Nutrition and Climate Policies in the European Union: Friends or Enemies?

Abstract

The European Union (EU) Green Deal and its Farm to Fork Strategy are intended to promote sustainable food systems to achieve EU climate-neutrality by 2050. The Farm to Fork action plan also foresees the introduction of a harmonized mandatory front-of-pack nutrition labelling scheme in 2023. The EU countries have yet to reach agreement on the nutrition labelling scheme, which will also have environmental impacts. This article raises the question of whether at the European level, countries should seek agreements on both climate mitigation and nutrition policies (full agreement as in the case of the Green Deal) or should negotiate separate climate and nutrition policy agreements (as for the nutritional labelling). To address this question, this paper develops a game-theoretic model with multiple countries where each country implements a climate policy and a nutrition policy. We compare the consequences in terms of total emissions, the level of the nutrition policy and the welfare under different institutional arrangements of a non-cooperative equilibrium, full agreement, and three alternative agreements. Our results show in particular that full agreement always leads to the lowest total emissions at the expense of the level of nutrition policy in some cases. In an extension of our analysis, we show that agreements that include cooperation over nutrition policies do not necessarily imply formation of a larger coalition of signatory countries, even if a nutrition policy has positive or negative impacts on emissions.

Appendices

Appendices

A Second-Order Conditions

In Nash equilibrium, the Hessian matrix of the second derivative of the payoff function $H^{N}$ is given by:

$$\begin{aligned} H^{N}=\left( \begin{array}{cc} \frac{\partial ^{2}U_{i}}{\partial e_{i}^{2}} &{} \frac{\partial ^{2}U_{i}}{ \partial e_{i}\partial f_{i}} \\ \frac{\partial ^{2}U_{i}}{\partial f_{i}\partial e_{i}} &{} \frac{\partial ^{2}U_{i}}{\partial f_{i}^{2}} \end{array} \right) = \begin{pmatrix} B_{ee}-D_{EE} &{} -\alpha D_{EE} \\ -\alpha D_{EE} &{} \gamma A_{ff}-C_{ff}-\alpha ^{2}D_{EE} \end{pmatrix} \end{aligned}$$

(A1)

The first determinant of $H^{N}$, $D_{1}=B_{ee}-D_{EE}$, is negative by assumption a), and the second $D_{2}=Det(H^{N})=B_{ee}\left( \gamma A_{ff}-C_{ff}-\alpha ^{2}D_{EE}\right) -D_{EE}(\gamma A_{ff}-C_{ff})$ is positive by assumptions a) and b). Thus, $H^{N}$ is defined as positive, and $U_{i}$ is strictly concave. Then, there is a unique solution to the optimization program (2), $(e^{N},f^{N})$ defined by equations 5 and 6.

At the full agreement, the Hessian matrix of the second derivatives of the welfare function, $H^{C}$, is a symmetric matrix of size 2n with $\frac{ \partial ^{2}W}{\partial e_{i}^{2}}=B_{ee}-nD_{EE}$, $\frac{\partial ^{2}W}{ \partial e_{i}\partial e_{j}}=-nD_{EE}$, $\frac{\partial ^{2}W}{\partial e_{i}\partial f_{j}}=n\alpha D_{EE}$, $\frac{\partial ^{2}W}{\partial f_{i}^{2}}=\gamma A_{ff}-C_{ff}-n\alpha ^{2}D_{EE}$, and $\frac{\partial ^{2}W}{\partial f_{i}\partial f_{j}}=-n\alpha ^{2}D_{EE}\quad \forall i,\,j.$ Assumptions a) and b), ensure that all the eigenvalues of the matrix $H^{C}$ are negative; therefore, the welfare function is quasi-concave. As a result, there is a unique solution to the optimization program (23 ), $(e^{C},f^{C})$ defined by equations 24 and 25.

At the climate policy agreement, the Hessian matrix of the second derivatives of the welfare function is given by $H^{P}$, a symmetric matrix of size n with $\frac{\partial ^{2}W}{\partial e_{i}^{2}}=B_{ee}-nD_{EE}$, $\frac{\partial ^{2}W}{\partial e_{i}\partial e_{j}}=-nD_{EE}\quad \forall i,\,j.$ Assumptions a) and b) ensure that all the eigenvalues of the matrix $H^{P}$ are negative; therefore, the welfare function is quasi-concave. As a result, there is a unique solution to the optimization program (13), $(e^{P},f^{P})$ defined by equations 16 and 17.

B Proof of Proposition 1

Here, we investigate the reaction functions in Nash equilibrium.

The total differential of Equation (3) is:

$$\begin{aligned} (B_{ee}-D_{EE})de_{i}-\alpha D_{EE}df_{i}=D_{EE}dET_{-i} \end{aligned}$$

(B1)

The total differential of Equation (4) is:

$$\begin{aligned} -\alpha D_{EE}de_{i}+(\gamma A_{ff}-C_{ff}-\alpha ^{2}D_{EE})df_{i}=\alpha D_{EE}dET_{-i} \end{aligned}$$

(B2)

Equations (B1) and (B2) can be written in matrix form:

$$\begin{aligned} \begin{pmatrix} B_{ee}-D_{EE} &{} -\alpha D_{EE} \\ -\alpha D_{EE} &{} \gamma A_{ff}-C_{ff}-\alpha ^{2}D_{EE} \end{pmatrix} \times \begin{pmatrix} de_{i} \\ df_{i} \end{pmatrix} = \begin{pmatrix} D_{EE} \\ \alpha D_{EE} \end{pmatrix} dET_{-i} \end{aligned}$$

(B3)

$$\begin{aligned} \Leftrightarrow \begin{pmatrix} de_{i} \\ df_{i} \end{pmatrix} =\frac{1}{Det(H^{N})} \begin{pmatrix} \gamma A_{ff}-C_{ff}-\alpha ^{2}D_{EE} &{} \alpha D_{EE} \\ \alpha D_{EE} &{} B_{ee}-D_{EE} \end{pmatrix} \times \begin{pmatrix} D_{EE}dET_{-i} \\ \alpha D_{EE}dET_{-i} \end{pmatrix} \end{aligned}$$

(B4)

• (i). Equation (B4) leads to $\frac{de_{i}}{dET_{-i}}=\frac{(\gamma A_{ff}-C_{ff})D_{EE}}{Det(H^{N})}<0$, since $A_{ff}<0$, $D_{EE}>0$, and $Det(H^{N})>0$.

• (ii). Equation (B4) leads to $\frac{df_{i}}{dET_{-i}}=\frac{\alpha B_{ee}D_{EE}}{Det(H^{N})}$, with the sign depending on that of $\alpha$, since $B_{ee}<0$, $D_{EE}>0$, and $Det(H^{N})>0$.

• (iii). The first-order conditions reduce to $\gamma A_{f}(f_{i})-C_{f}(f_{i})=\alpha B_{e}(e_{i})$. The total differential of this equation leads to $\frac{de_{i}}{df_{i}}=\frac{\gamma A_{ff}-C_{ff}}{ \alpha B_{ee}}$ $\Leftrightarrow sgn(\frac{de_{i}}{df_{i}})=sgn(\alpha )$ since $A_{ff}<0$, $C_{ff}>0$, and $B_{ee}<0$.

