Environmental taxation with endogenous choice of environmental quality Christos Constantatos Department of Economics, University of Macedonia
Christos Pargianas Department of Economics, Brown University
Eftichios S. Sartzetakis 1 Department of Accounting and Finance, University of Macedonia
Abstract The present paper examines the role of environmental policy in the presence of imperfect competition and endogenous quality choices, assuming environmentally conscious consumers. We compare the shortsighted policy intervention (the government takes the choice of environmental quality as given) to the second best maximization of social welfare, where the environmental tax is set taking into account both its long run effect on quality choices, as well as its short run effect on quantities. We find that ignoring long run effects leads to the introduction of less polluting product types, yet, higher environmental damage. Firms anticipate that the environmental tax will depend on their product’s emissions per unit of output, and increase the environmental quality of their product in order to enjoy a lower tax rate. Total emissions increase despite the reduction in emissions/unit and social welfare decreases relative to the second best. Our results imply that a short-sighted environmental policy may induce higher environmental quality but it might also be detrimental to the environment. JEL classifications: D62, H21, L13
1
Eftichios Sartzetakis, Department of Accounting and Finance, University of Macedonia, 156 Egnatia Str., Thessaloniki 54006, Greece. E-mail address:
[email protected]. The authors would like to acknowledge financial support from the Pythagoras I (Environment)-Funding of research groups in the University of Macedonia, Priority Action 2.6.1m, Action 2.6.1, Measure 2.6, to be implemented within the framework of the Operational Programme “Education and Initial Vocational Training II (EPEAEK) and co-financed by the European Union [3rd Community Support Framework, 75% financed by the European Social Fund 25% National Resources]
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1
Introduction
Output or product taxes are taxes imposed on products that are environmentally harmful either when used in production or when consumed. While often confused with Pigouvian taxes, output taxes differ from the latter in that they are imposed on output rather than on emissions. Pigouvian and output taxes are equivalent if there is a one-to-one correspondence between output level and emissions. While prices and quantities are more easily observable and verifiable, verification of emissions levels is very costly. So it is easier to impose a commodity tax/subsidy policy, relative to a Pigouvian tax/subsidy on emissions. Many papers consider output tax/subsidy policies rather than emission tax/subsidy policies. Petrakis et al. (2004) considers a model with two firms which produce two differentiated goods and compete in prices. In this model consumption generates two types of damages: damage to the individual consumer and an environmental externality. One good causes no damage at all and it is called the clean good, while the other, called the dirty good, generates damages both to the consumer using it and to the society. There are also two groups of consumers, those that have perfect knowledge of the negative effect of the dirty good’s consumption and those that have no knowledge at all. As expected, prices of both goods are higher after imposing the tax and the higher the tax rate, the more of the clean and less of the dirty good is consumed by both groups of consumers. There is, however, a problem due to the fact that the regulator cannot distinguish between the two groups of consumers: the tax rate being common for all consumers is set at a higher (lower) for the informed (uninformed) consumers relative to the social optimum, since the informed already take into account part of the externality as their individual damage. This means that at the equilibrium, the informed (uninformed) consumers purchase less (more) than their optimal quantity of the dirty good. Since the dirty good imposes external damages in addition to individual damages, the inability of the regulator to distinguish between the two groups of consumers allows uninformed consumers to partially free ride. Bansal and Gangopadhyay (2001) considers a model with output tax/subsidy policies in order to show how these policies affect the clean up levels. In their model two firms produce a physically homogenous good x and they play two-stage game. In the first stage they choose the clean up levels and in the second stage they choose prices. Consumers derive utility from the polluting good x and a composite good, 2
money. Each consumer buys only one unit of x. Consumers are environmentally conscious. They are willing to pay a higher price for x, if it is environmentally better. Production of the homogenous composite good has no environmental impact. In this work we examine the role of environmental policy, in the presence of imperfect competition and endogenous quality choices. We consider two different cases. In the first, the environmental tax is set taking into account both its long run effect on quality choices and its short run effect on prices. In the second, only the latter effect is taken into account, as is done in many cases. It turns out that results of the two models are completely different which in turn implies that environmental policy may be misleading when its effects on quality choices are not taken into account. The first of the aforementioned cases is examined using a model where the regulator chooses the tax level at the first stage, firms choose their qualities at the second and compete in prices at the third (Model A). In order to examine the second case we use a model where the firms choose their quality during the first stage, the regulator imposes a tax at the second stage, and firms compete in prices at the third (Model B). Several conclusions come out from our study: social welfare and taxes are higher when the regulator plays first. On the other hand, profits and both products environmental quality are higher when firms play before the regulator. Despite the higher environmental quality of both products in model B, total environmental damage is bigger in that model since quantities consumed are larger. The regulator’s environmental preferences are also very important. When the regulator cares more about the environment, optimal taxes are higher and the environmental problem abates, but the market distortion problem worsens: prices go up and quantities are reduced. Finally, we examine the regulator’s time consistency. We find that if the regulator could play twice, he would have no incentives to change the tax, since he could only marginally improve social welfare. Thus, we can say that the regulator is time consistent and the model that dominates is the long run model. The dissertation is organised in the following way: chapter 2 contains the theoretical background, chapter 3 reviews the available environmental policy instruments, chapter 4 presents the model and its main results, and chapter 5 concludes.
