metadata: authors: 0: author: Strand, Jon
1: author: Miller, Sebastian
2: author: Siddiqui, Sauleh
count: World
docna: 0: docna: Infrastructure investments under uncertainty
            with the possibility of retrofit : theory and simulations
docty: Policy Research Working Paper
subtopic: Climate Change Mitigation and Green House Gases,Energy Consumption,Energy Conservation & Efficiency,Environment and Energy Efficiency,Energy Demand,Energy and Environment,Energy and Mining,Global Environment
teratopic: Environment,Macroeconomics and Economic Growth,Energy
abstracts: cdata!: Investments in large, long-lived,
            energy-intensive infrastructure investments using fossil
            fuels increase longer-term energy use and greenhouse gas
            emissions, unless the plant is shut down early or undergoes
            costly retrofit later. These investments will depend on
            expectations of retrofit costs and future energy costs,
            including energy cost increases from tighter controls on
            carbon emissions. Simulation analysis shows that the
            retrofit option can significantly reduce anticipated future
            energy consumption as of the time of initial investment, and
            total future energy plus retrofit costs. The more uncertain
            are the costs, the greater the value of this option.
            However, the future retrofit option also induces more
            energy-intensive infrastructure choices, partly offsetting
            the direct effect of having the option on anticipated energy
            use. Efficient, forward-looking infrastructure investments
            have high potential for reducing long-term energy
            consumption. Particularly if energy prices are expected to
            rise, however, the potential for reduced energy consumption
            will be eroded if expectations of energy prices do not
            include environmental costs or future retrofit possibilities
            and technologies are not adequately developed.
repnme: repnme: Infrastructure investments under
            uncertainty with the possibility of retrofit : theory and simulations
display_title: Infrastructure investments under
            uncertainty with the possibility of retrofit : theory and simulations
listing_relative_url: /research/2011/01/13309896/infrastructure-investments-under-uncertainty-possibility-retrofit-theory-simulations
topicv3: Energy and Extractives m1326595 507,Energy Efficiency m1326852 533,Climate Change Mitigation m1327079 579,Environment, Natural Resources and Blue
            Economy m1327391 566,Population, Aging and Health m1327736 640
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Environment,Macroeconomics and Economic Growth,Energy
no_of_pages: 63
model_extractions: WPS5516 Policy Research Working Paper 5516 Infrastructure Investments under Uncertainty with the Possibility of Retrofit Theory and Simulations _Jon Strand Sebastian Miller Sauleh Siddiqui_ The World Bank Development Research Group Environment and Energy Team January 2011
Policy Research Working Paper 5516

# **Abstract**

Investments in large, long-lived, energy-intensive infrastructure investments using fossil fuels increase longer-term energy use and greenhouse gas emissions, unless the plant is shut down early or undergoes costly retrofit later. These investments will depend on expectations of retrofit costs and future energy costs, including energy cost increases from tighter controls on carbon emissions. Simulation analysis shows that the retrofit option can significantly reduce anticipated future energy consumption as of the time of initial investment, and total future energy plus retrofit costs. The more uncertain are the costs, the greater the value of this option. However, the future retrofit option also induces more energy-intensive infrastructure choices, partly offsetting the direct effect of having the option on anticipated energy use. Efficient, forward-looking infrastructure investments have high potential for reducing long-term energy consumption. Particularly if energy prices are expected to rise, however, the potential for reduced energy consumption will be eroded if expectations of energy prices do not include environmental costs or future retrofit possibilities and technologies are not adequately developed.

This paper is a product of the Environment and Energy Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at jstrand1@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Infrastructure Investments under Uncertainty with the Possibility of Retrofit: Theory and Simulations***

## **By**

**Jon Strand Development Research Group Environment and Energy Team The World Bank** **<u>Jstrand1@worldbank.org</u>**

## **Sebastian Miller Research Department Inter-American Development Bank**

**and**

**Sauleh Siddiqui Department of Mathematics University of Maryland, College Park**

> ***** _This paper was prepared as part of a larger research program in the Development Economics Group on infrastructure and its relation to greenhouse gas mitigation.  Financial support (to Strand) was provided by the Government of Norway; and (to Miller and Siddiqui) by the Bank’s Research Support Budget. The views expressed in the paper are the author’s alone and not necessarily those of the World Bank or its member countries._
# **1. Introduction**

This paper presents an analytical model framework for choice of energy consumption in relation to energy-demanding infrastructure investments, and simulations to illustrate properties of this framework. An important premise for the analysis is that infrastructure investments, sunk at an initial time of decision, “tie up” energy consumption for a long future period. In our stylized and simple model, time is divided into two discrete periods: an initial period 1, “the present”; and a “future” period 2, which may be much longer. All relevant variables are assumed to be commonly known at the start of the period, _for that period_ . Several key cost variables in period 2, including energy and environmental costs and certain technological costs (so-called “retrofit” costs, to be explained below), are unknown in period 1, but ex ante statistical distributions, for their period 2 realizations, are assumed to be known (or knowable) at the time when the initial infrastructure investment is sunk in period 1.<sup>1</sup> We assume that the infrastructure persists throughout both periods, but may be (deliberately, and at no additional cost) shut down at the start of period 2. The energy consumption tied to the infrastructure is by assumption based on fossil fuels, at least initially.

The motivation for our analysis is that many types of infrastructure lead to considerable climate policy “inertia”, in that they establish levels of fossil-fuel consumption that may be difficult to reduce later. As is increasingly recognized in the literature, including Ha-Duong et al (1996), Wigley (1996), Ha-Duong (1998), Lecocq et al (1998), and Shalizi and Lecocq (2009), the presence of such an established infrastructure may form a major _ex post_ obstacle to effective mitigation policy, for a long future period, possibly 50-100 years or more. This holds regardless of whether the initial infrastructure investment is “optimal” (in an ex ante sense), or not. The particular problem with infrastructure investment in this context is that costs of mitigation or abatement, to reduce emissions to desirable levels, may, at least in some cases, be very high ex post after the infrastructure has been established.

Below we discuss reasons why the infrastructure investment may turn out not to be optimal. The bias is then typically in the direction of too high fossil energy consumption and carbon emissions. Potential reasons are many and include systematic under-valuation of future energy costs; failures to incorporate true (current and future) social carbon emissions costs; and excessive discounting.

In our analysis we use a very stylized model of infrastructure investment, introducing two potential mechanisms by which the fossil-fuel consumption can be modified “ex post” (in period 2).

The first is to “retrofit” the infrastructure in period 2, at a cost. After a “retrofit”, we assume, infrastructure operation causes no emissions of greenhouse gases (GHGs) from then on. A “retrofit” can be interpreted in several ways. First, infrastructure may be viewed as run without any use of fossil fuels from then on. This is particularly relevant when fossil fuels are replaced by alternative (non-fossil) energy sources, and the use of these sources reduces or eliminates the emissions of GHGs from the “normal” operation of the infrastructure. But this

> 1 This of course is a simplification at least relative to some presentations where the distribution of future energy and environmental cost is assumed to be unknown at an initial stage; this can lead to some serious problems of inference as argued e g by Weitzman (2009). Here, we may use a standard Bayesian approach to justify our position; namely, as the “best initial assessment” of the distribution given our current knowledge. See also Geweke (2001) and Schuster (2004) for formal justifications of this approach.

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interpretation is also relevant when applied to cases where the overall energy demand of the infrastructure is (dramatically) reduced. In our stylized model, emissions are also in such cases assumed to be eliminated completely. With either interpretation, the existence of a potential retrofit option in no way implies that retrofit is necessarily optimal; in many cases, exercising the option of retrofit would be prohibitively costly.