C Proof of Proposition 2

Here, we investigate the reaction functions at the climate agreement solution. The total differential of Equation (14) is:

$$\begin{aligned} (B_{ee}-nD_{EE})de_{i}-n\alpha D_{EE}df_{i}=nD_{EE}dET_{-i} \end{aligned}$$

(C1)

The total differential of Equation (15) is:

$$\begin{aligned} -\alpha D_{EE}de_{i}+(\gamma A_{ff}-C_{ff}-\alpha ^{2}D_{EE})df_{i}=\alpha D_{EE}dET_{-i} \end{aligned}$$

(C2)

Equations (C1) and (C2) can be written in matrix form:

$$\begin{aligned} \begin{pmatrix} B_{ee}-nD_{EE} &{} -n\alpha D_{EE} \\ -\alpha D_{EE} &{} \gamma A_{ff}-C_{ff}-\alpha ^{2}D_{EE} \end{pmatrix} \times \begin{pmatrix} de_{i} \\ df_{i} \end{pmatrix} = \begin{pmatrix} nD_{EE} \\ \alpha D_{EE} \end{pmatrix} dET_{-i} \end{aligned}$$

(C3)

$$\begin{aligned} \Leftrightarrow \begin{pmatrix} de_{i} \\ df_{i} \end{pmatrix} =\frac{1}{Det(H^{P})} \begin{pmatrix} \gamma A_{ff}-C_{ff}-\alpha ^{2}D_{EE} &{} n\alpha D_{EE} \\ \alpha D_{EE} &{} B_{ee}-nD_{EE} \end{pmatrix} \times \begin{pmatrix} nD_{EE}dET_{-i} \\ \alpha D_{EE}dET_{-i} \end{pmatrix} \end{aligned}$$

(C4)

• (i). Equation (C4) leads to $\frac{de_{i}}{dET_{-i}}=\frac{n(\gamma A_{ff}-C_{ff})D_{EE}}{Det(H^{P})}<0$, since $A_{ff}<0$, $C_{ff}>0$, $D_{EE}>0$, and $Det(H^{P})>0$.

• (ii). Equation (C4) leads to $\frac{df_{i}}{dET_{-i}}=\frac{\alpha B_{ee}D_{EE}}{Det(H^{P})}$, with the sign depending on that of $\alpha$, since $B_{ee}<0$, $D_{EE}>0$, and $Det(H^{P})>0$.

• (iii). The first-order conditions reduce to $n(\gamma A_{f}(f_{i})-C_{f}(f_{i}))=\alpha B_{e}(e_{i})$. The total differential of this equation leads to $\frac{de_{i}}{df_{i}}=\frac{n(\gamma A_{ff}-C_{ff})}{ \alpha B_{ee}}$ $\Leftrightarrow sgn(\frac{de_{i}}{df_{i}})=sgn(\alpha )$ since $A_{ff}<0$, $C_{ff}>0$, and $B_{ee}<0$.

D Table of Results

Equilibria	With emissions from nutrition policy
No agreement	$B_{e}(e^{N})=D_{E}\left( n(e^{N}+\alpha f^{N})\right)$
	$\gamma A_{f}(f^{N})-C_{f}(f^{N})=\alpha D_{E}\left( n(e^{N}+\alpha f^{N})\right)$
Agreement on climate policy	$B_{e}(e^{P})=nD_{E}\left( n(e^{P}+\alpha f^{P})\right)$
	$\gamma A_{f}(f^{P})-C_{f}(f^{P})=\alpha D_{E}\left( n(e^{P}+\alpha f^{P})\right)$
Full agreement	$B_{e}(e^{C})=nD_{E}(n(e^{C}+\alpha f^{C}))$
	$\gamma A_{f}(f^{C})-C_{f}(f^{C})=\alpha nD_{E}(n(e^{C}+\alpha f^{C}))$
Agreement on nutrition policy	$B_{e}(e^{V})=D_{E}\left( n(e^{V}+\alpha f^{V})\right)$
	$\gamma A_{f}(f^{V})-C_{f}(f^{V})=\alpha nD_{E}\left( n(e^{V}+\alpha f^{V})\right)$
Separate agreements	$B_{e}(e^{S})=nD_{E}(n(e^{S}+\alpha f^{S}))$
	$\gamma A_{f}(f^{S})-C_{f}(f^{C})=0$

E Proof of Proposition 3

1.1 E.1 Part 1 of the Proof

We will use the method of proof by contradiction to compare the levels of the variables between the full agreement and the non-cooperative solution.

• Suppose that $\alpha >0$ and $e^{N}\le e^{C}$; this implies that

$$\begin{aligned} B_{e}(e^{N})\ge & {} B_{e}(e^{C})\quad \text { since }B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\ge nD_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (24) and (5)} \\\Rightarrow & {} \alpha D_{E}(n(e^{N}+\alpha f^{N}))\ge \alpha nD_{E}(n(e^{C}+\alpha f^{C})) \\\Rightarrow & {} G_{f}(f^{N})\ge G_{f}(f^{C})\quad \text { from Eq. (25) and (6) with }G(f)=\gamma A(f)-C(f) \\\Rightarrow & {} f^{N}\le f^{C}\,\text { since }G_{ff}(f)=\gamma A_{ff}(f)-C_{ff}(f)<0 \\\Rightarrow & {} n(e^{N}+\alpha f^{N})\le n(e^{C}+\alpha f^{C}) \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\le nD_{E}(n(e^{C}+\alpha f^{C})) \text { which is a contradiction; thus, we have }e^{N}>e^{C}\text {.}\\ e^{N}> & {} e^{C}\text { implies }B_{e}(e^{N})<B_{e}(e^{C})\quad \text { since } B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))<nD_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (24) and (5)} \\\Rightarrow & {} \alpha D_{E}(n(e^{N}+\alpha f^{N}))<\alpha nD_{E}(n(e^{C}+\alpha f^{C})) \\\Rightarrow & {} G_{f}(f^{N})<G_{f}(f^{C})\quad \text { from Eq. (25) and (6) with }G(f)=\gamma A(f)-C(f) \\\Rightarrow & {} f^{N}>f^{C}\,\text { since }G_{ff}<0 \\\Rightarrow & {} n(e^{N}+\alpha f^{N})>n(e^{C}+\alpha f^{C})\Rightarrow ET^{N}>ET^{C} \end{aligned}$$

• Suppose that $\alpha <0$ and $e^{N}\le e^{C}$; this implies that

$$\begin{aligned} B_{e}(e^{N})\ge & {} B_{e}(e^{C})\quad \text { since }B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\ge nD_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (24) and (5)} \\\Rightarrow & {} \alpha D_{E}(n(e^{N}+\alpha f^{N}))\le \alpha nD_{E}(n(e^{C}+\alpha f^{C})) \\\Rightarrow & {} G_{f}(f^{N})\le G_{f}(f^{C})\quad \text { from Eq. (25) and (6) with }G(f)=\gamma A(f)-C(f) \\\Rightarrow & {} f^{N}\ge f^{C}\,\text { since }G_{ff}<0 \\\Rightarrow & {} n(e^{N}+\alpha f^{N})\le n(e^{C}+\alpha f^{C}) \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\le D_{E}(n(e^{C}+\alpha f^{C}))\\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\le nD_{E}(n(e^{C}+\alpha f^{C})) \text { which is a contradiction; thus, we have }e^{N}>e^{C}\text {.}\\ e^{N}> & {} e^{C}\text { implies }B_{e}(e^{N})<B_{e}(e^{C})\quad \text { since } B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))<nD_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (24) and (5)} \\\Rightarrow & {} \alpha D_{E}(n(e^{N}+\alpha f^{N}))>\alpha nD_{E}(n(e^{C}+\alpha f^{C})) \\\Rightarrow & {} G_{f}(f^{N})>G_{f}(f^{C})\quad \text { from Eq. (25) and (6) with }G(f)=\gamma A(f)-C(f) \\\Rightarrow & {} f^{N}<f^{C}\,\text { since }G_{ff}<0 \\\Rightarrow & {} n(e^{N}+\alpha f^{N})>n(e^{C}+\alpha f^{C})\Rightarrow ET^{N}>ET^{C} \end{aligned}$$

We will now use the method of proof by contradiction to compare the levels of the variables between the full agreement and the climate policy agreement.