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2
The model
Assume that there are two single-product firms, 1 and 2, producing a differentiated good and competing in prices. The representative consumer’s utility depends on the consumption of the two products and of the numeraire good Y , according to the function: 1 (1) U = (a + vh )qh + (a + vg )qg − (qh2 + qg2 + 2γqh qg ) , 2 with, a > 0, 0 < γ < 1, vh , vg ≥ 0, where qh , qg represent the consumed quantity of each firm’s product. The utility function in (1) exhibits a taste for brand name variety in that the representative consumer prefers to consume both firms’ product. On the other hand, the parameters vh , vg increase the representative consumer’s willingnessto-pay for the corresponding product, they can therefore be considered as a product quality parameter. Therefore, the model allows for both brand name and quality differentiation. When γ = 1 and vh = vg products 1 and 2 are completely homogeneous and the only thing that matters is their total quantity consumed. On the other hand, when γ = 0 there is no substitutability at all between the two products. When vh 6= vg , there is an element of vertical differentiation between the two goods in the model. The cost of each producer is, k (2) Ci = cqi + vi2 , 2 with, i = h, g, and c, k > 0. Total cost consists of two parts, variable cost, cqi , and fixed cost, k2 vi2 . The former is assumed to be the same for the two firms and, for simplicity, it is also set equal to zero. The latter is increasing in the environmental quality of each good.
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Environmental policy
Consumption of the good in question generates environmental damages. The per unit damage generated by each firm’s product is inversely related to that product’s quality v. This specification implies either that high qualities pollute less, or that consumers partly (but not fully) internalize the externality emanating from their own consumption. The regulator maximizes, therefore, the following social welfare function 4
k SW = U − n(v − vh )qh − n(v − vg )qg − (vh2 + vg2 ) , (3) 2 with n, v > 0, vh , vg ≤ v. The term v represents a quality level that creates no pollution, while n can be interpreted as either a technical parameter translating quality deviations from v into social damage, or as the importance society accords to the environmental problem. In order to correct for the environmental externality, the regulator can impose a per unit output tax. In what follows we consider two three-stage games, A and B. At the first stage of game A the regulator chooses a tax rate that is imposed on each firm’s product, at the second stage both firms choose the quality of their product while at the third stage they compete in qualities. In game B firms choose their quality at the first stage while at the second stage the regulator chooses the tax rates applying on each firm’s product. Price competition takes place at stage 3 of model B as well. The two games represent equivalent types of intervention. Obviously, for a given quality level a tax on output corresponds to a tzx on emissions in an one to one basis. However, in game A the regulator when choosing the tax rates knows that qualities will not be unaffected by its decision and takes this influence into account. On the contrary, in model B the regulator considers qualities as given and determines the tax rates accordingly. Firms, on the other hand, in model B anticipate the impact their quality decision may have on the tax rate they will face later on. By appropriately manipulating their quality, they can eventually obtain a more favorable treatment by the regulator. One may say that in model A the regulator considers all the long run consequences of his choice while in model B the regulator considers only short term consequences.