Alternatively, a “retrofit” could mean that the consumption of fossil fuels in operating the infrastructure is unchanged, but that the carbon is removed from these fuels (through carbon capture and storage, CCS, or similar technologies or processes). It can however be a more problematic interpretation in our model given that the post-retrofit operating period T is variable, since the retrofit cost is then “periodized” and assumed constant “per time unit” within period 2. When T is variable, retrofit costs with a given distribution G will correspond to a variable total retrofit cost (as it would be proportional to a variable T). When the retrofit cost in fact represents a given sunk cost initially in period 2, the G function will need to be amended when T changes.

Both “retrofit” and basic energy/climate cost in period 2 are assumed to be uncertain in period 1, but with initially known joint distribution, assuming that the two costs are non-negatively correlated. Retrofit costs could easily be more uncertain than energy and climate costs, since the nature of future (period 2) retrofit technologies is likely to be unknown at the start of period 1 when initial infrastructure investments must be made.

The second way to avoid energy consumption related to existing infrastructure in period 2 is to abandon it, “close it down”. Since this investment is then rendered worthless, this is a wasteful, and painful, option since the initial infrastructure investment is costly to establish. It can still be attractive and rational ex post, given that energy and/or retrofit costs of continued operation both turn out to be _both very high_ : the lower of the two costs must be higher than the utility value of continued operation. The abandoned infrastructure will then, presumably, be replaced by alternative, less energy-intensive, infrastructure in period 2. The probability of such a simultaneous occurrence is greater when energy and retrofit costs are positively correlated. Our closedown alternative represents a “benchmark” case with zero emissions and energy consumption; and we do not specify any particular replacement alternative.<sup>2</sup> We characterize ex ante strategies for establishing energy and emissions intensity associated with the initial infrastructure investment; ex post strategies for retrofitting and operating the infrastructure at a “later” stage (in period 2); and interactions between these strategies. We characterize optimal infrastructure investment, and (important from a climate policy standpoint) identify factors behind inefficient (too energy and emissions intensive) infrastructures being established. We also study whether, and to what extent, an initially high energy intensity level can be modified in later periods through retrofit or closedown, in cases where energy and environmental costs are high.

“Optimal” infrastructure choice is defined for given current prices, and given distributions of future prices. “Optimality” can be established either for a private agent making the infrastructure decisions, or for a social planner; which if these will be invoked in the following will depend on the context. The decision-making agent could be private, but in most cases a public-sector entity (a local or national government). A social planner, taking a

> 2 One possible interpretation of this case is that energy consumption and emissions in the closedown alternative serve as a reference “zero” point, relative to the “business-as-usual” and retrofit alternatives.

3
national, regional or global perspective, will tend to incorporate prices, costs, discount rates etc. given at the respective (national, regional or global) level, and, we assume, optimally from that particular point of view. A fundamental problem with this, in a climate policy context, is that such a view tends not to be correct, even when formulated at the national level. As a key feature of the GHG emissions control problem, a global view is needed, where the marginal externality cost at the _global_ level is incorporated. Local decision makers are likely not to behave in a globally optimal way, except when international agreements dictate that globally optimal (emissions and energy) prices be applied. Little today indicates that such “optimal” prices will be applied in the near or intermediate future; thus a discrepancy between the ideal, global, social planner and the practical decision maker will be the order of the day. One objective of our analysis is to study how much such a decision maker is likely to deviate from a “socially optimal” decision (from a global point of view).<sup>3</sup> Increased availability of retrofit and closedown options affect expected fossil-fuel consumption and GHG emissions over the lifetime of a given infrastructure in two opposing ways. First, they reduce expected fossil-fuel consumption and emissions through the option to avoid such consumption and emissions ex post, in states where emissions and energy costs are particularly high, where the infrastructure can then instead be retrofitted or closed down. On the other hand, an anticipated increased availability of such options serves to increase the chosen energy intensity embedded in the infrastructure when established. A greater availability of the two additional options make it less risky for the decision maker to choose a high initial (fossil-fuel) energy efficiency.

Simulations, in section 3 below, show that a higher variance on retrofit and/or energy costs (for given unconditional expectations of these costs) reduces expected future costs, both in energy terms, and (energy plus retrofit) costs terms. With more uncertainty about both energy and retrofit costs (for given unconditional expectations), the retrofit option is exercised in more cases, and to greater benefit (more states with “gainful” retrofit).<sup>4</sup> Greater variances are beneficial as they open up for more low-cost alternatives, which are exploited under an optimal ex-post policy.<sup>5</sup> The gains from greater cost variability are reduced when costs are positively correlated (assumed in some of our simulations).

The distribution of retrofit costs in period 2 depends on technological retrofit possibilities, which in turn are affected by R&D efforts to develop such technologies. While R&D efforts are not modelled explicitly in the paper, we study exogenous shifts in the distribution for such costs. When such cost reductions are correctly anticipated at the time infrastructure investments are made, they lead to increased energy intensity of the initial infrastructure investment choice. Overall expected energy consumption over the lifetime of the project may then either increase or decrease, depending on which of two effects is stronger: this initial effect on infrastructure energy intensity, which results in higher energy consumption in “business-as-usual” states (from the higher energy intensity of the initially established infrastructure); or the lower energy use ex post, due to “business-as-usual” states being fewer and the infrastructure expected to be retrofitted in more states (in period 2 of our model). Simulations, in section 3 and appendix B, show that the latter factor tends to dominate in our model; except possibly in a special case where the initial energy consumption is very sensitive

> 3 See Strand (2009) for further elaboration on these issues.

> 4 This conclusion holds when the decision maker is risk neutral as assumed in our presentation; it may need qualification under risk aversion.

> 5 This result holds when decision makers are risk neutral, which is assumed here. Under risk aversion, the utility effect of greater uncertainty could here go either way.

4
to future energy costs (when the utility function for infrastructure services takes a CobbDouglas form). Overall expected energy use and climate impacts are reduced with more energy technology R&D, resulting in lower expected retrofit costs. Perhaps equally reasonably, however, (major) reductions in expected retrofit costs will not be fully anticipated at the time infrastructure investments are made (or, at least, are not fully built into these infrastructure decisions). In such cases, reductions in expected infrastructure costs will reduce expected energy costs by more, as they are not accompanied by offsetting effects (in terms of higher energy intensity) for the initially established infrastructure.

Overall expected retrofit costs may in such cases increase or decrease. The factor contributing to a decrease is the very drop in cost. The factor contributing to an increase is the retrofit option being exercised in more states of the world (thus avoiding energy expenditures in more ex post states). Typically here also, the first factor seems to dominate.

More interesting from a policy perspective is what happens in response to an upward shift in the ex ante distribution of period 2 energy costs, here taken to include any climate or other environmental costs resulting from energy use; a key issue in a climate policy context. We study the system’s response to increases in the future cost of carbon emissions and/or energy that may or may be not fully anticipated at the time when major infrastructure investments are made. Our model provides answers to both the following two questions: What is the fossil energy use (and carbon emissions) response, when these future cost increases are fully anticipated, and expected to be incurred? And what is the response when they are not expected to be incurred (as may be the case for some climate-related costs)?

When an increase in expected future energy cost is _not anticipated at time of investment_ , the embedded energy intensity in these investments will be excessive. Our simulations here indicate that when a doubling of the energy/environmental cost can be anticipated, and this is not considered in making the infrastructure investment, it could result in an overall increase in fossil energy consumption by 30-60 percent, even when the ex post response to the energy price increase is optimal. It is here still the case that expected energy use and emissions will be reduced ex post in proportion to the increased use of the retrofit option. Most of this reduction is realized in states with high energy costs, thus leading to (perhaps substantial) expected energy cost savings.

