Constant marginal utility of pollution and multiple clean technologies 77

This example shows how the incentives to invest in dirty versus clean technologies are influenced by the possibility of underutilization. This issue was left out in the first specification for computational simplicity. The utility function

U = c^1−ψ/(1 − ψ) − dZ

has the constant marginal disutility of pollution d to obtain a two-state problem. There is a continuum of clean technologies C on [0, ¯C] with aggregate capital R

CKCdC. The marginal product function

Q(x) = Q^′′+ (Q^′− Q^′′)(1 + vx) exp(−vx), 0 ≤ Q^′′ < Q^′,

decreases in the aggregate input x in clean technology investment. Along this frontier there are decreasing returns to scale in using aggregate clean technology capital in loca-tions of varying productivity. These localoca-tions can be occupied with capital. In general, the new aggregate clean technology capital here is a function of x on geographic sites with distributed marginal product.³⁵ The equilibrium and efficient motion of aggregate capital in clean technologies is R

CK_C(t + 1)dC = R^R_CxC(t)dC

0 Q(z)dz if all capital units of clean technologies are fully utilized.³⁶ In addition, then there is a value function in the

35Physical capital is chosen in Section 2.4.1.

36One may define the mass S ⊆ J^′ of a continuum J^′ of technologies with capital in the current period, S = {j | Kj > 0}, whose utilization rate is optimally either zero or one, with proper subsets S \ S⁻ = {j | u_j = 1} and S⁻= {j | u_j = 0}. The law of motion of this set is S(t + 1) = S⁻(t) ∪ S⁺(t) when investment occurs in technologies S⁺⊆ J^′\ S⁻. The law of motion of aggregate capital in these technologies is R

jKj(t + 1)dj = [R

j∈S(t)γj(1 − uj(t))dj/R

j∈S(t)dj]R

jKj(t)dj +R

j∈S⁺(t)Qjdj. Let J^′ = {j | d_j= 0} and j = C. Upon change of variable the new capital stock isR

C∈S⁺QCdC =Rx⁺(t)

0 Q⁺(x)dx.

0 200 400 600 800 1000 1200 1400 1600 1800 0

5000 10000 150000

500 1000 1500 2000 2500 3000 3500 4000 4500

K Z B

KC

Figure 2.5: Trajectories of optimal (solid) and constrained optimal or laissez-faire (dotted) pollution Z, dirty capacity KB, and clean capacity R

CKCdC (squares on curves show values in successive periods).

states dirty technology capital and aggregate clean technology capital, (KB,R

CKCdC), since both utility U and absorption A are linear in pollution. I assume scale-dependent relative advantage of dirty and clean technologies, Q^′ = 1.1×Q_B = 20×Q^′′, and emissions from investment, ρB = ρC = (βQB)^−1/ψQB(1/20)(1/30) all C. Further parameters are d = 7.4 × 10⁻⁷, v that solvesR

CKC(0)dC = 1000/2.4, B = 1.2, and b = (19/20)(1/30).

The horizontal dashed line runs at θB.

The trajectories in Figure 2.5 emanate from states in the laissez-faire equilibrium that is initialized at Z = KB = 0, R

CxC(0)dC = y, and R

CKC(0)dC = Ry

0 Q(x)dx such that Q(y) = QB. In the laissez-faire equilibrium aggregate clean capacity is R

CKC(0)dC if dirty technology investment is below ¯KB = 14442 in the preceding period. The laissez-faire economy does not generate sufficient output for optimal underutilization of dirty technology capital when the pollution stock is below its long-term level. In the first example, underutilization occurs at such states.

0.4 0.6 0.8 1

Figure 2.6: Shadow return and cost of polluting, and time paths of dirty and clean tech-nology savings.

The policy of underutilization diminishes the incentives to invest in expensive clean technologies. The right panel in Figure 2.6 shows the time paths of savings. Unused dirty technology capital (1 − uB)KB contributes to the savings ((1 − γB/QB)(1 − uB(t))KB(t) + KB(t + 1)/QB) out of capacity. Clean technology investment R

CxCdC equals its savings because clean technology capital is fully utilized following Proposition 2.6. In the con-strained optimum the incentives to invest are aligned in both dirty and clean technologies.

In optimum with underutilized dirty technology capital the incentives to invest in clean technologies are relatively lower early than in the constrained optimum because underuti-lization helps mitigating pollution effects on society. Clean technology investment peaks in the period in which both investment and underutilization in the dirty technology are optimal, when the state is in R2, below the initial level in the constrained optimum at the same initial state. This explains the V-shaped path of the shadow return. The shadow return and the cost of polluting are positively related on intervals with underutilized dirty technology capital, and negatively related on intervals with full utilization, which the curve in the upper left panel of Figure 2.6 depicts. The dotted trajectory shows a con-strained optimal path. The initial values of pollution and dirty capacity are (876, 7790)

and (1500, 12254) in this Figure. The former values initialize the constrained optimal path in the upper left panel. The cost of polluting is plotted over time in the lower left panel. The dashed lines are at the levels (B/dB) and θB.

2.3.3 Summary

The simulation yields the following five insights. (i) Delay of government policy. Emis-sions are an externality. The longer emisEmis-sions are unpriced the greater dirty technology capacity is built relative to the efficient long-term dirty technology output. This means that delayed government policy leads to efficient underutilization of dirty capacity. (ii) Location of regions of underutilization of dirty capacity in the state space. Intuitively, the society can afford underutilization well when capital is large given pollution and desires it when pollution is large given capital. In fact in the first example pollution Z and dirty capacity KB relate negatively on the boundary of R1 and R2. In the example with con-stant marginal utility of pollution a concon-stant KB given the optimal choice KC forms the boundary of R1 and R2. Then affordability is the dominant force behind underutilization.

The slope of this boundary in the state plane (Z, K_B) is minus one (not shown) if utility is quadratic in both consumption and pollution and absorption is linear in pollution. In these examples thus underutilization of capital can be optimal when current pollution is smaller (environmental quality is greater) than its long-term stabilization level. The Propositions 2.3 and 2.4 do not tell for what initial levels of pollution dirty capacity use is postponed. (iii) Timing of emissions. The optimal and constrained optimal emissions on optimal paths that start at the same state in R2 or R3 differ substantially early. Vari-able utilization allows to start with low emissions followed by greater emissions. Fixed full utilization necessitates emissions decreases early on in the constrained optimum to approach lower capital values. (iv) Cost of polluting. The cost of polluting can differ substantially between an optimal plan with chosen utilization and a constrained optimal plan with assumed fully utilized capital subject to the same initial condition. The sub-stantive difference lies in the early periods with optimal underutilization in the Figures 2.4 and 2.6. Underutilization mitigates societal effects of pollution and thus achieves a lower cost. (v) Clean technology investment. Expensive clean technologies are not needed when dirty technology capital utilization can be varied.