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Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Biases from Overlooking Price Discrimination
And A Solution
Benjamin R. Shiller and Shuran Zhang
(Brandeis University)
Presented at the IIOC
April 25, 2015
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline for section 1
1
Introduction
2
Biases
3
Background and Data
4
Method
5
Estimation
6
Results
7
Conclusion
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
The Problem
Many papers estimate demand using aggregate data
Assume all consumers pay average price
Rev
Q
This simplification biases demand estimates:
Some consumers pay below average price
Might buy less at average price. Assuming paid average price
overestimates their demand
Some consumers pay above average price
Their demand weakly underestimated
Direction of bias is not immediately obvious
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Preview of Theory Results
If price discrimination is present, and use standard models:
Estimated demand at given price biased upwards
Estimated absolute slope of demand biased upwards
Slope of inverse demand too flat
Standard IV methods are of no help!
And problem may not be recognized
Out-of-sample testing typically uses data with same problems.
May falsely seem to validate
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline for section 2
1
Introduction
2
Biases
3
Background and Data
4
Method
5
Estimation
6
Results
7
Conclusion
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
General Proof
We show Price Discrimination (PD) only profitable
if more
bought under PD than at average price P¯
Implies demand biased weakly upwards when PD ignored
Brief Explanation of Proof
Revenue under PD = RevPD ∗
QPD
QPD
= P¯ ∗ QPD
If same quantity sold when charging everyone P¯ , revenues
same
If PD incurs ǫ > 0 cost, then PD strictly lower profits
¯
If more sold at P:
Implies some Pˆ > P¯ s.t. quantities (and production costs)
same
But then profits higher if uniformly charging Pˆ
PD would not be a profit maximizing strategy
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Slope of Demand Biased Too
If charge price high enough
so only one person buys,
avg. = only price charged
P
Biased demand and unbiased
demand curves should coincide
at demand choke price
At lower prices, demand
upward biased
Implies steeper demand
function
Less steep inverse demand
function
(P¯ , Q(P¯ ))
(P¯ , QBiased)
0
Q
Yields upward bias in demand’s slope magnitude opposite direction of the typical bias arising from
measurement error in predictor variables!
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Instrumental Variable Methods Do Not Address Biases
Standard methods link
changes in demand to
exogenous price
shocks
S
P
S′
But problem arises
from measurement
error in quantity
(P¯ , Q(P¯ ))
Measurement error
in quantity persists
after instrumenting
for average price
(P¯ , QBiased)
Q
Using IV merely isolates movements along BIASED demand curve
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline for section 3
1
Introduction
2
Biases
3
Background and Data
4
Method
5
Estimation
6
Results
7
Conclusion
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Data
Data source: a well-established certified Volvo dealer in the
Jiangsu Province of China
Time span: July 2006 (dealer entered) to July 2010
Key variables for each Volvo model S40 sold:
Final sale’s price
Cost to dealer
Date car was sold, arrived on lot, and was ordered by dealer
Model and trim
Dealer’s inventory (constructed from sale & arrival dates)
VIN number
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Price Residual (in 10,000 Yuan)
−4
−2
0
2
By contrast is 36% for
airlines [Borenstein and
Rose, (1994)]
−6
On average consumer
pay quality-adjusted
prices that differ by
10, 570 Yuan ≈ 4% avg.
