Chapter 11, Problem 12PAA

To determine

The profits that result from charging client with given per unit price is to be ascertained. Also to construct the report including recommendation resulting in higher profits.

Explanation of Solution

Demand function $Q^d=300-0.2P$

Fixed Cost = $15000

Marginal Cost = $1000

Inverse demand function

  $0.2P=300-Q$

  $P=1500-5Q$

Under monopoly market, profit maximizing condition is written as:

  $MC=MR$

  $TR=P\times Q$

  $TR=\left(1500-5Q\right)\times Q$

  $TR=1500Q-5Q^2$

  $MR=\frac{dTR}{dQ}$

  $MR=1500-10Q$

So, putting $MC=MR$

  $1000=1500-10Q$

  $500=10Q$

  $50units=Q$

  $P=1500-5Q$

  $P=1500-5\times 50$

  $P$ = $1250.

  $Profit=\left(1250-1000\right)\times 50-15000$

  $\mathrm{Pr}ofit =250\times 50-15000$

  $\mathrm{Pr}ofit =-\$2500$

Profits are -$2500.

Theprofits earned by charging client $1450 for first 10units and for additional units amount charged is $1225 is ascertained as follows:

When for the first 10 units amount charged is $1450 and for additional units amount charged is $1225. This is the case of second-degree price discrimination.

Quantity demanded at price $1450 can be calculated as

  $Q^d=300-0.2P$

  $=300-0.2\times 1450$

  $=300-290$

  $=10units.$

Quantity demanded at a price $1225 can be calculated as

  $Q^d=300-\text{0}.2P$

  $=300-0.2\times 1225$

  $=300-245$

  $=55units$

Total units demanded at $1225 is 55 units but 10 units were sold at $1450. So, additional 45 units will be sold at $1225.

Profit is calculated as total cost subtracted by the total revenue.

  $Profit=\left[\left(1450-1000\right)\times 10+\left(1225-1000\right)\times 45\right]-15000$

  $Profit=[\left(450\times 10+225\times 45\right]-15000$

  $Profit=\left[4500+10125\right]-15000$

  $Profit=14625-15000$

  $Profit=-\$375$

Therefore, the profit earns is -$375.

The strategy that can be used for higher profits isexplained below:

An optimal and feasible recommendation will be Two-part pricing strategy.

Under this strategy, there would be a fixed fee plus a per unit fee for each unit of the software installed and maintained.

Per unit fee will be equal to $1000 which is a marginal cost

Quantity demanded at this price will be calculated as

  $Q^d=300-0.2P$

  $=300-0.2\times 1000$

  $=300-200=100units$

Optimal fixed fee = $\frac{1}{2}\left(1500-1000\right)\times 100$ (when Q= 0, P = 1500)

  = $\frac{1}{2}\times 500\times 100$

  $=\$25000$

Profit under two-part pricing strategy will be $25000 - $15000

  = $10000.

Profit under two-part pricing strategy is $10000.