• Suppose that $\alpha >0$ and $e^{P}\ge e^{C}$; this implies that

$$\begin{aligned} B_{e}(e^{P})\le & {} B_{e}(e^{C})\quad \text { since }B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{P}+\alpha f^{P}))\le D_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (24) and (16)} \\\Rightarrow & {} D_{E}(n(e^{P}+\alpha f^{P}))\le nD_{E}(n(e^{C}+\alpha f^{C})) \\\Rightarrow & {} \alpha D_{E}(n(e^{P}+\alpha f^{P}))\le \alpha nD_{E}(n(e^{C}+\alpha f^{C})) \\\Rightarrow & {} G_{f}(f^{P})\le G_{f}(f^{C})\quad \text { from Eq. (25) and (17) with }G(f)=\gamma A(f)-C(f) \\\Rightarrow & {} f^{P}\ge f^{C}\,\text { since }G_{ff}<0 \\\Rightarrow & {} n(e^{P}+\alpha f^{P})\ge n(e^{C}+\alpha f^{C}) \\\Rightarrow & {} D_{E}(n(e^{P}+\alpha f^{P}))\ge nD_{E}(n(e^{C}+\alpha f^{C})) \text { which is a contradiction; thus, we have }e^{P}<e^{C}\text {.} \\ e^{P}< & {} e^{C}\text { implies }B_{e}(e^{P})>B_{e}(e^{C})\quad \text { since } B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{P}+\alpha f^{P}))>D_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (24) and (16)} \\\Rightarrow & {} n(e^{P}+\alpha f^{P})>n(e^{C}+\alpha f^{C}) \\\Rightarrow & {} ET^{P}>ET^{C} \\\Rightarrow & {} f^{P}>f^{C} \end{aligned}$$

• Suppose that $\alpha <0$ and $e^{P}\ge e^{C}$; this implies that

$$\begin{aligned} B_{e}(e^{P})\le & {} B_{e}(e^{C})\quad \text { since }B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{P}+\alpha f^{P}))\le D_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (24) and (16)} \\\Rightarrow & {} D_{E}(n(e^{P}+\alpha f^{P}))\le nD_{E}(n(e^{C}+\alpha f^{C}))\\\Rightarrow & {} \alpha D_{E}(n(e^{P}+\alpha f^{P}))\ge \alpha nD_{E}(n(e^{C}+\alpha f^{C})) \\\Rightarrow & {} G_{f}(f^{P})\ge G_{f}(f^{C})\quad \text { from Eq. (25) and (17) with }G(f)=\gamma A(f)-C(f) \\\Rightarrow & {} f^{P}\le f^{C}\,\text { since }G_{ff}<0 \\\Rightarrow & {} n(e^{P}+\alpha f^{P})\ge n(e^{C}+\alpha f^{C}) \\\Rightarrow & {} D_{E}(n(e^{P}+\alpha f^{P}))\ge nD_{E}(n(e^{C}+\alpha f^{C})) \text { which is a contradiction; thus, we have }e^{P}<e^{C}\text {.}\\ e^{P}< & {} e^{C}\text { implies }B_{e}(e^{P})>B_{e}(e^{C})\quad \text { since } B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{P}+\alpha f^{P}))>D_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (24) and (16)} \\\Rightarrow & {} n(e^{P}+\alpha f^{P})>n(e^{C}+\alpha f^{C})\Rightarrow ET^{P}>ET^{C} \\\Rightarrow & {} f^{P}<f^{C} \end{aligned}$$

We will now use the method of proof by contradiction to compare the levels of the variables between the climate policy agreement and the non-cooperative solution.

• Suppose that $\alpha >0$ and $e^{N}\le e^{P}$; this implies that

$$\begin{aligned} B_{e}(e^{N})\ge & {} B_{e}(e^{P})\quad \text { since }B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\ge nD_{E}(n(e^{P}+\alpha f^{P}))\quad \text { from Eq. (16) and (5)} \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\ge D_{E}(n(e^{P}+\alpha f^{P})) \\\Rightarrow & {} \alpha D_{E}(n(e^{N}+\alpha f^{N}))\ge \alpha nD_{E}(n(e^{P}+\alpha f^{P})) \\\Rightarrow & {} G_{f}(f^{N})\ge G_{f}(f^{P})\quad \text { from Eq. (17) and (6) } \\ \text {with }G(f)= & {} \gamma A(f)-C(f) \\\Rightarrow & {} f^{N}\le f^{P}\,\text { since }G_{ff}<0 \\\Rightarrow & {} n(e^{N}+\alpha f^{N})\le n(e^{P}+\alpha f^{P})\text { which is a contradiction } \\ \text {with the condition }D_{E}(n(e^{N}+\alpha f^{N}))\ge & {} D_{E}(n(e^{P}+\alpha f^{P}))\text {, } \\ \text {thus we have }e^{N}> & {} e^{P}\text {.} \\ e^{N}> & {} e^{P}\text { implies }B_{e}(e^{N})<B_{e}(e^{P})\Leftrightarrow \\ D_{E}(n(e^{N}+\alpha f^{N}))< & {} nD_{E}(n(e^{P}+\alpha f^{P})) \\ \text {Let suppose that }n(e^{N}+\alpha f^{N}))< & {} n(e^{P}+\alpha f^{P}) \\\Rightarrow & {} f^{N}<f^{P}\Leftrightarrow G(f^{N})>G(f^{P})\Leftrightarrow \\ \alpha D_{E}(n(e^{N}+\alpha f^{N}))> & {} \alpha D_{E}(n(e^{P}+\alpha f^{P})) \\\Rightarrow & {} n(e^{N}+\alpha f^{N})>n(e^{P}+\alpha f^{P})\text { which is a contradiction,} \\ \text { thus we have }n(e^{N}+\alpha f^{N}))> & {} n(e^{P}+\alpha f^{P})\Leftrightarrow ET^{N}>ET^{P} \\ \text {Let us suppose that }f^{N}> & {} f^{P} \\\Rightarrow & {} G_{f}(f^{N})<G_{f}(f^{P})\Leftrightarrow \alpha D_{E}(n(e^{N}+\alpha f^{N}))<\alpha D_{E}(n(e^{P}+\alpha f^{P})) \\\Rightarrow & {} n(e^{N}+\alpha f^{N})<n(e^{P}+\alpha f^{P})\text { which is a contradiction; } \\ \text { thus, we have }f^{N}< & {} f^{P} \end{aligned}$$

• Suppose that $\alpha <0$ and $e^{N}\le e^{P}$; this implies that

$$\begin{aligned} B_{e}(e^{N})\ge & {} B_{e}(e^{P})\quad \text { since }B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\ge nD_{E}(n(e^{P}+\alpha f^{P}))\quad \text { from Eq. (16) and (5)} \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\ge D_{E}(n(e^{P}+\alpha f^{P})) \\\Rightarrow & {} \alpha D_{E}(n(e^{N}+\alpha f^{N}))\le \alpha nD_{E}(n(e^{P}+\alpha f^{P})) \\\Rightarrow & {} G_{f}(f^{N})\le G_{f}(f^{P})\quad \text { from Eq. (17) and (6) } \\ \text {with }G(f)= & {} \gamma A(f)-C(f) \\\Rightarrow & {} f^{N}\ge f^{P}\,\text { since }G_{ff}<0 \\\Rightarrow & {} n(e^{N}+\alpha f^{N})\le n(e^{P}+\alpha f^{P})\text { which is a contradiction } \\ \text {with the condition }D_{E}(n(e^{N}+\alpha f^{N}))\ge & {} D_{E}(n(e^{P}+\alpha f^{P}))\text {, } \\ \text {thus we have }e^{N}> & {} e^{P}\text {.} \\ e^{N}> & {} e^{P}\text { implies }B_{e}(e^{N})<B_{e}(e^{P}) \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))<nD_{E}(n(e^{P}+\alpha f^{P})) \\ \text {Let suppose that }n(e^{N}+\alpha f^{N}))< & {} n(e^{P}+\alpha f^{P}) \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))<D_{E}(n(e^{P}+\alpha f^{P})) \\\Rightarrow & {} \alpha D_{E}(n(e^{N}+\alpha f^{N}))>\alpha D_{E}(n(e^{P}+\alpha f^{P})) \\\Rightarrow & {} G_{f}(f^{N})>G_{f}(f^{P}) \\\Rightarrow & {} f^{N}<f^{P} \\\Rightarrow & {} n(e^{N}+\alpha f^{N})>n(e^{P}+\alpha f^{P})\text { which is a contradiction,} \\ \text { thus we have }n(e^{N}+\alpha f^{N}))> & {} n(e^{P}+\alpha f^{P})\Rightarrow ET^{N}>ET^{P}\\ \text {Let us suppose that }f^{N}\le & {} f^{P} \\\Rightarrow & {} G_{f}(f^{N})\ge G_{f}(f^{P})\Leftrightarrow \alpha D_{E}(n(e^{N}+\alpha f^{N}))\ge \alpha D_{E}(n(e^{P}+\alpha f^{P})) \\\Rightarrow & {} D_{E}(n(e^{N}+\alpha f^{N}))\le D_{E}(n(e^{P}+\alpha f^{P}))\\\Rightarrow & {} n(e^{N}+\alpha f^{N})\le n(e^{P}+\alpha f^{P})\text { which is a contradiction; } \\ \text { thus, we have }f^{N}> & {} f^{P} \end{aligned}$$