3.1
The pricing stage
We start by examining the third stage which is similar in the two models. Maximizing (1) yields the demand functions for the two goods: qi =
a(1 − γ) + vi − γvj − pi + γpj , 1 − γ2
(4)
k Πi = (pi − ti )qi (pi ) − vi2 , 2
(5)
where i, j = h, g and i 6= j. The profit functions for the two firms, 1 and 2, are:
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with, k > 0 and where ti is the tax that the regulator imposes on each firm. Maximizing each of the above functions with respect to its corresponding price yields the reaction functions: a(1 − γ) + γpj + ti + vi − γvj . 2 Solving the above system yields equilibrium prices: pi =
pi =
(6)
a(2 − γ − γ 2 ) + 2ti + γtj + 2vi − γ 2 vi − γvj . 4 − γ2
(7)
a(2 − γ − γ 2 ) − (2 − γ 2 )(ti − vi ) + γ(tj − vj ) (4 − γ 2 ) (1 − γ 2 )
(8)
qi2 (1 − γ 2 )
(9)
Substituting the prices from (7) back into the demand (4) and profit (5) functions yields quantities and profits as function of qualities and taxes: qi =
Πi =
i i < 0, and ∂p > 0, as expected, which means that when taxation Notice that: ∂q ∂ti ∂ti is higher the quantity of the good is lower and its price is higher because the cost is higher.
3.2
Model A (t-v-p)
In this model the regulator imposes a tax in the first stage, thus affecting both environmental quality choices and prices. In other words this model focuses on both the short run effect (price choice) and the long run effect (quality choice) of a tax. At the second stage both firms maximize their profit functions (9) in order to choose the quality of their product. This yields reaction functions in the (vh , vg ) space, from which equilibrium qualities as functions of taxes and other parameters are obtained: . vi =
2(2 − γ 2 ) [Ψa − (2 − γ 2 ) [(4 − γ 2 )k − 2] ti + γ(4 − γ 2 )ktj ] , 4(2 − γ 2 )2 − 4(4 − γ 2 )(2 − γ 2 )2 + (1 − γ 2 )(4 − γ 2 )3 k2
(10)
where, Ψ = [(2 − γ)(1 − γ)(2 + γ)2 k − 2(2 − γ 2 )] > 0. The second order conditions of the profit maximization ensure that the denominator of (10) is positive, and therefore i i < 0 and ∂v > 0. An increase in the rate of taxation on a the following is true: ∂v ∂ti ∂tj 6
product reduces that product’s quality, while the opposite is true for an increase in the rate of taxation on the other product. This happens because t is not function of v, which means that the regulator gives firms no incentive to increase the environmental quality of their product, and, at the same time, a higher tax reduces marginal revenue. Substituting the v’s from (10) back into (8) we derive quantities as functions of taxes only, qi = qi (a, γ, k, ti , tj ), which we then substitute into (3) to derive social welfare as function of taxes, SW = SW (a, γ, n, v, k, th , tg ). At the first stage, the regulator maximizes SW in order to choose the tax or subsidy to be imposed on each firm. Since firms are completely symmetric, both taxes are the same,2 Φnv − Xa , (11) D where, Φ = (4 − γ 2 ) [(2 − γ)2 (1 + γ)(2 + γ)k − 2(2 − γ 2 )] > 0, X = [16(k + 2n) − 4(6k + 6n − 1)γ 2 + (9k + 4n − 2)γ 4 − kγ 6 ] > 0, and D = 16 [(1 + γ)k − 2n − 1] + γ 2 [8 + 4(6 − γ 2 )n − (1 + γ)(8 − γ 2 )k] > 0. Depending on the values of the parameters, the government could either subsidize or tax the two firms. When the environmental externality is not prominent, that is, nv is relatively small, the government will subsidize production to decrease the existing market distortion. We then substitute taxes from (11) into the expression (10) in order to obtain equilibrium qualities, t∗A =
2(4 − γ 2 )(2 − γ 2 )(a − nv) . (12) D Because of the symmetry assumption, both firms choose the same quality at the equilibrium. The higher is the environmental externality, the lower is the quality that firms choose to offer, since the tax is increasing on the level of environmental externality. By substituting the optimal qualities from (12) into qi = qi (a, γ, k, ti , tj ) and 2 )2 (a−nv) , SW = SW (a, γ, n, v, k, th , tg ), we obtain the equilibrium values qA∗ = k(4−γ D k(4−γ 2 )2 (a−nv)2 ∗ ∗ = (a − nv)qA . and SWA = D vA∗ =
3.3
Model B (v-t-p)
In this model, the regulator attempts to correct the environmental externality given that the firms have chosen dualities at an earlier stage, anticipating government’s 2
The subscript A denotes equilibrium values in the first model.
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intervention. Therefore, the government cannot take into account the fact that the tax affects quality choices in the long run. Instead, it focuses only on the short run effects of tax which are the effects on prices. At the second stage the government chooses the tax level given the firms’ quality choice. We substitute optimal quantities from (8) into the social welfare function (3), to obtain social welfare as function of qualities and taxes, SW = SW (a, γ, n, v, k, vh , vg , th , tg ). Maximization of this social welfare function yields the optimal tax or subsidy as function of the qualities chosen by the firms, ti = (2 − γ)nv − (1 − γ)a − (1 + 2n)vi + γ(1 + n)vj .
