On the Timing of Climate Agreements

Abstract

A central issue in climate policy is the question whether long-term targets for greenhouse gas emissions should be adopted. This paper analyzes strategic effects related to the timing of such commitments. Using a two-country model, we identify a redistributive effect that undermines long-term cooperation when countries are asymmetric and side payments are unavailable. The effect enables countries to shift rents strategically via their R&D efforts under delayed cooperation. In contrast, a complementarity effect stabilizes long-term cooperation, because early commitments in abatement induce countries to invest more in low-carbon technologies, and create additional knowledge spillovers. Contrasting both effects, we endogenize the timing of climate agreements.

Appendix

This appendix collects the formal proofs of the lemmas and propositions.

Proof of Lemma 1

Due to the convexity of $C_i$, it is to show that the right-hand side of (5) is non-negative. In the presence of an abatement externality, we have $\beta _{-i} >0$ so that it remains to be shown that $\partial a^n_{-i}/\partial r_i \ge 0$. To determine the sign of $\partial a^n_{-i}/\partial r_i$, we use the first-order conditions (4) of stage 2 and insert the functions $a^n_1(r_1,r_2)$ and $a^n_2(r_1,r_2)$ that describe the equilibrium outcome:

$$\begin{aligned} \frac{\partial C_i(a^n_i(r_1,r_2),r_1,r_2)}{\partial a_i} = b_i. \end{aligned}$$

Applying the Implicit Function Theorem, we obtain (after rearranging):

$$\begin{aligned} \frac{\partial a_{-i}^n}{\partial r_i} = - \left( \frac{\partial ^2 C_{-i}}{\partial a_{-i} \partial r_i}\right) \bigg / \left( \frac{\partial ^2 C_{-i}}{\partial a^2_{-i}}\right) . \end{aligned}$$

Because by assumption $\partial ^2 C_{-i}/\partial a_{-i}\partial r_{i} \le 0$, the numerator is non-positive, and due to $\partial ^2 C_i/\partial a^2_i >0$ the denominator is positive. Hence, $\partial a_{-i}^n/\partial r_i \ge 0$, which proves that the right-hand side of (5) is non-negative. $\square$

Proof of Lemma 2

To show that the overall strategic effect in stage 1 of the early cooperation game increases total welfare, note that countries could neglect this effect in the cooperative stage 1. Hence, the effect raises welfare whenever it affects the final outcome, and (by (7)) it affects the final outcome in the presence of R&D spillovers.

To show the second claim, note that the functions $r^e_1(a_1,a_2)$ and $r^e_2(a_1,a_2)$ that describe the outcome of R&D competition in stage 2, are implicitly defined by the system $\partial C_i(a_i, r^e_1(a_1,a_2)$, $r^e_2(a_1,a_2))/ \partial r_i =0$ for $i=1,2$ [see (8)]. Applying the Implicit Function Theorem, we obtain (after rearranging):

$$\begin{aligned} \frac{\partial r^e_i}{\partial a_i}&= {\left( -\frac{\partial ^2 C_i}{\partial r_i \partial a_i} \frac{\partial ^2 C_{-i}}{\partial r_{-i}^2} \right) }\bigg /{ \left( \frac{\partial ^2 C_i}{\partial r_i^2} \frac{\partial ^2 C_{-i}}{\partial r_{-i}^2}- \frac{\partial ^2 C_i}{\partial r_i \partial r_{-i}} \frac{\partial ^2 C_{-i}}{\partial r_i \partial r_{-i}} \right) }; \\ \frac{\partial r^e_{-i}}{\partial a_i}&= {\left( \frac{\partial ^2 C_i}{\partial r_i \partial a_i} \frac{\partial ^2 C_{-i}}{\partial r_i \partial r_{-i}} \right) }\bigg /{ \left( \frac{\partial ^2 C_i}{\partial r_i^2} \frac{\partial ^2 C_{-i}}{\partial r_{-i}^2}- \frac{\partial ^2 C_i}{\partial r_i \partial r_{-i}} \frac{\partial ^2 C_{-i}}{\partial r_i \partial r_{-i}} \right) }. \end{aligned}$$