1.2 E.2 Part 2 of the Proof

1.2.1 E.2.1 For f

From first-order conditions on f ((6), (17), (25) and (11)) and

• if $\alpha >0$, we obtain that $\gamma A_{f}(f^{k})-C_{f}(f^{k})>\gamma A_{f}(f^{S})-C_{f}(f^{S})=0$, for $k=N,\,P,\,C.$

  Since $G_ {ff}(f)<0$, $f^k<f^S$ and $0<\alpha f^k<\alpha f^S$, with $k=N,\,P,\,C.$

• if $\alpha <0$, we obtain that $\gamma A_{f}(f^{k})-C_{f}(f^{k})< \gamma A_{f}(f^{S})-C_{f}(f^{S})=0$, for $k=N,\,P,\,C.$

  Since $G_ {ff}(f)<0$, $f^k>f^S$ and $\alpha f^k<\alpha f^S<0$, with $k=N,\,P,\,C.$

We obtain that

$$\begin{aligned} \begin{array}{rr} f^{C}>f^{N}>f^{P}>f^{S} &{} \quad \text { when }\alpha<0 \\ f^{C}<f^{N}<f^{P}<f^{S} &{} \quad \text { when }\alpha >0 \end{array} \end{aligned}$$

1.2.2 E.2.2 For e

We will use the method of proof by contradiction to compare the levels of the variables between different institutional arrangements.

Suppose that $e^{k}\le e^{S}$ for $k=P,\,C$; this implies that

$$\begin{aligned} B_{e}(e^{k})\ge & {} B_{e}(e^{S})\quad \text { since }B_{ee}\le 0 \\ \Rightarrow D_{E}(n(e^{k}+\alpha f^{k}))\ge & {} D_{E}(n(e^{S}+\alpha f^{S}))\quad \text { from Eq. (24), (16) and (9)} \\ \Rightarrow e^{k}+\alpha f^{k}\ge & {} e^{S}+\alpha f^{S} \\ \text {But }\alpha f^{k}<\alpha f^{S},&\text {then }&e^{k}>e^{S}\text { which is a contradiction. Thus, we have }e^{C}>e^{S}\text { and }e^{P}>e^{S}. \end{aligned}$$

Since $e^{N}>e^{P}$ and $e^{C}>e^{P}$, we obtain $e^{S}<e^{P}<e^{C}<e^{N},\, \forall \alpha$.

1.2.3 E.2.3 For ET

From the ranking of individual emissions $e^{S}<e^{P}<e^{C}$, we obtain:

$$\begin{aligned} B_{e}(e^{S})= & {} nD_{E}(n(e^{S}+\alpha f^{S}))>nD_{E}(n(e^{P}+\alpha f^{P}))=B_{e}(e^{P}) \\ B_{e}(e^{S})= & {} nD_{E}(n(e^{S}+\alpha f^{S}))>nD_{E}(n(e^{C}+\alpha f^{C}))=B_{e}(e^{C}) \end{aligned}$$

which imply $ET^{S}>$ $ET^{P}$ and $ET^{S}>ET^{C}$, respectively.

We will use the method of proof by contradiction to compare $ET^{N}$ and $ET^{S}$.

When $\alpha <0$, suppose that $ET^{N}\le ET^{S}$. This implies that $\alpha f^{N}\ge \alpha f^{S}$ since $e^{S}<e^{N}$. This in turn implies that $f^{N}\le f^{S}$, which is not true (see part (2) of the proposition). Thus, we have $ET^{N}>ET^{S}$.

When $\alpha >0$, suppose that $ET^{N}\le ET^{S}$. This implies that $\alpha f^{N}\ge \alpha f^{S}$ since $e^{S}<e^{N}$. This in turn implies that $f^{N}\ge f^{S}$, which is not true (see part (2) of the proposition). Thus, we have $ET^{N}>ET^{S}$.

1.3 E.3 Part 3 of the Proof

Let $G(f)=\gamma A(f)-C(f)$.

In the case of the agreement on nutrition policy, the first-order conditions are given by:

$$\begin{aligned} B_{e}(e^{V})=D_{E}(n(e^{V}+\alpha f^{V})) \end{aligned}$$

(E1)

$$\begin{aligned} G_{f}(f^{V})=\alpha nD_{E}(n(e^{V}+\alpha f^{V})) \end{aligned}$$

(E2)

Recall that the first-order conditions for the Nash equilibrium are:

$$\begin{aligned} B_{e}(e^{N})&=D_{E}(n(e^{N} +\alpha f^{N})) \end{aligned}$$

(E3)

$$\begin{aligned} G_{f}(f^{N})&=\alpha D_{E}(n(e^{N} +\alpha f^{N})) \end{aligned}$$

(E4)

The first-order conditions for the full agreement are:

$$\begin{aligned} B_{e}(e^{C})&=nD_{E}(n(e^{C}+\alpha f^{C})) \end{aligned}$$

(E5)

$$\begin{aligned} G_{f}(f^{C})&=\alpha nD_{E}(n(e^{C}+\alpha f^{C})) \end{aligned}$$

(E6)

1.3.1 E.3.1 For e

We will use the method of proof by contradiction to compare the levels of the variables between different institutional arrangements.

Suppose that $e^{V}\le e^{N}$; this implies that

$$\begin{aligned} B_{e}(e^{V})\ge & {} B_{e}(e^{N})\quad \text { since }B_{ee}\le 0 \\\Rightarrow & {} D_{E}(n(e^{V}+\alpha f^{V}))\ge D_{E}(n(e^{N}+\alpha f^{N}))\quad \text { from Eq. (E10) and (5)} \\\Rightarrow & {} n(e^{V}+\alpha f^{V})\ge n(e^{N}+\alpha f^{N}) \\ \text {If }\alpha> & {} 0\text {, then }\alpha nD_{E}(n(e^{V}+\alpha f^{V}))>\alpha D_{E}(n(e^{N}+\alpha f^{N})) \\\Rightarrow & {} G_{f}(f^{V})>G_{f}(f^{N})\quad \text { from Eq. (E11) and (6)} \\ \text {Since }G_{ff}(f)< & {} 0,f^{V}<f^{N}\text { and }e^{V}+\alpha f^{V}<e^{N}+\alpha f^{N}\text { which is a contradiction with } \\ n(e^{V}+\alpha f^{V})> & {} n(e^{N}+\alpha f^{N})\text {.} \\ \text {If }\alpha< & {} 0\text {, then }\alpha nD_{E}(n(e^{V}+\alpha f^{V}))<\alpha D_{E}(n(e^{N}+\alpha f^{N})) \\\Rightarrow & {} G_{f}(f^{V})<G_{f}(f^{N})\quad \text { from Eq. (E11) and (6)} \\ \text {Since }G_{ff}(f)< & {} 0,\text { }f^{V}>f^{N}\text { and }e^{V}+\alpha f^{V}<e^{N}+\alpha f^{N}\text { which is a contradiction with} \\ \text { }n(e^{V}+\alpha f^{V})> & {} n(e^{N}+\alpha f^{N})\text {.} \\ \text {Thus we have }e^{V}> & {} e^{N}. \end{aligned}$$

With the previous results, we obtain that

$$\begin{aligned} e^{S}<e^{P}<e^{C}<e^{N}<e^{V},\,\forall \alpha . \end{aligned}$$

1.3.2 E.3.2 For f

Since $e^{N}>e^{C}$, we have $e^{V}>e^{C}.$

• if $\alpha >0$, then $f^{V}<f^{C}$. On the contrary, suppose that $f^{V}>f^{C}$. Since $G_{ff}(f)<0$, this implies that $G_{f}(f^{V})<G_{f}(f^{C})$, and $D_{E}(n(e^{V}+\alpha f^{V}))<D_{E}(n(e^{C}+\alpha f^{C}))$ from Eq. (E2) and (17). Thus, we have $e^{V}+\alpha f^{V}<e^{C}+\alpha f^{C}$, which is a contradiction with $e^{V}>e^{C}$ and $f^{V}>f^{C}$. Thus, we have $f^{V}<f^{C}$.

