The question
What is the optimal carbon tax when countries do not cooperate — and how does trade change it?

Simple formulas for the Social Cost of Carbon (SCC) exist, but only for a single global economy. In reality, countries set carbon taxes unilaterally, and international transfers to enforce a global optimum have not materialised after thirty years of climate summits. This paper derives tractable closed-form solutions for the SCC, optimal carbon taxes, and abatement policies in a multi-country differential game with endogenous international trade — and shows that trade significantly redistributes the SCC across countries.

Three main contributions
①
Closed-form SCC in a Nash game
The first closed-form solution for the SCC, optimal carbon taxes, and abatement in a non-cooperative multi-country game with trade. Nests DICE as a special case and replicates its numerical results analytically.
②
Trade decomposes the SCC
The country-specific SCC splits cleanly into a domestic part and a foreign part driven by trade volumes. On average 24% of the global SCC is generated by international trade linkages.
③
North-South redistribution
Heterogeneous damages and trade shift the SCC from high-damage southern countries to low-damage northern ones — the north implements a higher tax than in autarky, reducing global emissions even without cooperation.
The model — key equations
Multi-country IAM with endogenous trade and differential game
Output and capital accumulation
\[Y_{nt} = A_n K_{nt},\qquad dK_{nt} = K_{nt}\bigl[g_n(t,\chi_{nt},\mu_{nt}) - \xi_n T^{\mathrm{at}}_t\bigr]\,dt\]
Linear (AK) production with TFP \(A_n\). The growth rate \(g_n\) depends on the consumption share \(\chi_n = C_n/Y_n\), abatement \(\mu_n\), and investment adjustment costs. The term \(\xi_n T^{\mathrm{at}}_t\) captures the growth-rate impact of global temperature: higher temperatures erode growth at rate \(\xi_n\), calibrated to match DICE level damages.
Budget constraint and Cobb-Douglas trade
\[Y_{nt} = I_{nt} + A_{nt} + C_{nt}, \qquad \mathcal{C}_{nt} = \prod_{\ell=1}^{N}\!\bigl(C^n_{\ell t}\bigr)^{\beta^n_\ell}\]
Output is split between investment, abatement expenditure, and consumption. Country \(n\) derives utility from a Cobb-Douglas bundle of all countries' goods; the weight \(\beta^n_\ell\) measures how much country \(n\) values the good produced in country \(\ell\). A balanced trade account ensures \(\mathrm{EX}_{nt}=\mathrm{IM}_{nt}\).
Abatement costs and emissions
\[A_{nt} = a_n(t)\,\mu_{nt}^{b_n} Y_{nt}, \qquad E_t = \sum_{n=1}^{N}\!\bigl[Y_{nt}\,\sigma_n(t)(1-\mu_{nt}) + E^{\mathrm{land}}_n(t)\bigr]\]
Abatement costs are convex in the emission control rate \(\mu_n \in [0,d_n]\) and proportional to output; \(a_n(t)\) declines with technology. Carbon intensity \(\sigma_n(t)\) also falls exogenously over time. Global emissions feed into a three-layer carbon cycle and two-layer temperature system (following DICE).
Logarithmic preferences and Nash equilibrium
\[J^\pi_n(t,x) = \int_t^\infty e^{-\delta_n(s-t)} \log \mathcal{C}_{ns}\,ds, \qquad J_n(0,x) = \sup_{\pi_n} J^{(\pi_n|\pi^*_{-n})}_n(0,x)\]
Each region maximises discounted log utility over consumption bundles. An open-loop Nash equilibrium obtains when no country can gain by deviating given the optimal strategies of all others. The HJB system is solved in closed form via a separation-of-variables conjecture.
Key result — SCC decomposition
Trade splits the social cost of carbon into a domestic and a foreign part

The closed-form solution delivers a country-specific SCC that decomposes naturally into what the country suffers at home and what it internalises through trade:

SCC decomposition
Total SCC
\[\mathrm{SCC}_{nt} = \frac{Y^*_{nt}\chi^*_{nt}}{M^{\mathrm{PI}}\,\delta_n}\sum_{\ell=1}^{N}\beta^\ell_n\xi_\ell \cdot \Psi_n\]
Proportional to domestic GDP \(Y^*_{nt}\). The climate factor \(\Psi_n\) aggregates the carbon-cycle and temperature-system parameters (see below). The SCC is independent of abatement costs \(a_n\) and carbon intensity \(\sigma_n\).
=
Domestic part
\[\mathrm{SCC}^{\mathrm{dom}}_{nt} = \frac{C^n_{nt}\,\xi_n}{M^{\mathrm{PI}}\,\delta_n}\cdot\Psi_n\]
Driven solely by own consumption \(C^n_{nt}\) and own damage parameter \(\xi_n\). Internalises only damages to the domestic economy — the standard autarky result.
+
Foreign part
\[\mathrm{SCC}^{\mathrm{for}}_{nt} = \frac{1}{M^{\mathrm{PI}}\,\delta_n}\!\sum_{\ell\neq n}\!P^\ell_n C^n_{\ell t}\,\xi_\ell\cdot\Psi_n\]
Driven by the value of imports \(P^\ell_n C^n_{\ell t}\) from each partner times that partner's damage \(\xi_\ell\). Larger trade volumes force countries to internalise more foreign climate damage.
North-South example
How heterogeneous damages and trade redistribute the SCC

Consider two regions with heterogeneous damage parameters \(\xi_1 < \xi_2\): the low-damage north (carbon-intensive, industrialised) and the high-damage south. The SCC decomposition implies a striking redistribution once trade is introduced.

Global North — low damage (\(\xi_1 < \xi_2\))
In autarky the north ignores the south's higher climate vulnerability entirely. With trade, the foreign SCC component forces the north to account for \(\xi_2 > \xi_1\) in proportion to its import value \(P^2_1 C^1_{2t}\). The north's optimal carbon tax rises above the autarky level. Since the north is the more carbon-intensive region, this directly reduces global CO₂ emissions — even without any international agreement.
Tax vs. autarky: higher
Abatement: more stringent
Global South — high damage (\(\xi_2 > \xi_1\))
In autarky the south bears the full burden of its high damage parameter in its SCC. With trade, the south also internalises the north's lower damage parameter \(\xi_1\) via its import share — partially offsetting its high domestic SCC. The net effect is a shift of the SCC from south to north: the south's relative share of the global SCC falls, while the north's rises. Global emissions nonetheless fall because it is the north that abates more.
SCC share vs. autarky: lower
Global emissions: reduced
Note: The global SCC is invariant to trade — only its regional distribution changes. The key mechanism is that trade volumes act as weights that determine how much each country internalises foreign climate damages.
Key results — SCC and abatement
What the closed-form solution reveals
SCC is proportional to GDP — always
The country-specific SCC is proportional to domestic GDP \(Y^*_{nt}\) in both cooperative and non-cooperative games, and regardless of trade volumes. This mirrors Golosov et al. (2014) but now holds in a full multi-country, open-economy setting.
More countries → less abatement, same SCC
As the number of non-cooperating countries grows, the individually optimal abatement approaches zero (Prisoner's dilemma). Yet the initial SCC remains unchanged — breaking the tight SCC–abatement link that holds in cooperative settings.
Abatement costs don't affect the SCC
The SCC is determined solely by the climate-damage and macroeconomic system, not by \(a_n\) or \(b_n\). Cheaper abatement technology raises the quantity of abatement but leaves the shadow value of carbon — and hence the Pigouvian tax — unchanged.
Trade raises global abatement — without a deal
When the north's economy is more CO₂-intensive, the trade-induced rise in the northern carbon tax reduces global emissions even in the absence of any international climate agreement. Trade acts as an implicit coordination mechanism.
Why it matters
Practical relevance for policy makers, negotiators, and carbon-market practitioners
Carbon pricing without global consensus
The model shows that unilateral carbon taxes can still reduce global emissions if trading partners internalise each other's damages. This justifies border carbon adjustments (e.g., the EU CBAM) as a mechanism to make trading partners account for foreign climate costs — even without a global climate treaty.
A tractable formula for national carbon taxes
Governments and regulatory bodies can implement the rule directly: set \(\tau^*_n \propto Y_n \cdot \sum_\ell \beta^\ell_n \xi_\ell \cdot \Psi_n\). The formula is transparent, auditable, and can be updated as trade volumes or damage estimates evolve — unlike black-box integrated assessment models.
Equity and climate negotiations
The 24% trade-mediated share of the global SCC quantifies what is at stake in trade-linked climate negotiations. High-damage regions (Global South) can point to a concrete, model-consistent number when arguing that northern countries should bear a larger share of the global carbon bill — driven by trade alone, not transfers.
Investors and stranded-asset risk
The model predicts that non-cooperative optimal carbon taxes are 30–50% below the cooperative optimum. This gap is a direct measure of policy ambition deficit: investors in carbon-intensive sectors can use the SCC difference to gauge the additional tightening risk if international coordination improves — a key input for climate scenario analysis and stress testing.
JEL Q5, Q54, C73  ·  Keywords: social cost of carbon · non-cooperative game · differential game · international trade · carbon tax · DICE