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Modeling Inattentive Consumer’s Behavior under
Nonlinear Pricing in Residential Water Market
Xiangrui Wangab, Jukwan Leea, Jia Yana and Gary Thompsonb
Abstract
We propose a structural demand model based on “perceived price” for inattentive consumer’s
residential water consumption under increasing block pricing. We first test whether consumers in our
sample respond to marginal price or “perceived price”. The test results suggest that consumers are
inattentive and tend to respond to “perceived price”. We then construct a structural demand model based
on consumer’s indirect utility using “perceived price” (current bill and lagged bill) as main decision
variable. We estimate both our model and Discrete/Continuous Choice (DCC) Model (assuming
consumers are capable of reacting to marginal price) and use the simulated data to compare the
performance. The results show that our model out-perform DCC model for low and high users, while
DCC model performs better for mid-level users.
JEL classification: Q21; Q25; L95; D12;
Keywords: Water Demand; Increasing Block Pricing; Inattentive Consumers; “Perceived Price”
a School of Economics Science, Washington State University
b Department of Agricultural and Resource Economics, University of Arizona
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1. Introduction
Discrete/Continuous Choice (DCC) model is a dominant modela in recent water demand
literature to structurally characterize consumer’s behavior under increasing block pricing (IBP
hereafter) (Hewitt & Hanemann, 1995; Pint, 1999; Olmstead, Hanemann, & Stavins, 2007). It
originates from literature associated with non-linear budget set (Burtless & Hausman, 1978;
Hausman, 1985; Moffitt, 1986; 1990) and discrete/continuous demand estimation (Hanemann,
1984; Dubin & McFadden, 1984). Assuming all consumers understand IBP and respond both
timely and correctly, DCC model using IBP structure, such as marginal price and block
thresholds, to capture consumer’s behavior. For consumers, each block of IBP is a discrete
choice, which associated with different marginal price and budget set. Within each discrete
choice, consumers then decide usage on a continuous interval.
However, DCC model’s assumption on consumers’ full understanding and correct reaction of
IBP is rather strong. For residential energy consumption, consumer may be inattentive since the
energy bill accounts for a small proportion of household monthly expenditure. Also, the
information consumers possessed is very likely to be limited due to the complexity of IBP
structure. Reacting to IBP requires consumer not only understands the IBP structure, i.e.
marginal price and quantity for block thresholds, but also knows the energy consumption of her
appliances. These demanding requirements are not met by all consumers. As a result, using
DCC model to capture inattentive consumers’ behavior may not be appropriate. This paper
explores the decision making mechanism of inattentive consumers and proposes a structural
model to characterize their behavior in residential energy consumption.
a Another popular water demand model is introduced recently by Strong and Smith (2010).
2
Recent development in behavioral economics suggests that inattentive consumers may make
“imperfect optimization” when facing complex decisions (Kling, Congdon, & Mullainathan,
2011; Madrian, 2014). The imperfect optimization arises because consumer have limited
attention and cannot possess all the information for decision making. Instead, consumers tend to
apply simplified heuristics for complicated choice problem, which leads to outcomes deviated
from optimal rational decision. For residential energy consumption, inattentive consumers may
not make decisions based on marginal price of IBP, but other easily-acquired “perceived price”.
A stream of recent literature test whether consumers respond to marginal price or other
“perceived price” in residential energy consumption (Shin, 1985; Borenstein, 2009; Nataraj &
Hanemann, 2011; Ito, 2013; 2014; Wichman, 2014). The “perceived price” has been tested are
average price and expected marginal price. Average price is popular since it can be calculated
directly from bill amount and usage amount appeared on monthly bills. The expected marginal
price is a result of probability weighted marginal prices from different blocks. The probability
weights measure the chance that consumers being on different blocks. Shin (1985) uses a
specific price perception variable to isolate marginal price and average price to conduct a test.
The empirical results support that consumers respond to average price. Borenstein (2009) utilize
the change of IBP schedule to estimate the price elasticity using combination of three competing
price variables. The results suggest marginal price is the least useful of three measures in
gauging consumer response. Ito (2014) exploit price variation of electricity service areas to test
consumer’s response under IBP. A “bunching” test, i.e. testing if consumers are distributed
around the kink points of IBP, and an encompassing tests of alternative prices have been
conducted. Both results indicate consumers respond to average price instead of marginal price.
Similar results can also been found at Ito (2013) in the context of water consumption. Wichman
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(2014) shows that water consumers tend to react to average price in a quasi-experiment when the
IBP is introduced. Only in Nataraj and Hanemann (2011), high volume water consumers seem
to react to the marginal price.
Test results from recent literature in general indicate that inattentive consumers may make
decision through other mechanism based on “perceived price” instead of DCC model built on
marginal price. In this paper, we propose a structural demand model with behavioral factors to
capture decisions of inattentive consumers. Our model assumes consumers have no information
about IBP so that the decision making relies on “perceived price” instead of marginal price. We
use the bill and lagged bill as the “perceived price”, since they are the most direct information
appeared on the monthly bill statement. In our model, consumer makes a two-step decision on
future bill based on current and lagged bill. By the end of each month, consumer receives
current bill statement. Comparing current bill to lagged bill, consumer first makes a “direction
decision” on whether she should use more, remain same, or use less for future billing cycle. In
the future billing cycle, consumer then adjust her usage from current usage, conditioning on the
direction choice. Our model relates to an emerging stream of literature by incorporating
behavior factors in modeling to understand consumer behavior (Chetty, 2015).
