Game of Regional Environmental Policy: Europe and US

Abstract

We analyse a different timing implementation of environmental regulation and compare the effects on the markets from a policy innovation perspective. The paper addresses the question: Should a regulator try to commit to a policy (ex-ante regulation) or rather adapt its policy to a firm’s decisions (ex-post)? The findings are of interest towards improving regulatory effectiveness and economics efficiencies, e.g. for the transatlantic regional relationship between EU and USA. Our findings highlight differences in policy timing between markets may be harmful. The transatlantic regulators should consider the timing of the policy innovation for the achievement of mutual benefits.

Appendix

1.1 Ex-ante Scenario - Monopoly

1.1.1 Stage 2: Firm Decides on Output

We calculate the First Order Conditions (FOC)¹⁶

$$ \frac{ \partial {\Pi}_{1}}{ \partial q} =a -2 q (1 +c) = 0 $$

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and solving with respect to q the firm’s level of production and the analogous profits are

$$ \bar{q}_{1} =\frac{a}{2 (1 +c)} $$

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$$ {\Pi}_{1}^{ \ast } =\bar{q}_{1}^{2} (1 +c) -(k -1)^{2} $$

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where the subscript 1 indicates the first scenario (ex-ante).

1.1.2 Stage 1: Regulator Decides for Technology

After the substitution of Eq. 6 into Eq. 1 we have

$$ S W_{1}^{ \ast } =(b -1) \left[(k -1)^{2} -\frac{a^{2}}{4 (1 +c)} \right] +\frac{a b (a -4 (1 +c) e k)}{8 (1 +c)^{2}} $$

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and from the FOC we obtain¹⁷

$$ \frac{\partial S W_{1}^{ \ast }}{ \partial k} = 2 (b -1) (k -1) -\frac{a b e}{2 (1 +c)} = 0 $$

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yielding the optimum technology

$$ \bar{k}_{1} = 1 -\frac{a b e}{4 (1 -b) (1 +c)} $$

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where $\frac { \partial \bar {k}_{1}}{ \partial b} <0$ thus, the technological choice is reducing in b or the higher the value of the parameter b the less polluting the technological choice k is. We substitute all the results to the initial equations and therefore we have the results for the ex-ante case. Particularly, the profits are given by

$$ \bar{{\Pi} }_{1} =\frac{a^{2} \left( 1 -\frac{b^{2} e^{2}}{4 (b -1)^{2} (1 +c)}\right)}{4 (1 +c)} $$

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and the price of good in the market is

$$ \bar{p}_{1} =a -\frac{a}{2 (1 +c)} $$

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After substituting the technology to the Social Welfare (8) we obtain:

$$ \bar{SW}_{1} =\frac{ -a A +a^{2} b^{2} e^{2}}{16 (b -1) (1 +c)^{2}} $$

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where A = (b − 1)(2a(2c(b − 1) + b − 2) + 8b(1 + c)e).

Also, the firm’s emissions are

$$ \bar{y}_{1} =\frac{a (4 (b -1) (1 +c) +a b e)}{8 (b -1) (1 +c)^{2}} $$

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where $\frac { \partial \bar {y}_{1}}{ \partial b} <0$ thus, emissions’ level is decreasing in b. Finally, the damage to the environment is equal to $\bar {DF}_{1} =e \bar {y}_{1}$.

1.2 Ex-post Scenario - Monopoly

1.2.1 Stage 2: Regulator Decides for Technology

The regulator decides on the technological level in order to maximize the social welfare. The FOC is equal to:¹⁸

$$ \frac{ \partial S W}{ \partial k} = 2 (1 -b) (1 -k) -b e q = 0 $$

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and solving with respect to k we have

$$ k_{2}^{ \ast } = 1 -\frac{b e q}{2 (1 -b)} $$

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where the subscript 2 indicated the ex-post case.

1.2.2 Stage 1: Firm Decides on Output

After the substitution of k^∗ into the profit’s equation we have ${\Pi }_{2} =(a -q) q -c q^{2} -\frac {b^{2} e^{2} q^{2}}{(2 b -2)^{2}}$. As usual we calculate the FOC which is¹⁹

$$ \frac{ \partial {\Pi}_{2}}{ \partial q} =a +\frac{1}{2} \left( -4 -4 c -\frac{b^{2} e^{2}}{(b -1)^{2}}\right) = 0 $$