Given the assumptions $\partial ^2 C_i/\partial r^2_i>0$, $\partial ^2 C_i/\partial r_i\partial r_{-i} \ge 0$, and $\partial ^2 C_i/\partial r^2_i > \partial ^2 C_i/\partial r_i\partial r_{-i}$, the denominator is always positive. Using $\partial ^2 C_i/\partial a_i\partial r_i < 0$ and $\partial ^2 C_i/\partial r_i\partial r_{-i} \ge 0$, we find that ${\partial r^e_i}/{\partial a_i} > 0$ and ${\partial r^e_{-i}}/{\partial a_i} \le 0$. Using the condition $\partial ^2 C_i/\partial r^2_i > \partial ^2 C_i/\partial r_i\partial r_{-i}$, we find ${\partial r^e_i}/{\partial a_i} > \left| {\partial r^e_{-i}}/{\partial a_i}\right|$. If countries are symmetric, then ${\partial C_{-i}}/{\partial r_i} = {\partial C_{i}}/{\partial r_{-i}}$. The overall strategic effect is, then, always negative, which [by (8) and (7)] implies higher abatement and higher R&D levels than in the absence of the strategic effect. $\square$

Proof of Lemma 3

Using $\beta \equiv \beta _1=\beta _2$ and inserting the function $a^l_i(r_1,r_2)$ into (10) yields

$$\begin{aligned} \frac{\partial C_i \left( a^l_i(r_1,r_2),r_1,r_2 \right) }{\partial a_i}=b_i+\beta . \end{aligned}$$

Applying the Implicit Function Theorem, we obtain after rearranging:

$$\begin{aligned} \frac{\partial a^l_i}{\partial r_{i}} = -\left( \frac{\partial ^2 C_i}{\partial a_i \partial r_{i}} \right) \bigg / \left( \frac{\partial ^2 C_i}{\partial a^2_i} \right) \quad \text{ and } \quad \frac{\partial a^l_i}{\partial r_{-i}} = -\left( \frac{\partial ^2 C_i}{\partial a_i \partial r_{-i}} \right) \bigg / \left( \frac{\partial ^2 C_i}{\partial a^2_i} \right) . \end{aligned}$$

Using the regularity conditions $\partial ^2 C_i/\partial a^2_i>0$, $\partial ^2 C_i/\partial a_i \partial r_i < 0$, $\partial ^2 C_i/\partial a_i \partial r_{-i} \le 0$, and $\left| \partial ^2 C_i/\partial a_i \partial r_i \right| \ge \left| \partial ^2 C_i/\partial a_i \partial r_{-i} \right|$, we find that ${\partial a^l_i}/{\partial r_{i}}$ is positive and ${\partial a^l_i}/{\partial r_{-i}}$ non-negative, and ${\partial a^l_i}/{\partial r_{i}} \ge {\partial a^l_i}/{\partial r_{-i}}$. Hence, we obtain for the overall strategic effect $\beta \left( {\partial a^l_{-i}}/{\partial r_i} - {\partial a^l_i}/{\partial r_i} \right) \le 0$, which [by (11)] implies lower R&D efforts than in the absence of the strategic effect. The reduction in R&D-levels aggravates the problem of under-investments in R&D (relative to the full cooperation case) and, therefore, reduces total welfare. $\square$

Proof of Proposition 1

Without R&D externalities we have $\partial C_{-i}/\partial r_i=0$. Using (2) and (3), the cooperative solution $(a_1^f,a_2^f,r_1^f,r_2^f)$ is with $i=1,2$ characterized by

$$\begin{aligned} \frac{\partial C_i}{\partial a_i}=b_i+\beta _i \quad \text{ and } \quad \frac{\partial C_i}{\partial r_i}=0. \end{aligned}$$

According to (8) and (7), early cooperation $(a_1^e,a_2^e,r_1^e,r_2^e)$ is characterized by the system

$$\begin{aligned} \frac{\partial C_i}{\partial r_i}=0 \quad \text{ and } \quad \frac{\partial C_i}{\partial a_i}=b_i+\beta _i. \end{aligned}$$

The result follows immediately because the two systems of optimality conditions coincide. $\square$

Proof of Proposition 2

Without abatement externalities we have $\beta _1=\beta _2=0$. Using (4) and (5), the solution without cooperation, $(a_1^n,a_2^n,r_1^n,r_2^n)$, solves for $i=1,2$

$$\begin{aligned} \frac{\partial C_i}{\partial a_i}=b_i \quad \text{ and } \quad \frac{\partial C_i}{\partial r_i}=0. \end{aligned}$$

Using (10) and (11), we find that the late cooperation outcome $(a_1^l,a_2^l,r_1^l,r_2^l)$ is characterized by the same system of optimality conditions. Hence, the outcome without cooperation and with late cooperation are identical.

