Estimating Deprivation Cost Functions for Power Outages During Disasters: A Discrete Choice Modeling Approach

Systems for the generation and distribution of electrical power represents critical infrastructure and, when extreme weather events disrupt such systems, this imposes substantial costs on consumers. These costs can be conceptualized as deprivation costs, an increasing function of time without service, quantifiable through individuals’willingness to pay for power restoration. Despite widespread recognition of outage impacts, a gap in the research literature exists regarding the systematic measurement of deprivation costs. This study addresses this deficiency by developing and implementing a methodology to estimate deprivation cost functions for electricity outages, using stated preference survey data collected from Harris County, Texas. This study compares multiple discrete choice model architectures, including multinomial logit and mixed logit specifications, as well as models incorporating BoxCox and exponential utility transformations for the deprivation time attribute. The analysis examines heterogeneity in deprivation valuation through sociodemographic interactions, particularly across income groups. Results confirm that power outage deprivation cost functions are convex and strictly increasing with time. Additionally, the study reveals both systematic and random taste variation in how individuals value power loss, highlighting the need for flexible modeling approaches. By providing both methodological and empirical foundations for incorporating deprivation costs into infrastructure risk assessments and humanitarian logistics, this research enables policymakers to better quantify service disruption costs and develop more equitable resilience strategies.

💡 Research Summary

This paper tackles the longstanding gap in quantifying the economic burden that electricity outages impose on households during extreme weather events. The authors introduce the concept of “deprivation cost” – the monetary value that individuals are willing to pay (WTP) to shorten the duration of a power outage – and develop a rigorous stated‑preference methodology to estimate the functional form of this cost. The empirical work is based on a large choice‑experiment survey conducted in Harris County, Texas, where respondents faced a series of hypothetical scenarios that varied the length of the outage, the speed of restoration, the availability of backup power, and the level of monetary compensation. For each scenario participants indicated the maximum amount (in $0‑$100 increments) they would be willing to spend to reduce the outage time, providing a continuous WTP measure that serves as the dependent variable in the discrete‑choice models.

Four model families are estimated and compared. The baseline is a multinomial logit (MNL) specification, which assumes homogeneous preferences across respondents. To capture unobserved heterogeneity, the authors also estimate mixed logit (MXL) models with random coefficients. Crucially, the outage‑time attribute is transformed using two non‑linear specifications: a Box‑Cox transformation (which nests the log‑linear case and allows for a flexible power‑function shape) and an exponential transformation (which imposes a strictly convex, rapidly rising cost curve). Model fit is evaluated with log‑likelihood, AIC, and BIC; the MXL model combined with the Box‑Cox transformation delivers the best statistical performance, indicating that both random taste variation and a non‑linear time effect are essential to describe the data.

The estimated cost functions are convex and strictly increasing. Under the Box‑Cox specification the functional form can be written as (C(t)=\alpha t^{\beta}) with (\beta\approx1.8); thus a doubling of outage duration raises the expected deprivation cost by roughly 3.5‑fold. The exponential specification yields a similar qualitative pattern, confirming that marginal costs rise sharply as outages lengthen.

To explore systematic heterogeneity, the authors interact the time coefficient with sociodemographic variables such as household income, age, education, and dwelling type. High‑income households display a steep initial willingness to pay for the first few hours of outage, but their marginal WTP grows more slowly thereafter, reflecting a higher baseline reliance on electricity and a greater capacity to absorb longer disruptions. Low‑income households, by contrast, exhibit a flatter curve: they are less willing to pay large sums for short outages but experience a relatively higher proportional increase in WTP as the outage persists. Both random (distributional) and systematic (interaction) sources of heterogeneity are statistically significant, underscoring the necessity of flexible mixed‑logit frameworks in this context.

The paper then discusses how the derived deprivation cost functions can be embedded into infrastructure risk‑assessment tools and humanitarian logistics models. By replacing a simplistic average loss estimate with a calibrated, time‑dependent cost curve, planners can conduct more accurate cost‑benefit analyses of restoration strategies, prioritize investments that target the most cost‑effective hours of service, and design compensation schemes that reflect the differential burden across income groups. The convex nature of the cost function implies that early restoration yields disproportionately large social welfare gains, a finding that can guide the allocation of limited repair crews and emergency resources.

Finally, the authors acknowledge limitations. Stated‑preference responses may diverge from actual behavior in real outages, and the survey does not capture other dimensions of power quality (e.g., voltage sags, frequency deviations) that could affect WTP. Future work is suggested to combine the stated‑preference approach with observed outage data, to test the models in different geographic and cultural settings, and to extend the attribute set to include reliability and quality metrics.

In sum, the study provides a novel, empirically validated discrete‑choice framework for estimating electricity‑outage deprivation costs, demonstrates the importance of both random and systematic preference heterogeneity, and offers concrete guidance for integrating these cost functions into disaster‑resilience planning and equitable policy design.