• if $\alpha <0$, then $f^{V}>f^{C}$. On the contrary, suppose that $f^{V}<f^{C}$. Since $G_{ff}(f)<0$, this implies that $G_{f}(f^{V})>G_{f}(f^{C})$, and $D_{E}(n(e^{V}+\alpha f^{V}))<D_{E}(n(e^{C}+\alpha f^{C}))$ from Eq. (E2) and (17) because $\alpha <0$. Thus, we have $e^{V}+\alpha f^{V}<e^{C}+\alpha f^{C}$, which is a contradiction with $e^{V}>e^{C}$and $f^{V}<f^{N}$. Thus, we have $f^{V}>f^{C}$.

With the previous results, we obtain that

$$\begin{aligned} \begin{array}{rr} f^{V}>f^{C}>f^{N}>f^{P}>f^{S} &{} \quad \text { when }\alpha<0 \\ f^{V}<f^{C}<f^{N}<f^{P}<f^{S} &{} \quad \text { when }\alpha >0 \end{array} \end{aligned}$$

1.3.3 E.3.3 For ET

Equation (E1) and (5) imply $D_{E}(n(e^{V}+\alpha f^{V}))<D_{E}(n(e^{N}+\alpha f^{N}))$. Thus, we have $ET^{V}<ET^{N}$ $\forall \alpha$.

We now compare $ET^{V}$ and $ET^{C}$.

From the ranking of nutrition policy $f^{V}<f^{C}$ when $\alpha >0$, we obtain:

$$\begin{aligned} G_{f}(f^{V})> & {} G_{f}(f^{C})\text { since }G_{ff}(f)<0 \\ \alpha nD_{E}(n(e^{V}+\alpha f^{V}))> & {} \alpha nD_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (E2) and (17)} \\\Rightarrow & {} ET^{V}>ET^{C}\text { since }\alpha >0. \end{aligned}$$

This is also confirmed in the case of $\alpha <0$. From the ranking of nutrition policy $f^{V}>f^{C}$, when $\alpha <0$, we obtain:

$$\begin{aligned} G_{f}(f^{V})< & {} G_{f}(f^{C})\text { since }G_{ff}(f)<0 \\ \alpha nD_{E}(n(e^{V}+\alpha f^{V}))< & {} \alpha nD_{E}(n(e^{C}+\alpha f^{C}))\quad \text { from Eq. (E2) and (17)} \\\Rightarrow & {} ET^{V}>ET^{C}\text { since }\alpha <0. \end{aligned}$$

Thus, we have $ET^{V}>ET^{C}$ $\forall \alpha$.

The comparison of $ET^{V}$ with $ET^{P}$ and $ET^{S}$ is not possible in the case of general functional forms.

F Quadratic Model

The quadratic functional forms should respect the model assumptions:

• $\left| \alpha f_{i}\right| <e_{i}.$ When $\alpha >0$, this condition is equal to $\left( e_{i}-\alpha f_{i}\right) >0$. When $\alpha <0$, this condition is equal to $\left( e_{i}+\alpha f_{i}\right) >0$.

• $D_{E}=d(ET)>0$ and $D_{EE}=d>0$.

• $A_{f}=a_{1}-a_{2}f_{i}>0$ and $A_{ff}=-a_{2}<0$.

• $C_{f}=cf_{i}>0$ and $C_{ff}=c>0$.

• $B_{e}=b_{1}-b_{2}e_{i}>0$ and $B_{ee}=-b_{2}<0$.

1.1 F.1 Nash Equilibrium

The reactions functions are given by:

$$\begin{aligned} e_{i}&=\frac{b_{1}-d(\alpha f_{i}+E_{-i}+\alpha F_{-i})}{b_{2}+d} \nonumber \\ f_{i}&=\frac{\gamma a_{1}-\alpha d(e_{i}+E_{-i}+\alpha F_{-i})}{\gamma a_{2}+c+\alpha ^{2}d} \nonumber \\ f_{i}&=\frac{\gamma a_{1}+\alpha b_{2}e_{i}-\alpha b_{1}}{\gamma a_{2}+c} \end{aligned}$$

(F1)

The solution is given by:

$$\begin{aligned} e^{N}&=\frac{b_{1}(\gamma a_{2}+c)+\alpha dn(\alpha b_{1}-\gamma a_{1})}{ \gamma a_{2}(b_{2}+dn)+b_{2}(\alpha ^{2}dn+c)+cdn} \\ f^{N}&=\frac{\gamma a_{1}(b_{2}+dn)-\alpha b_{1}dn}{\gamma a_{2}(b_{2}+dn)+b_{2}(\alpha ^{2}dn+c)+cdn} \\ ET^{N}&=\frac{n\left[ \gamma (a_{1}\alpha b_{2}+a_{2}b_{1})+b_{1}c\right] }{ \gamma a_{2}(b_{2}+dn)+b_{2}(\alpha ^{2}dn+c)+cdn} \end{aligned}$$

The total payoff function then is given by:

$$\begin{aligned} W^{N}=nU^{N}=n\left[ \gamma \left( a_{1}f^{N}-\dfrac{a_{2}}{2} f^{N^{2}}\right) +\left( b_{1}e^{N}-\dfrac{b_{2}}{2}e^{N^{2}}\right) -\dfrac{ c}{2}f^{N^{2}}-\dfrac{d}{2}ET^{N^{2}}\right] \end{aligned}$$

1.1.1 F.1.1 Proof of Proposition 4 for the Nash Equilibrium

Here, we investigate how parameter $\gamma$ affects the variables in Nash equilibrium, defined by the Eq. F1.

The effect of parameter $\gamma$ on the level of nutrition policies is given by:

$$\begin{aligned} \frac{df^{N}}{d\gamma }=\frac{(nd+b_{2})\left[ nd(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+a_{1}c)+a_{1}b_{2}c\right] }{\left( \gamma a_{2}(b_{2}+dn)+b_{2}(\alpha ^{2}dn+c)+cdn\right) ^{2}}>0\quad \forall \alpha \end{aligned}$$

The numerator is positive thanks to the condition $A_{f}=a_{1}-a_{2}f^{N}>0$, which is given by $\left[ nd(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+a_{1}c)+a_{1}b_{2}c\right] >0.$

The effect of parameter $\gamma$ on $e^{N}$ depends on $A_{f}=a_{1}-a_{2}f^{N}>0$ and on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{de^{N}}{d\gamma }=\frac{-\alpha nd\left[ nd(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+a_{1}c)+a_{1}b_{2}c\right] }{\left( \gamma a_{2}(b_{2}+dn)+b_{2}(\alpha ^{2}dn+c)+cdn\right) ^{2}} \end{aligned}$$

Similarly, the effect of parameter $\gamma$ on the total level of emissions depends on $A_{f}=a_{1}-a_{2}f^{N}>0$ and on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{dE^{N}}{d\gamma }=\frac{\alpha nb_{2}\left[ nd(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+a_{1}c)+a_{1}b_{2}c\right] }{\left( \gamma a_{2}(b_{2}+dn)+b_{2}(\alpha ^{2}dn+c)+cdn\right) ^{2}} \end{aligned}$$

