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Then we define aggregate demand Q t as the sum of Q bkt over all customer classes andbilling cycles. Recall that one fifth of the population is on each billing cycle:Q t =KX 4XKXQ bkt = A + ·k=1 b=0k=14XB k ∆P k bt + ε t . (6)b=0Given N t customers at time t, the average demand equation per consumer (q t ≡ Q t /N t )is a function of 5K price deviations:KXq t = α + ·k=14Xβ k ∆P k bt + ξ t , (7)b=0where ξ t may depend on shocks to the economy, weather, etc. Demand is normalized bythe number of customers so we can compare SDG&E with our control group in section 4.This can be further simplified. As noted above, the changes in rates were roughlycommon across customers and rate classes: ∆P k bt = ∆P bt for all k. Let ∆P bt be the(unweighted) average of the wholesale price shocks (∆P wt+b+l−4) overthefive weeks from4Xl =0to 4: ∆P bt = 1 ∆P5 wt+b+l−4. We can write the model as:l=0Ã K!ÃX 4X!q t = α +k=1β k ∆P bt + ξ t (8)b=0Ã!14X 4X= α + β ∆P wt +b+l−4 + ξ t .5b=0 l=0Aggregate demand depends on four lagged prices, the current price, and four forwardprices. We further simplify the model by looking at how aggregate consumption respondsto an average price. This price shock differs slightly from the true price each customerpays.We estimated a model with the five distinct price averages as implied by (7). The dataare highly correlated; the correlation between this week’s PX price and last week’s is 0.82and the correlation among bills is even greater. Due to the high degree of multi-collinearity,none of the coefficients were significant. For this reason, we use a single price measure inthe model. Therefore, we model (8) as:q t = α + βP t + ξ t , (9)where P t is a weighted average of price calculated four separate ways as described in thetext.21