Self-Fulfilling Prophecies in the Transition to Clean Technology

Layer 1: Overview

This paper by Smulders and Zhou challenges the standard lock-in narrative for the slow green transition. The conventional explanation — path dependency in directed technical change (DTC) — is hard to reconcile with forward-looking investors who anticipate an eventual move to clean technology. The authors propose an alternative: strategic investment complementarities among innovators can produce self-fulfilling prophecies that delay the low-carbon transition even when all agents foresee it will ultimately occur.

The framework is a continuous-time general equilibrium DTC model in the tradition of Acemoglu et al. (2012), modified in two key ways: patents last forever (rather than one period), and labor is mobile between production and R&D. The economy has a clean and a dirty final-goods sector with substitution elasticity σ between them. A continuum of monopolistic intermediate goods suppliers in each sector invest in R&D to improve product quality. The key mechanism is a demand externality: when goods are gross substitutes (σ > 1), innovation in a sector reduces the relative price of that sector’s output, shifting consumer expenditure toward it. This raises the return to all innovation in the sector. For σ > 2, this demand externality outweighs the intra-sector business-stealing effect, making within-sector innovations strategic complements — each firm’s R&D raises the payoff to R&D for all others in the same sector. The threshold σ > 2 is necessary and sufficient for a coordination problem to arise in the unregulated economy.

The paper establishes three steady states: two saddlepath-stable corner steady states (one with innovation only in the clean sector, one only in the dirty sector) and an unstable interior steady state with simultaneous R&D. When σ > 2, there exists a range of initial clean market shares θc,0 (the “overlap”) from which both corner steady states are reachable under rational expectations. The overlap grows with σ and shrinks with impatience ρ (Proposition 3). Furthermore, for any initial condition within the overlap, multiple transition paths to the same corner steady state exist: a “fast” path with immediate concentration of R&D in one sector, and “delayed” paths in which firms temporarily innovate in the competing sector before finally converging. For higher σ values, these delays may involve regime switches between the clean-only and dirty-only innovation regimes (σ ∈ [σ-bar, σ-bar-bar)) or even stagnation periods with zero R&D (σ > σ-bar-bar), producing non-monotonic patterns of clean innovation — rises followed by falls before eventual clean dominance (Proposition 4).

The welfare-maximizing path always leads to the clean steady state: a dirty steady state violates the transversality condition on the carbon stock because unbounded climate damages accumulate. The paper calibrates to 2019 data: initial clean sector share θc,0 = 0.177 (matching the 17.7% renewable energy share in global final energy consumption), world GDP per capita of $11,019 (constant 2015 USD), per capita carbon emissions of 1.22 metric tons, emission intensity ad = 0.198 tonnes per thousand USD, and σ = 1.5. Under this calibration, three distinct equilibrium paths coexist under an optimal Pigouvian carbon tax — one with clean-only innovation from the start and two involving temporary dirty R&D — all converging to the clean steady state but at different speeds and with different amounts of stranded dirty assets.

The central policy finding (Proposition 7) is that a Pigouvian carbon tax set equal to the social cost of carbon at all times eliminates the dirty steady state but does not pin down a unique transition path. Multiple equilibria with different durations of dirty innovation persist under the first-best carbon tax. Effective coordination requires a second instrument that directly controls relative innovator profitability: a minimum clean revenue guarantee, an emission cap, a dirty R&D tax, or a contingent super-Pigouvian carbon tax all qualify. A clean R&D subsidy works but is an inferior device because it distorts labor allocation between production and research. Crucially, commitment is required: unless the government commits to maintaining the coordination instrument until the economy exits the multiple-equilibria region, delayed transitions remain possible.

Layer 2: Deep Dive

What is the core mechanism generating multiple equilibria, and why does it require σ > 2?

Intermediate good monopolists in each sector earn profits proportional to their sector’s expenditure share, which rises with relative quality when σ > 1 (demand shift effect). But a firm’s share of sector profits falls as rivals innovate (business-stealing effect). From equation (24), the relative marginal profit of clean versus dirty innovation scales as (Qc/Qd)^(σ-2). The demand shift effect dominates the business-stealing effect if and only if σ > 2. When σ > 2, innovations within a sector are strategic complements: any firm’s R&D raises all other firms’ marginal return to R&D in the same sector. This complementarity means beliefs about which sector will be large in the future become self-reinforcing: if investors expect the clean sector to grow, clean innovation is profitable, and the expectation is validated.

How do the two modifications from Acemoglu et al. (2012) affect the results?

First, infinite (rather than one-period) patents allow future expected profits to influence innovation decisions, giving expectations a more direct role. Second, labor mobility between production and R&D makes the speed of innovation endogenous alongside its direction. However, the paper shows (OA3.2 and Section 3.3) that neither modification is necessary for the qualitative result: the overlap and strategic complementarity arise even with finite patent length and segmented labor markets. Longer patent length has an effect similar to lower impatience — it increases the overlap. OA4 shows that a segmented labor market model has essentially identical dynamics but requires a third state variable (an effective savings-rate proxy), so it is no simpler than the baseline.

