Math 165: Elasticity

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If a quantity x is changed by an amount ∆x, the relative change in x is the ratio [(∆x)/x]. The percentage change in x is 100 [(∆x)/x]. Note that there are no units for the ratio of two quantities with the same units.

Suppose the quantity q and the price p are related, e.g., by a relation of the form q = D(p),

To understand the price elasticity of demand , take the ratio

relative change inq relative change in p = ([(∆q)/q])/([(∆p)/p]) = [p/q] [(∆q)/(∆p)] →  [p/q] [dq/dp]

as ∆p → 0. Define

E(p) = price elasticity of demand ≡ p q   dq dp .

Note that E(p) should be negative, and in general will depend on the value of the price p.

A practical interpretation of elasticity is that for every 1 percent increase in the price p, the demand q decreases by approximately |E(p)| percent.

What is the significance of elasticity? The revenue R = p ·q, and

dR dp = 1 ·q + p · dq dp = q ⎛ ⎝ 1 + p q   dq dp ⎞ ⎠ = q ( 1 + E(p)).

For q > 0, the sign of [dR/dp] is the same as the sign of 1 + E(p).

There are three cases (Hoffmann, p. 246):

1.

Elastic Demand: |E(p)| > 1, 1 + E(p) < 0, [dR/dp] < 0, R is decreasing with respect to p. Demand is relatively sensitive to changes in price.

2.

Inelastic Demand: |E(p)| < 1, 1 + E(p) > 0, [dR/dp] > 0, R is increasing with respect to p. Demand is relatively insensitive to changes in price.

3.

Demand is of Unit Elasticity: |E(p)| = 1, 1 + E(p) = 0, [dR/dp] = 0, R has a critical number at p which is a likely relative maximum. The percentage changes in price and demand are approximately equal.

Exercises Section 3.4

23.

D(p) = − 1.3 p + 10, p = 4.

dq dp = − 1.3, E(p) = − 1.3 p/q, E(p)|p = 4 = − 1.08. dR dp = − 2.6 p + 10 dR dp ⎢ ⎢ p=4  = − .4

|E(4)| = −0.59 < 1, Inelastic Demand, R is decreasing with respect to p.

25.

D(p) = 200 − p², p = 10.

dq dp = − 2p, E(p) = − 2 p2/q, E(p)|p = 10 = − 2. dR dp = 200 − 3p2 dR dp ⎢ ⎢ p=10  = −100

|E(4)| = 2 > 1, Elastic Demand, R is decreasing with respect to p.

40.

When an electronics store prices a certain brand of stereos at p hundred dollars per set, it is found that q sets will be sold each month, where q² + 2 p² = 41.

a.

Find the elasticity of demand for the stereos.

Using implicit differentiation, 2 q  [dq/dp]+ 4 p = 0, so [dq/dp] = [(− 2 p)/ q], and E(p) = [(− 2 p²)/(q²)].

b.

For a unit price of p = 4 ($400), is the demand elastic, inelastic, or of unit elasticity?

E(4) = [(−2·4²)/(3²)], Elastic Demand, R is decreasing with respect to p.

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On 23 Aug 2014, 01:54.