Early cooperation satisfies the system (8) and (7), which differs from the other system of equations due to the presence of strategic effects. By ignoring them, in stage 1 countries can assure a total welfare at least as large as under late or no cooperation. $\square$

Proof of Proposition 3

The outcome under late cooperation is characterized by the system (10) and (11). According the Lemma 3, the overall strategic effect in stage 1 [see (11)] is welfare-reducing if $\beta _1=\beta _2$. The outcome under early cooperation satisfies the system (8) and (7), which differs from (10) and (11). According to Lemma 2, the overall strategic effect in stage 1 [see (7)] is welfare-enhancing. The claim, thus, follows immediately from Lemma 2 and Lemma 3. $\square$

Proof of Proposition 4

To show that cooperation in the countries’ choices of abatement targets fails when there is a unilateral abatement externality but no R&D externality, note that the country that generates the abatement externality can achieve its own welfare maximum without any cooperation. Hence, this country can never gain from cooperating.

To show that cooperation fails when there is a unilateral R&D externality but no abatement externality, note first that by Proposition 2 late cooperation in abatement yields no welfare gains. Furthermore, early cooperation fails, because the country that generates the R&D externality has no gains from it. It is assigned a higher abatement level in the cooperative stage (stage 1) in order to trigger additional R&D investments by this country in stage 2. This reduces the welfare of this country that can achieve its own welfare maximum without any cooperation. $\square$

Proof of Proposition 5

To show the first claim, note that the country that generates both externalities achieves its individual welfare maximum without any cooperation. Hence, whenever cooperation affects the final outcome, it negatively affects this country’s welfare.

To show that late cooperation fails if each of the unilateral externalities affects a different country, note that for fixed R&D levels, the country that generates the abatement externality suffers from cooperation in stage 2. For any given R&D levels, it attains its individual welfare maximum when ignoring the abatement externality. Hence, it rejects late cooperation.

To show that early cooperation can succeed if both externalities are of comparable strength and each affects a different country, it suffices to demonstrate that there exists an example where this holds. Such an example is provided at the end of Sect. 4.2. $\square$

Proof of Lemma 4

Straightforward calculations yield the following results:

• 1. No cooperation:

  $$\begin{aligned} a_i^n&= 1 \text{ and } r_i^n=1;\nonumber \\ \Pi ^n_i&= 1+\beta _{-i}+\rho _{-i};\nonumber \\ \Pi ^n&= 2+\beta _1+\beta _2+\rho _1+\rho _2. \end{aligned}$$

  (16)

• 2. Early cooperation:

  $$\begin{aligned} a_i^e&= \frac{3+2 \beta _{i}+\rho _{i}}{3} \quad \text{ and }\quad r_i^e=\frac{3+ \beta _{i}+\rho _{i}/2}{3} < r^f_i;\nonumber \\ \Pi ^e_i&= 1+\beta _{-i}+\rho _{-i}+\frac{ (2\beta _{-i}+\rho _{-i})^2}{6}- \frac{(2\beta _{i}+\rho _{i})^2}{12};\nonumber \\ \Pi ^e&= 2+\beta _1+\beta _2+\rho _1+\rho _2 + \frac{(2\beta _1+\rho _1)^2}{12} + \frac{(2\beta _2+\rho _2)^2}{12}. \end{aligned}$$

  (17)

• 3. Late cooperation:

  $$\begin{aligned} a_i^l&= 1+ \beta _{i}/2 \text{ and } r_i^l=1=r^n_i;\nonumber \\ \Pi ^l_i&= 1+\beta _{-i}+\rho _{-i}+\beta _{-i}^2/2 -\beta _{i}^2/4;\nonumber \\ \Pi ^l&= 2+\beta _1+\beta _2+\rho _1+\rho _2+\beta _1^2/4 +\beta _2^2/4. \end{aligned}$$

  (18)

A direct comparison yields $\Pi ^e \ge \Pi ^l \ge \Pi ^n$.