1.2 F.2 Full Agreement

The solution is given by:

$$\begin{aligned} e^{C}&=\frac{b_{1}(\gamma a_{2}+c)+\alpha dn^{2}(\alpha b_{1}-\gamma a_{1}) }{\gamma a_{2}(b_{2}+dn^{2})+b_{2}(\alpha ^{2}dn^{2}+c)+cdn^{2}} \nonumber \\ f^{C}&=\frac{\gamma a_{1}(b_{2}+dn^{2})-\alpha b_{1}dn^{2}}{\gamma a_{2}(b_{2}+dn^{2})+b_{2}(\alpha ^{2}dn^{2}+c)+cdn^{2}} \nonumber \\ ET^{C}&=\frac{n\left[ \gamma (a_{1}\alpha b_{2}+a_{2}b_{1})+b_{1}c\right] }{ \gamma a_{2}(b_{2}+dn^{2})+b_{2}(\alpha ^{2}dn^{2}+c)+cdn^{2}} \end{aligned}$$

(F2)

The total payoff function then is given by:

$$\begin{aligned} W^{C}=nU^{C}=n\left[ \gamma \left( a_{1}f^{C}-\dfrac{a_{2}}{2} f^{C^{2}}\right) +\left( b_{1}e^{C}-\dfrac{b_{2}}{2}e^{C^{2}}\right) -\dfrac{ c}{2}f^{C^{2}}-\dfrac{d}{2}ET^{C^{2}}\right] \end{aligned}$$

1.2.1 F.2.1 Proof of Proposition 4 for the Full Agreement

Here, we investigate how parameter $\gamma$ affects the variables at the full agreement, defined by the equations F2.

The effect of parameter $\gamma$ on the level of nutrition policies is given by:

$$\begin{aligned} \frac{df^{C}}{d\gamma }=\frac{(n^{2}d+b_{2})\left[ n^{2}d(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+a_{1}c)+a_{1}b_{2}c\right] }{\left( n^{2}d(\alpha ^{2}b_{2}+a_{2}\gamma +c)+b_{2}(a_{2}\gamma +c)\right) ^{2}} >0\quad \forall \alpha \end{aligned}$$

The numerator is positive thanks to the condition $A_{f}=a_{1}-a_{2}f^{C}>0$, which is given by $\left[ n^{2}d(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+a_{1}c)+a_{1}b_{2}c\right] >0.$

The effect of parameter $\gamma$ on $e^{C}$ depends on $A_{f}=a_{1}-a_{2}f^{C}>0$ and on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{de^{C}}{d\gamma }=\frac{-\alpha n^{2}d\left[ n^{2}d(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+a_{1}c)+a_{1}b_{2}c\right] }{\left( n^{2}d(\alpha ^{2}b_{2}+a_{2}\gamma +c)+b_{2}(a_{2}\gamma +c)\right) ^{2}} \end{aligned}$$

Similarly, the effect of parameter $\gamma$ on the total level of emissions depends on $A_{f}=a_{1}-a_{2}f^{C}>0$ and on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{dE^{C}}{d\gamma }=\frac{\alpha nb_{2}\left[ n^{2}d(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+a_{1}c)+a_{1}b_{2}c\right] }{\left( n^{2}d(\alpha ^{2}b_{2}+a_{2}\gamma +c)+b_{2}(a_{2}\gamma +c)\right) ^{2}} \end{aligned}$$

1.3 F.3 Agreement on climate policy

The solution is given by:

$$\begin{aligned} e^{P}&=\frac{b_{1}(\gamma a_{2}+c)+\alpha dn(\alpha b_{1}-\gamma a_{1}n)}{ \gamma a_{2}(b_{2}+dn^{2})+b_{2}(\alpha ^{2}dn+c)+cdn^{2}} \nonumber \\ f^{P}&=\frac{\gamma a_{1}(b_{2}+dn^{2})-\alpha b_{1}dn}{\gamma a_{2}(b_{2}+dn^{2})+b_{2}(\alpha ^{2}dn+c)+cdn^{2}} \nonumber \\ ET^{P}&=\frac{n\left[ \gamma (a_{1}\alpha b_{2}+a_{2}b_{1})+b_{1}c\right] }{ \gamma a_{2}(b_{2}+dn^{2})+b_{2}(\alpha ^{2}dn+c)+cdn^{2}} \end{aligned}$$

(F3)

The total payoff function then is given by:

$$\begin{aligned} W^{P}=nU^{P}=n\left[ \gamma \left( a_{1}f^{P}-\dfrac{a_{2}}{2} f^{P^{2}}\right) +\left( b_{1}e^{P}-\dfrac{b_{2}}{2}e^{P^{2}}\right) -\dfrac{ c}{2}f^{P^{2}}-\dfrac{d}{2}ET^{P^{2}}\right] \end{aligned}$$

1.3.1 F.3.1 Proof of Proposition 4 for the Climate Policy Agreement

Here, we investigate how parameter $\gamma$ affects the variables at the climate agreement, defined by the equations F3.

The effect of parameter $\gamma$ on the level of nutrition policies is given by:

$$\begin{aligned} \frac{df^{P}}{d\gamma }=\frac{(n^{2}d+b_{2})\left[ nd(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+na_{1}c)+a_{1}b_{2}c\right] }{\left( nd(\alpha ^{2}b_{2}+n\gamma a_{2}+nc)+b_{2}(\gamma a_{2}+c)\right) ^{2}} >0\quad \forall \alpha \end{aligned}$$

The numerator is positive thanks to the condition $A_{f}=a_{1}-a_{2}f^{P}>0$, which is given by $\left[ nd(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+na_{1}c)+a_{1}b_{2}c\right] >0.$

The effect of parameter $\gamma$ on $e^{P}$ depends on $A_{f}=a_{1}-a_{2}f^{P}>0$ and on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{de^{P}}{d\gamma }=\frac{-\alpha n^{2}d\left[ nd(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+na_{1}c)+a_{1}b_{2}c\right] }{\left( nd(\alpha ^{2}b_{2}+n\gamma a_{2}+nc)+b_{2}(\gamma a_{2}+c)\right) ^{2}} \end{aligned}$$

Similarly, the effect of parameter $\gamma$ on the total level of emissions depends on $A_{f}=a_{1}-a_{2}f^{P}>0$ and on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{dE^{P}}{d\gamma }=\frac{\alpha nb_{2}\left[ nd(\alpha ^{2}a_{1}b_{2}+\alpha a_{2}b_{1}+na_{1}c)+a_{1}b_{2}c\right] }{\left( nd(\alpha ^{2}b_{2}+n\gamma a_{2}+nc)+b_{2}(\gamma a_{2}+c)\right) ^{2}} \end{aligned}$$

1.4 F.4 Agreement on Nutrition Policy

The solution is given by:

$$\begin{aligned} e^{V}&=\frac{b_{1}(\gamma a_{2}+c+\alpha ^{2}n^{2}d)-\alpha dna_{1}\gamma }{ \gamma a_{2}(b_{2}+dn)+b_{2}(\alpha ^{2}dn^{2}+c)+cdn} \nonumber \\ f^{V}&=\frac{\gamma a_{1}(b_{2}+dn)-\alpha b_{1}dn^{2}}{\gamma a_{2}(b_{2}+dn)+b_{2}(\alpha ^{2}dn^{2}+c)+cdn} \nonumber \\ ET^{V}&=\frac{n\left[ \gamma (a_{1}\alpha b_{2}+a_{2}b_{1})+b_{1}c\right] }{ \gamma a_{2}(b_{2}+dn)+b_{2}(\alpha ^{2}dn^{2}+c)+cdn} \end{aligned}$$

(F4)

The total payoff function then is given by:

$$\begin{aligned} W^{V}=nU^{V}=n\left[ \gamma \left( a_{1}f^{V}-\dfrac{a_{2}}{2} f^{V^{2}}\right) +\left( b_{1}e^{V}-\dfrac{b_{2}}{2}e^{V^{2}}\right) -\dfrac{ c}{2}f^{V^{2}}-\dfrac{d}{2}ET^{V^{2}}\right] \end{aligned}$$

1.4.1 F.4.1 Proof of Proposition 4 for the Agreement on Nutrition Policy

Here, we investigate how parameter $\gamma$ affects the variables at the agreement on nutrition policy, defined by the equations F4.