We apply our model to a residential water consumption monthly panel dataset secured from
water utility EPCOR. The data includes 5480 households’ monthly water consumption from
Feb.2005 to Dec.2010 from Sun City, a retirement community located in Phoenix, AZ. The
water meters in Sun City are not smart meter, which makes it difficult for residents to acquire
timely information such as accumulated usage and marginal price. As a consequence, we
conjecture that a majority of consumers to be inattentive in their water consumption. The IBP
rate changed once in Jun.2008. Before applying our structural demand model, we test that if
4
consumers in our sample respond to the marginal price from two perspectives. The first test is
examining if the distribution of consumers is “bunching” around the block break points of IBP.
The “bunching” test originates from literature on progressive tax (Saez, 2010; Chetty, Friedman,
Olsen, & Pistaferri, 2011) and it has been applied to residential energy market recently
(Borenstein, 2009; Ito, 2013; 2014). The rationale for “bunching” is that that a large number of
indifference curves would intersect the kink points of the IBP if consumers respond to marginal
price. We can’t find strong evidence of “bunching” in our sample. The second test is designed
by exploiting the IBP rate change. The change of IBP rate leads to heterogeneous marginal price
increases for consumers in different usage levels. We first group consumers into different usage
levels based on their consistent usage before the rate change, then test if the usage reduction
matches the heterogeneous marginal price increases. The result shows that a larger marginal
price increase does not necessarily lead to a higher usage reduction. These two tests suggest that
consumers in our sample may not respond to marginal price. As a result, demand model based
on “perceived price” instead of marginal price may suit our sample consumers better.
We then apply our behavioral model to the EPCOR data. The DCC model is also estimated
for comparison. Both models are estimated using data before the rate change. Using estimated
parameters, we simulate one-month ahead water consumptions of two models for all periods
after the rate change. The performance of two models is examined by comparing simulated post-
change usage of two models to the real usage. We first calculated monthly mean and quartiles of
both simulated data and real data for comparison. The results show that, after one or two months
adjustment following the IBP rate change, our behavioral model has closer statistics to the real
data than DCC model. Secondly, we again use the IBP rate change to design another test.
Grouping consumers into different usage level based on their pre-change consistent usage, we
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estimate the usage reduction caused by rate change with both real and simulated post-change
data. The performance of two models is determined by comparing coefficient of simulated data
from which model is closer to the real data for various usage levels. The results show that our
model predict better for the low and high users, while DCC model for the middle users. This
interesting results indicate that low and high users tend to make decisions based on “perceived
price”, whereas mid-level users tend to react to marginal price. From the behavioral perspective,
the extreme users have less incentive to respond to the IBP. For low users, the water
consumption is necessity. For high users, most likely associated with a higher income, the water
bill is trivial for them to respond. Only the mid-level users have the flexibility and incentive to
understand IBP and respond correctly and timely.
Our results have significant policy values for the pricing strategy in residential energy
market. The nonlinear pricing, especially IBP, is efficient pricing strategy only when consumers
understand it and respond correctly. The efficiency is realized by promoting conservation among
high-users and affordability for low-users. This efficiency is mitigated in facing inattentive
consumers, since the price signal based on marginal price is no longer transmitted appropriately.
Comparing the two models in our sample shows that some inattentive consumers respond to
“perceived price”, because they have limited information. The limited information are results of
either insufficient effort or complexity of IBP. To solve this problem, providing timely
information with smart meters is effective for consumers with insufficient efforts. Some recent
literature using randomized control trial shows that empowering consumers with more
information makes them more responsive to marginal price (Jessoe & Rapson, 2014; Kahn &
Wolak, 2013; Attari, Gowrisankaran, Simpson, & Marx, 2014). On the other hand, the
complexity of IBP should be addressed by simplifying the pricing strategy and introducing
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behavioral interventions based on inattentive consumers’ behavior. The famous OPOWER field
experiment shows the great policy value of understanding consumer’s decision making
mechanism and introducing policies based on social norm (Allcott & Mullainathan, 2010;
Allcott, 2011; Allcott & Rogers, 2014).
7
2. Data
This section describes the data and the price schedule. Before introducing our behavioral
demand model, we first formally examine if the consumers in our sample respond to marginal
price or not. We test consumer’s response through two different perspectives. Our first test is
“bunching test” based on usage distribution of consumers in whole community, i.e. testing if
consumers are distributed around the block thresholds. Secondly, the change of IBP rate is
exploited to design another test using individual usage data. The IBP rate change in our data
leads to heterogeneous increase in marginal price for consumers with various usage levels.
Consumers are grouped into different usage levels based on their pre-rate-change consistent
usage. We test if the usage reduction across different groups matches the heterogeneous increase
in marginal price. Results from both tests suggest that consumers in our sample tend to respond
to other “perceived price” instead of marginal price.
2.1. Data, Summary Statistics and Rate Schedule
Our primary data is a monthly balanced panel of residential water bill under IBP provided by
“EPCOR Utility” from Feb.2005 to Dec.2010. There are 5480 households in our sample from a
retirement community Sun City located in Phoenix, AZ. In Sun City, the monthly water usage is
not measured by smart meters. Instead, the EPCOR agents use tradition measurement to record
usage by the end of each month. As a consequence, residents are less likely to have a clear track
of their usage during each month. Understanding the marginal price decided by the accumulated
usage is also difficult. Therefore, we conjecture that residents in Sun City are inattentive in their
water consumption decision.
8
Table 1.
