (-1/4)*(5P^-5/4)*(P/Q) = (-1/4)*(5P^-5/4​​​​​​​)*(P/5P^-1/4)

Thus, ed = -1/4 (So, elasticity is constant here, and not a function of P)

ii) Notice that price elasticity of given demand is not a function of P, and thus we cannot set any range for P which influence elasticity as elasticity is independent of P.

Thus, with |ed| = |-1/4| = 0.25 < 1. Price elasticity in this case is always inelastic.

In other words, for all price levels (entire price range), price elasticity is inelastic.

iii) At P = 3, ed = -0.25 (as already noted, any level of price won't create any difference in the elasticity value)

This means that as price of good increase from 3 by 1%, the demand for the good falls by 0.25%, holding other determinants of demand constant.

(c) Q(P) = a - b*P

i) So, $\partial Q/ \partial P$ = -b

ed = -b*(P/Q) = -b*(P/(a-b*P))

Thus, ed = -b*P/(a - b*P)

ii) This elasticity is

Elastic if |ed| >1 implying |-b*P/(a - b*P)| >1. This gives P > a/2b

Unitary elastic if |ed| =1 implying |-b*P/(a - b*P)| =1. This gives P = a/2b

Inelastic if |ed| <1 implying |-b*P/(a - b*P)| <1. This gives P < a/2b

Thus, we have the following range for P: Inelastic if 0 < P < a/2b, unitary elastic if P = a/2b, Elastic if a/2b < P < a/b (since demand, Q cannot be negative)

iii) At P = 3, ed = -b*3/(a - b*3)  

This means that as price of good increase from 3 by 1%, the demand for the good falls by (3b/(a-3b))%, holding other determinants of demand constant.