When the increase in future expected energy cost is _correctly anticipated_ at time of infrastructure investment, lifetime expected energy use and emissions are reduced by more than when higher energy costs are not anticipated. The additional factor is that the energy intensity of the initial infrastructure is reduced. We show, mainly through simulations, that the compounded effect of these two factors can be substantial, and that greater uncertainty about energy and retrofit costs adds to this effect.

When an increased future energy and climate cost is _not expected to be incurred_ (but correctly anticipated) by the decision maker (be it a national or local government, or a private party), it has no effect on decisions, neither ex ante nor ex post. This implies policy failures in two different respects: both making the infrastructure excessively energy intensive; and allowing for too few retrofits ex post. When again there is a doubling of the energy price which is not considered, our simulations here show (see table 1 in the final section 4) that the degree to which energy consumption is excessive could be dramatic, and could exceed 200 percent, when compounding the effect of excessively energy-intensive infrastructure ex ante, and excessive energy consumption ex post.

5
The analytical and quantitative literature dealing with such issues is small. Arthur (1983), David (1992) and Leibowitz and Margulis (1995) provide background by defining and discussing the issue of path dependency and its implications for future actions. The more specific topic of infrastructure choice and its implications for mitigation policy is discussed only recently. Shalizi and Lecocq (2009) provide a discussion of infrastructure costs and constraints which is more applied and intuitive than that provided here. The persistent effects of infrastructure choice on energy consumption and carbon emissions are discussed also by Brueckner (2000), Gusdorf and Hallegatte (2007a,b), and Glaeser and Kahn (2008). In particular, Gusdorf and Hallegatte (2007a) study the energy intensity of urban infrastructure for given population density. They focus in particular on inertia resulting from established urban structure, in response to “low” initial energy prices, which may later rise. They show, through simulations, that a permanent energy price shock leads to a transition period that is long (20 years or more) and painful (with high energy costs, and carbon emissions), but that energy consumption eventually tends to fall toward a substantially lower steady-state level. Glaeser and Kahn (2008) by contrast focus on energy consumption implications of differences in population density, both within urban regions and when comparing urban and rural population patterns. In this context they seek to quantify relationships between energy consumption and spatial patterns of cities in the U.S. They find, in particular, much lower percapita energy consumption, and carbon emissions, in central cities than in suburbs. This bears on our analysis as it indicates that “compact” infrastructure (as found e g in central cities) is less energy demanding than “less compact” (found e g in suburbs).

The option to retrofit already established infrastructure, by removing either the initial energy requirement, or the carbon emissions associated with it (via CCS technology or replacing fossil fuels by renewables), reduces the inertia associated with infrastructure. This is a focus in this and two related papers, Strand (2010) and Framstad and Strand (2010). Strand (2010) considers different utility function representations and their implications, in a similar setting. Framstad and Strand (2010) study optimal infrastructure investment when future energy prices follow a continuous stochastic process, where a delayed retrofit decision has a positive option value. Implications of retrofit possibilities and costs are further discussed analytically by Jaccard (1997) and Jaccard and Rivers (2007). The latter paper studies three more specific types of demand-side infrastructure: urban structure; buildings; and equipment. The authors argue, based on simulations (and using a discount rate of 3 percent), that for buildings, and even more for urban structure, it is generally advantageous to make strong considerations for future emissions even when emissions prices start low and increase strongly over time (while this is often not the case for equipment where natural turnover provides sufficient flexibility). Shalizi and Lecocq (2009) provide a broader and more practically oriented discussion, with examples from both energy demand and supply; their overall argument is that energyintensive infrastructure involving supply is generally more rigid than that involving demand; but sometimes (but not always) more prone to complete retrofit.

Other literature connections need mention. One is the “low-carbon society” and ways to achieve it, treated by many; among others Strachan et al (2008a, b), and Hourcade and Cerassous (2008).<sup>6</sup> This basic idea is, as here that achieving a society with low GHG emissions (necessary for efficiency in the long run) requires a high concern for infrastructure investment design. Another connection is two World Development Reports, the WDR 2003

> 6 A very early champion of this line of thinking and discussion was Amory Lovins; see, in particular, Lovins (1977).

6
(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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retrofitted) due to its greater fossil-fuel energy requirement. Denote the current (per time unit) utility flowing from the infrastructure when being operated by _U(H)_ , where _U’(H)_ > 0, _U’’(H)_ < 0. We assume that _U(H)_ is given and constant and the same in both periods (and thus not subject to uncertainty).

Three alternative actions may be chosen in period 2:

1. _No new action, proceed with “business as usual.”_ In this case the full energy cost will be incurred in period 2. This is the optimal strategy when the energy cost in period 2 turns out to be lower than both the retrofit cost, and the period-2 utility of the infrastructure.

2. _Retrofitting the infrastructure_ . This is the optimal strategy when the retrofit cost in period 2 turns out to be lower than either the energy cost, or the period-2 utility of the infrastructure.

3. _Infrastructure closedown_ . This is optimal when environmental and retrofit costs both turn out higher than the period-2 utility of the infrastructure. Closedown is a drastic measure. It will typically require that other infrastructure be supplied, to replace the services lost by project closedown. This is not explicitly modelled here. Implicitly our model embeds such effects, via the absolute value of the utility flow provided by the infrastructure (which should be defined relative to a situation where the utility is missing; thus a “relatively drastic” alternative).

The problem of a decision maker in establishing the infrastructure in period 1 is to select an energy investment intensity _H_ so as to maximize where _E_ is the expectations operator and _W_ (2) is the (optimized) value function associated with the infrastructure in period 2 (embedding the optimal action, among alternatives 1-3 above).

_EW_ (2) embeds the decision maker’s optimal responses at the start of period 2 (assuming that no further changes occur during period 2). Define _F_ ( _q_ , _y_ ) = _Fqy_ as the (continuous ex ante, when viewed from period 1) cumulative bivariate distribution over _q_ and _y_ levels to be realized in period 2, with support [0, _qM_ ] × [0, _yM_ ], where _qM_ and _yM_ could be large.<sup>10</sup> Possibly, the marginal distribution _Fq_ for period 2 is shifted up by increased emissions in period 1. Here, _y_ represents retrofit costs (per unit of energy capacity to be retrofitted) in period 2. Retrofit costs cannot be negative, but could in principle be small in period 2, depending on the technology available for substituting out the fossil-fuel energy consumption or purging carbon from fossil fuels at that time. We assume that an infrastructure project, after a retrofit, incurs no energy costs nor any other current costs in period 2, apart from the retrofit cost itself (which in the model is “periodized” in the same way as energy cost).<sup>11</sup> In the

> 10 In simulations below we assume that F is bivariate log-normal, in which case F is not bounded above (it is however “thin-tailed”).

> 11 Alternatively, the retrofit cost could be interpreted to include some energy cost. This is unproblematic as long as the retrofit cost can be periodized.

