Break even analysis
The break even point is the point at which the revenues generated exactly balance the
costs incurred i.e. it is the point at which the profit is zero.
Revenue = Cost
or
Profit = 0.
Once the cost, revenue and profit equations have been established the break even point may
be obtained either by graphical means or by solving a mathematical equation.
Finding the Break even Point graphically
The break even point may be found by either
• plotting the cost and revenue functions accurately on a graph and finding the point at
which the two functions cross,
or
• plotting the profit function accurately on a graph and finding the point at which the
profit function is zero.
Finding the Break even Point using algebra
To determine break-even points algebraically either
• solve the equation
revenue = cost,
or
• solve the equation
profit = 0.
Task 1
A small firm manufactures widgets. Fixed costs are £1000 per week and in addition each
widget costs £8 to produce. The firms sells all of the widgets it can make for £10.50 each.
If q is the weekly output of widgets, what is the output required to break even on a weekly
basis?
Step 1
Determine the equations for cost, revenue and profit:
1
Step 2
Download the excel spreadsheet break.xls from the IT Skills website and complete the table
below:
q
0
250
500
750
1000
C
R
P
Step 3
Plot Cost and Revenue on a single graph and hence determine the breakeven point
Step 4
Plot Profit on a second graph and hence determine the breakeven point - you should find
you have the same answer as in step 3
Step 5
Solve the equation Cost = Revenue algebraically
Step 6
Solve the equation Profit = 0 algebraically
Task 2
A small firm manufactures items for sale in the retail industry. The firm has fixed costs of
£1875 per month arising from overheads and the variable costs are £4.00 per item produced.
The demand for the items depends upon the price. In order to sell x items the firm must
set the price at 14 − 0.01x pounds. By graphical means
i estimate the breakeven points,
ii determine the value of x which gives the maximum profit and
iii estimate the maximum profit.
A copy of the following table can be found in the Task 2 tab in the spreadsheet break.xls
x
C
R
P
0
200
400
2
600
800
1000
Task 3
Solve the following breakeven problems using Excel for plotting graphs, where required:
1 The total cost and total revenue functions for a firm are given by
C = 20 + 2.5q
and
R = 5q,
respectively. Determine the breakeven value of q graphically and algebraically.
2 A company manufactures lawnmowers and sells them for £390 each. The monthly
production costs are comprised of fixed costs of £6500 and variable costs of £340 for
each lawnmower the company produces.
(a) Construct the revenue, cost and profit functions.
(b) By solving an appropriate equation find the number of lawnmowers the company
must make each month in order to break-even.
(c) Machine constraints limit the number of machines which can be produced each
month to 570. What is the maximum possible monthly profit the company can
hope to achieve?
3 An electricity board offers customers a choice between two tariffs:
Tariff 1: A fixed charge of £30 per quarter plus a charge of 7.5 pence per unit consumed.
Tariff 2: A fixed charge of £40 per quarter plus a charge of 7.0 pence per unit consumed.
How many units would need to be consumed in a quarter for the overall cost under
both tariffs to be the same?
4 A manufacturer makes a certain product for which his fixed costs are £800 per month
and his variable costs are £1.20 per unit. His records show that he can increase the
quantity sold, x, per month by reducing price. The relationship being
Price = 3.6 − 0.001x.
Generate graphs of the Cost and Revenue for x between 0 and 2000, and use the graphs
to find the breakeven point. Obtain the value of which maximises profit and hence
obtain the maximum monthly profit.
Solutions
Task 1
C = 100 + 8q R = 10.50x
Break even point: q = 400
P = 2.5q − 1000
3
Task 2
C = 1875 + 4x R = (14 − 0.01x)x
Profit = Revenue - Cost = −0.01x2 + 10x − 1875
(i) break even points: 250, 750
(ii) max profit at x = 500
(iii) max profit £625
Task 3
1 q=8
2 (a) C = 6500 + 340x; R = 390x; P = 50x − 6500
(b) x = 130
(c) £22000
3 2000 units
4 breakeven points 400 and 2000; max profit when x = 1200; max profit= £640
October, 2008
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