(13)
As in the previous case, if the environmental externality is not very prominent the government will subsidize production. It is also clear from (13) that the higher is the ∂ti < 0. firm’s choice of quality, the lower is the tax levied on its product, that is ∂v i The higher is the quality chosen by the firm, the lower is the environmental damage ∂ti > 0, because of strategic and thus, the environmental tax is lower. Furthermore, ∂v j complementarity between the firms’ choice variables. At the first stage, the firms take taxes and chose qualities in order to maximize their profits. Substituting taxes from (13) into the profit functions (9) yields profits as functions of qualities only, that is, Πi = Πi (a, γ, n, v, k, vi , vj ). The two firms maximize their profit function and the system of the first order conditions yields optimal qualities,3 2(1 + n)(a − nv) . (1 + γ)k − 2(1 + n)2
(14)
[(2 + γ − γ 2 )k − 2(1 + n)] nv − [(1 − γ 2 )k + 2(1 + n)] a . (1 + γ)k − 2(1 + n)2
(15)
vB∗ =
Because of the symmetry assumption, both firms choose the same quality. Substituting optimal qualities from (14) into equation (13) yields the equilibrium value of tax, t∗B =
Because of the assumed symmetry between firms, the government levies the same tax, or gives the same subsidy to both of them. By substituting the optimal qualities from (14) and the optimal taxes from (15) into (8) and SW = SW (a, γ, n, v, k, vh , vg , th , tg ), 3
The subscript B denotes equilibrium values in the second model.
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we obtain the equilibrium values qB∗ = (1+γ)k−4(1+n)2 k
4
k(a−nv) , (1+γ)k−2(1+n)2
and SWB∗ =
[((1+γ)k−4(1+n)2 ]k(a−nv)2 [(1+γ)k−2(1+n)2 ]2
(qB∗ )2 .
Comparison of the two model
We start by comparing the optimal qualities that the firms choose under the two models. From (12) and (14) we obtain, vB∗ − vA∗ =
2Ξ1 (a − nv) , [(1 + γ)k − 2(1 + n)2 ] D
where Ξ1 = 8 [(1 + 2n)k − (1 + n)2n]+8(1+2n)kγ −2(4kn+k +2−n−6n2 )γ 2 −2(1+ 4n)kγ 3 + (2 + kn − 2n2 )γ 4 + nkγ 5 . This expression is positive for the range of values of the parameters that satisfy second order conditions and positivity constraints. Figure 1 illustrates the qualities firms choose in the two models, as functions of the cost of variety parameter k.4 We choose to present the comparison in terms of the parameter k because this parameter is critical in terms of second order conditions of the maximization problems. As can be seen in Figure 1, we are interested in values of k that exceed 2(1 + n)2 /(1 + γ)5 so that the second order condition of the profit maximization problem maxΠi (a, γ, n, v, k, vi , vj ) in the second model holds. vi
The quality chosen by the firms in the case that they choose quality before the imposition of the tax, exceeds the quality that they choose when the government chooses the tax level first. We then compare optimal taxes that the two different models yield. From (11) and (15) we obtain, t∗A − t∗B =
2Ξ2 (a − nv) , [(1 + γ)k − 2(1 + n)2 ] D
where Ξ2 = 8(k + 2n) + (4 − γ) [4 + (2 − γ)2 (1 + γ)(2 + γ)k + 4 − 2γ 2 ] n2 + [(2 − γ)2 (1 + γ)(2 + γ)k − 8γ 2 ] n−[14k − 4 − [[(3 − γ)(2 + γ)γ − 2] k − 2γ] γ] γ 2 > 0. Figure 2 illustrates the taxes of the two models as functions of the cost of variety parameter k. 4
In order to derive Figure 1 and all the rest of the figures in this Section we use the following values for the parameters: a = 60, γ = 0.5, v = 20, and n = 1.5. 5 This expression is equal to 8.333, as shown in Figure 1, for the values of the rest of the parameters that we have chosen.