8
analytical presentation, period 2 realizations of energy cost and retrofit cost are not assumed to be independent.<sup>12</sup> Consider the choice between the three alternatives lines of action 1) – 3) in period 2. We start with action 3), project closedown. Define total utility per energy unit for installed infrastructure by _U_ ( _H1_ )/ _H1_ = _y_ *, where _H1_ is energy intensity associated with the infrastructure investment chosen in period 1. Action 3) will then be chosen when the cost per unit of energy _q_ , and the retrofit cost y, both exceed _y_ *. The probability of this event, when viewed from period 1, is Assume 0 < _y_ * < min { _qM_ , _yM_ }, and 0 < _F_ ( _y_ *, _y_ *) < 1, implying _P_ (3) > 0. The probability _P_ (1) of inaction (action 1), is given by the following expression: (3) The probability _P_ (2) of retrofit (action 2) is complementary (equal to 1 – _P_ (1) – _P_ (3)), but can also be found in a similar way as _P_ (1), as follows: (4) Note that we can switch the order of any double integrals with an application of Fubini’s theorem.<sup>13</sup> Define the following (unconditional) expected unit cost variables in period 2, related to energy cost and retrofit cost respectively, by These are costs that would be realized given that, in the first place, “business as usual” energy use is applied in all states in period 2; and in the second place, retrofit is applied in all states in

> 12 Correlation of energy and retrofit costs is often realistic. Such correlation could in principle be either positive or negative. Negative cost correlation could occur when the energy cost in period 2 is anticipated during the period of R&D efforts to develop new retrofit technologies. A high anticipated energy cost could then make the development of retrofit technologies more urgent, and more effort expended for that purpose. This endogenous process could then lead to negatively correlated costs. On the other hand, common drivers may affect both costs in the same direction. This is relevant e g when energy cost is correlated with general production cost; when a retrofit involves some use of fossil energy; or when the subsequent use of renewable energy whose marginal production cost is positively correlated with the cost of fossil fuels. In such cases the two cost variables would tend to be positively correlated. This corresponds to our main assumption, for the simulations below. 13 See e g Royden (1988).

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period 2. These are not realized costs and instead merely benchmarks (both excessive), for two main reasons: the lower of the cost alternative is applied given that one of them is applied; and in some states the third alternative (closedown) implies lower costs.

Considering now actual realized costs, the expected “per time unit” period 2 energy and retrofit cost as viewed from period 1, given an optimal strategy for period 2, are respectively _E_ [ _CH_ (2)] expresses energy costs per “time unit” in period 2, while _E_ [ _CR_ (2)] similarly expresses retrofit costs when similarly periodized (counted per period unit), under an optimal decision rule for ex post factor choice (“normal operation”; retrofit; or closedown). _H1_ denotes (per-period) energy consumption associated with the infrastructure as established in period 1. Define _E_ [ _C_ (2)] = _E_ [ _CH_ (2)] + _E_ [ _CR_ (2)], as well as _Ec_ (2) = _E_ [ _C_ (2)]/ _H1_ . In particular, we must then have _Ec_ (2) < (1- _P_ (3))min[ _Eq_ (2), _Ey_ (2)], as long as all three ex post policy alternatives: “business as usual”, retrofit, and closedown, are actually exercised in period 2.

A factor not apparent from this derivation is that two alternative interpretations of retrofit costs are likely to have somewhat different implications for the analysis. The first is to view retrofitting simply as replacing a fossil fuel with a non-fossil fuel which gives rise to no GHG emissions. Retrofit costs are then incurred currently in the same way as regular (fossil) fuel costs. Under the second interpretation, retrofit costs represent an investment to remove the fuel need tied to the infrastructure, or the emissions associated with the fossil fuel. In (8), this implies that _E_ [ _CR_ (2)] must be “periodized” (given that _T_ is different from unity), and spread evenly across _T_ time units in period 2.

Expected (discounted) net utility from the infrastructure when operated in period 2 is denoted _EW_ (2), and equals the gross utility of the infrastructure, _TU_ ( _H1_ ) = _Ty_ * _H1_ , in states where it is not closed down in period 2 (thus with probability 1- _P_ (3)), minus total combined expected energy and retrofit costs, _T_ { _EC_ (2)} (= _T_ { _E_ [ _CH_ (2)] + _E_ [ _CR_ (2)]}), over states where the infrastructure is operated (without, or with, retrofitting). We then have The first-period decision problem is formulated as maximizing the expected utility of the infrastructure investment in period 1, considering an optimal strategy in period 2. Define 10
Assume for now that the distribution of period 2 energy (including environmental) costs is exogenous (and not affected by emissions arising from the infrastructure). The solution to the maximization problem in period 1 takes the form From (9) we find Using the definition of _EW_ (2), and setting the derivative in (11) equal to zero, we find the following implicit expression for the optimal energy intensity of the infrastructure: (13) 1+[1- _P_ (3)] _T_ equals the expected number of time units that the infrastructure will be operative during period 2. { _E_ ( _CH_ (2))+ _E_ ( _CR_ (2))} _T_ = { _EC_ (2)} _T_ is the ex ante expected (energy plus retrofit) cost in period 2, which divided by _H1_ is measured relative to energy intensity as established in period 1. { _EC_ (2)} _T_ represents total energy costs plus retrofit costs during the period 2 expected operation time [1– _P_ (3)] _T_ . The expression in the curled bracket in the numerator of (13) then denotes expected energy plus retrofit cost per unit of energy consumption defined by the established infrastructure. (13) can then be given a simple interpretation: the marginal utility of increased energy intensity associated with the infrastructure investment (the left-hand side of (13)) should equal the average energy plus retrofit costs incurred over the lifetime of the infrastructure capital (the right-hand side).

The optimal energy intensity is chosen according to average energy cost in operating the infrastructure, over its expected period of operation. Perhaps surprisingly, the extent of the operation period as such is not very important for the chosen intensity.<sup>14</sup> For e.g. a hypothetical case where the infrastructure is shut down in period 2 for certainty; the energy intensity would be determined simply by _U_ ’( _H1_ ) = _q_ 1. When average energy cost in period 2 exceeds _q1_ , _U_ ’( _H1) > q1_ , and _H1_ lower (since the required utility return per unit of infrastructure must be higher which requires a lower infrastructure mass).

> 14 This result is closely related to our initial assumption, that the size of the infrastructure investment is exogenously given.

11
For _y_ * (= _U_ ( _H1_ )/ _H1_ ) we find Here _y_ *- _U_ ’ must be positive and more so the more curved the utility function is in the neighborhood of _H1_ . Thus we can expect _yH_ * < 0.<sup>15</sup> This implies that a more energydemanding infrastructure will have a lower threshold for operation, and will be “closed down” (or rather, replaced) ex post in more cases in period 2, when energy and retrofit costs increase. This may appear reasonable; remember that in our model all infrastructure projects are considered to be “equally large” in the sense of requiring the same initial investment cost. What distinguishes projects is the ex post energy requirement for their operation (or put otherwise, their energy intensity).

Appendix A1-A2 deals with comparative statics for the above model, when there are period 2 shifts in the distribution of energy costs (A1), and in the distribution of retrofit costs (A2), for a simplified case with independent costs.

# **2.2 Analytical Specification of the U Function**

We will in this sub-section consider the main class of the utility function specification, _U(H1),_ that will be applied in the simulations in section 3 below. Our basic assumption is that _U(H1)_ belongs to the class of constant relative risk aversion utility (CRRA) functions. The (vNM) utility function then takes the following general form: where A and ρ are positive parameters, and K is a (non-negative) scaling parameter ensuring that U only takes non-negative values on its relevant ranges.<sup>16</sup> The (Arrow-Pratt) measure of relative risk aversion is given by ρ. Define _[q+Ec(2)T]/[1+(1-P(3))T] = Q_ as the expected total unit cost of operating the infrastructure over its lifetime, per expected time unit of operation (where T is the number of time units within period 2, and 1-P(3) is the probability of operation in period 2), and per unit of established infrastructure, H1. We have defined _Ec(2) = E(C(2))/H1_ (as the average expected realized cost per time unit in period 2, per unit of invested H1), which from (7)-(8) is constant for given distributions of q and y. Maximizing ex ante net utility associated with the initial infrastructure investment, given by _U(H1) – Q_ , we derive the first-order condition Here all magnitudes except _H1_ can be viewed as constants. Consider the effect on _H1_ when the cost variable _Q_ changes. This is found as

> 15 Another way of interpreting this can be found by considering that _y*_ is an expression of the average utility of infrastructure per unit of _H1. yH*_ < 0 then simply expresses that average utility of infrastructure must exceed its marginal utility at the point of indifference between operation and closedown.