Descriptive Statistics
Variable Description Units Mean Std. dev. Min. Max.Usage Monthly usage kgal 7.19 3.70 0.86 25.86Usage_before Monthly usage before rate change kgal 7.37 3.73 0.86 25.86Usage_after Monthly usage after rate change kgal 6.95 3.64 0.86 25.86Bill Monthly bill $ 14.30 4.91 6.30 51.89Bill_before Monthly bill before rate change $ 13.15 4.17 6.30 39.38Bill_after Monthly bill after rate change $ 15.79 5.37 8.72 51.89Weather Evapotranspiration mm 6.58 2.54 1.79 10.92Living area Living area size 100 ft2 15.30 3.56 7.88 43.82Yard area Yard area size 100 ft2 9.15 3.08 0.32 91.79
Income Assessed home value discounted by CPI $1,000 90.93 27.29 21.56 558.19
Notes: Number of Observation is 389,080. Before the rate change number of observation is 219,200. After the rate change number of observation is 169,880.
During our studying periods, the IBP rate in Sun City changed once on Jun.1st 2008 as
Figure 1. Before the change, Sun City has three blocks with two thresholds at 4 kgal and 18
kgal. The marginal prices per kgal are $0.72, $1.10 and $1.33 from block 1 (the lowest) to block
3 (the highest). The monthly fixed charge is $6.33. After the change, the new thresholds are
reallocated to 3 kgal and 10 kgal. The new marginal prices are $0.72, $1.32 and $1.69 from
block 1 to block 3, respectively. The monthly fixed charge increases to $7.99. The change in
marginal price and reallocating of thresholds together lead to larger raise of marginal price
between 3 to 4 kgal and 10 to 18 kgal.
According to 2010 census, Sun City has population 37,499. The Median household income
is $36,464. The percentage of senior population (over 65 years old) is 74.9%. Using the billing
address, we match the bill data with Maricopa County Assessors’ Office data to get real estate
information. These information includes assessed home value, living area size and yard area
size. We calculated the discounted assessed home value as income measure for these senior
citizens. Also, we acquire the weather data from the Arizona Meteorological Network
9
(AZMET). Following water demand literature, the weather index we use is evapotranspiration,
which is determined by solar radiation, wind speed, humidity and temperature to measure the
amount of water getting out of ground (Brown, 2014). The summary statistics of key variables
are summarized in Table 1. The average monthly bill rises after the IBP rate change from $13.15
to $15.79. Consequently, the average monthly usage drops from 7.37 kgal to 6.95 kgal.
Fig. 1. IBP Schedule
2.2. Test of Marginal Price Using Distribution (“Bunching”)
Suggested by recent literature in residential water (Ito, 2013), electricity (Borenstein, 2009;
Ito, 2014) and progressive tax (Saez, 2010; Chetty, Friedman, Olsen, & Pistaferri, 2011), we
should be able to observe “bunching” around the IBP thresholds if consumers react to marginal
price. The intuition is that a large number of indifference curves would intersect the thresholds,
which leads to a sudden increase of consumer percentage (Figure 2 of (Borenstein, 2009)). Also,
10
since the price rate changes in Jun.2008, the “bunching” should adjust accordingly if consumers
react to new marginal price timely. We calculate each consumer’s average usage before and
after the rate change and plot the histogram in Figure 2 to examine if there is “bunching”. The
summer (June, July and August) and winter (December, January and February) usage are
distinguished since water consumption is strongly affected by seasonality. If consumers respond
11
to IBP, we should be able to observe (1) histogram increase dramatically at the IBP thresholds,
and (2) histogram increase at the new thresholds after the rate change.
Based on Figure 2, we don’t have strong evidence to support “bunching”. Before the rate
change in Jun.2008, Panel (a) and Panel (b) shows that there are relatively larger increases of
consumer percentage occurred before and after the 4 kgal threshold. The 18 kgal cannot be
verified clearly due to a lack of observations. After the rate change, Panel (c) and Panel (d)
shows that the fraction of consumers increases most in the middle of two thresholds 3 kgal and
10 kgal, instead of at them. In summary, there is no clear “bunching” at the thresholds. The
absence of “bunching” suggests consumers either respond to marginal price with nearly zero
elasticity or respond to “perceived price” other than marginal price. The former conjecture may
not be true in our sample since the average usage drops after the rate change. To further validate
the second conjecture, we conduct another test based on the individual usage data.
2.3. Test of Marginal Price Using Individual Consumption Data
Using the IBP rate change, we design a second test based on consumers’ individual water
consumption data. The Jun.2008 rate change leads to disproportionate marginal price increase
for consumers with different usage levels. The marginal price remains same for consumers with
very low usage (1-3 kgal). For the rest consumers, the marginal price increases heterogeneously.
For consumers use 3-4 kgal, 4-10 kgal, 10-18 kgal, and beyond 18 kgal, the marginal price
increases 84.7%, 20.9%, 53.6% and 28%, respectively. If consumers respond to the marginal
price, the extent of usage reduction should agree with the heterogeneous marginal price increase.
12
We would expect that consumers use 3-4 kgal and 10-18 kgal reduce their consumption most in
the percentage.
To test if the usage reduction matches to the heterogeneous marginal price increase, we first
group consumers into different usage levels. Using usage data before the IBP rate change, we
calculate consumers’ average usage level. Then we assign each consumer to the integer usage
level closest to her average usage for grouping. Again, the summer (June, July and August) and
winter (December, January, February) has been studied separately due to strong seasonality. We
then estimate following equation for each usage group j to evaluate the usage reduction in a
“pre-change vs. post-change” version:
ln (c¿ )=After ⋅α +x¿ β+μi+ϵ¿ ,if i∈ j(1)
where ln (c¿ ) is the natural logarithm of consumer i‘s usage at time t . The After is the dummy
variable with value 1 if it is after rate change. The parameter α measures the percentage usage
reduction after the rate change. The x¿ is control variables includes weather and household’s real
estate information. The μi is household fixed-effect. Equation (1) is different from “Difference-
in-Difference” estimator, since there is no control group. Without a control group, variables
should be included to control the exogenous trends which might affect water consumption. In
our case, the weather and household’s real estate information (income measure) in x¿ are
introduced for this purpose. The detailed estimation results are included in the Appendix 2.
