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3Dynamic Oligopoly Competition with Public Environmental Information Disclosure1
Guiomar Martín-Herrán1, Abderrahmane Sokri2, Georges Zaccour3
1 Departamento de Economa Aplicada (Matematicas),
Universidad de Valladolid,, Spain 2 GERAD and IEA,
HEC Montréal, Canada 3 Chair in Game Theory and Management and GERAD HEC Montréal, Canada
Abstract. The main purpose of this paper is to study the impact of traditional and emergent environmental regulations on firms’ strategies and outcomes. The former corresponds to, e.g., emissions taxing, and the latter to emissions reporting. To do so, we consider a differential game between two polluting firms.
The market potential of each firm varies with its goodwill, which evolution depends on its advertising effort and on its emissions, as well as those of its competitor. We characterize the open-loop Nash equilibrium and contrast the results obtained under different regulatory regimes (laisser-faire, traditional regulation, emergent regulation and dual regulation). We also carry out a sensitivity analysis to assess the impact of some key model parameters on strategies, steady-state goodwill stocks and payoffs.
Keywords: Traditional Environmental Regulation; Public Disclosure Program; Pricing; Advertising; Goodwill; Differential games.
Introduction
Environmental issues, ranging from smog in cities to global warming, undoubtedly rank very high on the political and scientific agendas. Almost all decision-makers are
1 Research supported by NSERC, Canada, and FQRSC, Quebec. Research completed when the first author was a visiting professor at GERAD, HEC Montreal. The first author’s research
was partially supported by MEC under project SEJ2005-03858/ECON and by JCYL under project
VA045A06, co-financed by FEDER funds.
calling for at least some environmental regulation to limit the negative externalities of human activities, and to face the problem of market failure. Environmental regulations can be schematically classified into two categories. The first one, to which one usually refers as traditional regulation, consists in monitoring firms and enforcing reductions in their pollutant emissions. Instruments used by the regulator include, e.g., emissions quotas, taxes and subsidies. The second approach, emergent regulation, consists of forcing the firm to provide information on its environmental record, through what is known as a Public Disclosure Program (PDP). The rationale here is that the consumer and financial markets will react and make decisions that are consistent with the environmental performance of the firm. The increasing popularity of the PDP is due to its supposedly lower cost and higher effectiveness, as compared to traditional means.
When it comes to the impact of environmental regulation on firms’ profit and competitiveness, two opposite opinions are found in the literature:
• Supporters of the “win-win” paradigm, or what is referred to as the Porter hypothesis [Porter, 1991], suggest that a severe environmental regulation may have a beneficial effect on firms by stimulating innovation. Porter and van der Linde (1995) suggest that environmental regulation can create a double dividend by improving social welfare as well as firm profit. Taking stock of several case studies of firms, they argue that “properly designed environmental standards can trigger innovation that may partially or more than fully offset the costs of complying with them”. Mohr (2002) shows that endogenous technical change makes the Porter hypothesis feasible, but concludes that when the result is consistent with the hypothesis, the policy adopted is not necessarily optimal. Through the use of a principal-agent model, Ambec and Barla (2002) show that, by reducing agency costs, an environmental regulation may enhance firm benefits as well as innovation.
• Supporters of the traditional economic paradigm consider that environmental regulation, since it is a cost, harms profit and spoils competitiveness. For instance, Smith and Walsh (2000) report an experimental test of the Porter Hypothesis. They provide one explanation that the empirical difficulties in rejecting the Porter hypothesis are due to limitations in the economic methods of decomposing productivity. Gray (1987) shows that OSHA2 and EPA3 regulations may be responsible for about 30% of the reduction in productivity growth in the American manufacturing sector during the seventies. Dufour et al. (1998) state that the decline in productivity growth in the Quebec manufacturing sector during the eighties is at least partially attributed to environmental regulation. Stewart (1993) argues that imposing stringent environmental regulation and liability rules on firms may harm their international competitiveness.
2 Occupational Safety and Health Administration.
3 Environmental Protection Agency.
To conclude, empirical findings are clearly not unanimous regarding the impact of environmental regulation on firms’ profits and welfare. It also seems that, to date, there is no theoretical framework that fully accounts for the complex relationship between environmental regulations and competitiveness (see, e.g., Jaffe et al. (1995) for a discussion of this topic4). An alternative explanation of the “win-win” possibility may lie in the firm image (goodwill or reputation). By being greener, whether compelled by regulation or through voluntary actions, a firm may enhance its competitiveness, at least in market segments where consumers value the environment attribute. The main purpose of this paper is precisely to study the impact of a PDP and of traditional regulation on the long-term goodwill of competing firms, and on their equilibrium policies. For this, we adopt a differential game model in which two firms compete in prices and advertising. The latter, as well as the environmental records of both firms, feed the goodwill stock of each, which in turn determines its market potential5. Assuming that consumers prefer the products of greener firms, and by linking public disclosure actions to the firm’s profit via its goodwill, we extend to a competitive framework the monopoly setting studied in [Sokri and Zaccour, 2007]. As in that paper, we consider here that the regulator can use a tax/subsidy regulation and a PDP to curb pollution. We analyze and contrast different scenarios, namely a laisser-faire scenario where there is no regulation; a scenario with one regulation (either a traditional or emergent regulation); and finally, a dual-regulation scenario. We employ numerical experiments to illustrate the kind of insight that one can get from the model. The remainder of the paper is organized as follows. In Section 2 we set up the conceptual model. In Section 3 we characterize the open-loop Nash equilibrium strategies. In Section 4 the numerical results under the different regulatory regimes are interpreted and a sensitivity analysis on key model parameters is conducted. Finally, Section 5 makes some brief concluding remarks.