The effect of parameter $\gamma$ on the level of nutrition policies is given by:

$$\begin{aligned} \frac{df^{V}}{d\gamma }=\frac{(nd+b_{2})[a_{1}b_{2}c+nd(ca_{1}+\alpha n(\alpha a_{1}b_{2}+a_{2}b_{1})]}{\left( nd(\alpha ^{2}b_{2}n+\gamma a_{2}+c)+b_{2}(\gamma a_{2}+c)\right) ^{2}}>0\quad \forall \alpha \end{aligned}$$

The numerator is positive thanks to the condition $A_{f}=a_{1}-a_{2}f^{V}>0$, which is given by $[a_{1}b_{2}c+nd(ca_{1}+\alpha n(\alpha a_{1}b_{2}+a_{2}b_{1})]>0.$

The effect of parameter $\gamma$ on $e^{V}$ depends on $A_{f}=a_{1}-a_{2}f^{P}>0$ and on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{de^{V}}{d\gamma }=\frac{-\alpha dn[a_{1}b_{2}c+nd(ca_{1}+\alpha n(\alpha a_{1}b_{2}+a_{2}b_{1})]}{\left( nd(\alpha ^{2}b_{2}n+\gamma a_{2}+c)+b_{2}(\gamma a_{2}+c)\right) ^{2}} \end{aligned}$$

Similarly, the effect of parameter $\gamma$ on the total level of emissions depends on $A_{f}=a_{1}-a_{2}f^{V}>0$ and on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{dE^{V}}{d\gamma }=\frac{\alpha b_{2}n[a_{1}b_{2}c+a_{1}cdn+n^{2}\alpha d(\alpha a_{1}b_{2}+a_{2}b_{1})]}{\left( nd(\alpha ^{2}b_{2}n+\gamma a_{2}+c)+b_{2}(\gamma a_{2}+c)\right) ^{2}} \end{aligned}$$

1.5 F.5 Separate Agreements

The solution is given by:

$$\begin{aligned} e^{S}&=\frac{b_{1}(c+\gamma a_{2})-n {{}^2} d\alpha \gamma a_{1}}{(b_{2}+dn^{2})(c+\gamma a_{2})} \nonumber \\ f^{S}&=\frac{\gamma a_{1}}{c+\gamma a_{2}} \nonumber \\ ET^{S}&=\frac{n\left[ b_{1}(c+\gamma a_{2})+b_{2}\alpha \gamma a_{1}\right] }{(b_{2}+dn^{2})(c+\gamma a_{2})} \end{aligned}$$

(F5)

The total payoff function then is given by:

$$\begin{aligned} W^{S}=nU^{S}=n\left[ \gamma \left( a_{1}f^{S}-\dfrac{a_{2}}{2} f^{S^{2}}\right) +\left( b_{1}e^{S}-\dfrac{b_{2}}{2}e^{S^{2}}\right) -\dfrac{ c}{2}f^{S^{2}}-\dfrac{d}{2}ET^{S^{2}}\right] \end{aligned}$$

1.5.1 F.5.1 Proof of Proposition 4 for Separate Agreements

Here, we investigate how parameter $\gamma$ affects the variables at the separate agreements equilibrium, defined by the equations F5.

The effect of parameter $\gamma$ on the level of nutrition policies is given by:

$$\begin{aligned} \frac{df^{S}}{d\gamma }=\frac{a_{1}c}{(c+\gamma a_{2})^{2}}>0\quad \forall \alpha \end{aligned}$$

The effect of parameter $\gamma$ on $e^{S}$ depends on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{de^{S}}{d\gamma }=\frac{-\alpha a_{1}cdn {{}^2} }{(b_{2}+dn^{2})(c+\gamma a_{2})^{2}} \end{aligned}$$

Similarly, the effect of parameter $\gamma$ on the total level of emissions depends on the sign of parameter $\alpha$:

$$\begin{aligned} \frac{dE^{S}}{d\gamma }=\frac{\alpha a_{1}cb_{2}n}{(b_{2}+dn^{2})(c+\gamma a_{2})^{2}} \end{aligned}$$

G Coalition Stability

1.1 G.1 Quadratic Case

Here, we consider the quadratic model exposed in section 5.1.

The timing of the coalition formation game is as follows:

$\mathbf {Stage\ 1}$

All countries choose simultaneously whether to join coalition $P\subseteq N$ or to remain a singleton player. Countries $i\in P$ are called signatories, and countries $j\notin P$ are called non-signatories.

$\mathbf {Stage\ 2}$

All non-signatories $j\notin P\;$choose their economic strategies (levels of climate and nutrition policies) to maximize their individual payoffs, and all signatories $i\in P$ do so to maximize the aggregate payoff to all coalition members. The choices of all the players are simultaneous.

Stage 1 is the cartel formation game, which originates from the industrial organization literature (d’Aspremont et al. 1983) and has been widely applied to study international environmental agreements (e.g., Barrett 1994; Carraro and Siniscalco 1993). This is an open-membership, single-coalition game because by assumption, i) membership in coalition P is open to all players, and ii) players can choose only between joining coalition P or remaining a singleton. Stage 1 requires the internal and externality stability conditions to be satisfied. Internal stability means that each player i that announces its intention to join coalition $P^{*}$ will have no incentive to (unilaterally) change its strategy by leaving coalition $P^{*}$. External stability means that each player j that announces its intention to remain a singleton (not to join coalition $P^{*}$) will have no incentive to (unilaterally) change its strategy and join coalition $P^{*}$, given the equilibrium announcements of all the other players. As expected, the internal stability condition is more difficult to satisfy than the external stability condition.

The internal and external stability functions, which must be positive, can be written as follows:

$$\begin{aligned} IS(p)= & {} U^{s}(p)-U^{ns}(p-1) \\ ES(p)= & {} U^{ns}(p)-U^{s}(p+1) \end{aligned}$$

Stage 2 follows the standard assumption in the literature on coalition formation: the coalition acts as a unique player, internalizing the externality among its members, whereas non-signatories act selfishly, maximizing their own payoff. We also follow the standard assumption of this literature and assume that signatories and non-signatories choose their economic strategies simultaneously. The two-stage coalition formation game is solved by backward induction.

1.1.1 G.1.1 Full Agreement

For a coalition of size p, the levels of nutrition and climate policy are, respectively (ns denotes a non-signatory and s denotes a signatory):

$ET=\frac{\tfrac{n\gamma \alpha a_{1}}{\gamma a_{2}+c}+\tfrac{nb_{1}}{b_{2}} }{1+(n-p+p^{2})(\tfrac{\alpha ^{2}d}{\gamma a_{2}+c}+\tfrac{d}{b_{2}})};\,\,f^{s}=\frac{\gamma a_{1}-p\alpha dET}{\gamma a_{2}+c}$, $f^{ns}=\frac{ \gamma a_{1}-\alpha dET}{\gamma a_{2}+c}$, $e^{s}=\frac{b_{1}-dpET}{b_{2}}$ and $e^{ns}=\frac{b_{1}-dET}{b_{2}}$.

We search for the size of the coalition for which the internal and external stability conditions are satisfied. We do not write the expressions for the internal and external stability conditions here because they are cumbersome. They are available from the authors upon request. In the quadratic model considered here, the stable size of the coalition cannot be derived analytically, in contrast to the linear-quadratic case (see below). We thus determine it numerically by using our initial parameter set described in section 5.1. We consider both cases for parameter $\gamma$: its value is lower than 1 and larger than 1. We find that the stable size of the coalition is equal to 2, regardless of the sign of parameter $\alpha$, that is whether the nutrition policy decreases or increases the GHG emissions generated by a country.

1.1.2 G.1.2 Agreement on Nutrition Policy

The countries do not cooperate on emissions $e_{i}.$ Because we assume identical countries, individual emission levels are identical $e_{i}=e^{V}.$

The nutrition policy f is decided either cooperatively by the members, denoted by s as signatories, of the coalition of size p,  or individually by the non-members of the coalition denoted by ns as non-signatories.