Here, we plot the coefficient with its 95% confidence interval in Figure 3 with the percentage
change of marginal price for our test. We expect that the lower confidence bound for consumers
between 3-4 kgal and 10-18 kgal should be higher than the higher confidence bound of their
neighbors.
13
According to Figure 3, there is no clear evidence that consumer on average use 3-4 kgal, 10-
18 kgal has relative higher usage reduction compared to their neighboring counterparts. From
panel (a), the summer reduction for consumers at 3-4 kgal is not statistically different from 0.
For 10-18 kgal consumers, their coefficients are not well above their neighbors. From panel (b),
the winter usage reduction for 4 kgal is again not statistically different from 0. Compared to 4-
10 kgal users, the 10-18 kgal users reduce slightly more as expected. Since there are no enough
observations in the high usage group because of a lower winter usage, we can’t compare 10-18
kgal users to higher users. In summary, the higher increase in marginal price does not
necessarily leads to more usage reduction. We again verify the conjecture that consumers do
respond to the other “perceived price” instead of marginal price.
14
Panel (a)
Panel (b)
-.50
.51
Per
cent
age
Cha
nge
0 5 10 15 20Usage Level (kgal)
Usage Reduction % Usage Reduction 95% CIMarginal Price Change %
Marginal Price Change and Useage Reduction (Summer)
-.50
.51
Per
cent
age
Cha
nge
0 5 10 15 20Usage Level (kgal)
Usage Reduction % Usage Reduction 95% CIMarginal Price Change %
Marginal Price Change and Useage Reduction (Winter)
Fig. 3. Test of Marginal Price Using Individual Consumption Data
15
3. Model
The test results from Section 2 suggests that consumers in our sample tend to be inattentive
in water consumption. Instead of reacting to the marginal price, consumers react to other
“perceived price”. In order to construct a structural demand model for inattentive consumers, the
main decision variables should be “perceived price”. This section first presents our behavioral
model. Then we draw a comparison between our behavioral model and DCC model.
3.1. Assumptions and Model Timeline
Our behavioral model assumes that consumers are inattentive to IBP, so they are not
responding directly to the marginal price. Instead, they seek other easily-acquired “perceived
price” for decision making. The most direct information consumers have is the current and
lagged bills appeared on the monthly statement. We assume that consumers using two bills as
main decision variables instead of marginal price. The exact usage level remains unknown to
consumers.
We model consumer’s water demand as a two-step decision based on the timeline in Figure
4. In Figure 4, each increment is a one-month billing cycle. The decision is using information of
current bill and lagged bill to determine future bill. The first step is a direction decision. By the
end of current billing cycle, consumer first decides the direction for next billing cycle by
comparing current bill and lagged bill. The direction decision is either “use more”, “remain
same” or “use less”. For example, if current bill is way higher than lagged bill, consumer has
higher probability to use less in the coming billing cycle. The direction decision is affected by
consumer’s income level, which determines her sensitivity with bill. The second step is an
adjustment decision started in the new billing cycle. The adjustment is based on current usage,
16
conditioning on the direction decision. Since we assume consumers has no information about
the usage level, the adjustment for each direction takes a fixed step. Another reason for using a
fixed step is that water usage is normally induced by the appliances, so the usage adjustment is
somehow fixed without a major overhaul of appliances. For example, suppose that current much
bill is higher than lagged bill, consumers have incentive to cut water consumption. Without
tracking the exact usage level, it will be difficult for consumer to adjust based on exact usage
level, i.e. reduce 1.3 kgal. Easier adjustments are based on specific activities such as taking one
less shower or using dishwasher once per day instead of twice. As a result, the usage adjustment
takes fixed steps determined by appliances and activities. In our model, we also included
weather and real estate variables to gain a better control of the adjustment decision.
Fig. 4. Model timeline
17
3.2. Specification and Estimation
We model the first-step direction decision as a selection model in which consumer i at time t
has following indirect utility:
V ¿=α X ¿+u¿
In this indirect utility V ¿, X ¿ is the decision variable included difference between current bill and
lagged bill and discounted assessed home value as income measure. No constant is included in
X ¿ and u¿ is the error term for direction decision. Following most selection model, we assume
u∼N (0,1) to make the model identifiable. Then the direction decision is:
d i ,t={ 1 (usemore ) if −∞<V ¿<V2 (remain same ) if V <V ¿<V
3 (useless ) if V <V ¿
where V and V are the cutoff parameters to be estimated.
For the second-step adjustment decision, consumer’s future bill is an adjustment based on
current bill, the adjustment decisions are:
bil li ,t+1k =βk Y i ,t +ζ k Z i , t+1+ϵ¿
k
where k={1,2,3 } for d i ,t={1,2,3}, respectively. Y i ,t is current bill and Zi , t +1 includes weather
and real estate variables in the future billing cycle. No constant is involved so that the future bill
is now adjusted from the current bill instead of its unconditional mean. The ϵ k is adjustment
error for different directions. We assume that each ϵ k is distributed normally as ϵ k∼N (0 , σ k)
and there is no correlation between adjustment errors for different directions. The 0 expectation
means that consumer’s monthly adjustment errors from previous bill offset each other over a
long period time. Together, the direction error u and adjustment error ϵ are jointly distributed as:
18
[ uϵ 1
ϵ 2
ϵ 3]∼N ([0000] , [ 1⋯⋯⋯
ρ1 σ1
σ12
⋯⋯
ρ2 σ2
0σ 2
2
0
ρ3 σ3
00σ3
2 ])where ρk is the correlation between direction error and adjustment error at k direction. Since we
normalized direction error to 1, the adjustment error σ k are automatically normalized based on
direction error.