price
4
Evidence of Price Discrimination
Cumulative Order of Sales
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline for section 4
1
Introduction
2
Biases
3
Background and Data
4
Method
5
Estimation
6
Results
7
Conclusion
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Method Overview - Illustration 1
Observe quantity and
price by group k
defined by price
charged
P
(Pk , Qk )
If knew MC, could
infer slope which
implies the observed
price is the
profit-maximizing
price1
Q
1
Paper shows method still valid if group k comprised of subgroups i
separately set optimized prices (which happened to be same)
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Method Overview - Illustration 2
If knew curvature,
could extrapolate
group’s demand to
non-local prices
No way to estimate
aggregate demand
without extrapolation
P
(Pk , Qk )
Since charged
different prices,
can’t directly
observe amount
one group buys at
other’s price
Shiller and Zhang
(Pl , Ql )
Q
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Summing Up
We can yield an unbiased estimate of aggregate demand by:
Inferring a group’s demand from the price charged
Aggregating across groups
BUT, need to know:
1
Marginal cost
2
Curvature of groups’ demands
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Estimating Marginal Costs
C′ =
d P¯
dQ Q
+ P¯
Where:
d P¯
dQ
is biased estimate of inverse demand’s slope
Q and P¯ are total quantity and average price in period of
interest
It turns out that upward biases in magnitudes of (i)
(ii) Q offset in above formula
Shiller and Zhang
dQ
d P¯
and
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Estimating Groups’ Demand Curvature
Given curvature of group demands, can simulate biased
aggregate curve
For particular marginal cost
Compute optimal quantity and price to charge each group
(Qk (Pk∗ ), Pk∗ )
P
P
Q (P ∗ )∗P ∗
Aggregate: Q = k Qk (Pk∗ ), P = k k Q k k
This is one point on simulated biased aggregate demand
Trace out demand by repeating for vector of marginal costs
Find curvature which minimizes differences between
simulated and unsimulated biased aggregate demand
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline for section 5
1
Introduction
2
Biases
3
Background and Data
4
Method
5
Estimation
6
Results
7
Conclusion
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline of Steps
In Aggregate Data
1
2
¯ using cost-based instruments. Invert: P¯ = g (Q)
Estimate biased demand Q = f (P)
Infer marginal costs (Ct′ ) by plugging reciprocal of
Eq. 5
¯
df (P)
d P¯
obtained from step 1 into
In Disaggregated Data
dQk,t
dPk,t
for each group k and time t by plugging Ct′ from step 2 into Eq. 9
3
Find
4
Choose guess at parameter γ determining curvature of group demand functions
5
Extrapolate demand for each group k & period t using γ and
6
7
8
dQk,t
dPk,t
from step 3
Simulate biased inverse demand P¯ = g SIM (Q): find one point on inverse demand
∗
curve by averaging predicted optimal prices (Pk,t
) and then aggregating sales
∗
′
(Qk,t (Pk,t )) for an assumed marginal cost (Ct ). Repeat for vector of Ct′
2
P SIM
g (Q) − g (Q)
Calculate objective: obj =
Repeat steps 5-7 searching over γ until objective minimized
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline of Steps - Step 1
In Aggregate Data
1
2
¯ using cost-based instruments. Invert: P¯ = g (Q)
Estimate biased demand Q = f (P)
Infer marginal costs (Ct′ ) by plugging reciprocal of
Eq. 5
¯
df (P)
d P¯
obtained from step 1 into
In Disaggregated Data
dQk,t
dPk,t
for each group k and time t by plugging Ct′ from step 2 into Eq. 9
3
Find
4
Choose guess at parameter γ determining curvature of group demand functions
5
Extrapolate demand for each group k & period t using γ and
6
7
8
dQk,t
dPk,t
from step 3
Simulate biased inverse demand P¯ = g SIM (Q): find one point on inverse demand
∗
curve by averaging predicted optimal prices (Pk,t
) and then aggregating sales
∗
′
