Energy taxes and oil price shock

April 2012, revised September 2013

Abstract This paper examines if an energy price shock should be compensated by a reduction in energy taxes to mitigate its impact on consumer prices. It shows that the consumer price does not increase by as much as the producer price implying a small reduction in the energy tax in dollars. The energy tax rate, on the other hand, decreases sharply. This decline is primarily due to an adjustment in the Pigouvian component: A constant marginal social damage being divided by a higher producer price. The redistributive component of the tax remains at about 10% of the social cost of energy. JEL classi…cation: H21; H23. Keywords: Oil price shock, energy tax, Pigouvian tax, redistributive concerns.

1

Introduction

As energy is heavily taxed in most industrialized countries, an “oil shock” (a sudden and signi…cant increase in energy prices) often leads to political pressure, with various interest groups asking for tax reductions. Connecticut has capped its tax on the wholesale price of gasoline and Iowa has forsaken a previously planned state gas tax increase.1 The issue is regularly comes up in US presidential election campaign as it did during the 2008 campaign. Overseas, the Israeli government cut its gasoline taxes citing the increasing prices as the reason.2 There has been a similar debate in France with many interest groups (truck and …shery industry, agriculture, etc.) asking for an energy tax relief to mitigate the impact of the price increase. Reacting to the recent increases in energy prices, the socialist presidential candidate François Hollande has proposed a legal cap on consumer price of gasoline.3 This implies that any future increases in world energy prices will be absorbed by a one to one reduction in energy taxes. Among the possible justi…cations for energy tax reductions is the idea that consumers have di¢ culties to adjust to strong and sudden price shocks. This is because existing technologies and equipment limit the substitution possibilities in the short run. However, this argument would at best lead to a temporary reduction to smooth the transition. Another argument is based on the regressive character of rising energy prices. The share of energy consumption in total spending tends to decrease with income so that low-income individuals are a¤ected more heavily, relative to their income, than high-income individuals by an oil price shock. Redistributive concerns may then call for an energy tax reduction. This paper studies the validity of this redistributive argument using the model of 1

http://news.yahoo.com/malloy-signs-bill-capping-conn-wholesale-gas-tax-144423620.html http://www.desmoinesregister.com/article/20120331/NEWS/303310028/House-speaker-putskibosh-increasing-state-s-fuel-tax 2 http://www.jpost.com/NationalNews/Article.aspx?id=264228 3 http://www.bloomberg.com/news/2012-04-04/hollande-says-he-d-seek-eu-…scal-pact-revampfreeze-gas-prices.html

1

optimal emission taxation developed by Cremer et al.

(1998, 2003 and 2010).4 En-

ergy is used by households as a consumption good and by the productive sector as an input. This model is adapted to study the impact of an exogenous upward shock in the before-tax price of energy. We calibrate a modi…ed version of this model on US data and simulate the optimal energy taxes for di¤erent levels of this shock assuming a concomitant adjustment in optimal income taxes. We show that redistributive concerns call for a subsidy on energy goods equal to about 10% of its social cost (producer price plus the associated marginal social damage of emissions). Interestingly, simulations indicate that variations in the world price of energy has an almost negligible e¤ect on this percentage. On the other hand, the total tax rate on energy (redistributive plus Pigouvian) decreases sharply as the world price of energy increases. This arises purely as an arithmetic adjustment: A constant marginal social damage is divided by a rising producer price leading to a decline in the Pigouvian tax rate. Nevertheless, it is also true that the consumer price does not increase dollar for dollar with the producer price. As the world price of energy increases, the dollar amount of the subsidy increases too— albeit by small amounts.

2

The model

Consider an open economy wherein people consume two produced goods: a composite consumption good and “energy.” The composite consumption good is produced domestically using “energy inputs,”capital, and labor. Energy, whether used as a consumption good or as a factor input, is imported from overseas. Capital services are also rented from outside.5 Labor is the only factor of production which is supplied domestically. All imports are …nanced through exports of the portion of the general output that is 4

The major di¤erences between these paper and a large number of calibrated general equilibrium models (like Goulder, 1994) is that they do not use a representative agent setting. 5 The assumptions that energy and capital services are rented from outside are made to ensure that the world price of energy and returns to capital will not be a¤ected by domestic decisions.