We estimate the behavioral structural model with maximum-likelihood technique. The log-
likelihood for individual i at time t with direction decision k can be written as:
L Li , t+1=∑k=1
3
I {d i, t+1=k } log ~Li ,t+1k
~Li , t+1k =
1√2 π
∗exp (−(sk )2
2 )
σk∗[Φ (rk )−Φ (nk) ]
where
sk=bil li , t+1
k −βk Y i , t
σk, r j=
Bk−ρk sk
√1−ρk2
, n j=Ak− ρk sk
√1−ρk2
and
{ A1=−∞ , B1=V−~V i , t(Use More)A2=V −~V i ,t , B2=V−~V i ,t(Remain Same)
A3=V−~V i , t , B3=∞ (Use Less)
3.3. Comparison Between Behavioral Model to DCC Model
19
We compare our behavioral demand model to the Olmstead et al (2007) and Olmstead (2009)
version of DCC model. The construction of two models both follows a two-step procedure. In
our model, the first step is a direction decision based on comparing indirect utility of different
choice and the second step is an adjustment decision. For the DCC model, the first step is a
discrete choice on which block of IBP is to be chosen based on usage and the second step is to
choose usage level within the block. The specification and estimation approach are similar. The
main difference is the key decision variable. In our model, consumers decide water consumption
based on “perceived price”, since they don’t know the real-time usage and marginal price. For
the DCC model, consumers have clear information of IBP so that they respond to the marginal
price. We expect the performance of two models to vary case-by-case. Consumers’ information
level and context of different studies determine the performance.
20
4. Empirical Results
We applied both our behavioral model and DCC model to the subsample before the rate
change in Jun.2008. In this section, we first present the estimates of both models. The second
part of this section is to examine the performance of two models in our sample. The earlier test
results from Section 2 suggest that consumers in our sample tend to be inattentive in water
consumption. To verify this hypothesis, we simulated the one-month-ahead post-change usage
of two models using the pre-change parameters to compare to real data. Our verification relies
on two different approaches. First, we calculated the main statistics (mean and quartiles) of
simulated usage and real usage for comparison on a monthly basis. The results suggest that our
behavioral model is closer to the real usage, after one or two months adjustment. Secondly, we
design another test based on the change of IBP rate. We estimate the usage reduction caused by
IBP rate change using simulated data to replace the post-change real data. Then we compare
estimate from which model can better reflect the real data on various usage levels. The result is
interesting: our behavioral model performs better than DCC for consumers with low and high
usage, while DCC is better for the rest consumers with mid-level usage.
4.1. Estimation
The result of our behavioral structural model using pre-rate-change subsample is included in
Table 2. The parameters for direction decision and adjustment decision are jointly estimated.
For direction decision, the difference between current bill and lagged bill has significant positive
coefficient. For consumers have a higher bill in current month than previous month, they have a
higher probability to cut usage in the coming month, and vice versa. However, the income
measure is not significant for the direction decision. A potential reason is that the water
consumption is induced by consumers’ home appliances. The home appliances are durables so
21
that minor fluctuations in the income will not lead to a major overhaul. As a result, the income
coefficient is not significant. Also, consumers with dramatic change of income may move out
from Sun City. The vacancies are filled with new consumers with similar income level, which
mitigate the magnitude of the income coefficient. The cut off points for direction decision
behave normal as expected.
Table 2.
Behavioral model estimates
Variable Estimates Std. devDirection Regression
Lagged bill difference 0.102 *** (0.0022)
Income 0.000 (0.0002)
Main Regression (Bill goes up) Weather * Lagged bill 0.024 *** (0.0004)
Living area * Lagged bill -0.001 *** (0.0002)
Yard area * Lagged bill -0.004 *** (0.0002)
Lagged bill 1.109 *** (0.0058)
Main Regression (Bill stays) Weather * Lagged bill 0.003 *** (0.0001)
Living area * Lagged bill -0.004 *** (0.0001)
Yard area * Lagged bill 0.005 *** (0.0000)
Lagged bill 0.992 *** (0.0005)
Main Regression (Bill goes down) Weather * Lagged bill -0.002 *** (0.0003)
Living area * Lagged bill -0.001 *** (0.0003)
Yard area * Lagged bill -0.005 *** (0.0002)
Lagged bill 0.844 *** (0.0042)
Cutoff 1 -0.228 *** (0.0206)
Cutoff 2 0.377 *** (0.0237)σ 1 2.119 *** (0.0170)
σ 2 0.139 *** (0.0025)
σ 3 1.617 *** (0.0045)
ρ1 -0.013 (0.0284)
ρ2 0.022 *** (0.0053)
ρ3 0.338 *** (0.0652)
22
Notes: *** Significant at the 0.05 level.
For adjustment decisions, the variable of interest is the current bill. For consumers decide to
“use more”, “stay same” and “use less”, their future bills are approximately 110%, 99% and 84%
of current bill. Interestingly, consumers make larger adjustments when they decision to cut
water consumption. For other control variables, a drier weather (a higher evapotranspiration)
leads to a larger adjustment for “use more” and “stay same” scenarios as expected, but not for
“use less”. For consumers decide to cut water consumption, a dry weather hold back their
adjustments. The larger indoor living area size shrinks the usage adjustment. A larger indoor
living area means a relatively smaller outdoor area. The indoor water consumption is almost fix
all around the year, while the outdoor consumption varies. As a result, larger indoor living area
means relatively less flexibility in adjusting water usage. The outdoor yard area size has unclear
effect on the usage adjustment. We don’t have the details on consumers’ outdoor condition, such
as pool ownership or turf area, so we cannot explain these parameters clearly. Other estimates
like standard error and correlation behave normal as expected.