What types of transition delays are possible and how do they depend on σ?

Proposition 4 identifies three regimes of delay: (a) for 2 < σ < σ-bar, only temporary simultaneous R&D is possible as a delay; (b) for σ ∈ [σ-bar, σ-bar-bar), delay must include temporary regime switches between the clean-only and dirty-only innovation regimes; (c) for σ > σ-bar-bar, delay must include a stagnation period with no R&D at all. The numerical example shows that for σ = 2.5 and σ = 3, delayed paths involve a flat simultaneous-research segment (mc = 1/2). For σ = 5 and σ = 7, equilibrium paths involve switches between clean-only and dirty-only regimes. For σ = 8 and σ = 9, paths contain vertical stagnation sections and multiple regime switches, with clean innovation peaking, falling, then rising again before converging to the clean steady state.

What does the welfare analysis reveal about the costs of delayed transition?

Under the calibrated model (σ = 1.5, θc,0 = 0.177), three equilibrium paths coexist under the Pigouvian carbon tax, corresponding to no delay, short delay, and long delay in clean innovation. Paths with delay accumulate more dirty capital (Qd,∞ > Qd,0), creating more stranded assets in the long run. Figure 4 shows that, at calibrated emission intensity (ad = 0.198), the clean-only path dominates in welfare whenever multiple equilibria arise. However, at a counterfactually low pollution intensity (ad = 0.0198, one-tenth of calibrated), the planner may prefer some temporary dirty innovation when the clean sector starts small, because investment complementarities in the (larger) dirty sector generate higher short-run consumption growth that outweighs the smaller pollution cost.

Why does a Pigouvian carbon tax fail to coordinate the transition, and what instruments can succeed?

A Pigouvian tax changes the marginal cost of emissions and affects relative profitability, but it does not fully control relative innovation profitability because strategic complementarities within a sector persist: total innovation in a sector still raises marginal returns for all firms in it, and the complementarity can dominate the tax effect. An emission cap, by contrast, fixes the quantity of dirty output (given the Leontief emissions-to-output structure), which mutes the complementarity: expanding dirty productivity no longer pays if the quantity cap is binding. A minimum clean revenue guarantee sets a floor on clean firms’ profits that controls relative profitability directly without taxing the dirty sector. A dirty R&D tax raises the marginal cost of dirty research, shifting the innovation regime border and eliminating dirty equilibrium paths. A contingent super-Pigouvian carbon tax (above the social cost of carbon) that activates only when the economy innovates in the dirty sector also works. All of these require policy commitment over the duration of the multiple-equilibria region; without commitment they fail.

How does the paper relate to and differ from Acemoglu et al. (2012)?

The model starts from Acemoglu et al. (2012) but reaches a qualitatively different policy conclusion. Acemoglu et al. (2012) acknowledge the multiplicity of equilibria in their appendix but restrict their analysis to initial conditions and policies that make equilibrium unique, concluding that a Pigouvian tax combined with an R&D subsidy is sufficient for the optimal transition. This paper shows that when forward-looking expectations and investment complementarities are fully accounted for, the coordination failure is separate from the pollution and monopoly externalities, and a Pigouvian tax — even when optimal — does not resolve it. The paper also differs by using infinite patent length (vs. one-period) and an integrated labor market (vs. segmented), though Appendices OA3.2 and OA4 show the qualitative conclusions are robust to these modeling choices.

How does the paper relate to the stranded asset literature?

Van der Ploeg and Rezai (2020) and Kalkuhl et al. (2020) explain asset stranding through policy uncertainty, distributional effects, or disordered transition. This paper provides a complementary explanation: excess dirty investment and asset stranding can occur even under a committed, fully optimal Pigouvian tax — not because of uncertainty, but because of rational coordination failure. Firms continue investing in polluting technologies, knowing a clean steady state is inevitable, because strategic complementarities make the dirty sector temporarily attractive when the dirty sector is larger. The amount of stranded assets varies across equilibria: the longer the delay in clean innovation, the larger the accumulated stock of ultimately worthless dirty technology capital (Qd,∞ > Qd,0).

What role do knowledge spillovers and cross-sectoral knowledge externalities play?

The baseline model assumes knowledge spillovers within sectors (quality in sector j benefits from sector-wide average quality Qj). The Online Appendix (OA3) shows that inter-sectoral knowledge spillovers (parameter χ) do not affect complementarities at all, because knowledge stock is predetermined and current rival innovation cannot affect one’s own value through the knowledge channel. Learning-by-doing production spillovers (parameter ε) strengthen complementarities. The general condition for self-fulfilling prophecies in the extended model is ψ > max{0, -η}, where ψ = (1+ε)(σ-1)(1-α)/(1-ωα) - 1 and η measures own-sector knowledge advantage in innovation productivity. The baseline model (ε=0, ω=1) gives ψ = σ-2, recovering the σ > 2 condition.

