Video CPM as a function of Demand
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Experiment description and Analysis

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13 biggest client tactics representing about 9% of total video revenue were duplicated with all prior restrictions (geo, site, device etc.), additionally restricted to Tier I-III inventory and budget-density tested at 1X, 2X, 6X and 12X daily budget on 25, 12, 4 and 2 buckets respectively.
For each of these budget points we have the spend, cost and impressions in the TierI-III inventory.

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From other sources (Atul's presentation) we know that about 15% of all our impressions (50% of all avails) are in Tier I-III, so for all the video tactics which weren't restricted to TierI-III as part of this test, we uniformly apportioned 20% of spend and cost (assuming margin was same) and 15% impressions in the tested user buckets to TierI-III inventory.

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The above spend cost and impressions were added to those for the tested tactics,

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and CPM vs daily cost was plotted and fitted to a power law.

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Why a power law? Because cost (CPM) elasticity of demand (total cost) is the ratio of the logarithmic derivatives, which is just the exponent in a power law.

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How do I put in formulas?
dailySpend = AnnualSpend/365

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dailyCost = (1-margin)*dailySpend

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actualDailyCost = (spendFraction + (1-spendFraction)*costFraction) * dailyCost

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Cost Based CPM from fit to elasticity curve evaluated at actualDailyCost

Revenue CPM = CostCPM/(1-margin)