## Friday, October 18, 2013

### Simple pricing: the "stay even" quantity

The marginal analysis of simple pricing, i.e., price at the point where MR=MC, is sometimes implemented by using a version of break-even analysis.  Instead of asking which price maximizes profit, you instead ask "will a given price increase, e.g., 5%, be profitable?"  To answer the question, we

1. Compute the stay-even quantity, the quantity you can afford to lose and still break even

2. Predict (or guess) whether the actual quantity lost will be greater or less than the stay-even quantity.

3.  If the actual quantity lost is less than the stay-even quantity, then the price increase will be profitable.

Here is the derivation of the stay-even quantity:

The benefit of a price increase is the extra revenue you earn at the new (and lower) quantity, Benefit=dP*(Q+dQ)

The cost of a price increase is the margin on the lost sales, Cost=dQ(P-MC)

where dP=P1-P0, dQ=Q1-Q0, P0=initial price, P1=final price, Q0=initial quantity, and Q1=final quantity.

You compute the Quantity at which you are indifferent between raising price or not, and you get the formula:

(dQ/Q)=(dP/P)/[(dP/P)+m]

where m=(P-MC)/P, dQ/Q=% change in Q, and dP/P=% change in P.

EXAMPLE:

The stay even quantity for a 5% price increase for a firm with a 40% contribution margin is 11.1%=(5%)/[(5%)+(40%)].  If you expect to lose less than 11%, then a 5% price increase will be profitable.

REFERENCE:  Page 9 of
A Critical Analysis of Critical Loss Analysis,
Daniel P. O’Brien and Abraham L. Wickelgren, January 2003 (Published in Antitrust Law Journal) [PDF 236K]

1. This comment has been removed by the author.

2. This makes a lot of sense. This actually simplifies the decision to raise a price or not. This does not take into account customer loyalty or loosing repeat business but it simplifies the formula and train of thought for increasing the price. I just hope whoever uses the formula has good reasoning for believing that units sold will only drop a certain percentage.

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4. This perspective on simple pricing and price accuracy expands upon the concept of marginal analysis, which aids in the decision making process for setting price. In the text we learn that when Marginal Revenue is greater than Marginal Cost, one should lower the price in order to sell more, and that in cases where Marginal Cost is higher than Marginal Revenue, one should increase cost and sell less. This process as presented in the text is helpful when one is determining in which direction to go (increase price or decrease price), however it does complete the picture because the decision maker must still determine how far they should go in the direction they choose.
As described here, the process goes a step further by asking if a certain percentage increase will be profitable or not. This will help one determine not only the direction in which they should proceed, but how far as well. For the non-economist, this seems to be a quick and relatively easy way to make a determination on pricing as long as all the pertinent data is available.