We then estimate DCC model following exact procedures on Olmstead et al (2007) and
Olmstead (2009). The elasticity is simulated using non-parametric approach suggested by
Olmstead (2009). Results are summarized in Table 3. Following these two papers, the economic
explanation is based on the simulated price and income elasticities. The negative price elasticity
and positive income elasticity are as expected. The price parameter indicates that 1% increase in
the price reduce 0.22% in usage. The income parameter indicates that 1% increase in wealth
(measured by assessed value) will increase the usage by 0.11%. In other DCC applications, the
standard deviation of consumer heterogeneity error (σ η) is higher than the optimization error (σ ϵ ¿
. However, the standard deviation of consumer heterogeneity is very small in our sample. We
23
found similar results in Dale et al. (2009). The small magnitude may be caused by the fact that
our sample included a single community, whereas most other applications included multiple
communities located in different cities.
Table 3.
DCC model estimates and simulated elasticity
DCC Model Estimates Variable Coefficient Std. dev.ln(price) -1.522*** (0.080)ln (virtual income) 1.521*** (0.050)
Weather 0.028*** (0.001)Living area -0.065*** (0.002)Yard area -0.017*** (0.001)
Constant -3.911*** (0.160)σ η 0.014*** (0.004)σ ϵ 0.605*** (0.002)
Simulated Elasticity Variable Elasticity estimates Std. dev.Price -0.22*** (0.02)Income 0.11*** (0.01)
Notes: *** Significant at the 0.05 level.
4.2. Performance Test Based on Statistics
Using pre-change estimates of both models, we simulated one-month-ahead usage for the
post-change periods. The simulator of our behavioral model is discussed in the Appendix 1. For
DCC model, we use the simulator listed on Olmstead (2009) (Appendix). We then calculate the
statistics (mean and quartiles) of simulated usage to compare to the real usage for all periods
after the rate change. The results for different statistics are plotted in Figure 5. Our earlier test
24
56
78
910
Usa
ge (k
gal)
0 20 40 60 80Bill order
DCC_use_avg Behavioral_use_avgReal_use_avg
Summary Statistics of Predicted Usage (Mean)5
67
89
10U
sage
(kga
l)
0 20 40 60 80Bill order
DCC_use_med Behavioral_use_medReal_use_med
Summary Statistics of Predicted Usage (Median)
24
68
10U
sage
(kga
l)
0 20 40 60 80Bill order
DCC_use_p25 Behavioral_use_p25Real_use_p25
Summary Statistics of Predicted Usage (25th Percentile)
56
78
910
11U
sage
(kga
l)
0 20 40 60 80Bill order
DCC_use_p75 Behavioral_use_p75Real_use_p75
Summary Statistics of Predicted Usage (75th Percentile)
from Section 2 suggests consumers in our sample are inattentive in water consumption, so that
our behavior model should out-perform the DCC model.
The results from Figure 5 are as we expected. Generally, after one or two month adjustment
following the rate change, our behavioral model start to perform better than DCC in the rest
periods. The results are consistent for different statistical measures. It agrees with our earlier
conjecture that consumers in our sample are inattentive in water consumption. However, the
statistical measures only pick desired information from the pool of usage data every month, it
does not keep a clear track of each household’s heterogeneity. The result may be biased. To
further confirm our conjecture, we use the change of the IBP rate to design another test. The test
is using consumer’s individual usage while controlling household fixed effect.
Fig. 5. Performance Test Based on Statistics
25
4.3. Performance Test Based on Individual Usage Data
To address the concern that simply comparing statistical measure may lose track of each
consumer’s heterogeneity, we use change of IBP rate to design another test. The approach we
took is same as the one discussed in Section 2.3, i.e. the estimate equation (1) to get usage
reduction estimates caused by rate change for consumers with various usage levels. To examine
which model perform better, we estimated equation (1) after replacing post-change real usage by
simulated data first. Then the performance of two models is determined by comparing the
coefficient from which model is closer to the real data for consumers with different usage levels.
The equation (1) includes the consumer’s fixed effect, so consumer’s heterogeneity is under
control. The coefficients and their 95% confidence intervals are plotted in Figure 6 for various
usage levels. Our test is conducted by comparing which model has closer coefficient band to the
coefficient band of real usage. Summer and winter are distinguished due to the seasonality. The
details of estimation results are in included in the Appendix 2.
Figure 6 shows very interesting results. Our behavioral model fits real usage better for the
low and high users, while the DCC result fits the users in the middle better. It implies that
consumers with low and high usage tend to make decision based on “perceived price” while
middle usage consumers tend to respond to marginal price. For low users, the water
consumption is necessity. The basic amount of water consumption such as taking shower and
cleaning dishes cannot be dramatically affected by change of price. For high users, most likely
associated with a higher income, the water bill is trivial for them to respond. Also, reducing
water consumption means loss of utility generated from other intensive water using activities,
such as having a swimming pool or keeping a green turf. As a result, consumers refuse to cut
26
water consumption. Only the mid-level users have the flexibility and incentive to respond to
marginal price. The result is robust to seasonality.
27
Panel (a)
Panel (b)
-1-.5
0.5
11.
5R
educ
tion
(%)
0 5 10 15 20Usage (kgal)
Data Data_95%CIDCC DCC_95%CIBehavioral Behavioral _95%CI
Usage Reduction Percentage (Summer)
-1-.5
0.5
11.