What are the policy implications and their scope conditions?

The main policy implication is that a single Pigouvian carbon tax is insufficient for the optimal green transition even if credibly committed to; a coordination device is necessary as a second instrument. Scope conditions: (1) This conclusion holds whenever σ > 1 under optimal industry policy (which internalizes monopoly and spillover externalities) — the threshold is lower than σ > 2 in the unregulated economy. (2) The preferred coordination device (revenue guarantee, emission cap, dirty R&D tax, or contingent super-Pigouvian tax) depends on institutional constraints. (3) All coordination devices require policy commitment for the duration of the multiple-equilibria region. (4) The conclusion that the clean-only path is welfare-superior when multiple equilibria arise holds at calibrated emission intensity; at very low pollution intensity the planner might prefer some temporary dirty innovation. (5) The analysis abstracts from uncertainty, heterogeneous beliefs, large players, multiple abatement options, and physical capital — directions for future quantitative work.

What is the role of impatience (ρ) and patent length in the size of the coordination problem?

Proposition 3 shows that the overlap (the range of initial conditions admitting multiple equilibria) decreases with impatience ρ. When ρ is large, investors discount future profits heavily, limiting how far ahead expectations can drive current investment choices. In the limit of infinite impatience, only current profit matters and the game collapses to a static one-period coordination problem (Section 3.3). Shorter patent length, modeled as a Poisson patent infringement risk ι (OA3.2), acts identically to higher ρ in the equilibrium dynamics: the dynamics of the model with infringement risk ι are identical to the baseline with ρ replaced by ρ + ι. Hence shorter patents shrink the overlap, and policy must subsidize R&D to compensate for the excessively short investment horizon.

Key Concepts

Strategic investment complementarity: Within-sector R&D is a strategic complement when σ > 2: one firm’s innovation raises the return to other firms’ innovation in the same sector, because the demand shift effect (innovation increases sector expenditure share) outweighs the business-stealing effect (innovation dilutes rivals’ profit share). This is not a knowledge spillover but a demand externality operating through the market size of the innovating sector.

Overlap: The range of initial clean market shares θc,0 from which both the clean and dirty corner steady states can be reached in a rational expectations equilibrium. The overlap exists if and only if σ > 2 in the unregulated economy (σ > 1 under optimal industry policy), grows with the substitution elasticity σ, and shrinks with impatience ρ or shorter patent length.

Market valuation share (mc): The share of the clean sector in the total marginal value of innovation across sectors, defined as mc = Qcλc / (Qcλc + Qdλd). When mc > 1/2, the economy is in the clean-only innovation regime; when mc < 1/2, in the dirty-only regime; when mc = 1/2, simultaneous research is active. Because mc is a forward-looking, continuous variable, it captures investors’ collective expectation about future market conditions and directly determines the direction of technical change.

Self-fulfilling prophecy (in innovation): An equilibrium in which investors’ shared belief about the future direction of innovation is rational precisely because all investors, acting on that belief, make it come true. If all investors expect the dirty sector to remain large, they concentrate R&D there, the dirty sector grows, and the belief is confirmed. The same logic applies to clean beliefs. In the paper’s context, self-fulfilling prophecies extend to the speed of transition: even if firms agree the economy will eventually go clean, pessimistic beliefs about timing can rationally support periods of dirty innovation before the switch.

Delayed transition: An equilibrium path in which the economy ultimately converges to the clean steady state but investors temporarily concentrate R&D in the dirty sector before switching permanently to clean. The delay generates more stranded dirty assets (a higher terminal dirty technology stock Qd,∞) and higher short-run growth (via dirty-sector complementarities) relative to the fast-transition path. Multiple delayed paths may coexist, distinguished by the length of the dirty innovation period and the amount of accumulated dirty capital.

Coordination device: A policy instrument that directly controls the relative profitability of clean versus dirty innovation, thereby eliminating the undesired equilibrium paths without relying solely on price incentives. The paper identifies four classes: (1) minimum clean revenue guarantee, (2) emission cap (quantity-based), (3) dirty R&D tax or clean R&D subsidy, and (4) contingent super-Pigouvian carbon tax. All require government commitment for the duration of the multiple-equilibria region. A clean R&D subsidy is inferior because it distorts labor allocation toward innovation.

Stranded assets: In this paper, the dirty technology capital that becomes economically worthless in the clean steady state. The amount of stranding is determined by the dirty technology stock at the moment the economy permanently switches to clean innovation (Qd,∞). Different equilibrium paths — fast vs. delayed transitions — imply different terminal dirty stocks and hence different quantities of stranded assets. Excess stranding relative to the social optimum is a welfare cost of coordination failure.