> 16 This issue is relevant only for case c) below, where _U_ would otherwise take only negative values.

12
From (17), the “elasticity of demand for energy intensity”, _εH_ , is found as Thus the energy intensity demand elasticity with respect to total operating cost per unit of established infrastructure (but where the initial infrastructure cost is not included) is a constant.<sup>17</sup> All possible values of _ρ_ from zero to infinity are allowed by this formulation. The effect of an increase in _Q_ on total expected costs (for energy and retrofit combined) is We can now easily distinguish between three distinct classes of cases for (15), namely a) 0 < _ρ_ < 1: _U_ takes a Cobb-Douglas form. Then _d(H1Q)/dQ < 0_ . Overall expected ex ante (total) operating costs, including the response of the initial energy intensity (and in terms of both expected energy costs, and expected retrofit costs), are then _reduced_ in response to an increase in expected ex ante unit cost. The reason is that the energy intensity chosen in response to a change in the cost variable is then reduced relatively more than the change in unit costs.

b) _ρ_ = 1: _U_ takes a logarithmic form. Then _d(H1Q)/dQ = 0_ . Overall expected ex ante (total) cost, including the response of the initial energy intensity, is _constant_ in response when expected ex ante unit operating cost increases. The energy intensity is reduced in response to a change in the cost variable, so as to exactly offset the increase in unit expected ex ante cost.

c) _ρ_ > 1: _U_ is exponential. Then _d(H1Q)/dQ > 0_ . Overall expected ex ante operating cost, including the response of the initial energy intensity, here _increases_ in response to an increase in expected ex ante unit operating cost. The energy intensity is reduced but by relatively less than the change in energy costs.<sup>18</sup> Under this class of utility functions we have the following general expressions for _y*_ (expressing the ex post marginal cost beyond which the infrastructure will be shut down in period 2), and its change with _H1_ :

> 17 Remember here our initial assumption, that all possible infrastructure projects have the same given initial investment cost; infrastructures differ only in their dimensioning of _H1_ , and their utility (which is higher for higher _H1_ ).

> 18 Note also that the utility function in this case takes only negative values for K = 0 in (15), and tends to minus infinity as H1 tends to zero. A meaningful application of the model to this case (in particular, a meaningful definition of the ‘closedown” limit operation price y*) would require K > 0 and that we restrict attention to H levels

13
(21) <!-- Start of picture text --> dHdy *1  yH *  H 11  A 1  H 1   HK 1  .<br><!-- End of picture text --> As noted, under case c) one would here need to impose a minimum positive value of K.

The CRRA class, embedding all cases a) - c) where the coefficient of relative risk aversion is allowed to vary from zero to infinity, can be viewed as sufficiently wide for our purposes. In particular, it embeds all relevant possible values of the (crucial) elasticity of demand for energy intensity in infrastructure with respect to expected operating cost. The only effective constraint on this class is the constancy of this elasticity, for any chosen value of _ρ_ .

# **2.3 Optimality, Deviations from Optimality, and the Relation to This Analysis**

“Optimality” will here be defined alternately by a (global) social planner, or by the local decision maker choosing the initial infrastructure investment (who is typically not a global planner). It is here useful to identify different sources of deviations between the two concepts, with resulting inefficiency in deciding the initial infrastructure investment, when decisions are otherwise “optimal” (from the point of view of the decision maker). We next wish to identify implications of “non-optimal” choices by the actual (local) decision maker, or choices made under incorrect information. We distinguish between the following five issues (spelled out in more detail in Strand (2010)). This classification is particularly useful for interpreting the simulation results in section 3 below.

- A) The initially expected distribution of energy costs, relevant for making current infrastructure decisions, is lower than (or more precisely, down-shifted relative to) the true (correct) distribution. This could occur e.g. when the infrastructure decision is based on an  expectation of something close to current energy cost on average in period 2, while the correct distribution implies higher average energy costs.<sup>19</sup> In this case we may expect the infrastructure energy intensity to be chosen at a too high level, and fossil-fuel consumption in period 1 to be excessive. The period 2 realized expected energy consumption is however not obviously higher than optimal in consequence. This is because the closedown and/or retrofit options could then in response be exercised in more cases.<sup>20</sup>

- B) The distribution of energy costs facing the policy maker is correctly anticipated, but is down-shifted relative to the “optimal” energy cost distribution. This is relevant whenever the authorities, in the economy in question, implement emissions prices that are lower than “globally correct” prices. This case is highly relevant: as of today, hardly any country implements what most analysis would agree are “globally correct” emissions prices; nor seems to be willing to do so in the foreseeable future. In this case we can expect, unambiguously, excessive fossil-fuel consumption in both periods.

> 19 We argue that this could occur even in cases where the entity making the infrastructure decisions would face a “true” energy/environmental cost. One such case is where the administrative procedure for making public investments involves incorporation of future costs and benefits for a limited period (say, 20 years), while a correct investment decision would require a much longer period (say, 50 years or more). An increasing energy price (beyond the 20 year term) would add to the bias involved in such investment.

> 20 In our simulations below we however do not allow for this possibility at least not with respect to retrofit costs which are throughout assumed to be proportional to energy costs, ex post.

14
- C) Future retrofit costs are incorrectly anticipated. In particular, with a too optimistic view, the anticipated retrofit cost distribution will be lower than the correct distribution. This will bias the infrastructure decision in the direction of excessive energy intensity for the initially established infrastructure.  On the other hand, when the distribution of future energy costs is too optimistic, the retrofit cost distribution will be less important for the energy intensity decision, since the prior expectation is then that retrofits are necessary in fewer cases. Instead of “technology optimism” one could however have “technology pessimism” (a too pessimistic view of the distribution of future retrofit costs), which would tend in the opposite direction.

- D) Future retrofit costs are correctly anticipated, but are higher than socially optimal costs. Such a case is where the R&D effort in developing new energy technologies is suboptimal, including those for retrofits. Overall implementation costs (in particular, minimum of energy and retrofit costs) would then be shifted upward in period 2. Given correct perceptions in period 1, the choice of energy intensity of the infrastructure is in response _lower_ than optimal (efficient energy intensity is greater than the one chosen), while the probability of ex post business-as-usual operation of the infrastructure is _higher_ than optimal (as there are more states where the energy cost is below retrofit cost). This follows straightforwardly from the fact that the business-as-usual alternative will be chosen in more cases in period 2, and the retrofit alternative in fewer cases. With no other distortions, the energy intensity of the initially chosen infrastructure would then in fact be _suboptimal_ . On balance, the second factor is likely to dominate, leading to socially excessive energy consumption in period 2.

- E) The policy-relevant value of _T,_ call it _T1_ , is less than the optimal value, call it _T0_ . Reasons for this may be either excessive discounting (for the case where _T_ is interpreted as a discounted value), or that the initial policy decision undervalues the length of period 2. Since _T,_ as noted at the start of this section, can be interpreted as a discounted value of period 2 relative to period 1, we may have a discrepancy between the “socially correct” value _T0_ and the value _T1_ used when deciding on _H1. T1 < T0_ could then reflect excessive discounting. When average costs per operation period are greater in period 2 than in period 1 (as might be expected), this leads to a lower average operation cost for the infrastructure, and a more energy-intensive infrastructure.

