# How do you compute an indifference point?

The net economic impact provides an assessment which is valid under very specify circumstances. Indeed, it depends on the accuracy of forecasts and estimates used to compute it. If any of the underlying assumptions is not met, the conclusion it supports may be compromised. Since neither forecasts nor estimates are perfectly reliable, it is important to ask whether the conclusion is robust to slight variations of the underlying estimates. This is the idea behind both the indifference point and sensitivity analyses.

The indifference point is the value of an estimate (volume, price, unit variable cost or fixed costs) for which the net economic impact is null and therefore the decision maker is indifferent between the two alternatives, keeping the other estimates constant.

The basic idea of the indifference point is to ask for which volume, price, units variable costs, or fixed costs one decision alternative or course of action is strictly equivalent from an economic perspective to the other. This is called the indifference point because for this set of values the manager is indifferent between either course of action because they yield equivalent costs and benefits.

You already have seen a special case of indifference point: the break-even point. The volume at break-even is the volume at which the decision maker is indifferent between starting the activity and doing nothing, since profit is null. It is a special case in the sense that it considers all the costs and revenues as relevant, i.e. avoidable.

Solution.

Switching to virgin mojitos becomes advantageous with a minimum increase in volume of 167 units, so a new volume of 1,667:

\begin{aligned} 0 = & + \Delta Q \times (7-2.5) \\ & + (1,500 + \Delta Q) \times (6-7) \\ & - (1,500 + \Delta Q) \times (1.8-2.5) \\ & - 250 \\ \leftrightarrow \\ 0 = & 4.2 \times \Delta Q - 700 \\ \leftrightarrow \\ \Delta Q = & 166.66667 \end{aligned}

Switching to virgin mojitos is advantageous with a maximum decrease in price of 1.08 euros, so a new price of 5.92:

\begin{aligned} 0 = & + 200 \times (7-2.5) \\ & + 1,700 \times \Delta P \\ & - 1,700 \times (1.8-2.5) \\ & - 250 \\ \leftrightarrow \\ 0 = & + 1,700 \times \Delta P + 1,840 \\ \leftrightarrow \\ \Delta P = & -1.0823 \end{aligned}

Switching to virgin mojitos is advantageous with a minimum decrease in unit variable cost of 0.62 euros, so a new unit variable cost of 1.88:

\begin{aligned} 0 = & + 200 \times (7-2.5) \\ & + 1,700 \times (6-7) \\ & - 1,700 \times \Delta V_c \\ & - 250 \\ \leftrightarrow \\ 0 = & - 1,700 \times \Delta V_c - 1,050 \\ \leftrightarrow \\ \Delta V_c = & -0.61765 \end{aligned}

Finally, an increase in 250+140 = 390 of fixed costs is enough to make both options equivalent from en economic perspective. If the percentage change necessary to go from the scenario to the indifference point, we can see that these percentages are small except for the fixed costs. This suggests that the conclusion might not be very robust, since small changes are enough to make the decision maker indifferent:

Estimate Scenario Indifference point Difference
Volume 1,700.00 1,667.00 -1.94%
Price 6.00 5.92 -1.33%
Unit variable cost 1.80 1.88 4.44%
Fixed costs 250.00 390.00 56.00%

Computing indifference points is a form of sensitivity analysis because it helps the decision maker determine over what range of an estimate the decision is unchanged. If the initial set of assumptions is very far from the indifference points, the conclusion is robust to large changes in the estimates. If on the contrary the estimates are very close to the indifference points, a modest variation is enough to reverse the conclusion which is therefore highly risky. And here again you can see the parallel between indifference point and break-even point: operating leverage, therefore risk, is also maximal at break-even.

Note that which magnitude of the distance between the estimate and the indifference point makes the conclusion robust depends on the quality of the cost system. A crude costing system is likely to result in unreliable estimates, the true value of which may be sometimes far from what the cost system reports. Being far from the indifference point then gives a misleading impression of safety. On the contrary, if the cost system is very accurate, the estimate can be very close to the indifference point and the conclusion still robust.