2

not consumed domestically. Energy, both as a consumption good and factor input, is polluting; the composite consumption good is not. Labor is heterogeneous with di¤erent groups of individuals having di¤erent productivity levels and di¤erent tastes. Denote a person’s type by j, his productivity factor by nj , and the proportion of people of type j in the economy by

j

(where the population

size is normalized at one). Preferences of a j-type person depend on his consumption of non-polluting goods, xj ; consumption of polluting goods, y j , labor supply, Lj , and the total level of emissions in the atmosphere, E. This construct is based on Cremer et al.’s (2010) model. To make this paper selfcontained, we …rst review its main features.6

2.1

Preferences

Consumers’ preferences are nested CES, …rst in goods and labor supply and then in the two categories of consumer goods. All consumer types have identical elasticities of substitution between leisure and non-leisure goods, , and between polluting and nonpolluting goods, !. Di¤erences in tastes are captured by di¤erences in other parameter values of the posited utility function (aj and bj in equations (2)–(3) below). Assume further that emissions enter the utility function linearly. The preferences for a person of type j can then be represented by fj = U (x; y; Lj ; where

j

j

)

E;

re‡ects the “taste parameter,”

j = 1; 2; 3; 4;

(1)

represents the marginal social damage of

emissions assumed to be constant, and U (x; y; Lj ;

j

) =

Qj 6

=

bj Qj aj x

! 1 !

1

+ (1 + (1

bj )(1 aj )y

For more details, see Cremer et al. (1998, 2003 and 2010).

3

! 1 !

1

Lj ) ! ! 1

1

; :

(2) (3)

Consumers choose their consumption bundles by maximizing (1)–(3) subject to their budget constraints. These will be nonlinear functions when the income tax schedule is nonlinear. However, for the purpose of uniformity in exposition, we characterize the consumers’ choices, as the solution to an optimization problem in which each person faces a (type-speci…c) linearized and possibly truncated budget constraint. To do this, introduce a “virtual income,” Gj , into each type’s budget constraint. Denote the jtype’s net of tax wage by wnj . We can then write j’s budget constraint as pxj + qy j = Gj + M j + wnj Lj ;

(4)

where p and q are the consumer prices of x and y, Gj is the income adjustment term (virtual income) needed for linearizing the budget constraint (or the lump-sum rebate if the tax function is linear), and M j is the individual’s exogenous income. The …rst-order conditions for a j-type’s optimization problem are 1

aj xj aj yj (1

aj bj

1 !

q = ; p

(5)

bj ) xj =(1

h aj + (1

Lj )

aj )(xj =y j )

1

1 ! !

i

! (1 !)

=

wnj : p

(6)

Equations (4)–(6) determine xj ; y j and Lj as functions of p; q; wnj and Gj + M j .

2.2

Production technology

The production process, for the composite consumption good, uses three inputs: capital, K, labor, L and energy, D. The technology of production is represented by a nested CES, h O = O (L; K; D) = B (1 )L h i 1 1 = A K + (1 )D 4

1

1

+ ;

1

i

1

;

(7) (8)

where A and B are constants, L and

and

represent the elasticities of substitution between

and between K and D (given ) respectively. Substituting (8) in (7) yields, "

O = B (1

1

)L

+ A

1

h

K

1

+ (1

)D

1

i

( (

1) 1)

#

1

:

(9)

Aggregate output, O, is the numeraire. Capital services and energy inputs are imported at constant world prices of r and pD where the units of D is chosen such that initially pD = 1. Let w denotes the price of one unit of e¤ ective labor,

D

denotes the tax on energy input, and assume that there

are no producer taxes on labor and capital.7 The …rst-order conditions for the …rms’ input-hiring decisions are, assuming competitive markets, O L (L; K; D) = w;

(10)

O K (L; K; D) = r;

(11)

O D (L; K; D) = pD +

D:

(12)

Equations (9)–(12) determine the equilibrium values of O; L; K and D as functions of w; r and pD +

D

[where r and pD are determined according to world prices].