1. The Model
We consider a duopoly where two competing firms produce differentiated goods over an infinite time horizon [0, to). At each moment t G [0, to) the demand for
4 The Porter hypothesis was controversial and heavily criticized by certain economists. Bradford and Simpson (1996), for instance, assert that it is unlikely that environmental regulations enhance industrial advantage. Palmer et al. (1995) harshly criticize the theoretical foundation of this hypothesis and develop a simple static model of innovation in technology to prove that, on the contrary, an increase in the stringency of environmental regulation makes the polluting firm worse off. For Jaffe and Palmer (1997), the Porter hypothesis is hindered by ambiguity. The authors distinguish at least two different hypotheses:
• The weak version, which says only that regulation will stimulate certain kinds of innovation, and that the additional innovation comes at an opportunity cost that exceeds its benefits;
• The strong version, which suggests that firms do not necessarily find or pursue all profitable opportunities for new products or processes. The regulation induces innovation whose benefits outweigh its cost. For these authors, this strong form of Porter hypothesis is anecdotal and presents environmental regulation as a free lunch.
5 Surveys of goodwill models are found in [Jorgensen and Zaccour, 2004] for the competitive setting, and in [Feichtinger, 1994] for the monopolistic case.
brand i, qi(t), depends on this firm’s goodwill, Gi(t), the price of its product, pi(t), and the price of its competitor, pj (t). The demand function for brand i is assumed to be linear and given by
where a,bi > 0 and i G [0,1] measures the degree of substitutability between the two varieties. If i = 1, then the products are perfect substitutes; if i = 0, then the two brands do not compete against one other, and the firms behave as monopolists.
As an inevitable by-product, the production of brand i yields ei(t) as pollutant emissions. We assume that these emissions are proportional to production: ei(t) = = aqi(t), with 0 < a < 1.
In order to improve its goodwill, Gi(t), and, hence, its market potential measured by a+biGi(t), firm i can count either on its advertising effort, Ai(t), or on its capacity to reduce its emissions, ei(t). The latter impact will be endogenous in our model. The firm’s image is also affected by the emission behavior of its competitor, ej (t). We assume that the dynamics of firm i’s goodwill stock obeys the following law:
in which ^i,Yi,S > 0. This differential equation extends the standard Nerlove and Arrow (1962) setting by adding the term (yiej — ^iei) which expresses the effect of both firms’ environmental records on firm i’s goodwill. This term captures the idea of environmental reporting, that is the PDP, on the image of the firm. Parameter S is the decay rate of the goodwill stock, which is assumed equal for both firms.
Parameter i measures the marginal effect of the firm’s own emissions on goodwill. The more the firm emits, the more it hurts its image and, consequently, its market potential. Except for its take on the role of innovation, this assumption is firmly within the win-win point of view (see [Porter and van der Linde, 1995], [Blend and Ravenswaay, 1999], [Kristrom and Lundgren, 2003]). On the other hand, each firm benefits, image-wise, from its competitor’s emissions. The inclusion of the differential environmental records term, i.e. Yiej — ^iei, emphasizes that, when judging companies, consumers acknowledge the fact that emissions are unavoidable, and that their grading is comparative. We assume that the marginal effect of a firm’s own emissions on its goodwill is greater than or equal to the marginal effect of the competitor’s emissions, that is, i > Yi.6
Each firm has a constant cost of production, c, and pays a tax t per unit of emissions. The advertising cost of firm i is assumed to be quadratic and convex, i.e. C(Ai) = 1/2A2. Denoting by r the constant positive discount rate, the optimization problem of player i is given by
qi(t) = a + biGi(t) — pi(t)+ipj(t), i,j = 1, 2, i = j,
(1)
6 This assumption is not essential and can be relaxed.
subject to the dynamics in (2).
To recapitulate, by (3) and (2) we have described a two-player differential game with two state variables, the firms’ goodwill stocks Gi(t) and Gj(t), and two control variables for each player, the price and the advertising effort, pk(t),Ak (t), k = i, j. We assume that the game is played a la Nash and that both firms noncooperatively and simultaneously decide their prices and their advertising investments. We consider that the firms select open-loop strategies, which means that pricing and advertising strategies are functions of time.
From now on we assume that h = a — (1 — i)(c + aT) > 0, which guarantees that the firms produce a positive quantity when their goodwill stocks are zero. We eliminate the time argument when no confusion may arise.
2. Pricing and Advertising Strategies
Each firm seeks two control paths pk(•), Ak() so as to maximize its objective functional while considering the dynamics of both goodwill firms, Gk,k = i, j.
The following proposition characterizes the firms’ pricing and advertising strategies and outcomes at equilibrium.
