Question

Alpha Company has estimated that the demand curve for its product is represented by the equation Q=2,840-20P, where Q is the quantity sold per week and P is the price per unit.

a. Based on the estimated demand curve, write the equations for Alpha's

(1) Average revenue,

(ii) Total revenue, and

(iii) Marginal revenue.

b. What will be the maximum total revenue per week that Alpha can obtain from sales of its product? (Give the exact dollar amount and explain how you determine it.)

c. Calculate the point price elasticity of demand for Alpha's product when Q= 1,600. Is demand elastic or inelastic at this quantity? How do you know?

d. Calculate the arc price elasticity of demand for Alpha's product between Q== 1,000 and Q=1,100. Interpret your result and relate it to what will happen to total revenue if Alpha is initially at Q = 1,000 and decides to cut price to increase its sales from 1,000 to 1.100 units.

Answer 1

i. find P first

$1600 - 2840 = -20P, P = 62$

Point elasticity of $\frac{P}{Q} \times F'$ (demand function)

$F'(demand) = -20$

$\frac{62 }{1600 }\times (-20) = -0.775,$ thus inelastic but not completely inelastic

$Ed = 0$ Perfectly inelastic demand

$- 1 < Ed < 0$ Inelastic or relatively inelastic demand

$Ed = - 1$ Unit elastic, unit elasticity, unitary elasticity, or unitarily elastic demand

-$8 < Ed < - 1$ Elastic or relatively elastic demand

$Ed = - 8$ Perfectly elastic demand

ii. Total revenue:$P \times (2840 - 20P)$ (the last term stems from $q=2840-20p,$ revenue $= P \times Q)$

Total revenue: $-20P^2 + 2840P$

Average revenue: $\frac{-20P^2 + 2840P}{ Q}$ or it can be expressed as $\frac{(-20P^2 + 2840P)}{(2840-20P)}$

Marginal revenue = derivative of total revenue with respect to P

$F'(revenue) = -40P + 2840$

iii. maximize revenue

occurs when first derivative = 0

F'(total revenue) $= -40P + 2840,$ is equal to 0 when$P = \frac{2840}{40} = 71$ . Plus 71 into the first function from A)

Total revenue $= 71 \times (2840 - 20\times 71) = \$100,820$

b. The total revenue will be the highest when the marginal revenue=0

From $0=2840-40P$ we can get $P=71$

So the maximum total revenue $=2840\times 71-20\times 71^2=100820$

c. $E=-20\times \frac{62}{1600}=-0.775<1,$ so the point th point of demand is inelastic at this quantity.