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Quantity Uncertainty and Demand: The Case of Water Smart Reader Ownership

  • Aaron Strong EMAIL logo and Chris Goemans


There are a number of instances when consumers have imperfect information regarding the quantity they consume. This paper has two objectives: (1) to formally describe how quantity uncertainty is likely to affect consumer behavior and (2) to describe how these changes in behavior are likely to differ depending on how the quantity uncertain good is priced. We develop a theoretical model of consumer behavior under quantity uncertainty which we use to illustrate how different price structures and different locations within price structures matter for how information impacts behavior. We test these hypotheses using a unique panel data set containing information on water consumption habits of more than 88,000 households in the City of Aurora, Colorado. In 2005, Aurora subsidized the purchase of electronic devices for households to monitor water use. These devices provide households with real-time information on their water use. We find that, consistent with the aims of the program, households with the device decreased their water use during periods when they faced a constant marginal price; however, contrary to the aims of the program, their consumption increased during periods when they faced an increasing block rate pricing structure. These results are consistent with the predictions of the theoretical model developed herein.


Proof of Proposition 1:

We know that wp solves


If the consumer is risk neutral and has unbiased expectations, then b=0 and the expectation can be passed through the sub-utility function so that:


We can now rewrite this as:


Taking the first-order condition, we have


Since risk neutrality implies linear utility, vx(.)=v¯x, a constant. So uw(wp)=p_v¯x, the same first-order condition as in the certain case. ⃞

Proof of Proposition 2:

The first-order condition is defined by:


By assumption, vx(.) is a convex function, so by Jensen’s inequality on convex functions, we know



Which implies that uw(wp)uw(w). So that wwp. If νxxx(.)>0, then Jensen’s inequality applies with a strict inequality and the result follows. ⃞

Proof of Proposition 3:

We outline the proof corresponding to the case when households underestimate their actual use. Note when b is positive it shifts the distribution of actual water use to the right. Since the consumer is unaware of this bias, they make decisions under the incorrect assumption that the distribution of actual use is fwa;wb>0p instead of fwa;wp,b. The latter corresponds to the distribution of actual use for the unbiased consumer with wp=wb>0p+b.

For b>0:


However, for the case when the consumer is risk neutral, the marginal utility of consumption is constant so that


Therefore, uwwb>0p=uwwpand wb>0p=wp.

It follows that:

Corollary 1:

We outline the case for the risk averse consumer who underestimates actual water use. Once again this implies that for b>0:


For the risk averse consumer, vx>0 and it follows that


Therefore, uwwb>0p<uwwp and wb>0pwp<w.

Thus, Ewb>0a>Ewa<w

Proof of Proposition 4:

If we suppress the price structure, we can write the consumer’s problem under uncertainty as in eq. [6]:


Now, since preferences are risk neutral, we can rewrite this as:


Since preferences are risk neutral, we know vx(.)=v¯x, a constant. Hence, we can rewrite the first-order condition as:


Since p¯>p_, u(w˜p)>p_v¯x=u(w˜). Hence, w˜p<w˜ because of the concavity of v. ⃞

Proof of Proposition 5:

The proof follows similarly to that of Proposition 3 and noticing that since p¯>p_, uwp(wp*)=p¯νx(x)f(wa;wp*,b)dwa. ⃞


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  1. 1

    For a comprehensive review of the early literature, see Machina (1987).

  2. 2

    Although one may think of a mapping between quality and quantity, we draw a distinction from this literature in that the consumer does not know how much they will be billed for consumption. This is in stark contrast to the quality literature where, quality and, hence, utility derived from the good is unknown. Here, we assume that utility for the good is known, although residual income and therefore utility from a numeraire good is unknown at the time of purchase.

  3. 3

    Increasing block rate structures are characterized by different marginal prices for different levels of consumption. For example, a two-tiered increasing block rate structure may charge households $2 for each thousand gallons below 10,000 and $4 for each thousand gallons above 10,000. The main idea being that the marginal prices are increasing as one increases consumption.

  4. 4

    “Morgan Stanley…predicts that the worldwide smart grid market alone will grow…to $100 billion in 2030.” (Economist 2010, 8). Note, the consumer side of this is just one part.

  5. 5

    Jordan (1999) provides examples of how, even with water bill in front of consumers, it might be difficult to identify how much water a household has consumed.