Proposition 1. Assuming interior solutions, the firms’ pricing and advertising strategies at equilibrium satisfy:
(a+c+ar)(2 +^) + 25fcGfc + bipGi + 2a[(ikP+yk)^fcfc - (y + yip)^hi]
Pk ~ 4-M2
ap[(lh + yhp)^ik - (lip+Vi)^ii]
, k,l = i,j,k^l, (4)
4 — i2
Ak = 0kk, k = i,j. (5)
An open-loop Nash equilibrium is fully characterized by the following system of linear
first-order differential equations:
y(t) = My(t) + n, (6)
where
y(t) = (Gi(t), Gj(t), 0u(t), 0ij(t), jt), j(t))T,
0ki denotes the costate variable that firm k associates with the goodwill of firm l, and matrix M and vector n are given in the Appendix.
The initial conditions Gk(0) = Gko, k = i,j, and the transversality conditions
lim e rt^ik (t)Gk (t) = 0,k,l = i,j,
t—>-oo
have to be satisfied.
Proof.
See the Appendix.
The advertising strategy in (5) follows the familiar rule of marginal cost (Ak) equals marginal revenue (0kk), here, the shadow price of its own goodwill stock. The pricing strategy in (4) shows that each firm fixes its price depending on both firms’ goodwill stocks and all costate variables. The price increases with both firms’ goodwill stocks as well as with the shadow price that each firm assigns to its own goodwill stock. However, the price decreases with the shadow price associated with the competitor’s goodwill stock.
An open-loop Nash equilibrium of the differential game at hand is fully characterized by a solution to (6), which satisfies the initial and transversality conditions, (4) and (5). To obtain a solution to (6), first, we derive the fundamental solution of (6) and, second, we compute a particular solution that satisfies the initial and transver-sality conditions. As is usually done in infinite-horizon dynamic games to ensure that the transversality conditions are satisfied, we look for a solution to system (6) which converges to a steady state.
Unfortunately, we cannot obtain a closed-form expression of the six eigenvalues of the system matrix M (see the Appendix). However, it is well known (see, e.g. [Hirch and Smale, 1974]) that the general, asymptotically stable solution of the homogeneous system, y(t) = My(t), can be written
ygh(t) = 53 Ki^vi,
iei
where Ki is constant, I denotes the set of indices such that £i is a real negative eigenvalue of matrix M with £i = £j ,i,j G I,i = j and vi denotes the corresponding eigenvector.
The particular solution of the non-homogeneous system (6) is given by
yp(t) = —M-1n,
which corresponds to the steady state of the dynamical system, which is denoted, from now on, by
yss = (GSs ,G^,0H ,0j,j ,j )T.
Therefore, the general solution of system (6) can be expressed as
y(t) = ygh(t) +yp(t) = ^2 KiC^Vi + yss.
iei
The expression of the steady state is set out in the Appendix.
The following remark collects the steady-state values when the game is symmetric in all its aspects.
Remark. In the symmetric scenario, i.e. bi = bj = b, Yi = Yj = Y, yi = Vj = V,
the steady state reduces to:
h[b(r + ô)(o2(y - y>)(7/x - y) - 1) - a((r + S)2 (7 - y) + b2y)]
Den
hbô(r + S + bay)
Den hb2a^S Den
+ b2[bay + (r + S)(1 - a2(7 - y)(jM - y)) + a2S(^2 - y2)(1 - m)]-The steady-state level for the demands of both firms, qSs = qss, is given by
hS(r + S)
GSs = s
0SS = éss y33
éss = Y13 éSS y 31
where
Den =
j Den
Since h is assumed to be positive, in order to ensure a positive demand in the long run, expression Den has to be negative. Under this condition, the steady-state levels for the costate variables 0skk>0hS= i,j are also positive. Therefore, the advertising investments are also positive in the long run. The steady-state levels for the goodwill stocks are positive if and only if b(r + S)(a2(7 - y)(YM - y) - 1) -a((r + S)2(y - y) + b2y) > 0.
The symmetric steady-state prices of the products in the long run are
hS[(r + S + bay)2 - b2a2Y2]
PtS = PT =c + or---------------------—-----------------•
Since we are assuming Den < 0 and y > 7, the steady-state prices are greater than the total cost, c + ar.
3. Numerical Results
It is well known that, in infinite-horizon linear-quadratic optimization models, the time paths of the state (goodwill) and of the controls are exponential functions of time that converge towards the steady state, either from above or below depending on whether the initial condition is higher or lower than the steady state. The initial conditions for the goodwill stocks, and the relationship between the initial values for the state and costate variables, that are required to have a trajectory that converges towards the steady state, give us the initial conditions for the firms’ control variables: advertising investments and prices.
As previously stated, we cannot analytically derive the time paths for the goodwill stocks, and, hence, for the advertising efforts and prices, since the eigenvalues of the
6 x 6 system matrix M in (6) cannot be explicitly obtained. Therefore, in order to illustrate the behavior of the strategies and the outcomes, we provide some numerical examples. Being interested in the long-run behavior of the decision variables, the goodwill stocks, and the firms’ demands and profits, their values are shown at the steady state.
For all reported numerical simulations, the matrix associated with the state dynamics has two negative and four positive real eigenvalues. Therefore, the steady state has a two-dimensional stable manifold. In the tables qsks, n|s,k = i,j denote, respectively, the demand and the instantaneous profit at the steady state for firm k. We retain the following parameter values for the “base” case:
r = 0.1, a = 0.5, c =3, a = 7, m = 0.25, t = 0.2, S = 0.1, bk = 0.1, yk = 0.5, Yk = 0.25, k = i,j.