Moreover, $\Delta \Pi ^{el}_i \equiv \Pi ^e_i-\Pi ^l_i = (2\varepsilon _{-i}-\varepsilon _{i})/12$ so that $\Pi ^e_i<\Pi ^l_i$ if $\varepsilon _{i}>2\varepsilon _{-i}$. Furthermore, comparing (16) and (18), we find that $\Delta \Pi ^{ln}_i \equiv \Pi ^l_i-\Pi ^n_i = (2\beta ^2_{-i}-\beta ^2_{i})/4$ so that country $i$ prefers late cooperation over the non-cooperative outcome if $\beta _{i} < \sqrt{2} \beta _{-i}$. Finally $\Delta \Pi ^{en}_i \equiv \Pi ^e_i-\Pi ^n_i =\beta ^2_i(2\beta ^2_{-i}/\beta _i^2-1)/4+\varepsilon _i(2\varepsilon _{-i}/\varepsilon _i-1)/12$. $\square$

Before we state the proof of Proposition 6, we establish the following useful Lemma:

Lemma 6

If $\Delta \Pi ^{ln}_i(r_1,r_2) \equiv \Pi ^l_i(r_1,r_2) - \Pi ^n_i(r_1,r_2)$ is independent of $r_{-i}$ for $i=1,2$, then in the subgame LN, late cooperation succeeds iff $\Pi ^l_i(r_1^l,r_2^l) \ge \Pi ^n_i(r_1^n,r_2^n)$ for $i=1,2$.

Proof of Lemma 6

If $\Delta \Pi ^{ln}_i(r_1,r_2)$ does not depend on $r_{-i}$ for $i=1,2$, then country $i$’s decision whether to accept or reject the formation of a late coalition ($l_i$) is independent of $r_{-i}$. Hence, if (for any given $r_{-i}$) country $i$ prefers the cooperative outcome given $r_i=r_i^l$ over the non-cooperative outcome obtained under $r_i^n$, and the same holds true for the other country, then late cooperation succeeds, and the choices $(r_1^l,r_2^l)$ are optimal given this expectation. Conversely, if there is at least one country $i$ that (for any given $r_{-i}$) prefers the outcome under no cooperation given $r_i^n$ to the outcome obtained under late cooperation given $r_i^l$, then this country chooses $r_i=r_i^n$, planning to reject late cooperation, irrespective of the value of $r_{-i}$ chosen by the other country (which is fixed when countries choose $l_i$). Hence, country $-i$ cannot affect country $i$’s cooperation decision in the LN-subgame via its own R&D choice, and in anticipation of the failure of cooperation, countries choose $(r_1^n,r_2^n)$ accordingly. A necessary and sufficient condition for late cooperation to succeed is, thus, that $\Pi ^l_i(r_1^l,r_2^l) \ge \Pi ^n_i(r_1^n,r_2^n)$ holds for $i=1$ and $i=2$. $\square$

Intuitively, if $r_i$ affects country $-i$’s payoff under late cooperation, $\Pi _{-i}^l(r_1,r_2)$, in exactly the same way as it affects $\Pi _{-i}^n(r_1,r_2)$, then a change in $r_i$ has no effect upon country $-i$’s decision whether to enter a late coalition or not. In that case, it suffices to compare the payoffs under the no cooperation case (Sect. 3.2) with the payoffs in the late cooperation case (Sect. 3.4) to derive the equilibrium in the subgame LN.

Proof of Proposition 6

By (14), $r_{-i}$ enters linearly in country $i$’s payoff. Therefore, both under late and under no cooperation, country $i$’s abatement following the R&D stage, $a_i(r_1,r_2)$, is independent of $r_{-i}$. It follows immediately that $\Delta \Pi _i^{ln}(r_1,r_2)$ is also independent of $r_{-i}$, so Lemma 6 applies. Hence, to determine the endogenous success and the timing of cooperation in the overall game, it suffices to compare the equilibrium outcomes of the early, late, and no cooperation cases. Now we apply Lemma 4.

If $\beta ^r \in (\sqrt{2}/2,\sqrt{2})$ and $\varepsilon ^r \in (1/2,2)$ then it follows from the Lemma that for either country $i\in \{1,2\}$ we have the ordering $\Pi _i^e \ge \Pi _i^l \ge \Pi _i^n$. This leads to the unique equilibrium outcome of early cooperation.