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and solving for q we calculate the optimum level of output

$$ \bar{q}_{2} =\frac{2 a}{4 + 4 c +\frac{b^{2} e^{2}}{(b -1)^{2}}} $$

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We substitute $\bar {q}_{2}$ into $k_{2}^{ \ast }$ and then we have the optimum technology for this model

$$ \bar{k}_{2} = 1 -\frac{a (1 -b) b e}{4 (b -1)^{2} (1 +c) +b^{2} e^{2}} $$

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Hence, the final results for the ex post model are; profits equals

$$ \bar{{\Pi} }_{2} =\frac{a^{2} (b -1)^{2}}{4 (b -1)^{2} (1 +c) +b^{2} e^{2}} $$

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and price of the good is

$$ \bar{p}_{2} =a -\frac{2 a}{4 + 4 c +\frac{b^{2} e^{2}}{(b -1)^{2}}} $$

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After the necessary substitutions and calculation we get the Social welfare

$$ \bar{SW}_{2} =\frac{a (b -1)^{2} (X +Y)}{(4 (b -1)^{2} (1 +c) +b^{2} e^{2})^{2}} $$

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where X = − 8(b − 1)²b(1 + c) − 2b³e³ and Y = a(b − 1)(− 2(b − 1)(b − 2 + 2(b − 1)c) − 3b²e²).

The level of the emissions are given by

$$ \bar{y}_{2} = 2 a (b -1)^{2} \left( \frac{1}{L} +\frac{a (b -1) b e}{L^{2}}\right) $$

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where L = 4(b − 1)²(1 + c) + b²e². So, the analogous damage function is $\bar {DF}_{2} =e \bar {y}_{2}$.

1.3 Proof of Lemma 1

From the calculation of the difference we take $\bar {\mathbf {p}}_{1} -\bar {\mathbf {p}}_{2} = -\frac {\mathbf {a} \mathbf {b}^{2} \mathbf {e}^{2}}{2 (1 +\mathbf {c})(4 (\mathbf {b} -1)^{2} (1 +\mathbf {c}) +\mathbf {b}^{2} \mathbf {e}^{2}}$ and it is straightforward that it is a negative result since the numerator and the denominator is positive. Thus, the price is lower under the ex-ante case. Also, we calculate $\bar {\mathbf {q}}_{1} -\bar {\mathbf {q}}_{2} =\frac {\mathbf {a} \mathbf {b}^{2} \mathbf {e}^{2}}{2 (1 +\mathbf {c})(4 (\mathbf {b} -1)^{2} (1 +\mathbf {c}) +\mathbf {b}^{2} \mathbf {e}^{2}}$ which is positive since both numerator and denominator are positive.

1.4 Proof of Lemma 2

We calculate $\bar {k}_{1} -\bar {k}_{2} =\frac {a b^{3} e^{3}}{4 (b -1) (1 +c) (4 (b -1)^{2} (1 +c) +b^{2} e^{2})}$ where the denominator is negative since b ∈ (0, 1), thus the technology is “greener” or less polluting under the ex-ante case. Furthermore, $\bar {{\Pi } }_{1} -\bar {{\Pi } }_{2} = -\frac {a^{2} b^{4} e^{4}}{16 (b -1)^{2} (1 +c)^{2} (4 (b -1)^{2} (1 +c) +b^{2} e^{2})}$ which is negative (both numerator and denominator are positive) since there is a negative sing in front of the ratio. Hence, the level of the profitsis lower under the ex-ante case.

1.5 Ex-ante Scenario - Oligopoly

1.5.1 Stage 2: Firms Decide on Output

We calculate the First Order Conditions (FOCs)²⁰$\frac { \partial {\Pi }_{i}}{ \partial q_{i}^{ \ast }} =a -2 q_{i} (1 +c_{i}) -q_{j} = 0$ and solving simultaneously with respect to q_i we take

$$ \bar{q}_{i}^{E A} =\frac{a (1 + 2 c_{j})}{3 + 4 c_{j} + 4 c_{i} (1 +c_{j})} $$

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$$ {\Pi}_{i}^{E A} =(\bar{q}_{i}^{E A})^{2} (1 +c_{i}) -(k_{i} -1)^{2} $$

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where the superscript EA indicates the first scenario (ex-ante).