5R
educ
tion
(%)
0 5 10 15 20Usage (kgal)
Data Data_95%CIDCC DCC_95%CIBehavioral Behavioral _95%CI
Usage Reduction Percentage (Winter)
Fig. 6. Performance Test Based on Individual Usage Data
28
5. Conclusion
In this paper, we propose a behavioral demand model based on “perceived price” to
characterize inattentive consumer’s behavior in residential energy market. We start by testing
whether consumers in Sun City, AZ respond to marginal price or “perceived price” in water
consumption. The test results suggest that “perceived price” should be decision variable for our
sample consumers. Then we applied our behavioral model with DCC model built on marginal
price to our sample. The performance of two models suggest that consumers with low and high
usage more likely to respond to “perceived price”, while mid-level users tend to respond to
marginal price. Our approach can be generalized from water demand to other energy markets.
We understand that this paper is limited by the fact that there is only a single community under
studied. Also, the data we have does not included detailed real estate information other than the
indoor and outdoor size. These limitations are to be tested and corrected by future research.
This paper helps explore the possible decision making process of inattentive consumers in
the residential energy market. For inattentive consumers, they do not respond to the real time
marginal price in deciding energy consumption. The inattentiveness is caused by short of
information, which is fundamentally decided by two factors. First, the efforts for acquiring
information are so demanding that consumers refuse to spend time to collect. The monetary
reward seems also too limited to provide strong incentives, since the energy bill is overall a
relative small part of household monthly expenditure. Second, the nonlinear price, IBP included,
may be difficult for some consumers to understand. Without help of records from the smart
meters, keeping track of accumulated usage and marginal price is almost impossible. As a
consequence, the efficiency of IBP is mitigated in facing inattentive consumers, since the signals
29
based on marginal price can no longer be transmitted appropriately. A better understanding of
consumer behavior is helpful in recovering the efficiency loss.
From the policy point of view, our suggestions focus on how to recover the efficiency loss
when IBP meets inattentive consumers. For inattentiveness caused by consumer’s limited effort,
we suggest to provide smart devices to facilitate the information collection. Recent literature
using randomized control trial shows that empowering consumers with more information makes
them more responsive to marginal price (Jessoe & Rapson, 2014; Kahn & Wolak, 2013; Attari,
Gowrisankaran, Simpson, & Marx, 2014). On the other hand, for inattentiveness caused by the
complexity of IBP structure, designing a simplified pricing schedule with behavioral intervention
may solve the problem. The famous OPOWER field experiment shows the great policy value of
understanding consumer’s decision making mechanism and introducing policies based on social
norm (Allcott & Mullainathan, 2010; Allcott, 2011; Allcott & Rogers, 2014). From our opinion,
the design of a new pricing schedule integrated with behavioral intervention is a great choice for
future research.
30
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Appendix
Appendix 1. Analytical form for Simulating Expected Bill
A generalized derivation can be found in Chiburis and Lokshin (2007). Here, indexing consumer’s direction choice for “use more”, “stay same” and “use less” byk={1,2,3 }, the bill is
bil li ,t +1k = βk Y i ,t +ϵ ¿
k
and the direction decision is:
d i ,t={ 1 if −∞<ui ,t <V−V i , t
2 if V−V i , t<ui , t<V−V i , t
3 if V −V i ,t <ui ,t
We can then have
E (billi ,t+1|X i , t , Y i ,t )=∑k=1
3
β j' Y i ,t+ E (ϵ i ,t
k |X i ,t )=∑k=1
3
βk' Y i , t+ρk σ k λ i ,t
k
λ i ,tk =E (ui ,t|X i ,t )=
∫μk
μk +1
(V i ,t−α ' X i , t )ϕ (V i , t−α' X i ,t ) dV i , t
Φ ( μk+1−a' X i ,t )−Φ ( μk−a' X i ,t )=
−∫μk
μk+1
ϕ' (V i ,t−α ' X i , t )d V i ,t
Φ (μk +1−a ' X i , t )−Φ (μk−a' X i ,t )=
ϕ (μk−a' X i ,t )−ϕ(μk+1−a' X i ,t)
Φ (μk +1−a ' X i , t )−Φ (μk−a' X i ,t )
where
{μ0=−∞μ1=Vμ2=Vμ3=∞
After simulated the bills, we convert the results into the usage for further analysis. In this
simulation, there is about 1% of expected bill is lower than the fix charge, so we cannot convert
these observations. Instead, we set their usage as missing value in model comparison.
33
Usage (kgal) Coefficient Std. Dev P for T-test P for F-test Nobs1 0.16 0.12 0.19 0.28 902 0.06 0.03 0.07 0.00 15843 0.02 0.02 0.15 0.00 44644 0.01 0.01 0.25 0.00 75065 -0.05 0.01 0.00 0.00 106026 -0.05 0.01 0.00 0.00 113227 -0.07 0.01 0.00 0.00 111428 -0.11 0.01 0.00 0.00 119529 -0.10 0.01 0.00 0.00 9720
10 -0.13 0.01 0.00 0.00 847811 -0.15 0.01 0.00 0.00 619212 -0.14 0.01 0.00 0.00 464413 -0.17 0.01 0.00 0.00 354614 -0.16 0.02 0.00 0.00 248415 -0.16 0.02 0.00 0.00 178216 -0.16 0.02 0.00 0.00 136817 -0.17 0.03 0.00 0.00 84618 -0.16 0.03 0.00 0.00 41419 -0.22 0.04 0.00 0.00 32420 -0.18 0.06 0.00 0.00 9021 -0.16 0.04 0.00 0.00 90
Real Data (Summer) Estimates of Equation (1) -- Only Reporting the Parameter alpha
Usage (kgal) Coefficient Std. Dev P for T-test P for F-test Nobs1 0.32 0.09 0.00 0.00 1022 0.10 0.02 0.00 0.00 23123 0.05 0.01 0.00 0.00 62224 0.02 0.01 0.09 0.00 99285 -0.02 0.01 0.00 0.00 136856 -0.03 0.01 0.00 0.00 152837 -0.05 0.01 0.00 0.00 136688 -0.05 0.01 0.00 0.00 110339 -0.11 0.01 0.00 0.00 7786
10 -0.14 0.01 0.00 0.00 510011 -0.18 0.01 0.00 0.00 350212 -0.16 0.02 0.00 0.00 202313 -0.22 0.02 0.00 0.00 117314 -0.18 0.03 0.00 0.00 64615 -0.30 0.04 0.00 0.00 35716 -0.29 0.05 0.00 0.00 22117 -0.34 0.09 0.00 0.00 3418 -0.31 0.08 0.00 0.00 51
Real Data (Winter) Estimates of Equation (1) -- Only Reporting the Parameter alpha
Appendix 2. Marginal Price Test Results in Details
The following results are the estimates of equation (1) for real data, DCC simulated Data and
our behavioral model’s simulated data. The summer and winter are separated due to the
seasonality.