If $\beta ^r \in (\sqrt{2}/2,\sqrt{2})$ and $\varepsilon ^r \not \in (1/2,2)$ then for either country $i\in \{1,2\}$ it holds $\Pi _i^l>\Pi _i^n$, while there exists one country $j$ for which $\Pi ^e_j<\Pi ^l_j$. This leads to the late cooperation outcome.

If $\beta ^r >\sqrt{2}$ and $\varepsilon ^r >2$ then $\Pi ^e_1<\Pi ^l_1$ and $\Pi ^l_1<\Pi ^n_1$, and country 1’s most preferred outcome is $\Pi ^n_1$. Because country $1$ can obtain this outcome by rejecting both early and late cooperation, there is no cooperation in any subgame perfect equilibrium. By symmetry the same holds for $\varepsilon ^r <1/2$ and $\beta ^r <1/\sqrt{2}$.

If $\beta ^r >\sqrt{2}$ and $\varepsilon ^r \in (1/2,2)$ then $\Pi ^e_2 > \Pi ^l_2>\Pi ^n_2$, while $\Pi ^e_1> \Pi ^l_1$ and $\Pi ^l_1<\Pi ^n_1$. Hence, country 2’s preferred outcome is early cooperation. In the case that $\Pi ^e_1>\Pi ^n_1$, also county 1’s preferred outcome is early cooperation and early cooperation arises in equilibrium. If $\Pi ^e_1<\Pi ^n_1$, county 1’s preferred outcome is no cooperation, which it can always reach by rejecting any form of cooperation. As a result, the unique equilibrium outcome is no cooperation. In the Proof of Lemma 4, it is shown that $\Pi ^e_i-\Pi ^n_i =\beta ^2_i(2\beta ^2_{-i}/\beta _i^2-1)/4+\varepsilon _i(2\varepsilon _{-i}/\varepsilon _i-1)/12$ so that the difference is decreasing in $\beta ^r$. As a result the former case arises for $\beta ^r>\sqrt{2}$ but sufficiently small (close to $\sqrt{2}$), and the latter one for $\beta ^r$ sufficiently large. By symmetry, the same holds for $\beta ^r <1/\sqrt{2}$ and $\varepsilon ^r \in (1/2,2)$. Finally, if $\varepsilon ^r <1/2$ and $\beta ^r >\sqrt{2}$ then it follows from Lemma 4 that $\Pi ^e_1> \Pi ^l_1$, $\Pi ^l_1<\Pi ^n_1$, $\Pi ^l_2> \Pi ^e_2$, and $\Pi ^l_2>\Pi ^n_2$. Again, late cooperation is never obtained as country 1 prefers the non-cooperative over the late cooperation outcome. Hence, either early or no cooperation is obtained in equilibrium. $\square$

Proof of Lemma 5

Straight-forward calculations yields the following results:

• 1. No cooperation:

  $$\begin{aligned} a_1^n&= \frac{(b+\delta )(3b^2-\delta ^2)}{8\gamma }, \quad a_2^n=\frac{(b-\delta )(3b^2-\delta ^2)}{8\gamma }, \quad r_1^n=\frac{3b^2+2b\delta -\delta ^2}{8\gamma }, \quad \nonumber \\ r_2^n&= \frac{3b^2-2b\delta -\delta ^2}{8\gamma };\nonumber \\ \Pi ^n_1&= \frac{27b^4+12b^3\delta -22b^2\delta ^2-4b\delta ^3+3\delta ^4}{64\gamma }, \quad \nonumber \\ \Pi ^n_2&= \frac{27b^4-12b^3\delta -22b^2\delta ^2+4b\delta ^3+3\delta ^4}{64\gamma };\nonumber \\ \Pi ^n&= \frac{27b^4-22b^2\delta ^2+3\delta ^4}{32\gamma }. \end{aligned}$$

  (19)

• 2. Early cooperation:³⁴

  $$\begin{aligned}&a^e_i=\frac{216b^3}{125\gamma } =1.728b^3/\gamma , \quad r_i^e=\frac{18b^2}{25\gamma } =0.72b^2/\gamma ;\nonumber \\&\Pi ^e_1=\frac{108 b^3(b+4\delta )}{125\gamma }=108\Pi ^f_1/125, \quad \Pi ^e_2=\frac{108 b^3(b-4\delta )}{125\gamma }=108\Pi ^f_2/125;\nonumber \\&\Pi ^e=\frac{216b^4}{125\gamma }=108\Pi ^f/125. \end{aligned}$$