The analysis is done in two steps. First, we assume that the game is symmetric in all aspects. Second, we study some scenarios in which the game is symmetric in all its aspects but one7.
3.1. Reference Cases
Our aim is to assess the impact on the long-term firm policies of the different regulatory regimes. To start, we wish to contrast the following reference scenarios:
1. Laisser-faire policy: yk = Yk = t = 0,k = i, j;
2. Traditional regulation: yk = Yk =0,k = i,j and t = 0.2;
3. Emergent regulation: t = 0, yk = 0.5, Yk = 0.25, k = i, j;
4. Dual regulation: t = 0.2, yk = 0.5,Yk = 0.25, k = i, j.
In the above scenarios, the regulation parameters (i.e. tax and PDP parameters) assume either their “base” case values or zero when the parameter(s) is (are) not a part of the considered scenario. In all four scenarios, the other parameters are kept at their base case levels. The steady-state values are collected in Table 1 and allow for the following comments:
• Comparing the results of the traditional regulation scenario to their laisser-faire policy counterparts shows a reduction in emissions and demand, an increase in price to consumer, lower advertising and goodwill levels, and finally, less profit. Therefore, the regulation achieves its environmental goal of forcing the firms to pollute less. Being an additional cost to the firms, the regulation increases their prices. Consumers are paying more and the firms are making less profit.
7 Note that the same qualitative effects as those collected in the tables below have been obtained for numerical simulations carried out for different base values of the model’s parameters.
Table 1: Steady-state results for the three special regimes
Laisser-faire Traditional R. Emergent R. Dual R.
Variables
G" 19.00 18.70 12.34 12.14
Pk 6.80 6.84 6.67 6.71
Ms 1.90 1.87 1.64 1.61
4s 1.90 1.87 1.61 1.59
qis 3.80 3.74 3.23 3.18
nr 12.64 12.24 10.52 10.19
• Comparing the results of the emergent regulation scenario to the laisser-faire one leads to the same observations as above, regarding emissions, advertising, goodwill and profit. Surprisingly, the price under a PDP is lower than under a laisser-faire policy. One possible explanation is that, by damaging the reputation of the firm, the PDP is shrinking the market size of the (symmetric) firms, and the firm’s response is to reduce prices. This leads to lower revenues, and hence, less funds are available to invest in advertising to counter the negative effect of emissions on goodwill and profits.
• Comparing the two regulatory regimes shows that the PDP is more efficient in reducing emissions than a tax (15.26% versus 1.58%). Also, the long-term goodwill value is lower under a PDP than under a tax mechanism. Given that the results are obtained with some particular parameter values, one should not consider them as definitive. Nevertheless, we note that in all the numerical simulations carried out for different values of t and y, the impact of a PDP is more pronounced than that of the traditional regime. Actually, contrasting the results of a dual regulation to the results with one regulation shows that adding a PDP to a tax regulation has a much higher impact than adding a tax to a PDP. One possible explanation for this greater impact is that, while a tax hurts “only” the cost side, a PDP has a dual effect. Indeed, it directly hurts the goodwill dynamics (and, hence, the cost of advertising), and, indirectly, the revenues of the firm.
To summarize, both regulations achieve lower emissions, and a dual regulation is more effective than any single one. This result was also obtained in the monopoly case in [Sokri and Zaccour, 2007]. Both of the regulations lower the firms’ profits, its goodwill and demand. Interestingly, the two regulations differ in terms of their impact on price to consumer. Whereas the traditional regulation leads to a higher price, the emergent one decreases the price to consumer. Comparing the regulatory regimes, it seems that a PDP has a more pronounced effect than taxing emissions do. Finally, note that in all simulations the goodwill steady-state values happen to be positive, which is not a requirement in our model.
3.2. Sensitivity Analysis
We turn now to a sensitivity analysis, to assess the impact of key parameters on the equilibrium results in the dual regulatory regime. Recall that the model has 13 parameters, namely:
Demand and cost parameters: a, bi, bj, i, a, c,
Regulation parameters: t, yi, yj, Yi, Yj,
Other parameters: r, 5.
All parameters that are common to the two players have a rather intuitive and straightforward impact on the equilibrium’s control and state values. We, therefore, do not include the results8. For instance, the parameter a has a positive scale effect, i.e. any increment in its value leads to an increase in any steady-state value. Conversely, a greater unit cost c (or a higher decay rate 5) implies a decrease in the steady-state values of the goodwill stocks, advertising, emission levels, and demand and profit levels. Further, increasing a amounts to an additional cost, and the impact is the same as the production cost increasing. Finally, varying the discount rate r has a pure translation effect on the results.
Since the focus of this paper is environmental regulation, we make an exception for the common tax rate. Table 2 collects the results of varying t. Actually, it can be shown analytically that an increase in t leads to lower steady-state values of the goodwill stocks, the advertising efforts, the emission levels, and the firms’ demand and profit levels. In order to compensate for this loss of profits, the firm reduces its advertising effort. This in turn negatively affects the firm’s goodwill. A lower goodwill leads to lower demand, and, therefore, lower emission levels. Further, the higher the tax, the higher the price the consumer pays.