As di¤erent types of people have di¤erent productivities, labor is an heterogeneous factor of production. When a j-type person with productivity nj works for Lj hours, P his e¤ective labor is nj Lj resulting in aggregate supply 4j=1 j nj Lj . Equating this with aggregate demand gives,

L=

4 X

j j

n Lj :

j=1

7

It is not optimal to tax capital in this setting. With the world rental price of capital …xed, a tax on capital income will have to be totally borne domestically. This can only distort the production decisions of domestic producers and is thus suboptimal. Formally, a tax on capital income does not a¤ect the economy’s resource constraint (15). Consequently, the …rst-order condition (A6), and its simpli…ed version (A20), in Appendix A, which characterizes the optimal usage of capital, remains the same with a tax on capital as without. That is, O K (L; K; D) = r characterizes the optimal condition for hiring capital by the economy. As shown by equation (11), this same relationship determines the equilibrium condition for hiring capital by the …rms in the absence of capital income taxes.

5

We choose the units of emissions so that a unit of energy, consumed or used as an input, results in one unit of emissions. Total emissions are then given by, E=

4 X

j j

y + D;

j=1

where

2.3

j

is the proportion of people of type j in the economy.8

Optimal tax policy

The optimal tax policy maximizes an iso-elastic social welfare function W = where

1 1

4 X

j

(fj )1

6= 1

j=1

and

is the “inequality aversion index”. The value of

redistribution in the economy: The higher is here we retain a relatively low value,

0

< 1;

(13)

dictates the desired degree of

the more the society cares about equality,

= 0:1.9

The feasibility of tax instruments depends on information available to the tax administration. Generally, this information allows for linear commodity taxes and non-linear income tax. This is why we restrict our analysis to this case even if other possibilities could be considered.10 Under linear commodity taxation, all consumers face the same commodity prices. The social welfare function (13) must thus be written as a function of the prices of goods. Denote cj the after-tax income (outlay) of a j-type household. Maximizing, the utility function (1) with respect to the budget constraint pxj + qy j = cj ; P The population size is normalized to 1, consequently 4j=1 j y j represents total households’energy consumption. 9 As is well-known, = 0 implies a utilitarian social welfare function and ! 1 a Rawlsian. The value we use is chosen according to the observed degree of redistribution of existing tax systems; see Bourguignon and Spadaro (2000). 10 In Cremer et al. (2008) di¤erent possibilities are examined including the case where all taxes are non-linear. 8

6

j

we obtain the demand functions for xj and y j as xj = x p; q; cj ;

and y j = y p; q; cj ;

j

Substituting these equations in the j-type person utility function (1), we have V

p; q; cj ;

Ij ; wnj

j

x p; q; cj ;

=U

j

; y p; q; cj ;

j

;

Ij ; wnj

j

;

where Ij

wnj Lj :

We have four feasible tax instruments in our model: two commodity taxes, an input tax and an income tax. As the demand functions for goods and the labor supply function are all homogeneous of degree zero, there is no loss of generality when setting one tax rate to zero. Since energy consumption creates an externality we choose to impose a zero tax on non-energy goods. The optimal tax structure is derived as the solution to 2 4 4 j X X 1 j4 j I j j max V p; q; c ; ; y p; q; cj ; j j j 1 wn q;c ;I ;K;D;w j=1

31

D5

j

j=1

(14)

under the resource constraint, O (L; K; D)

4 X

j

x p; q; cj ;

j

2 4 X pD 4

rK

j=1

j

y p; q; cj ;

3

+ D5

j

j=1

the incentive compatibility constraints, V

p; q; cj ;

Ij ; wnj

j

V

p; q; ck ;

Ik ; wnj

j

;

R

0; (15)

(16)

the endogeneity of wage condition, w with L=

O L (L; K; D) = 0; 4 X

j j

j

n L =

j=1

4 X j=1

7

jI

(17)

j

w

:

.