  (20)

• 3. Late cooperation³⁵:

  $$\begin{aligned}&a_i^l=b^3/\gamma , \quad r_1^l=\frac{b^2+2b\delta }{2\gamma }, \quad r_2^l=\frac{b^2-2b\delta }{2\gamma };\nonumber \\&\Pi ^l_1=b^2\frac{3b^2 + 4b\delta - 4\delta ^2}{4\gamma }, \quad \Pi ^l_2=b^2\frac{3b^2 - 4b\delta - 4\delta ^2}{4\gamma };\nonumber \\&\Pi ^l=b^2\frac{3b^2-4\delta ^2}{2\gamma }. \end{aligned}$$

  (21)

A direct comparison yields $\Pi ^e \ge \Pi ^l \ge \Pi ^n$.

Given these results, it is straight-forward to verify that in the relevant range, country 2 prefers early over late cooperation ($\Pi ^e_2 \ge \Pi ^l_2$) iff $\delta \le \delta _1(b) \equiv (307-2 \sqrt{21781})b/250 \approx 0.047b$.

In the subgame LN, for given R&D levels $(r_1,r_2)$, late cooperation succeeds if both countries achieve a higher welfare than in the absence of cooperation, hence, if $\Delta \Pi ^{ln}_1(r_1,r_2) \ge 0$ and $\Delta \Pi ^{ln}_2(r_1,r_2) \ge 0$. Under no cooperation (for fixed R&D levels $(r_1,r_2)$), country $i$’s abatement is determined by condition (4): $\partial C_i / \partial a_i=b_i$. Using $C_i=\frac{a^2_i}{r}+\gamma r^2_i$, $b_1=b+\delta$, and $b_2=b-\delta$, this yields

$$\begin{aligned} a_1^n(r_1,r_2)=(b+\delta )r/2 \quad \text{ and } \quad a_2^n(r_1,r_2)=(b-\delta )r/2, \end{aligned}$$

and, hence: $a=br$. Countries’ payoffs, thus, become:

$$\begin{aligned} \Pi ^n_1(r_1,r_2)=(3b^2+2b\delta -\delta ^2)r/4 - \gamma r^2_1, \quad \Pi ^n_2(r_1,r_2)=(3b^2-2b\delta -\delta ^2)r/4 - \gamma r^2_2. \end{aligned}$$

Under late cooperation (again for fixed R&D levels), country $i$’s abatement is determined by condition (10): $\partial C_i / \partial a_i=b_i+\beta _i$. Using $\beta _i=b_{-i}$, we, thus, obtain $a_1^l=a_2^l=br$, and $a=2br$. This yields the following expressions for welfare:

$$\begin{aligned} \Pi ^l_1(r_1,r_2)=(b^2+2b\delta )r-\gamma r^2_1, \quad \Pi ^l_2(r_1,r_2)=(b^2-2b\delta )r-\gamma r^2_2. \end{aligned}$$

This implies

$$\begin{aligned} \Delta \Pi ^{ln}_1(r_1,r_2) = (b^2+6b\delta +\delta ^2)r/4 > 0 \quad \text{ and } \quad \Delta \Pi ^{ln}_2(r_1,r_2) = (b^2-6b\delta +\delta ^2)r/4. \end{aligned}$$

Hence, for any fixed ($r_1,r_2$), country 1 is always willing to cooperate. Country 2 cooperates if $b^2-6b\delta +\delta ^2 \ge 0$, which, for the relevant interval $\delta \in [0,b/2]$, yields the critical value $\delta _2(b) \equiv b(3-2\sqrt{2}) \approx 0.17b$. If $\delta$ is below this value, it holds that $\Delta \Pi ^{ln}_2(r_1,r_2)>0$. As $\delta _2(b)$ is independent of $r_1$ and $r_2$, late cooperation succeeds in the subgame LN iff $\delta \le \delta _2(b)$. $\square$

Proof of Proposition 7

It is straight-forward to show [using (20) and (21)] that country 1 always prefers the early over the late cooperation outcome. The result, thus, follows immediately from Lemma 5. $\square$