# Finding optimal price and output level

You work for a drug manufacturing company that holds a patent on Hair Grow, the world most effective drug for restoring hair. Your job is to analyze the pricing and investment decisions facing the firm. Your marketing group estimates that Hair Grow has the following demand curve:

P = 101-0.00002Q

where P is measured in dollars and Q is measured in the number of pills a year. You have this patent for another five years.

a. Your marginal cost for producing a Hair Grow pill is $1. What is the profit-maximizing price and quantity? What is your profit?

b. Suppose that your production facility can only produce 1,200,000 pills per year. What is your optimal price and quantity given the production constraint? What are your profits (assume there is no fixed cost)?

c. Assume that your production capacity is currently 1,200,000 pills as in (b). Now, suppose that you could increase the capacity of your plant to 3,000,000 pills per year for a cost of $30,000,000. However, the construction of the new plant takes a full year, and during the year while your factory is under construction, you have to shut down the production facility, and cannot produce Hair Grow. Should you undertake the investment (for simplicity, assume you can borrow the funds for the expansion at a 0 percent interest rate)?

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#### Solution Preview

a.Your marginal cost for producing a Hair Grow pill is $1. What is the profit-maximizing price and quantity? What is your profit?

Given

Marginal Cost=MC=$1

P=101-0.00002Q

Total Revenue=P*Q = 101Q-0.00002Q^2

Marginal Revenue=MR=dTR/dQ = 101-0.00004Q

Put MR=MC for profit maximization

101-0.00004Q = 1.00

Q= 2,500,000

P=101-0.00002*2500000 =$51.00

Total Revenue=P*Q=51*2500000=$127,500,000

Total Cost=1*2500000=$2,500,000 (assume zero fixed costs)

Profit =127,500,000-2,500,000=$125,000,000

b. ...

#### Solution Summary

Solution describes the steps to find out optimal output and price level in the given case. It also calculates optimal profit and profit in case maximum production capacity is 1,200,000 pills.