In four of these cases (all except D), the overall expected fossil-fuel energy consumption (and GHG emissions) over the potential lifetime of the infrastructure would tend to be excessive from a social point of view, in the sense that it is higher than the expected fossil-fuel consumption and emissions for the case where all global externalities are optimally considered and anticipated.<sup>21</sup>

> 21 One might in addition argue that case D is less likely in practice: when actually realized retrofit costs tend to be higher than their socially optimal level, this will, probably, tend not to be anticipated. If not, the end result will be excessively energy intensive infrastructure, and overall excessive emissions.

15
# **3. Simulations**

We will in this section illustrate properties of our model through four sets of simulations: 1) ex ante expected probability of energy use or retrofit in period 2; 2) ex ante expected energy and retrofit costs in period 2; 3) energy intensity of the infrastructure chosen in period 1; and 4) overall expected energy use, which combines the energy intensity of infrastructure (3) with the probability of subsequent energy use (1). Under our (we think, reasonable) numerical cases the closedown option turns out to play a very small role for the overall results, and will not be emphasized in the following; but must still be kept in mind.<sup>22</sup> All simulations are made under the assumption that energy and retrofit costs are jointly log-normally distributed, and calculations are done using the scientific program Matlab. Most of the simulations for expected energy cost in period 2 (section 3.2) depart directly from the expected cost expressions (7), and (8) for expected retrofit cost in period 2. For probabilities of using energy and retrofitting respectively (section 3.1) we use (3) and (4), and for energy intensity of infrastructure (section 3.3), we use (13). For expected overall energy use section 3.4), we apply the solution value for _H1_ together with the ex ante probability of energy use, (3).

In all cases where nothing is otherwise stated, the (unconditionally) expected energy and retrofit cost (per unit of energy use and retrofit investment respectively) are both kept constant, at levels _E(q(2))_ = 2, and _E(y(2))_ = 3. ( _E(q(2_ )) is the unconditionally expected energy/environmental cost in period 2; it would be the _actual_ expected cost given no retrofit or closedown. A similar interpretation holds for _E(y(2)_ ).) While both energy and retrofit costs are uncertain, ex ante energy costs are thus generally lower _in expectation_ . The distributions of energy and retrofit costs are both assumed to be jointly log-normal, and either independent, or have a positive correlation coefficient of 0.5. In sections 3.3-3.4, simulations are done for all variants a) – c) of the utility function (15).

# **3.1 Ex Ante Probability of Energy Use or Retrofit in Period 2**

This first set of simulations considers implications of various parametric changes in the model, for the expected value of (per-period) energy costs during period 2. It is useful to first study period 2 behavior; this will be a benchmark for subsequent analysis of period 1 investment and overall energy use, in sections 3.3-3.4 below.

Figure 1 describes the probability that the firm in period 2 will continue using energy at the level initially established in period 1, given a parametric change in expected energy cost, for different degrees of uncertainty about both energy and retrofit cost.<sup>23</sup> In all examples used in figure 1, _Ey_ = 3. We see, as should be obvious, that a low (high) _q_ implies a high (low)

> 22 Two other sets of assumptions in the following also potentially be attacked. One is the log-normal distribution assumption. Log-normality is however a rather robust assumption in this context; see e g Schuster (1984). The other is our assumption on the utility function of energy intensity in infrastructure, where we assume a set of functions parameterized by the demand elasticity with respect to price (thus constant for a given specified function; the CRRA class of functions). While this general class of functions can be viewed as robust, criticism can be raised in particular to the Cobb-Douglas and log specifications; see below.

> 23 Note that the probability of closedown is also accounted for in these calculations, but its probability turns out to be too small to matter for the results. (Throughout, the value of continued use is set at 10, implying that this option will be exercised only when min( _q, y_ ) > 10.)

16
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# **3.3 The Choice of Infrastructure Investment**

We now simulate the initial infrastructure investment decision, _H1_ , as given in (13) (jointly with factors determining the joint distribution of _q_ and _y_ ). The reported simulations all assume _T_ = 5. We are here thus assuming that period 2 is 5 times as long as period 1 (in expected discounted value terms). Throughout we assume that the probabilities of period 2 action are not affected by _H1_ .<sup>25</sup> The number of time periods is however found to affect _H1_ (the balance between periods 1 and 2 then changes, and expected costs are generally different in the two periods).

The form of the utility function _U(H1)_ , going into (13), here matters greatly. Figures 7-8 report such results for our three main alternative utility function specifications (all based on CRRA assumptions): a) The Cobb-Douglas case (with exponent _ρ_ = 0.5). In this case the elasticity of _H1_ with respect to total expected ex post cost (after investment is sunk) is _εH = 1/ρ_ = 2. b) The log-linear case, in which _εH_ = 1. c) The exponential specification exponent 0.5; here we assume _ρ_ = 1.5, and thus _εH_ = 2/3. Among these specifications, the exponential function provides the least sensitive choice of _H1_ in response to an increase in _Eq_ ; and the Cobb-Douglas function the most sensitive, with the log-linear specification as an intermediate case. This follows naturally from section 2.2 above, where we showed that in the log-linear case, an increase in ex ante unit cost (combining energy and retrofit costs) leaves total ex ante cost constant; the reason is that a reduction in _H1_ exactly offsets the expected unit cost increase in this particular case. In the Cobb-Douglas case the offsetting change in _H1_ is greater; and in the exponential case, smaller.

While our simulations thus cover a range of possible cases, the realism of some of the alternatives may be questioned. In particular, for some alternative specifications it might be argued that the scope for change in H1 is, perhaps, too great. In particular, under the CobbDouglas specification this scope may appear excessive relative to what is seen empirically in most cases. The exponential specification (with _εH_ = 2/3 under our parametric example) may then perhaps seem as the most realistic case. This is useful to have in mind in interpreting our numbers in the following.

> 25 Conceivably, there may be such effects, which will in case modify our conclusions somewhat; see section 4 below for a discussion.

22
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A few cautionary notes must be recognized when interpreting these simulations. In particular, the model has the (perhaps unrealistic) feature that when the closedown option is exercised, there is no energy consumption or any emissions. The same holds when the retrofit option is exercised. This assumption could however, without much loss of generality, be weakened by assuming a certain (minimum) level of energy use and emissions in this case.

# **4. Summary and Final Comments**

The main purpose of this paper has been to analyze the implications of two interacting sets of decisions concerning infrastructure investments with long lifetimes and that commit society to potentially high levels of energy use, and carbon emissions, for a long future period after the investments have been made. One of these decisions is the (basic and initial) energy intensity of the infrastructure investment. In section 2 above we studied factors behind such investments. With particular reference to the class of so-called constant relative risk aversion (CRRA) utility functions for characterizing the relationship between energy intensity and the public’s utility from the infrastructure, we noted that the energy intensity of the initially established infrastructure can respond more or less strongly to a change in its future expected cost of operation (considering different infrastructure types that all have the same initial investment cost). We discussed three classes of cases: a) one where initial energy intensity is increased more than proportionately to a given relative increase in ex post operation costs (when the utility function takes a Cobb-Douglas form); b) the case where energy intensity increases proportionately to the reduction in overall cost (the logarithmic case); and c) a set of cases where it increases less than proportionately (the exponential case). We indicated that case c), which implies that the price elasticity of demand for energy through infrastructure energy intensity is less than unity in absolute value, seems as the most empirically reasonable of the three cases. We indicated that the implication of a change in ex post costs for initial energy intensity, and as a result overall expected lifetime energy costs and carbon emissions, could be substantial.

