Researchers propose a unified economic model that links household consumption, financial decisions, and the choice to adopt energy-efficiency measures under uncertainty. The paper builds a decision problem in which a household chooses how much to spend on heating, how much non-energy goods to consume, how to invest in financial assets, and when (if ever) to retrofit a home to a more efficient heating technology. The authors also introduce formal, welfare-based definitions of the rebound and backfire effects and use the model to study optimal subsidy design to correct energy-use externalities.

At the heart of the model are two consumption goods: the energy service (heating) and a non-energy good. Heating output comes from fuel use c scaled by an efficiency factor η (so heating s = η c). A retrofit raises that efficiency to a higher level e·η at cost K, which the household finances with a loan paid as a constant flow ρK. The household can invest in a risk-free bond and a risky index fund and chooses an allocation share in the risky asset. The model allows for uncertainty in asset returns and in the household’s planning horizon, which is represented by a random stopping time with an exponential distribution; this leads the household to maximise the expected discounted utility with an effective discount rate δ+λ.

The authors obtain explicit solutions for key decision rules. They derive a closed-form adoption threshold and the household’s optimal consumption and investment after adoption. A central finding is that this adoption threshold depends explicitly on household wealth. That creates a new channel through which financial conditions and borrowing constraints influence whether and when households upgrade their heating. The paper also proposes an approximation scheme to estimate welfare effects, shows that adoption is generally welfare improving, and studies how a subsidy policy can steer aggregate energy consumption.