Hence, the firms are simply asking the consumer to pay at least part of the regulation bill. A relevant result to point out is that the impact of the tax is rather low. Indeed, multiplying the tax rates by four (from 0.1 to 0.4) leads to a variation of less than 10% in the steady-state values of the strategies, the goodwill, the demand and profit. We now turn to assessing the impact on the strategies and outcomes of varying the parameters that affect goodwill stocks, i.e. bk, Yk and yk. Tables 3,
5, 7 collect the results when these changes preserve the symmetric structure of the game; that is, each modification affects both firms identically. In all cases the other parameters are fixed as in the base case. Afterwards, three asymmetric scenarios are considered:
1. Firms are asymmetric with respect to parameter bk (bi = 0.1,bj e {0.08,0.09,0.1,0.11,0.12});
2. Firms are asymmetric with respect to parameter yk (yi = 0.5, yj e {0.25, 0.5, 0.75,1,1.25,1.5});
8 The results are available from the authors upon request.
Table 2: Sensitivity of steady-state results for changes in t
r = 0 r = 0.1 r = 0.2 r = 0.3 r = 0.4
Variables
Gsk 12.34 12.24 12.14 12.04 11.95
Pk 6.67 6.69 6.71 6.74 6.76
Ms 1.64 1.62 1.61 1.60 1.59
4s 1.61 1.60 1.59 1.57 1.56
qis 3.23 3.20 3.18 3.15 3.13
nr 10.52 10.35 10.19 10.03 9.87
3. Firms are asymmetric with respect to parameter (Y = 0.25, Yj € {0, 0.15, 0.25, 0.35, 0.5});
Tables 4, 6 and 8 collect the results of these numerical simulations. Recall that in all tables the base case corresponds to a symmetric scenario, and, therefore, symmetric steady-state values are attained.
3.3. Impact of bk
Table 3 shows that an increment in parameter bk moves up the steady-state values of all model variables. In our framework, increasing bk amounts to an expansion of the firm’s market potential and an increase in its prestige, which reduces the impact of price on demand. Therefore, it is not surprising that the firms increase their prices with bk. The fact that consumers are becoming less sensitive to price results in higher demand, and consequently, more pollution. The firms increase their advertising levels to compensate for, and even offset, the damage caused by the PDP to their goodwill. The ultimate consequence is a higher profit in the long run. In Table 4 the two
Table 3: Sensitivity of steady-state results for changes in bk
bk = 0.08 bk = 0.09 Base case bk =0.11 bk = 0.12
Variables
G%‘ 8.12 10.01 12.14 14.62 17.56
Pk 6.33 6.50 6.71 6.98 7.31
A'k 1.18 1.38 1.61 1.88 2.21
esk 1.45 1.51 1.59 1.69 1.81
qsk 2.91 3.03 3.18 3.37 3.62
nr 8.68 9.33 10.19 11.31 12.82
columns to the left of the base case correspond to scenarios in which parameter bj is lower than parameter bi = 0.1, whereas the two columns to the right of the base case represent the opposite situation (bj > bi). A quick inspection of the results leads to the following comments:
• The firm enjoying a higher impact of goodwill on demand (higher bk) achieves better long-term results in terms of goodwill stock, demand and profits. It invests more in advertising, charges a higher price to consumers and pollutes the environment more. This qualitative behavior is the same as the one seen in the symmetric case.
• The increase in the emission level of firm j pushes up the goodwill stock of its competitor, Gi. In turn, this shifts upward the demand function, which results in a greater emissions level and a higher price. Both effects trigger a rise in the firm’s profits.
Table 4: Sensitivity of steady-state results for changes in bj
bj = 0.08 bj = 0.09 Base case bj = 0.11 bj = 0.12
Variables
Gf 11.72 11.91 12.14 12.43 12.78
Gf 8.47 10.22 12.14 14.29 16.75
v\a 6.65 6.68 6.71 6.76 6.81
pT 6.38 6.53 6.71 6.93 7.20
Ar 1.58 1.60 1.61 1.63 1.66
AT 1.20 1.39 1.61 1.86 2.15
ef 1.56 1.57 1.59 1.61 1.63
ef 1.48 1.53 1.59 1.66 1.76
q!s 3.12 3.15 3.18 3.22 3.27
9? 2.96 3.06 3.18 3.33 3.52
nss 9.82 9.98 10.19 10.44 10.75
n®s 8.99 9.52 10.19 11.03 12.10
To wrap up, in both the symmetric and asymmetric cases, a higher bk results in good news for the firm, and bad news for consumers (at least price-wise) and the environment (more emissions). The moral here is crystal clear: if the consumer values the firm’s reputation more, by increasing his willingness-to-pay and buying more then the firm takes note of this and satisfies the higher demand, with the unavoidable consequence of polluting more.