A second main aim of the paper has been to study impacts of two types of “ex post” policy interventions, that may be applied after an infrastructure investment has been sunk. The first is a “retrofit” of the infrastructure (by making an additional later investment that then removes the energy demand and/or emissions while retaining its utility value to the public). The second policy option is simply to close the infrastructure facility down (which is a more drastic alternative, as the utility value of the infrastructure is then is lost, together with energy use and emissions). Most of the focus here has been on the retrofit alternative, and on its ability to reduce subsequent (energy and environmental) costs, and its effect back on energy intensity of the initial infrastructure.

These objectives of the paper must be viewed against the 5 types of market failure, discussed in section 2 above, that can result in inefficient infrastructure choices. Roughly, these explanations can be classified into two groups. The first group is related to insufficient or faulty general climate-related or energy policies (including insufficient emissions pricing and technology support); while the second group is related to inefficiencies and incompleteness related to expectations about future policies. Points B and D fall largely into the former category, while points A and C fall mainly into the latter. Point E might conceivably fall into either category.

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It is easy to understand why policies of individual governments hosting infrastructure projects are often faulty: namely, the basic lack of incentives of governments to address the problems of mitigation, at least in the absence of a comprehensive and binding agreement requiring the governments to behave in an optimal fashion. The second group of explanations has a more diverse set of explanations, which are however all related to either policy incompleteness or “irrationality” (or “behaviorism”) in the policy process.

We now discuss the results from our simulations, presented in section 3 above and in appendix B, in light of these 5 points.

- A) _The initially expected distribution of energy costs, relevant for making current infrastructure decisions, is down-shifted_ relative to the true (correct) distribution that will be realized in period 2. This causes infrastructure decisions to be incorrect, and chosen with too high energy intensity. Figure 7 in section 3.3 then represents the degree of distortion (as the scaling of the initial infrastructure, which is proportional to final lifetime energy demand for the infrastructure, is in this case not optimal and generally excessive). Ex post decisions for given, already established, infrastructure are however assumed to be socially optimal. The energy consumption choice, in response to an increase in “true” energy cost which is not anticipated nor ignored, is then valid only for period 2. It is represented by simulations in figure 1 (figure 1.1 in appendix B for correlated costs). In our examples, the scope for ex post avoidance of costs could be substantial as expected energy costs increase (most for the CobbDouglas specification and least for the exponential specification). The fully optimal response, involving also optimal ex ante infrastructure choice, is characterized by figure 10 and is generally greater (in some cases far greater). To provide an example, assume that the correct value of Eq is 4, while the assumed value is 2. Focusing now on the log and exponential specifications, we find that the chosen energy intensity of infrastructure is in the range 30-50 percent higher than the optimal level in this case. This is then also the degree of excessive carbon emissions in this case.

- B) _Energy (including environmental) costs facing the policy maker are lower than globally optimal costs_ . The distribution of energy costs facing the policy maker is however correctly anticipated. The energy cost distribution applied by the policy maker is then down-shifted relative to the “optimal” energy cost distribution (the charge to the project, for combined energy and environmental costs, is lower than the global cost); and project energy costs are not responsive to increases in true environmental costs. In this case, there is (in the limit with no increase in the local energy cost in response to a “true” cost increase) no response whatsoever to an increase in energy/environmental costs, neither ex ante nor ex post. The degree of distortion to lifetime energy consumption can then be found directly from figure 10, in section 3.4. This is the maximal distortion to this variable found here; both the initial energy intensity, and the ex post choice of energy consumption versus retrofit, are distorted upward for a large effect. The distortive effect can be read out of figure 10 by comparing values at “correct” versus “accounted for” values of _Eq_ . Considering the same example as under case A (where the true expected energy cost is 4, while the expected cost facing policy makers is 2; and again considering the log and exponential specifications), the degree of excess consumption of fossil fuels is now more dramatic, between 2 and 3.5 times the optimal level. This reflects the inefficient choices at two levels, namely both the initial investment level, and the ex post level at which the investment is applied.

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- C) _Expectations about future retrofit costs are incorrect at the stage of infrastructure investment._ Such mistakes could go both ways (retrofit costs could be either over- or under-estimated), and thus lead to either excessive or inefficiently low overall energy consumption. Assume first that true retrofit costs are under-valued. Referring to our simulations, we must then consider the infrastructure decision which is biased by this distortion. Figure 8 (and figure 3.2 in the appendix, for the correlated cost case) provides the clue to answering this question. In particular, under the log-linear or exponential utility cases, with an under-valuation of the ex ante expected retrofit cost, set at 2 instead of a correct value of 3, infrastructure is too energy-intensive, but not dramatically so: energy intensity is excessive by about 10 percent or less in all examples. With an excessive assessment of the retrofit costs, the mistake goes in the opposite direction (so that infrastructure has too low energy intensity), but again only slightly.

- D) _Future retrofit costs are correctly anticipated, but are higher than socially optimal costs._ The idea here is that future retrofit costs are endogenous and affected e g by investments made in technology to reduce these costs; and such investments may be under-provided. In this case we need to consider lifetime energy consumption (both initial investments, and ex post energy use), and its response to changes in _Ey_ , described in figure 11 (and figure 4.2 in the appendix). Two factors then affect overall energy use, and in different directions. First, the higher than optimal expected future cost (implied by the higher retrofit cost) leads to an inefficiently low ex ante energy intensity of the initial infrastructure. Secondly, the ex post probability of energy use is excessive, since retrofit is used in too few cases. We find, from figure 11, that the balance between these two factors depends heavily both on the degree of cost uncertainty, and on the utility function specification. We first note that total energy consumption is rising with the level of the retrofit cost, in all cases of the simulations under the log and exponential utility specifications. We also note that when cost uncertainty is low, the ex post effect, representing the ex post probability of continued energy use, always dominates, and overall expected energy use increases under all utility function specifications. With greater variances on the cost components, however, the overall probability of energy use changes much less; and the two factors more or less cancel out in the simulations.

- E) _The policy-relevant value of T, T1, is below its optimal value, T0._ Such effects are less directly represented by our simulations. It may be represented via a too low weight to period 2 and its ensuing costs, when making the infrastructure decision. The main effect is that the energy intensity of the initial infrastructure, and in consequence energy consumption, is excessive. There is however less reason why the ex post retrofit decision should be distorted.

Policy incompleteness (applying to point E above) could arise as an issue when policies are based on ad hoc rules that are outcomes not of deliberate optimisation, but instead a simplified process that may lead to systematic biases in a climate context. One such case is when the rate of discount for evaluation of public projects with climate impacts is determined administratively for large classes of projects (typically at a high rate), and not aligned with optimality rules relevant for (long-run) climate-related projects. Another case of policy incompleteness is when the returns to public projects are accounted for only over a limited horizon (say, 20 years), or the project is not based on explicit cost-benefit calculations.

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Considering implications for initial infrastructure design of problems related to categories A- E above, we have seen, supported by our simulations, that all categories except D overwhelmingly tend to illustrate cases where energy consumption is excessive over the lifetime of an established infrastructure. In cases A-B, this occurs in two complementary ways: through excessive energy intensity of the initial infrastructure; and through excessive ex post “business-as-usual” operation of the investment in period 2. Under case C, initial energy intensity is excessive, while “business-as-usual” operation is not affected. On balance expected energy use is excessive. The fourth category (D) is different in that it tends to make energy intensity of the initial investment _lower than optimal_ ; while the retrofit option is used ex post in too few cases. Our simulations indicate that these two factors may balance out for overall energy consumption; or (when variances on energy and retrofit costs are low) lead to excessive overall energy consumption as the latter factor dominates. The fifth category, E, is similar to C as the ex ante infrastructure is biased in the direction of too high energy intensity, while the ex post decision is not generally biased.