  6. 6

    For households in our sample, prior to 2005, there was no legal way to easily acquire such information as meters are buried underground and not accessible.

  7. 7

    Note the similarities between the IBR pricing schemes used by many utilities and the way in which cellular phone minutes are often priced. Many cellular phone plans offer a fixed number of minutes at zero price, for consumption beyond this point consumers face a large per minute charge.

  8. 8

    If we relax this assumption of additive separability, the results would be affected by the degree to which consumption of these two goods interact. Obviously, if there is a complementary relationship between w and x, our results would be exacerbated and if w and × are substitutes the results would be muted.

  9. 9

    In one sense, we may think of wa as the result of a collection of decisions the individual makes over the course of a billing cycle.

  10. 10

    The anticipated quantity could differ from the actual quantity, because households do not know the rate at which appliances use water or the rate at which households utilize these appliances. It may be difficult to monitor either of these two types of uncertainty even if an individual constitutes the household.

  11. 11

    It is worth noting that the setup of the quantity uncertain model is similar to the setup for a precautionary savings model as in Leland (1969) or Ritchken and Huo (1988). In their framework, the decision involves choices in a two-period model with distinct utility functions in each period (corresponding to our separability assumption), with no discounting, and an uncertain income which is dependent upon decisions made in the first period. Hence, if an individual chooses to consume less than they would have in a certain world, we may think of that as simply precautionary savings for numeraire consumption.

  12. 12

    Note that under the assumption that wa is normally distributed with mean wp+b:


    As a result, the first-order condition can be rewritten as


    Integrating by parts we have



  13. 13

    Note that, without significant loss of generality, we have set the price of the first block equal to the price charged under the constant marginal pricing structure in order to better compare the results between the two pricing structures.

  14. 14

    It is important to remember that we are operating in the short term, under the assumption that capital is fixed.

  15. 15

    Together, these two attributes are behind the growing popularity of increasing block rate pricing structures. Increasing the per unit price for consumption beyond wˆ allows policy makers to target “high consumption” households without affecting the demand of those households who consume less than the threshold, where high consumption would be determined by the water service provider.

  16. 16

    Again, over time we might expect any differences between the consumer’s perception of the block they consume in and the actual block they consume in to disappear. However, this might not be the case for some consumers given the complicated nature of IBR structures, water bills, and so on. Thus, it is possible that a household who regularly, but unknowingly, consumes in the second block behaves as if the marginal price they expect to face is p_ and reduces their target level of consumption due to fear of crossing the threshold.

  17. 17

    This is a subset of the total records for the City of Aurora. We only consider households who do not move within Aurora in the sample period. This provides us with an unbalanced panel data set of households with a fixed location as some households move into and out of the data set over time.

  18. 18

    It is worth noting that IBR pricing was primarily, but not exclusively used during the irrigation season over the period of study.

  19. 19

    It is not clear if customers would have appreciated all the details of the price structure. For example, see Chicoine and Ramamurthy (1986) or Nieswiadomy (1992).

  20. 20

    Both this specification and a standard difference-in-difference model including a control for whether or not households eventually purchased the device were estimated. Both models produced estimates that were consistent in both sign and magnitude for all variables. We present the FE version as it not only controls for potential unobserved differences between WSR and non-WSR households, but other time-constant differences that might exist across households.

  21. 21

    We have also estimated the model using a lagged average price as well as instrumenting for the price, and the results for the other coefficients are similar in magnitude.

  22. 22

    See Hewitt and Hanemann (1995) for a discussion of the use of the discrete–continuous choice model as applied to water demand estimation under block rate structures. In a follow-up paper (Strong and Goemans 2013), we have estimated the discrete–continuous choice model in order to understand how price responsiveness changes with information. But, this does not allow us to tease out the effects of the block rate structure and WSR independently as well as jointly.

  23. 23

    Since we have a three-block structure, if a household consumes in the first block, we label them below a block boundary. If the household is consuming in the second block above the mid-point of the block, we also label them as consuming below the block boundary. If households are consuming in third block or in the first half of the second block, we label them above a block boundary.

Published Online: 2014-5-3
Published in Print: 2014-7-1

©2014 by Walter de Gruyter Berlin / Boston

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