3.4. Impact of yk and Yk
We now turn to an assessment of the impact of the PDP parameters, yk and Yk, on the strategies, goodwill stocks and payoffs. These parameters represent how consumers are reacting to the firms’ environmental records, which are made available thanks to the public disclosure program. The simulations presented here are conducted under the assumption that the firm’s goodwill is more sensitive to its environmental record than to its competitor’s. This translates in setting yk higher than Yk. (Clearly, the opposite can be also considered, which would lead to results that mirror those discussed below). Recalling that the parameter yk is fixed at 0.5 in the base case, we, therefore, allow Yk to vary from zero (i.e. the dynamics of the
Table 5: Sensitivity of steady-state results for changes in Yk
Yk = 0 Yk = 0.15 Base case Yk = 0.35 Yk = 0.5
Variables
GV 7.33 10.12 12.14 14.31 17.90
Psk 6.40 6.58 6.71 6.85 7.08
Ms 1.47 1.55 1.61 1.68 1.79
4s 1.47 1.54 1.59 1.65 1.74
Qk 2.93 3.07 3.18 3.29 3.48
nr 8.60 9.51 10.19 10.94 12.25
Table 6: Sensitivity of steady-state results for changes in Yj
Yi = 0 Yi = 0.15 Base case Yi = 0.35 Yi = 0-5
Variables
Gts 11.73 11.98 12.14 12.31 12.56
Gr 7.42 10.23 12.14 14.09 17.07
PV 6.66 6.69 6.71 6.74 6.77
Pf 6.44 6.60 6.71 6.83 7.00
Af 1.58 1.60 1.61 1.62 1.64
Ay 1.48 1.56 1.61 1.66 1.75
ef 1.56 1.58 1.59 1.60 1.62
ef 1.48 1.55 1.59 1.63 1.70
q!s 3.12 3.16 3.18 3.20 3.24
q? 2.97 3.09 3.18 3.26 3.39
nss 9.87 10.06 10.19 10.33 10.53
uss 8.81 9.62 10.19 10.79 11.73
goodwill stock is not affected by the competitor’s emissions) to 0.5 (i.e. both players’ emissions have the same effect on the evolution of the goodwill stock). Table 5 presents the results in the symmetric scenario and Table 6, those resulting from the asymmetric case. In Table 6 the middle column corresponds to the base case, where Yj = Yi = 0.25. The first two columns represent scenarios in which the marginal impact of firm Vs emissions on firm j’s goodwill stock is lower than the marginal impact of firm j’s emissions on firm i’s goodwill stock (that is Yj < Yi = 0.25).
The last two columns consider opposite scenarios (Yj > Yi). Tables 7 and 8 report the results of varying <pk in symmetric and asymmetric ways, respectively.
Tables 5-8 allow for the following remarks:
• As can easily be noticed from Tables 5-8 (and should actually be expected), increasing <pk has the same qualitative impact as decreasing Yk. A higher value of yk, meaning that consumers are more sensitive to the environmental record of the firm, leads to a lower goodwill stock, and consequently, to lower demand and prices. This means less revenues, and, therefore, less available funds for advertising.
• The lowest emissions levels are registered when yk > 0 and Yk = 0, i.e. when consumers judge a firm’s environmental record on its own merit and not on a comparative basis (see Table 5).
• The symmetric variations in y have a pronounced, but less than proportional, impact on goodwill stocks, and a modest impact on all other variables. For instance, from Table 5 we see that moving from Yk = 0.15 to Yk = 0.5 (i.e. more than tripling the value of this parameter) leads to a 75% increase in the goodwill and a variation of less than 50% on emissions, prices, advertising and profits. The symmetric variations in y have a much greater effect (see Table 7). Note that the goodwill steady-state value can even be negative, meaning that
Table 7: Sensitivity of steady-state results for changes in yk
ipk = 0.25 Base case ipk = 0.75 Pk — 1 ipk = 1.25 ipk = 1.5
Variables
GV 18.27 12.14 7.30 3.38 0.14 -2.58
Pk 6.97 6.71 6.51 6.25 6.22 6.10
Ay 1.83 1.61 1.44 1.30 1.19 1.09
ey 1.80 1.59 1.42 1.28 1.18 1.08
ipii 0.11 0.09 0.08 0.07 0.06 0.05
qsks 3.60 3.18 2.84 2.57 2.35 2.16
nr 12.26 10.19 8.67 7.52 6.62 5.90
the market potential is lower than what is normally expected (a case where the market potential is given by a).
To summarize the findings on y and y, if the values assumed in this exercise were empirical, then the message to each firm would be twofold: first, a PDP can be quite damaging in terms of reputation; and, second, what really matters is each firm’s environmental record rather than a comparative one.
4. Conclusion
In this paper we explored the relationships between environmental regulations and the goodwill of competing firms. One result is that regulation, regardless of the type, seems to correspond to bad news for the firms. Therefore, green goodwill seems not to be sufficient to achieve the “win-win” outcome that was hoped for. Our numerical results allow us to note two differences between traditional and emergent regulations. First, whereas a higher tax rate induces a higher price, an increase in the impact of information reporting leads to a lower price. In the tax regulation, the firms shift to the consumer any cost increase that results from the higher tax rate, which violates the “polluter pays” principle on which the tax mechanism was founded. In the PDP scenario the firms lower their prices in an attempt to compensate for the decrease in demand resulting from a loss of brand image that would be associated with a more stringent impact of a PDP, as measured by yk. Second, our simulations tend to show
Table 8 Sensitivity of steady-state results for changes in pj
ipj = 0.25 Base case ifij = 0.75 n = i ipj = 1.25 ifij = 1.5
Variables
GY 12.74 12.14 11.66 11.25 10.91 10.62
GT 17.58 12.14 7.73 4.07 0.98 -1.65
pY 6.77 6.71 6.67 6.63 6.59 6.57
pT 6.91 6.71 6.56 6.43 6.32 6.23
Ar 1.64 1.61 1.59 1.57 1.56 1.54
AT 1.80 1.61 1.46 1.34 1.23 1.14
eY 1.16 1.59 1.57 1.55 1.54 1.53
ess eJ 1.77 1.59 1.44 1.32 1.21 1.12
q!s 3.23 3.18 3.14 3.11 3.08 3.05
9" 3.54 3.18 2.88 2.63 2.43 2.25
n Y 10.51 10.19 9.93 9.72 9.54 9.39
n r 11.88 10.19 8.90 7.88 7.06 6.38
that a PDP seems to be more efficient in curbing emissions than taxation is. Not surprisingly, two regulations achieve better environmental results than any one taken separately.
