price-volume variance analysis

this should be a fairly simple question: So assume you’re trying to analyze a company’s revenues year over year. You have figures for 2 years: R1 = Q1 * P1 for year 1 (where R is Rev, Q is quantity, and P is price) and R2 = V2 * P2 for year 2. Say you’re trying to analyze the change in revenue due to each of the components. It seems to me, you could represent it as follows (d=delta): dR = dQ*P1 + dP*Q1 + dQ*dP, where “dQ*P1” is dR due solely to change in quantity, “dP*Q1” represents dR due solely to change in price and dQ*dP represents dR due to the combination of the delta Q & P. Simple, right? Well… it seems from what I could find on the internet that the accounting standard might be to represent the components as follows: dR due to dP = dP * Q2 dR due to dQ = dQ * P1 I KNOW that purely mathematically this is wrong… but perhaps there’s a rationale behind using this way in accounting. So my question is this: could someone please, either show me evidence that my way is right or the rationale to use the other way in accounting/finance? Thanks guys.

What accounting standard are you talking about? And mathematically why do we have the “dQ*dP” term in there (since both P and Q are smooth enough for this to be 0 in any situation I can easily think of).

I’m talking about discrete variables here… year over year change. say P goes from $5/unit to $6/unit and Q goes from 2500 units to 2700 units. Therefore revenues go from $12,500 to $16,200… and total increase of $3,700. We would like to be able to break down this $3,700 into the change driven by change in price and the change driven by change in quantity. regarding accounting standards… that’s what I’m trying to figure out. I’ve searched online… and the examples I’ve found seem to be erroneous on purely mathematical sgrounds. So I’m trying to figure out if there IS an accounting standard that someone knows of regarding this and/or if there’s any rationale for this method.

P2.Q2 - P1.Q1 = (P2-P1)*Q1 + P2 (Q2-Q1) = dp.Q1 + P2.dq so in your example ==> dp.Q1 = 1 * 2500 = 2500 P2.dq = 6 ( 200 ) =1200 change = 3700

this overstates the effect of price change and understates the effect of quantity change. Inherent in “P2.dq” which represents effect of change in quantity is ALSO the change in price: P2 (Q2-Q1) can be written as: P1(Q2-Q1) + (P2-P1)(Q2-Q1) where the 2nd term also takes into account gains/losses from the change in price. for instance, you could break it down in the following manner, also: P2*Q2 - P1*Q1 = (Q2-Q1)*P1 + Q2 (P2-P1) = dq.P1 + Q2.dp dq*P1 = 200 (5) = 1000 Q2*dp = 2700 (1) = 2700 This is because in both cases you’re deciding to lump in dp*dq into the equation for price effect or the equation for quantity effect. Thinking back to calculus, I believe this is the correct equation: for R = P*Q dR = dq*P + dp*Q + dq*dp so in our case, dR = dq*P1 + dp*Q1 + dq*dp = (200)(5) + (1)(2500) + (200)(1) = 3700 So again, the question is… can someone think of an accounting rationale behind representing effect of quantity as dq*P1 and effect of price as dp*Q2. This what I’ve seen being done in some examples of Revenue variance analysis. Thanks guys… I know it’s a theoretical question, but I figure someone in AF land will be able to shine some light on the issue.