34
Usage (kgal) Coefficient Std. Dev P for T-test P for F-test Nobs1 1.22 0.13 0.00 0.00 902 0.90 0.03 0.00 0.00 15843 0.65 0.01 0.00 0.00 44644 0.48 0.01 0.00 0.00 75065 0.32 0.01 0.00 0.00 106026 0.19 0.01 0.00 0.00 113227 0.10 0.01 0.00 0.00 111428 0.01 0.01 0.04 0.00 119529 -0.07 0.01 0.00 0.00 9720
10 -0.14 0.01 0.00 0.00 847811 -0.22 0.01 0.00 0.00 619212 -0.25 0.01 0.00 0.00 464413 -0.35 0.01 0.00 0.00 354614 -0.39 0.01 0.00 0.00 248415 -0.44 0.02 0.00 0.00 178216 -0.49 0.02 0.00 0.00 136817 -0.48 0.02 0.00 0.00 84618 -0.60 0.03 0.00 0.00 41419 -0.55 0.03 0.00 0.00 32420 -0.61 0.06 0.00 0.00 9021 -0.62 0.06 0.00 0.00 90
DCC Simulated Data (Summer) Estimates of Equation (1) -- Only Reporting the Parameter alpha
Usage (kgal) Coefficient Std. Dev P for T-test P for F-test Nobs1 1.48 0.06 0.00 0.00 1022 1.08 0.01 0.00 0.00 23123 0.77 0.01 0.00 0.00 62224 0.50 0.01 0.00 0.00 99285 0.28 0.01 0.00 0.00 136856 0.12 0.00 0.00 0.00 152837 -0.03 0.01 0.00 0.00 136688 -0.16 0.01 0.00 0.00 110339 -0.28 0.01 0.00 0.00 7786
10 -0.41 0.01 0.00 0.00 510011 -0.47 0.01 0.00 0.00 350212 -0.58 0.01 0.00 0.00 202313 -0.66 0.01 0.00 0.00 117314 -0.72 0.02 0.00 0.00 64615 -0.83 0.02 0.00 0.00 35716 -0.87 0.03 0.00 0.00 22117 -0.97 0.05 0.00 0.00 3418 -0.96 0.06 0.00 0.00 51
DCC Simulated Data (Winter) Estimates of Equation (1) -- Only Reporting the Parameter alpha
35
Usage (kgal) Coefficient Std. Dev P for T-test P for F-test Nobs1 0.53 0.14 0.00 0.00 872 0.32 0.03 0.00 0.00 15493 0.28 0.02 0.00 0.00 44104 0.24 0.24 0.00 0.00 74725 0.13 0.01 0.00 0.00 105616 0.09 0.01 0.00 0.00 112937 0.06 0.01 0.00 0.00 111148 0.00 0.01 0.72 0.00 119169 -0.01 0.01 0.30 0.00 9690
10 -0.04 0.01 0.00 0.00 843311 -0.07 0.01 0.00 0.00 616012 -0.07 0.01 0.00 0.00 460313 -0.13 0.01 0.00 0.00 351214 -0.14 0.02 0.00 0.00 243615 -0.13 0.02 0.00 0.00 174716 -0.16 0.02 0.00 0.00 131817 -0.16 0.03 0.00 0.00 82318 -0.18 0.01 0.00 0.00 38319 -0.21 0.04 0.00 0.00 30320 -0.22 0.07 0.00 0.00 8221 -0.13 0.05 0.01 0.00 81
Behavioral Model Simulated Data (Summer) Estimates of Equation (1) -- Only Reporting the Parameter alpha
Usage (kgal) Coefficient Std. Dev P for T-test P for F-test Nobs1 0.41 0.09 0.00 0.00 1022 0.19 0.02 0.00 0.00 23113 0.16 0.01 0.00 0.00 62174 0.10 0.01 0.00 0.00 99215 0.07 0.01 0.00 0.00 136776 0.06 0.01 0.00 0.00 152647 0.03 0.01 0.00 0.00 136338 0.03 0.01 0.00 0.00 110039 -0.03 0.01 0.00 0.00 7758
10 -0.08 0.01 0.00 0.00 507811 -0.12 0.01 0.00 0.00 348212 -0.11 0.02 0.00 0.00 199513 -0.17 0.02 0.00 0.00 116314 -0.19 0.03 0.00 0.00 62215 -0.24 0.04 0.00 0.00 35116 -0.29 0.05 0.00 0.00 20817 -0.26 0.06 0.00 0.00 3418 -0.22 0.07 0.00 0.00 50
Behavioral Model Simulated Data (Winter) Estimates of Equation (1) -- Only Reporting the Parameter alpha
36
