Strictly concave utility in pollution and one clean technology

Clean capacity is constant at ¯KC = QC × 1000 all time because investment in clean technology has a greater marginal product than investment in dirty technology, QC = 1.1 × QB > QB = (1.02)²⁰, and this is affordable, KC(0) = ¯KC. There is one clean technology, or equivalently other clean technologies have a low rate of return on investment that does not make investment worthwhile, and have zero stock at the initial date. This leads to a problem with two state variables. The utility function is

U (c, Z) = [((1 + ξZ) exp(−ξZ)c)^1−ψ]/(1 − ψ)

with constant index ψ = 2 of relative risk aversion and parameter ξ = 1/442.5. Marginal utility of pollution is finite for all pollution levels so there is no catastrophe level of pollution.³³ The values of other parameters are β = 0.9675, B = 1, b = 1/30, and ρB = ρC = 0. This example abstracts from emissions in the investment sector.

The economy subject to no government policy experiences increases in pollution and

33The utility function U is strictly concave in Z ∈ (0, ξ⁻¹) and yields the simple expression (−∂U/∂Z)/(∂U/∂c) = (ξc)(ξZ)/(1 + ξZ). Acemoglu et al. (2012) use a similar function whose dif-ferential with respect to pollution becomes −∞ for some finite pollution level.

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Figure 2.2: Trajectories of optimal (solid) and constrained optimal or laissez-faire (dotted) pollution Z and dirty capacity KB (squares on curves show values in successive periods).

dirty capacity along the dotted upward-sloping trajectory in Figure 2.2 for the initial state (Z, KB, KC) = (0, 0, ¯KC). The squares on curves starting on this laissez-faire curve for different delay dates t^∗ depict optimized values of state variables in six successive periods.

A date t^∗ corresponds to some initial date that is indexed zero in the planner problem subject to initial values of pollution and capital stocks equal to the date t^∗ values. Such states on solid curves arise in the global optimum with chosen utilization. These states on dotted curves are solutions to the planner problem that sets the utilization rate of capital to one, uB(t) = uC(t) = 1 all t ≥ t^∗ in the constrained optimum. States in further periods lie on the respective trajectory. All optimized trajectories converge to the same steady state (448, 0.82 ¯KC, ¯KC) as time goes to infinity. The constrained optimal trajectories starting at points in the designated region R1 in which dirty capacity is optimally fully utilized, such as path A, are globally optimal because these trajectories remain in this region. Underutilization of dirty capacity is efficient to smooth dirty capacity early on.

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Figure 2.3: Time paths of optimal (solid) and constrained optimal (dotted) utilization rate uB of dirty technology, emissions buBKB and pollution Z.

This underutilization occurs when the pollution stock is below or above its long-term level. Capital of the dirty technology in the laissez-faire equilibrium becomes so high that optimization at date t^∗ > 0 calls for its underutilization. The simulation shows that given large upper bound ¯KB there may be a minimum delay date t^′ such that for all t^∗ ≥ t^′ underutilization is optimal for any initial state at date zero.³⁴

The cost of polluting is θB = 15.46 in all states (Z, KB, ¯KC) in region R2 of joint investment and underutilized capital. The appendix shows how this parametric value was useful in finding the region R2 and the region R3 without investment and with underutilized capital in the dirty technology. For sufficiently large scale ¯KB of the dirty technology a region R3 exists.

The optimal plans B to E with initial states in R2 and R3 in Figure 2.2 obtain lower emissions in early periods and lower pollution in all periods relative to the constrained optimal plans B’ to E’ with same initial states. The underutilization of dirty capacity avoids current emissions that occur given assumed full utilization of capital. Figure

34This may not be true in a model in which the laissez-faire economy converges to a state with KB < ¯KB.

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Figure 2.4: Time paths of consumption c and cost of polluting θ in optimal (solid) and constrained optimal (dotted) plans.

2.3 shows the time paths of utilization rate, emissions, and pollution for the two initial states (Z, KB) = (176, 5130) in the plans C and C’ in the upper graphs, and (Z, KB) = (331, 8180) in the plans D and D’ in the lower graphs, rounded. The utilization rate in the initial period can be read off from the time series plots of emissions by dividing the optimal emissions amount by the constrained optimal emissions amount. These rates are roughly 0.21 and 0.14 for the paths in the upper panel and lower panel, respectively. The utilization rate increases over time until it reaches one. The utilization rate uB in the simulation of C is not one after it has been one. The utilization rate in the simulation of D is not one in the fourth period. However from the states I know that this is not optimal.

The method used approximates the solution. Comparison of 0.21 and 0.14 suggests that the utilization rate is expected to be smaller in the first period of an optimal allocation the longer the delay beyond the smallest delay level at which dirty capacity is underutilized.

Consumption decreases initially in all optimal and constrained optimal plans that start at states in the regions R2 and R3. Plans with uB(t) = uC(t) = 1 all t ≥ t^∗ by assumption are constrained optimal. This is not surprising because these plans involve a decrease in output, and the savings rate moves monotonously, so that consumption and output comove. The consumption in Figure 2.4 corresponds to the plans C, C’, D, and D’ for two

initial states with greater capital of dirty technology than its long-term level. Apparently underutilization lowers the rate of decrease in consumption relative to a constrained optimum. Figure 2.4 presents time paths of θ, that the summary after the next example interprets. The value of θ(0) in constrained optimum, 142.6 and 492.6, respectively, is so large that it is outside the picture at the given scale.

2.3.2 Constant marginal utility of pollution and multiple clean