Calculate Maximizing Profit by charting demand curve, revenue, and costs

Posted by Professor Cram in pecosjack

The Assigned Homework Problem

War Game, Inc. produces games that simulate historical battles. The market is small but loyal, and War Game is the largest manufacturer. It is thinking about introducing a new game.

Based on historical data regarding sales, War Game management forecasts demand for this game to be P = 50 – 0.002 Q, where Q denotes unit sales per year, and P denotes price in dollars. The cost of manufacture (based on royalty payments to the designer of the game, and the costs of printing and distributing) is C = 140,000 + 10Q.

Calculate the following, assuming that War Game maximizes profit:

a. Quantity

b. Price

c. Total cost

d. Total revenue

e. Total profit

Price Demand Curve: Plot Price and Quantity

All of the formulas and factors are interrelated, so it can be approached like solving simultaneous equations in algebra. Another approach is to chart the revenue and expenses at different levels of production. Calculate the profit at each level, looking for the maximum profit and answer the questions at that point. Let’s start with the demand curve as it relates price to quantity. Price demand starts at $50 at quantity zero and falls with every unit of production until it reaches zero price at a quantity produced of 25,000. It looks like this:
Demand Curve - War Game Inc.

We obviously don’t want a negative price, so the relevant range for this exercise is for quantities from 0 to 25,000.

Plot Total Revenue

With price at each quantity, we can calculate revenue.

Total revenue equals price times quantity (TR = P*Q).

It looks like this for War Game Inc.:

Plot Total Revenue for War Game Inc.

Plot Total Costs

We are given the formula for total costs as $140,000 (fixed cost component) plus $10 per unit produced (variable cost component) as

C = 140,000 + 10Q

It plots like this:

Total Costs for War Game Inc.

Plot Total Costs and Total Revenue together

Here we see Total Revenue and Total Costs plotted together.

Total Costs and Total Revenue - War Game Inc.

That area below revenue that is above costs is the area where profit is possible. We can narrow the range down to profit starting at 5,000 and ending by 15,000.

Plot Total Profit

Here is the plot of Total Profit = Total Revenue – Total Cost:

Total Profit - War Game Inc.

Calculating the answers

To calculate and plot these, I set up an Excel spreadsheet with a column for each of the 5 answer components. In the column for quantity, I started at zero and incremented up. Price is a function of quantity, and you provided that formula. Cost is also a function of quantity. Revenue is quantity times price. Profit is revenue minus costs. It all flowed from quantity using the formulas provided. Profit is at the highest point at a quantity of 10,000.

Quantity

Price

Total Costs

Total Revenue

Total Profit

10,000

$30

$240,000

$300,000

$60,000

It is possible to increment the spreadsheet in any amount, so we could have tested production at every quantity from 1 to 25,000. This sort of approach will provide the solution, but is not very satisfactory. I took you through the steps of graphs to demonstrate the concepts, but it would be better to be able to “solve” for the solution. We will take that approach to the second problem.

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