Question

Your pricing team has run an A/B test and determined that when the price of your product is $300 the quantity demanded is 100 units. However, when the price is $200 quantity demanded is 150 units.

Your procurement and warehousing team has also provided a best estimate of your costs. The fixed cost for rent is $1,000 / month. The variable cost to procure and ship your product is 8Q + 2Q²

Answer the following questions:

1. Write out your demand and inverse demand functions.

2. Write out your profit function.

3. Using calculus, what is the quantity that maximizes profit (prove using derivatives that it is indeed a maximum)?

4. Using Excel, create a graphshowing TR, TC, and Profit from Q = 50 to Q = 200 by increments of 10.

5. At P=84

1. How much quantity is demanded?
   

2. What is the elasticity of demand?
   

3. The company is debating increasing price by 5%. What will happen to revenue if price increases by 5%? (Revenue goes down or goes up? Why?)

Answer 1

1)∆Qd/∆P=(150-100)/(200-300)=50/-100=-0.5

Qd at p=0, = 150+200*0.5=150+100=250

Demand function:Qd=250-0.5p

Inverse demand:p=500-2Q

2)

Total revenue=Q*(500-2Q)=500q-2*Q^2

Total cost=1000+8Q+2*Q^2

Profit=TR-TC

Profit=500q-2q^2-1000-8q-2q^2=492q-4q^2-1000

3)

Derivative of profit function with respect to Q to find profit maximizing quantity.

∆Profit/∆Q=492-8q

492-8Q=0

Q=492/8=61.5

At p=84

A)Qd=250-0.5*84=250-42=208

B)Elasticity of demand={∆Qd/∆p)*(P/Q)

Elasticity of demand=(-0.5)*(84/208)=-0.2

C)Because magnitude of elasticity of demand is lower than one ,it means demand is inelastic at this price,which means increase in price will lead to very low Decrease in quantity demanded,so as a result total Revenue will increase.