1.5.2 Stage 1: Regulator Decides for Technology

The Social Welfare is given by SW^EA = b(CS^EA − DF^EA) + (1 − b)PS^EA where $C S^{E A} =\frac {1}{2} \overset {2}{ \sum _{i = 1}} \bar {q}_{i}^{E A}$, $D F^{E A} =e (k_{i} \bar {q}_{i}^{E A} +k_{j} \bar {q}_{j}^{E A})$ and $P S^{E A} =\overset {2}{ \sum _{i = 1}} {\Pi }_{i}^{E A}$ and from the FOC we obtain²¹

$$ \frac{ \partial S W^{E A}}{ \partial k_{i}} = 2 (b -1) (k_{i} -1) -\frac{b (a + 2 a c_{j}) e}{3 + 4 c_{j} + 4 c_{i} (1 +c_{j})} = 0 $$

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yielding the optimum technology

$$ \bar{k}_{i}^{E A} = 1 +\frac{a b (1 + 2 c_{j}) e}{2 (b -1) (3 + 4 c_{j} + 4 c_{i} (1 +c_{j}))} $$

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After the necessary substitutions the profits are

$$ \bar{{\Pi} }_{i}^{E A} =(\bar{q}_{i}^{E A})^{2} \left( \left( \frac{4 (b -1)^{2} (1 +c_{i}) -b^{2} e^{2}}{4 (b -1)^{2}}\right)\right) $$

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and the price is

$$ \bar{p}_{i}^{E A} =\frac{a (1 + 2 c_{i}) (1 + 2 c_{j})}{3 + 4 c_{j} + 4 c_{i} (1 +c_{j})} $$

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The emissions are given by $\bar {y}_{i}^{E A} =\bar {q}_{i}^{E A} \bar {k}_{i}^{E A}$ and $D F^{E A} =e \bar {y}_{i}^{E A}$. As usual we substitute the above equation into the SW^EA and we take the final $\bar {SW}^{EA}$.

1.6 Ex-post Scenario

1.6.1 Stage 2: Regulator Decides for Technology - Oligopoly

Like in the basic model, the regulator will decide on the level of the technology which maximises the social welfare. The FOC is equal to:²²

$$\begin{array}{@{}rcl@{}} \frac{ \partial S W}{ \partial k_{i}} & = & 2 (b -1) (k_{i} -1) -b e q_{i} = 0 , \\ \frac{ \partial S W}{ \partial k_{j}} & = & 2 (b -1) (k_{j} -1) -b e q_{j} = 0, \end{array} $$

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and solving simultaneously we have

$$ k_{i}^{E P} = 1 +\frac{b e q_{i}}{2 (b -1)} ,k_{j}^{E P} = 1 +\frac{b e q_{j}}{2 (b -1)} $$

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where the superscript EP indicates the ex-post case.

1.6.2 Stage 1: Firms decide on output

After the substitution of $k_{i}^{E P}$ into the profit’s equation we have ${\Pi }_{i}^{E P} =q_{i} (a +(-1 -c_{i} -\frac {b^{2} e^{2}}{4 (b -1)^{2}}) q_{i} -q_{j})$ and the FOCs are²³

$$ \frac{\partial {\Pi}_{i}^{E P}}{ \partial q_{i}} =a +\left( -2 -2 c_{i} -\frac{b^{2} e^{2}}{(b -1)^{2}}\right) q_{i} -q_{j} = 0 $$

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where the optimum level of output is

$$ \bar{q}_{i}^{E P} =\frac{4 a G^{4} (1 + 2 c_{j}) + 2 a G^{2} b^{2} e^{2}}{4 G^{4} F + 4 G^{2} b^{2} (2 +c_{i} +c_{j}) e^{2} +b^{4} e^{4}} $$

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where G = b − 1, F = 3 + 4c_j + 4c_i(1 + c_j). Moreover, the optimum level of technology (after the substitution) equals $\bar {k}_{i}^{E P} =\frac {2 a G^{3} b (1 + 2 c_{j}) e +a G b^{3} e^{3}}{4 G^{4} F + 4 G^{2} b^{2} (2 +c_{i} +c_{j}) e^{2} +b^{4} e^{4}}$ and similar we take the level of the emissions $\bar {y}_{i}^{E P} =\bar {q}_{i}^{E P} \bar {k}_{i}^{E P}$, the damage equation $D F^{E P} =e \bar {y}_{i}^{E P}$, the price $\bar {p}_{i}^{E P} =a -\bar {q}_{i}^{E P} -\bar {q}_{j}^{E P}$, the profits ${\Pi }_{i}^{E P} =\bar {q}_{i}^{E P} (a +(-1 -c_{i} -\frac {b^{2} e^{2}}{4 (b -1)^{2}}) \bar {q}_{i}^{E P} -\bar {q}_{j}^{E P})$ and the social welfare $\bar {SW}^{EP}$ (the last results are characterised by complex and long equations).