(Sustainable Development in a Dynamic World; World Bank (2003)), and the most recent WDR 2010 (Development and Climate Change; World Bank (2009)). In both, “inertia in physical capital” (echoing our analysis of infrastructure) is a main theme.

# **2. Two-Period Model with Uncertain Retrofit Costs**

# **2.1 The Basic Optimality Problem**

Consider a world existing for two “periods”. Infrastructure investment is made at the start of period 1, and can be “retrofitted” at the start of period 2.<sup>7</sup> As long as it is operated and not retrofitted, a given infrastructure gives rise to a given energy consumption per unit of time, determined at the time of initial investment. Energy supply costs and environmental/climaterelated costs are uncertain at the time of establishment in period 1, but are revealed at the start of period 2. We assume that when retrofitted, the infrastructure is purged of all fossil-fuel energy content and/or all its carbon emissions. The infrastructure however still provides the same utility services to the public as it did before the retrofit. “Retrofits”, we assume, are not available in period 1: they represent a new technology, developed and available at the start of period 2.

Period 1 has unit length, while T is the “length” of period 2. T can in principle be given two alternative interpretations. First, it could simply be interpreted as the time elapsing during period 2, relative to the initial (unit) period. To invoke this interpretation in the following, it would need to be coupled with an assumption that decision makers choose a zero discount rate. Alternatively, T could embed discounting, in which case it would represent the discounted value of period 2 relative to that of period 1.<sup>8</sup> Under this interpretation, heavier discounting would lead to reduced T for given period length.

We also assume that the infrastructure can in principle be shut down at the start of period 2. Such action will be taken when the total utility of operating the infrastructure is less than the minimum of the energy cost of operation, and the retrofit cost, in period 2.

In period 1, the unit energy cost is _q1_ (given and constant).<sup>9</sup> The policy maker decides on an infrastructure investment with given capital cost _K_ . For simplicity and to focus on other issues than investment size, assume that all relevant infrastructure projects have the same investment cost. Infrastructure type is identified by a given energy intensity _H_ , where we assume that all energy consumption associated with the infrastructure is fixed once the infrastructure is established, and until it is possibly retrofitted. Considering only economically viable projects, we focus on one particular trade-off only: an infrastructure project with higher energy content must give higher immediate utility, but will be more costly to operate (as long as not being

> 7 In the model as it otherwise stands, the assumption that a retrofit can be done only at the start of period 2, and not during this period, is no limitation as, we assume, no new information (nor any new or better retrofit technology) will be forthcoming during period 2.

> 8 More precisely, when unity represents the present discounted value of a current income flow of one dollar throughout period 1, T would in this case represent the present discounted value of a current income flow of one dollar throughout period 2, as evaluated from the start of period 1.

> 9 In the continuation, when we say “energy cost”, we mean the combined energy and environmental cost associated with (fossil-fuel) energy use. This would be unproblematic when all environmental costs are charged to energy use in the form of energy taxes and quota prices. It is more problematic when this is not the case; this issue is elaborated in the final section.

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