This section explains why a unique technology is chosen in the theory of substitutable capital and emissions yet there is investment in multiple clean technologies in the theory here in evaluating climate change, examines the incentives to invest in clean technology under one assumed dirty technology versus multiple dirty technologies, and discusses time paths with unconstrained dirty capacity choice. First I show that there is a unique stationary point generally, and how the local stability analysis for two technologies applies to more than two technologies.
14The feasibility frontier χ in pollution-consumption space tilts down and the preference curve φ shifts in so that at the new intersection consumption is smaller than before and pollution is smaller than at the intersection of the old φ and the new χ.
Proposition 1.8 There is a unique stationary point, if the cost of pollution reduction in terms of consumption units θj differs over all dirty technologies j ∈ J , and all clean technologies j ∈ J with emissions of investment, ρj > 0.
Proof. The distinct relative cost θj and Lemma 1.1 imply that capital can be interior, Kj ∈ (0, ¯Kj) only for one technology j ∈ J . There is a unique intersection of the curves χ(c, Z) formed by c/B = P
j(1 − 1/Qj)Kj and A(Z) = P
j(dj + ρj/Qj)Kj, and φ(c, Z) given (1.5), at Kj = ¯Kj all j with θj < θ varying one other technology’s investment.
Else consumption and emissions are unique if Kj = ¯Kj all j in which investment occurs, Kj > 0. There are no two clean technologies C′ and C′′ with ρC′ = ρC′′ = 0 and QC′ = QC′′ = β−1. Q.E.D.
Proposition 1.4 applies consequently to a continuum or a discrete set of dirty or clean technologies holding a mass or sum of input in investment constant. The sums of constant capital amounts must be replaced by integrals for a continuum of technologies.15
Clean technology frontier versus dirty technology frontier.—In the theory without com-mitment Stokey (1998) interprets the unique intratemporal emission control as a unique technology chosen at the date of production. The choice set can be thought of as a dirty technology frontier, where there is a trade-off between the productivity of capital and labour and the emission intensity of output. Nordhaus (2009) views emission control in this theory as a mix of clean and dirty technologies and energy efficiency choices that can be adjusted in each period.16 With commitment there is no unique technology choice if the marginal product QC of clean technologies ranges from QB to below one on small individual scales ¯KC, consistent with data on the dollar cost of avoiding emissions that Nordhaus (2009) uses to calibrate the mapping between productivity and emission inten-sity in his model DICE. Given this wide range, and roughly equal emission inteninten-sity ρj of the input in investment for all technologies j ∈ {B, C1, C2, . . .} the condition (1.8) implies that investment in some types of clean technology is worthwhile if there was investment in a dirty technology.
One dirty technology versus multiple dirty technologies.—I consider clean technology investment when the capacity choice of multiple dirty technologies is unconstrained. The
15In the canonical system Z(t+1) = (d+ρ)K(t)+P
jρ(Kj−c(t)/B)+P
j(ρj−ρ)Kj/Qj+Z(t)−A(Z(t)) extends (A-1) in the appendix, and K(t + 1) = Q(K(t) +P
j(1 − 1/Qj)Kj − c(t)/B) is the resource constraint for multiple technologies, holding constant each Kj.
16The factor efficiency choice seems a viable interpretation.
incentives to invest in a given clean technology may become greater or smaller from introducing multiple dirty technologies. As will be shown below the effect depends on the clean technology productivity relative to the discount factor and the cost of polluting relative to its stationary level.