Arguably, cases A-B are keys to understanding the implications of (perceived and actually realized) energy prices on the path of future energy consumption following from particular infrastructure investment projects. Table 1 below sums up some main results from our simulations in section 3, which are done under the assumption of uncorrelated energy and retrofit costs. We present two sets of figures. A) Those based on “policy error ex ante only”, which can be found from figure 7. These are excessive rates of fossil energy consumption over the lifetime of the infrastructure, that result from the investment decision itself being wrong; but where adjustments to correct realized energy price are made ex post. The only source if excessive energy consumption is thus that the infrastructure investment is based on wrong expectations about future energy prices.

Disregarding the Cobb-Douglas case (which seems less realistic than other cases), our simulations show that when the actual energy/environmental cost has expectation 3 (while the ex ante anticipated expectation is 2), the degree of excessive energy consumption due to “wrong” infrastructure is roughly between 20 and 35 percent; when the correct expectation is 4, this degree of excessive energy consumption is 30 and 60 percent.

When there is in addition a policy error ex post, so that the correct energy/environmental price is not implemented but instead the lower price is used, the overall error is far greater. Focusing also now on the log and exponential cases, the overall policy error (taking the compounded effect of the excessively energy intensive infrastructure, and the excessive energy use ex post) is now from about 65 percent to more than 100 percent of the efficient level when the correct Eq level is 3 (while the energy price actually applied is 2). When correct Eq = 4, this error is dramatically higher, most so for the case of low variances.<sup>26</sup>

> 26 This particular result has a natural explanation, namely, that for these simulations we assume throughout that Ey = 3. Since it here makes a big difference for realized energy consumption whether Eq is smaller or greater than Ey, there is a big jump in realized energy consumption as Eq goes from a level below to a level above Ey. This jump could be unrealistically high in particular when variances are small.

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**Table 1: Relative policy error, in the form of excessive carbon emissions over the lifetime of the infrastructure, in percent as a result of incorrect a) ex ante expected energy/environmental cost (=2); and b) ex post expected realized energy/environmental cost (=2), for different correct Eq, based on simulations in figures 7 and 10.**

|Type of<br>li|Utility<br>fti|Low v<br>|ariances<br>||Mediu<br>|m varianc<br>|es<br>|High v<br>|ariances<br>||
|---|---|---|---|---|---|---|---|---|---|---|
|pocy<br>error|uncon<br>specification|Eq=<br>2.5|Eq=3|Eq=4|Eq=<br>2.5|Eq=3|Eq=4|Eq=<br>2.5|Eq=3|Eq=4|
|Policy<br>error ex|Cobb-<br>Douglas|39|75|124|40|79|146|40|81|158|
|ante only|Logarithmic|18|32|50|18|34|57|18|35|61|
||Exponential|12|21|31|12|21|35|12|22|37|
|Policy<br>error|Cobb-<br>Douglas|68|177|693|64|154|446|62|147|403|
|both ex<br>|Logarithmic|42|109|430|39|90|248|37|84|213|
|ante and<br>expost|Exponential|34|91|363|31|72|199|30|66|168|

A main objective of this paper has been to study impacts of options to retrofit an established infrastructure on total expected costs, expected energy (including environmental) costs, and initial energy intensity of infrastructure. Our simulations provide a rich set of examples of how overall realized (conditional) expected costs vary in response to variations of certain key parameters (expectations and variances), and given that ex post energy and retrofit costs are both log-normally distributed (and independent). They indicate that some fraction of expected future energy use related to infrastructure can always be avoided by optimally exercising either the retrofit or close-down option at a later stage, given that exercising such options is ex ante optimal. This degree of cost avoidance can be large, under plausible assumptions; and larger when there is more uncertainty about energy and retrofit costs. In certain parametric cases more than half of the (ex ante expected) potential energy consumption is avoided through optimally exercising the retrofit alternative ex post. Expected total (energy plus retrofit) costs are then reduced, in some cases substantially.

Note that with a long expected time from infrastructure investment to availability of a relevant alternative option (retrofit, or closedown), and when the decision maker discounts too heavily (excessively), the options will also be discounted too heavily and given too little weight in the infrastructure decision problem. This factor would then serve as a partial counterweight to those emphasized here, that tend to reduce energy (and climate) cost below socially efficient levels, and lead to too energy intensive infrastructure choices.

Several limitations of our analysis must be pointed out. First, it is based on ex ante distributions of energy and retrofit costs that are both log-normal and known. Secondly, we assume only two periods, “the present”, and “the future”, which allows for only one decision point beyond that of infrastructure investment (at the start of period 2). Our choice of assumptions here was guided by a concern for generality in the distributional assumptions, while still permitting a tractable analysis. An extension of the current framework to three or more periods would make the analysis less tractable, but ought to be pursued in follow-up work. Relevant costs (of energy, emissions, and related to the retrofit technology), as well as benefits (the current utility value of the infrastructure technology), all in reality evolve 32
continuously through time making the two-period framework less accurate. An obvious development would then be to assume that retrofits could be carried out at several points of time; and with separate developments for energy and retrofit costs. Such extensions are being considered in a companion paper to the current one, Framstad and Strand (2010), where energy and environmental costs are assumed to evolve continuously over time; in other respects however assumptions are much simpler in that paper, in particular, a fixed retrofit cost is assumed.<sup>27</sup> Our analysis, both theoretical and for simulations, assumes that the scaling of the initial infrastructure does not interact with the ex post decision to retrofit. This follows from our basic assumption that retrofit costs are proportional to energy consumption. When comparing overall retrofit costs with overall energy costs ex post for a given project, project size then does not matter; only energy and retrofit prices matter. This is, admittedly, a special case, which could be generalized. With such a generalization, the ratio of overall retrofit costs to overall energy costs would likely be a decreasing function of project size (unit retrofit costs would be reduced for larger projects). Additional effects would then arise. In particular, larger ex ante projects would be favoured more as these would, effectively, permit more cost avoidance later on (through more frequent retrofitting). It is less clear how overall energy consumption over the project’s lifetime would be affected; the initial scale-up of projects, and the subsequent more frequent retrofitting, would work in opposite directions.

Our choice of utility functions for simulations might need some comment. One of our applied specifications, the Cobb-Douglas formulation, can be argued to yield unrealistic results as the response of energy intensity in infrastructure, to changes in expected lifetime project costs, is too great (with an elasticity for energy intensity in infrastructure to ex post cost which is greater than one in absolute value). This formulation is thus likely to be less credible than the other two utility function formulations, in yielding respectively a unitary (the log formulation) and less than unitary price elasticity (the exponential formulation).

A further extension would be to consider (partial) retrofits where neither fossil energy use nor carbon emissions are reduced to zero, which may be more cost efficient in some cases. Also, in our model, upon a possible infrastructure closedown, “nothing happens”. A more satisfactory analysis would involve a replacement infrastructure taking over the flow of services lost by closedown; which would require specific assumptions about costs (of new, future, infrastructure investment) and benefits (flowing from the new, instead of the old, infrastructure). We seek to pursue such extensions in future work.

> 27 A basic result here is that continuous development of costs produces an “option value” of waiting which serves to delay the retrofit decision; this lowers expected total costs but increases environmental costs.

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