Question

# A doughnut shop determines the demand function q=D(p)= 300/(p+3)^5 for a dozen doughnuts where q is...

A doughnut shop determines the demand function q=D(p)= 300/(p+3)^5 for a dozen doughnuts where q is the number of dozen doughnuts sold per day when the price is p dollars per dozen.
A.) Find the elasticity equation.
B.) Calculate the elasticity at a price of \$9. Determine if the demand elastic, inelastic, or unit elastic?
C.) At \$9 per dozen, will a small increase in price cause the total revenue to increase or decrease?

Solution:

A.) Demand function is given as: q = D(p) = 300/(p+3)5 or 300*(p + 3)-5

Price elasticity of demand, ed = (dq/dp)*(p/q)

dq/dp = 300*(-5)*(p + 3)-5-1*1 = -1500*(p + 3)-6

So, ed = (-1500*(p + 3)-6)*(p/q)

ed = (-1500*(p + 3)-6)*(p/(300*(p + 3)-5))

ed = -5p/(p + 3)

B.) At price of \$9, the demand can be found as:

ed = -5*9/(9 + 3)

ed = -45/12 = -3.75

The value of elasticity of demand is greater than 1 in absolute terms (3.75 > 1), so, demand is elastic.

C.) When the demand is elastic, price and total revenue are negatively related. So, with a small increase in price, the total revenue will fall.

As the price increases even slightly, the demand would falls substantially for the elastic demand, so with decrease in demand dominating increase in price, the total revenue decreases.