Let fix the ratio 4α = QB/dB of the productivity in creating dirty technology capital and the emission intensity in using this capital for some productivity levels QB ∈ [Q′′, Q′] subject to large scale ¯KB all B. The optimal interior choice Q = 2Bα/θ on a continuum of technologies inversely relates to the cost of polluting so that Q = 2β−1 in a stationary point if ρB = 0 all B. I disregard the emissions from investment here to focus on tech-nology choice per se. Suppose that this techtech-nology was the only dirty techtech-nology and 2β−1 ∈ (Q′′, Q′). This enables comparison of clean technology investment for a narrow choice set with one dirty technology and a wider choice set with more and less productive dirty technologies. The critical level for technology switching is θ∗ = θ(2−βQC) given the stationary level θ = Bαβ if there is one dirty technology. Indifference levels of the cost of polluting between investing in dirty technologies versus a given clean technology are de-rived as follows. The level γ∗ = Bα/QC of the cost of polluting equates the rates of return on investing in any of the dirty technologies with productivity QB ∈ [Q′′, Q′] as an un-constrained choice and in the clean technology with productivity QC below its maximum scale. A clean technology does not have an environmental cost here, ρC = 0 all C. The optimal dirty technology choice for θ(t+1) ≤ θ′ = 2Bα/Q′ and θ(t+1) ≥ θ′′= 2Bα/Q′′is investing in Q′ and Q′′ at t, respectively. If γ∗ < θ′ then clean technology investment oc-curs for θ(t+1) > θ′. If γ∗ > θ′′ then clean technology investment occurs for θ(t+1) > θ′′. Else it occurs for θ(t + 1) > γ∗.
(i) QC < β−1. There can be investment in a clean technology, though it would not be optimal if only one dirty technology was available, at θ greater than its stationary level. Then the switching level if there is only one dirty technology is greater than θ′′. In other cases the incentives to invest in the given clean technology are smaller in an extended choice set for θ ∈ (θ∗, min[γ∗, θ′′]) than under one dirty technology, because a dirty technology with smaller emission intensity than of this technology is feasible.
(ii) QC = β−1. The switching level equals the stationary level when there is one dirty technology. Thus the incentives to invest in clean technology are weaker for θ ∈ (θ∗, min[γ∗, θ′′]) because investing in a dirty technology with productivity smaller than
R(t + 1) θ(t)
Qgas Qoil Qcoal
B/dgas
B/doil
B/dcoal
O
b
b b
Figure 1.5: Shadow return and cost of polluting given investment in coal, oil, and gas technolo-gies.
the stationary level is optimal in this range.
(iii) QC > β−1. The hurdle to invest in a given clean technology is greater for low θ relative to its stationary level when a dirty technology with a large emission intensity is feasible, because its investment is optimal for low θ ∈ (θ∗, max[γ∗, θ′]).
Unconstrained dirty investment options.—Investment switches between technologies over time when the pair (θ(t), θ(t + 1)) leaves the region of investment in some technology if there is a continuum or discrete array of more than two technologies characterized by (Qj, dj, ρj) each on a large scale. The rationale is the same as with two technologies.
Among coal, petroleum, and natural gas energy conversions in an economy with the relative importance of their prime uses for the single factor in the model and emission intensity in the same order (trivially) a more dirty technology is more productive so that dB and QB relate positively all B.17 Then intersections of the curves QB/R(t + 1) − 1 = (ρB + ε(t + 1)dBQB)θ(t)/B in R(t + 1)-θ(t) space are indifference points regarding investment at date t. This equation results from combining (1.8) and θ(t + 1) = R(t + 1)ε(t + 1)θ(t). Figure 1.5 plots it for three technologies. Investment in a technology
17The prime uses are stationary motor drive and light using electricity for coal, mobile energy for petroleum, and space and water heating for natural gas. There is scientific research in reversing this relationship for fossil fuels, for example, producing clean coal.
with maximum R(t + 1) given θ(t) is optimal. Let ρB be equal all B and suppose that the long-term allocation is based on oil. Then it is optimal to move from a coal-based economy (high dBand QB) to an oil-based economy (medium dBand QB) when θ increases toward its long-term level—the initial energy production capacity is small relative to its long-term capacity. In contrast, investment switches from natural gas technologies (low dB and QB) to oil technologies, when θ decreases toward its long-term level—the initial energy production capacity is greater than its long-term capacity. I have used the term energy-carrier based economy. Multiple energy services may be served by different energy carriers in a given period.