This study is the first attempt to study in a dynamic and competitive framework the relationships mentioned above. Therefore, we made two simplifying assumptions that are worth relaxing. First, it would be interesting to adopt the more conceptually appealing feedback-information structure and contrast the resulting Nash equilibrium with the open-loop equilibrium obtained in this paper. Clearly, there is no hope of analytically fully characterizing the feedback equilibrium, and, therefore, we would need to resort to numerical methods. Second, it would also be interesting to introduce abatement capital, and, hence, fully participate in the debate surrounding the Porter hypothesis.
Appendix
Proof of Proposition 1.
We have to prove that the following system of linear first order differential equations completely characterizes an open-loop Nash equilibrium:
y(t) = My(t) + n, (7)
where
y(t) = (Gi(t),Gj (t),0u (t),^ij (t),^ji(t),^jj (t))T,
_ (ah(li ~ Vi) oMlj ~ <Pj) hh bjh \T
\ 2 — jj, ’ 2 — jj, ’ 2 — /j,’ ’ ’ 2 — jj,J ’
with
h = a — (1 — ¡i)(c + ar),
mii mi2 mi3 mi4 mi5 mi6
m2i m22 m23 m24 m25 m26
m3i m32 m33 m34 m35 m36
0 0 m43 m44 0 0
0 0 0 0 m55 m56
m32 m62 m63 m64 m65 m66
M
and mrs, r,s = 1,..., 6 are the following constants: abi (iip — 2^i)
mn =
mi3 =
abj (2y — Wi)
■7^5---------*■ "■■= = ——:
4 - ¡a,2 + a2(7»M + y>»)(7+ (2 ~ M2)y»)
4 — yU,2 ’
mi4 = —
mi5 =
a2(Tj + /xy>j)(7»/x + (2 - M2)^) 4 — yU,2 ’
q2(7i + - /j,2) +
4 — yLt2 ’
mi6 = —
«2(7jM + y,j)(7»(2 ~M2) +m) 4-M2 ’
m3i = —
_2b2_ A- ¡j2
m32 =
m34 =
bibj i abi((2 — \i2)^i — 2^ii)
-----J-^r, rriss = r + 0 H--------------------------------5------------,
,2 4 — i2
+ №i)
A — fj? ’
4 — i2
abi(2fj,<pj - Yj(r2 - /j,2)) A-fj,2 ’
m35 =
abinijjH + tpj)
m36 =--------------¡5---, m43 = -abjji, m44 = r + d + abjtpj.
4 — i2
The other constants can be obtained following next rule:
mii ^ m22, mi2 ^ m2i, mi3 ^ m26, mi4 ^ m25, mi5 ^ m24, mi6 ^ m23,
m3i ^ m62, m33 ^ m66, m34 ^ m65, m35 ^ m64, m36 ^ m63,
m43 ^ m56, m44 ^ m55,
where the arrow indicates that in each case the subscripts i and j have been interchanged. The current Hamiltonian of firm k can be written as:
Hk(Gk, Gi, Aj.,pk, V’fcfc, V’fci) = (Pk — c)qk — — Ak — rek +
+ ^kk (Ak — fk ek + Yk ei — SGk) + 0kt (Ai — ipi ei + Yiek — SGi),
where 0ki,k,l = i,j, denote the firm’s k costate variables associated with the state variables Gi,l = i,j. Replacing the expression of the emission levels in terms of the production and rearranging terms, the current Hamiltonian of firm k can be rewritten as:
Hu{Gk, Gi, Ak,Pk, V’fcfcj V’fci) = (Pfc — c — Ta)qk — 2^k +
0kk (Ak - a(yk qk + Ykqi) - àGk ) + 0ki (Ai - a(yi qi + Yiqu) - àGi )•
Substituting the demand functions of both firms given by (1) the Hamiltonian of firm k reads:
Hk(Gk, Gi, Ak,Pk, V’fcfc, tpki) = (Pk — c — ra)(a + b]~G]~ — pk + npi) — +
+0kk[Ak + a(a(Yk -yk) + (yk + MYk)Pk - (ykM+Yk)pi + YkbiGi) - (aykbk + à)Gk] + +0ki[Ai + a(a(Yi - yi ) + (yi + MYi)Pi - (yiM+ Yi )Pk + Yibk Gk )- (ayi bi + 5)Gi ]•
Assuming interior solution, the first order necessary optimality conditions for manufacturer k are given by:
dH
~ A (•) =i)kk — Ak = 0, (8)
dAk
dHfa
——(•)=a+c+ar+6fcG'fc-2pfc + /ufcpi4«[(7fc/Ui + yfc)V’fcfc-(7i + /Uiyi)V’fci] = 0 dpk
Gk = Ak +a [a( Yk - y k ) + (y k + MYk )Pk -yk M+Yk)pi +YkbiGi ] -(ay k bk +à)Gk,
Gk (0) = Gk0,
dHfa
V’fcfc = ripkk - -w^r(-) = (r + 0 + bkaip^ipkk + bk(c+Q.T - Pk - aYl'tpki),
OGk
dHfa
ipki = r'tpki - -ktt(-) = (r + 0 + biaipi)ipki - biQ.Yk^kk,
OGi
lim e-rt0kk(t)Gk(t) = 0, lim e-rt0ki(t)Gi(t) = 0, k = l• t—— t——
From equation (8) we get the expression of the optimal advertising investment given in (5). The advertising effort of each firm coincides with the costate variable that this firm associates to its own goodwill stock. To obtain the optimal expression of prices Pi and pj in (4), we have to solve simultaneously the two equations system given by (8) for k, l = i,j,i = j. Once the optimal advertising investments and prices in (5) and (4), respectively, have been replaced in the dynamics of the goodwill stocks and of the costate variables, the differential equations appearing in the first order optimality conditions can be rewritten equivalently in matrix notation.