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=
v
60
40
20
k 5
10
15
20
25
30
-20
-40
Figure 1: Comparison of equilibrium qualities in the two models. t
60
40
20
k 5
10
15
20
25
-20
-40
Figure 2: Comparison of equilibrium taxes in the two models. 10
30
q 100
75
50
25
k 5
10
15
20
25
30
-25
-50
-75
-100
Figure 3: Comparison of equilibrium quantities in the two models. Therefore, the government chooses a higher tax if it moves first. We move to the comparison of quantities next. The difference between qB∗ and qA∗ is, qB∗ − qA∗ =
2kΞ3 (a − nv) , [(1 + γ)k − 2(1 + n)2 ] D
where, Ξ3 = 8 − 4γ 2 + γ 4 + 4(4 − γ 2 )n + (4 − γ 2 )n2 > 0. Figure 3 illustrates the quantities chosen by the firms at the equilibrium of the two models, as functions of the cost of variety parameter k. Since the tax rat is always lower when the government takes the qualities as given (second model), it is clear that the quantities chosen by the firms will always be higher when the government moves second. Therefore, when the government takes qualities as given and firms anticipate taxation, they choose a higher quality, so as to force the government to levy a lower tax, which it actually does. As a consequence firms choose to produce a higher level of output. We can now move and compare total welfare under the two models. The difference between SWB∗ and SWA∗ is,
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SW 2000
1000
k 5
10
15
20
25
30
-1000
-2000
Figure 4: Comparison of equilibrium welfare in the two models.
SWA∗ − SWB∗ =
4kΞ4 (a − nv)2 , [(1 + γ)k − 2(1 + n)2 ]2 D
where, Ξ4 = 4k +8[k +2n(1+n)2 ]n+4k(1+2n)γ −2[k(1+3n)−4(1−n)(1+n)2 n]γ 2 − 2k(1 + 3n)γ 3 + [1 + n(k − 2n + n3 )]γ 4 + knγ 5 > 0.Figure 4 illustrates the equilibrium values of welfare in the two models, as functions of the cost of variety parameter k. Although firms choose higher quality and output in the case that they move first, total welfare is higher in the case that the government moves first. This is not surprising since in the latter case the government chooses the optimal taxation to correct the existing distortions. As can bee seen in Figure 5 that illustrates total environmental damage T D = n(v − vh )qh + n(v − vg )qg , despite the fact that firms choose higher qualities when they move first, total environmental damage is higher in this case because firms produce also higher quantities. It is the difference in total environmental damage between the two models that yields higher total welfare in the case that the government moves first. It is important to note that the difference between the two models is decreasing for higher values of k. When the cost of choosing higher quality is very high, the two
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TD
4000
2000
k 5
10
15
20
-2000
-4000
Figure 5: Comparison of equilibrium environmental damage in the two models. modes sigklivouv. Table 1 reports simulation results of models A and B (see the corresponding columns), for selected values of the parameters. SW represents the social welfare, v is the environmental quality, t is the tax per unit of product, p is the price of the good, q is the quantity that each firm produces, Π is the total profits of each firm, and is the total environmental damage that is caused by the production of the good. As we can see (see Table 1) SW is higher in model A than in model B which means that SW is higher when the regulator plays first in the three stage game. On the other hand the profits are higher in the second model, which means that firms prefer to commit to a product quality before the regulator imposes the tax. Moreover, both the environmental quality and the total quantity consumed are higher in model B, while prices and taxes are higher in model A. The C column presents results of a hybrid model: firms choose qualities before taxes are chosen (like in model B) but instead of choosing their profit maximizing quality they choose the quality dictated by model A. Comparing models A and C one can say that model C is like a four-stage model with the following order of moves: stages 1 and 2 are like in model A and stages 3 and 4 like in model B, i.e., at the 13
first stage the regulator chooses a tax rate, at the second stage firms choose their quality level, at the third stage the regulator re-considers his tax choice and at the last stage both firms set their price. This of course raises the question the regulator’s credibility when setting a tax rate at the beginning of the game. If we compare SW in model A and model C we observe that the difference in taxes between these models is sometimes significant while the difference in SW is almost zero. This means that the regulator indeed changes the tax during the third stage in model (t-v-t-p) but that only decreases total environmental damage, and, on the other hand, does not affect SW at all, so, it is quite difficult to say that the tax in model A is time inconsistent. The regulator chooses tax by maximizing SW and so we can say that the regulator does not have any incentives to change the taxation because this change does not affect SW. Let examine the role of n; which shows how important is the environmental problem for the regulator. We can say that n is a kind of weight which reveals the regulator’s preferences about the environment. We observe that the social welfare function can be separated in two parts: the first is net utility form consumption ( ), while the second is related to the environmental externality ( ). The environmental part is weighted with n and the other part with 1. When n=1, the two parts are weighted equally, when n>1, the environmental problem prevails, while when n