For a coalition of size p, the levels of nutrition policy and emissions are, respectively: $ET=\frac{\tfrac{n\gamma \alpha a_{1}}{\gamma a_{2}+c}+ \tfrac{ nb_{1}}{b_{2}}}{1+(n-p+p^{2})(\tfrac{\alpha ^{2}d}{\gamma a_{2}+c})+ \tfrac{nd }{b_{2}}}$, $f^{s}=\frac{\gamma a_{1}-p\alpha dET}{\gamma a_{2}+c}$, $f^{ns}=\frac{\gamma a_{1}-\alpha dET}{\gamma a_{2}+c}$, $e^{s}=e^{ns}= \frac{ b_{1}-dET}{b_{2}}$.

We search for the size of the coalition for which the internal and external stability conditions are satisfied. As for the full agreement, we do not write the expressions for the internal and external stability conditions here because they are cumbersome. They are available from the authors upon request.

In the quadratic model, the stable size of the coalition cannot be derived analytically, in contrast to the linear-quadratic case (see below). We thus determine it numerically by using our initial parameter set described in section 5.1. We consider both cases for parameter $\gamma$: its value is lower than 1 and larger than 1. For this parameter set and when $\alpha <0$, there is no stable coalition. Keeping the other parameter constellations unchanged, we now modify the value of parameter $a_{2}:$ $a_{2}$ now moves from 0.1 to 0.2 by 0.1, instead of from 1 to 2 by 1. In this case, we obtain the same result as for the full agreement. The stable size of the coalition is equal to 2, regardless of the sign of parameter $\alpha$, that is whether the nutrition policy decreases or increases the GHG emissions generated by a country.

Table 2 $a_{2}=1;2$: number of cases with a coalition size of 2

Table 3 $a_{2}=0.1;0.2$: number of cases with a coalition size of 2

1.2 G.2 Linear-Quadratic Case

We consider the following payoff function:

$$\begin{aligned} U_{i}= & {} G(f_{i})+B(e_{i})-D\left(\sum _{j=1}^{j=n}e_{j}+\alpha f_{i}\right) \\ \text {with }G(f_{i})= & {} \gamma a_{1}f_{i}-\frac{c}{2}f_{i}^{2} \\ B(e_{i})= & {} b_{1}e_{i}-\frac{b_{2}}{2}e_{i}^{2} \\ D\left(\sum _{j=1}^{j=n}e_{j}+\alpha f_{i}\right)= & {} d\left(\sum _{j=1}^{j=n}e_{j}+\alpha f_{i}\right) \end{aligned}$$

with $a_{1,}c,b_{1},b_{2},d,$ and $\gamma$ positive constants. We note that the function $G(f_{i})$ is at the maximum for $f_{i}=\frac{\gamma a_{1}}{c}$, and the function $B(e_{i})$ is at the maximum for $e_{i}=\frac{b_{1}}{b_{2} }$.

1.2.1 G.2.1 Full Agreement

For a coalition of size p, the levels of nutrition and climate policy are, respectively (ns denotes a non-signatory and s denotes a signatory): $f^{s}=\frac{\gamma a_{i}-p\alpha d}{c}$, $f^{ns}=\frac{\gamma a_{i}-\alpha d }{c}$, $e^{s}=\frac{b_{1}-dp}{b_{2}}$ and $e^{ns}=\frac{b_{1}-d}{b_{2}}$. Thus, we note that regardless of the sign of $\alpha$, the larger the coalition size is, the smaller the G and B functions.

This is an established finding for the B function in the literature on international environmental agreements but is a more novel finding for the G function. If $\alpha >0$, the negotiated nutrition policy decreases with p because of its negative impact on the environment; if $\alpha <0$, the nutrition policy increases with p because of its favourable impact on the environment, but it becomes increasingly costly as the coalition size increases, with the result that G decreases.

The stability function, which must be positive to respect internal stability, can be written as follows:

$$\begin{aligned} S(p)= & {} U^{s}(p)-U^{ns}(p-1)=-\gamma a_{1}\frac{\alpha d(p-1)}{c}+\frac{ \alpha d(p-1)}{2}\left( \frac{2\gamma a_{1}-(p+1)\alpha d}{c}\right) \\{} & {} -\frac{b_{1}}{b_{2}}d(p-1)+\frac{d(p-1)}{2}\left( \frac{2b_{1}-d(p+1)}{ b_{2}}\right) \\{} & {} +2d(p-1)(\frac{2\alpha ^{2}}{c}+\frac{d}{b_{2}}) \end{aligned}$$

As is usual, the term in the second line related to climate policy is decreasing in p due to free-riding incentives (positive externality effect). The term in the third line $2dp\frac{2\alpha ^{2}}{c}$ comes from consideration of the negative externality of GHG emissions on payoffs. It is positive and increasing in p, as the negative externality is internalized by more countries.

What are more interesting and new are the terms in the first line associated with the nutrition policy (the terms related to $\alpha$).

The decrease in the G and B functions as p increases means that the terms in $\alpha$ in the first and second lines, $-\gamma a_{1}\frac{\alpha d(p-1)}{c}+\frac{\alpha d(p-1)}{2}\left( \frac{2\gamma a_{1}-(p+1)\alpha d}{c }\right)$ are negative. For the nutrition policy (thus the terms in $\alpha$), we need to know whether the term $2d(p-1)\frac{2\alpha ^{2}}{c}$ is larger (in absolute value) than the expression $-\gamma a_{1}\frac{\alpha d(p-1)}{c}+\frac{\alpha d(p-1)}{2}\left( \frac{2\gamma a_{1}-(p+1)\alpha d}{c }\right)$. We note that the first term grows linearly with p but the second term decreases quadratically with p. Thus, the difference in the terms in $\alpha$ is negative, and so the nutrition policy reduces the stability function. In other words, the nutrition policy makes formation of a large coalition more difficult.

The three terms ultimately simplify to the following expression for the stability function:

$$\begin{aligned} S(p)=\frac{-d^{2}(p-1)(p-3)(\alpha ^{2}b_{2}+c)}{2b_{2}c}<0 \end{aligned}$$

The stability function is strictly positive for $p=2$ and null for $p=1$ and $p=3$. For $p\ge 3$, external stability ($S(p+1)<0)$ is respected. The stable size of the coalition is thus equal to 3.

1.2.2 G.2.2 Agreement on Nutrition Policy

The countries do not cooperate on emissions $e_{i}.$ Because we assume identical countries, individual emission levels are identical $e_{i}=e^{V}.$ The nutrition policy f is decided either cooperatively by the members, denoted by s as signatories, of the coalition of size p,  or individually by the non-members of the coalition denoted by ns as non-signatories.

The first-order conditions for the agreement on nutrition policy with a coalition of size p are then:

$$\begin{aligned} B_{e}(e^{V}) =D_{E}(\alpha ((n -p)f_{NS}^{V} +pf_{S}^{V})) +ne^{V}) \end{aligned}$$

(G1)

$$\begin{aligned} G_{f}(f_{NS}^{V}) =\alpha D_{E}(\alpha ((n -p)f_{NS}^{V} +pf_{S}^{V})) +ne^{V}) \end{aligned}$$

(G2)

$$\begin{aligned} G_{f}(f_{S}^{V}) =\alpha pD_{E}(\alpha ((n -p)f_{NS}^{V} +pf_{S}^{V})) +ne^{V}) \end{aligned}$$

(G3)

For a coalition of size p, the levels of nutrition policy and emissions are, respectively: $f^{s}=\frac{\gamma a_{i}-p\alpha d}{c}$, $f^{ns}=\frac{ \gamma a_{i}-\alpha d}{c}$, $e^{s}=\frac{b_{1}-dp}{b_{2}}$ and $e^{ns}=\frac{ b_{1}-d}{b_{2}}$.

We can then express the stability function, which must be positive to respect the internal stability condition:

$$\begin{aligned} S(p)=U^{s}(p)-U^{ns}(p-1)=\frac{-\alpha ^{2}d^{2}}{2c}(p-3)(p-1) \end{aligned}$$

(G24)

The stability function is strictly positive for $p=2$ and null for $p=1$ and $p=3$. For $p\ge 3$, external stability ($S(p+1)<0)$ is respected. The stable size of the coalition is thus equal to 3.