Steady state
The steady state of the dynamical system (6) has been computed using Mathe-matica 5.2. as follows. From the differential equations for the costate variables 0ij
and 0ji, the following relationships linking the values of the costate variables at the steady state are obtained:
0ki{0kk) = k I = i j k ^ I. (9)
r + S + bta^i
Replacing these expressions into the differential equation for the costate variables 0ii and 0jj and equating to zero, the steady-state values of Gi and Gj in terms of
0ii and j are given by: Gk (0kk, 0n) =
2(r + 5f + a(r + 6)(2bi<pi + bk{ifk - 7k/x)) + a2bkbi((pk(pi - YkYi) , _
b2k(r-+ 5 + biacpi) kk
(r + S)2 + a(r + S)(biw + bkWk) + a2bkbi(^kwi — YkYi)
bkbi(r + S + bk awk) (a — (1 — j))(c + a.T)
J0U —
k, l = i,j,k = l.
bk
Substituting Gi and Gj given in (11) into the differential equations for these variables
in the dynamical system (6), equating to zero and solving for 0ii and j- the steady-
state values 0SS and are obtained:
,ss _ Num(rksk) ( ,
DenMV’ (10)
where
Num(0skk) = —bkSh(r + S + bi awi ){(r + S)(bk ajk + S(2 + j))(r + S + bk awk) —
—b2((r + S)(1 + a2Yi^i) + bka^k) + b*a[(r + S)((r + S(2 + j))wi — Yi Sj)+
+bk a((r + S(2 + j))wk Wi) — (rj + S(1 + 2j))Yk Yi ]},
Den(0kl) = a2WiWj bfbf + Bibfb2 + Bj b2b3 + C^bj + Cj jbi + Dtf + Dj b?+ +Eb2b2 + Fib2 + Fjbj + Hibfbj + Hjbjbi + Ibibj + Jibi + Jjbj + S2(r + S) (4 — j2),
Bi = a[(r + S)wi(1 + a2(Yij^i — w2)) + a2 Swj (Yi Yj — Wi Wj)],
Ci = a2(r + S)[a2(r + 3S)jWiYi (YiYj — WiWj) — (r + 4S)wiWj + Yi Yj S],
Di = —aS(r + S)2^i(2 + 3a2Yij^i),
E = (r + S)2 — a4 (YiYj — Wi Wj )[(r + 2S)2(Yi Yj J2 + WiWj) — S2(Yi Yj + WiWj J2)] + + a2(r2 + 3rS + 2S2)((Yij — Wi)Wi + (Yj J — Wj )Wj),
Fi = —S(r + S)2 [2(r + S)+ a2 Wi(2Yij(2r + 3S) — Wi(2r + S(4 — j2)))],
Hi = —a(r + S){(r + S)2(Wj — a2(Yi J — Wi)(YiYj — WiWj)) — S[r(Yj (J(1 +
+ a2Yi2) — a2YiWi(1 — 3J2)) + (4a2Wi(Wi — YiJ) - 3)Wj) + S(Yj(J(1 + 2a2Yi2) —
— a2 YiWi (4 — 5j2)) — (3 + a2 Wi (5YiJ — 7Wi + 2j2Wi))Wj)]},
I = —a2(r + S)2[(r + S)2 (YiYj — Wi Wj) — S(Yi (Yj (2rj2 + 5Sj2 — 4S) — (2r +
+ 3S)jWj) + Wi( — (2r + 3S)jYj + Wj (6r + 15S — 4Sj2)))],
Ji = —a(r + S)3(Yi j(r + 3S) — 2Wi(r + S(4 — j2))).
Constants Bj, Cj, Dj, Fj, Hj and Jj can be obtained replacing the subscript i by j, and vice versa.
The final expressions of the steady-state values Gkk, Gjk, and 0jk can be easily
obtained replacing (12) in (11) and (10).
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