(Qk,t (Pk,t )) for an assumed marginal cost (Ct ). Repeat for vector of Ct′
2
P SIM
g (Q) − g (Q)
Calculate objective: obj =
Repeat steps 5-7 searching over γ until objective minimized
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline of Steps - Step 1
Dependent Variable is Logged Sales
OLS
IV
First Stage
Lag Dealer Inventory
Exchange Rate
Second Stage
Log (Avg. Price)
Year Fixed Effects
0.634
(0.712)
-0.976
(2.325)
-7.517
(3.280)
Y
Y
Month Fixed Effects
-0.0033
(0.0015)
-0.00010
(0.00076)
-0.000841
(0.000847)
-0.704
(0.0823)
-0.183
(0.0631)
-0.183
(0.0458)
-0.0948
(0.930)
-7.248
(5.589)
-7.715
(4.642)
Y
Y
Y
Constant
0.470
5.541
28.40
2.799
(2.280) (7.809) (11.21)
(2.976)
N
51
51
51
50
R2
0.0159
0.135
0.395
.†
Standard errors in parentheses - Price in 10,000 Yuan
Shiller and Zhang
Y
26.47
(18.75)
50
0.0122
29.00
(15.85)
50
0.400
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline of Steps - Step 2
In Aggregate Data
1
2
¯ using cost-based instruments. Invert: P¯ = g (Q)
Estimate biased demand Q = f (P)
Infer marginal costs (Ct′ ) by plugging reciprocal of
Eq. 5
¯
df (P)
d P¯
obtained from step 1 into
In Disaggregated Data
dQk,t
dPk,t
for each group k and time t by plugging Ct′ from step 2 into Eq. 9
3
Find
4
Choose guess at parameter γ determining curvature of group demand functions
5
Extrapolate demand for each group k & period t using γ and
6
7
8
dQk,t
dPk,t
from step 3
Simulate biased inverse demand P¯ = g SIM (Q): find one point on inverse demand
∗
curve by averaging predicted optimal prices (Pk,t
) and then aggregating sales
∗
′
(Qk,t (Pk,t )) for an assumed marginal cost (Ct ). Repeat for vector of Ct′
2
P SIM
g (Q) − g (Q)
Calculate objective: obj =
Repeat steps 5-7 searching over γ until objective minimized
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Dealer’s Cost (in 10,000s Yuan)
15
20
25
30
Outline of Steps - Step 2
2006m7
2007m7
2008m7
Date
Marginal Cost Point Estimate
Shiller and Zhang
2009m7
2010m7
Dealer’s Cost of Base Model
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline for section 6
1
Introduction
2
Biases
3
Background and Data
4
Method
5
Estimation
6
Results
7
Conclusion
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Estimated Bias in Demand
Biased Q(P)
10th (24.51)
19.51
(27.25)
Price Percentile (Price in 10,000 Yuan)
25th (26.4) 50th (29.55) 75th (32.7)
11.60
4.95
2.19
(4.71)
(2.17)
(1.16)
90th (34.59)
1.38
(0.27)
Unbiased Q(P)
18.49
(24.97)
10.89
(3.98)
4.54
(2.08)
1.95
(0.94)
1.21
(0.32)
Difference
1.02
(2.29)
0.71
(0.74)
0.41
(0.13)
0.24
(0.24)
0.17
(0.32)
5.51%
6.52%
9.03%
12.14%
(4.94)
(4.22)
(4.52)
(16.49)
Standard errors, in parentheses, calculated via the delta method
13.84%
(17.16)
% Difference
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Estimated Bias in Slope of Demand
Biased
10th (24.51)
−5.39
(24.93)
dQ
dP
Unbiased
dQ
dP
Price Percentile (Price in 10,000 Yuan)
25th (26.4) 50th (29.55) 75th (32.7)
−3.18
−1.32
−0.55
(3.92)
(0.07)
(0.55)
90th (34.59)
−0.33
(0.51)
−5.18
(23.68)
−3.05
(3.43)
−1.25
(0.07)
−0.51
(0.61)
−0.30
(0.54)
−0.21
(1.25)
−0.13
(0.50)
−0.07
(0.02)
−0.04
(0.06)
−0.03
(0.03)
3.97%
4.24%
5.63%
8.36%
(3.76)
(5.12)
(1.07)
(18.71)
Standard errors, in parentheses, calculated via the delta method
10.64%
(35.66)
Difference
% Difference
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Outline for section 7
1
Introduction
2
Biases
3
Background and Data
4
Method
5
Estimation
6
Results
7
Conclusion
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution
Introduction
Biases
Background and Data
Method
Estimation
Results
Conclusion
Conclusion
Paper:
Proved that when PD exists, yield upward biased estimates of:
Amount demanded at given price
Absolute slope of demand
Show inferred marginal costs not biased
Provided method for unbiased demand estimation using
disaggregated data
Shiller and Zhang
Biases from Overlooking Price Discrimination And A Solution