Elasticity of Demand

Question

Suppose a manufacturer can sell f(p)= -65p+5200 units of a product when the price is p dollars per unit.

a. Determine the elasticity of demand, E(p), when the price is p=20 dollars.

b. Is demand elastic, inelastic, or unit elastic for p=20? Justify your response.

c. Given the elasticity you noted above, to increase revenue, should the price be lowered or raised?

Answer 1

f(p)= f = -65p +5200, where p=price and f= quantity demanded

point elasticity of demand = (df/dp)(p/f) =-65p/f, although convention is to ignore the negative sign

as for normal goods, as price goes up, quantity demanded goes down

so E = |-65p/f|= 65p/f

when price $20, f = -65(20)+5200) = 5200-1300=3900

and E = |-65(20)/3900|= |-1300/3900|= |-1/3|= 1/3 which is inelastic as 1/3 <1=unit elastic

increase revenue by raising price

f = -65p +5200

revenue = R = fp = -65p^2 +5200p

R' = -130p +5200 = 0

p = 5200/130 = 520/13 = 40

maximum revenue is when price = $40

at p=20, revenue will increase as price is raised, up to when p=40

at p=40, E=1 = unit elasticity

E= 65p/f

E(40) = 65(40)/(-65(40)+5200) = 2600/(-2600+5200) = 2600/2600 = 1

another way of looking at the problem is graph f(p) = -65p +5200

with intercepts (0,5200) and (80,0)

take the midpoint of the line segment connecting the intercepts

= (40,2600) = ($40, $2,600) = (revenue maximizing price, max revenue)

for a linear demand curve, that midpoint is the point with unit elasticity

Area of the rectangle with vertices (0,0), (40,2600), (0,5200), (40,0) = max revenue

it may help to sketch the line, vertices and rectangle

or graph the Revenue function. It's a downward opening parabola with vertex = maximum point with coordinates = the revenue maximizing price and max revenue