The analytical results of Cremer et al. (2010) can easily be extended to show that the optimal tax on energy inputs (

D)

is Pigouvian and equal to its marginal social

damage of emissions. The optimal tax on the consumption of energy, on the other hand, is generally di¤erent from its Pigouvian level. See Appendix A.

3

Data and calibration

To solve our model numerically, one must know the values of the parameters of the utility functions ( , !, aj , bj , aj , ), and the values of the parameters of the production function ( , ,

,

, A, B). The data sources are the PSID (Panel Study of Income

Dynamics, web site: http://psidonline.isr.umich.edu), US Bureau of Labor Statistics (web site: http://www.bls.gov/) and the US Bureau of Economic Analysis. The two …rst one gives data on households’consumption, income and labor. The latter reports macroeconomic data from the EUKLEMS data base on capital, labor and energy. The calibration process follows the one we have used in our previous paper (see Cremer et al. (2008)).11 The data allows us to identify four types of households, “managers and professionals”(type 1), “technical sales and clerical workers”(type 2), “service workers, operators, fabricators and laborers”(type 3) and “construction workers and mechanics” (type 4). Table 1 provides a summary of the data and parameter values. Finally, observe that our optimal tax calculations are based on the assumption that the government’s external revenue requirement (share in GDP of expenditures on non-transfer payments) remains unchanged. The details of the calibration method are summarized in Appendix B (B1 and B2). 11

However, the actual calibration is completely di¤erent because we use US data in the current paper, while the earlier papers presented applications based on French data.

8

I px qy n L t G M a b

Managers & professionals (Type 1) 35.18 % 68712 51134 3051 1.33620 0.50731 28.0 % 9797 -5085 0.99997 0.53201

p = 1.00000 = 0.66490 A = 1.28395

Technical sales

Service workers, operators, fabricators & laborers (Type 3) 28.86 % 31887 29155 2520 0.71472 0.44015 15.0 % 2280 2290 0.99989 0.39438

Construction workers & mechanics (Type 4) 7.06 % 44111 37498 3100 0.88815 0.48998 15.0 % 2363 741 0.99991 0.46747

Type-independent …gures q = 1.00000 = 0.8 ! = 0.26892 = 0.98662 B = 0.74215

= 0.42141 = 0.54242

& clerical workers (Type 2) 28.90 % 40147 34742 2612 0.90094 0.43961 15.0 % 2195 1034 0.99993 0.39970

Table 1: Calibrated parameters (monetary …gures are in US dollars).

9

4

Results

Optimal energy taxes/subsidies are determined by solving the calibrated version of our model. The forces at work in their determination are twofold. One is Pigouvian in nature. To correct for the marginal social damage of emissions, one wants to impose a correcting tax on energy. In case of energy inputs, this is the only force at work. Another force comes into play in case of energy consumption goods. This arises because of the distributional considerations. Because the share of energy expenditures tends to decrease with one’s income, one may want to subsidize energy consumption goods to o¤set this regressive bias. It is true that an optimally designed income tax mitigates this regressive bias, but in a world of asymmetric information (where …rst best lumpsum taxes are unavailable), it cannot eliminate it completely (as long as Atkinson and Stiglitz Theorem does not apply so that Pareto-e¢ cient tax structures include commodity taxes). There still remains a role for energy subsidies; see Cremer et al. (1998), (2003) and (2010). Let

pig

denote the Pigouvian tax (equal to the marginal social damage of emission)

as de…ned by Cremer et al. (1998),12 2 4 X Ij pig j4 V p; q; cj ; ; wnj

4 X

j

j=1

j

y p; q; cj ;

j=1

In the case of inputs, the optimal energy tax is equal to

pig ;

j

3

D5

:

(18)

see Proposition A2 and

equation (A19) in Appendix A. On the other hand, in the case of energy consumption goods, both forces are at work in determining the energy tax, q

pD : This is also shown

in Appendix A where we prove that the optimal tax on energy goods is given by n o 4 P P kj k Ij Vc q; cj ; wn y q; cj ; j y q; cj ; k k; q

pD =

pig

+

j=1 k6=j

4 P

j=1 12

This is an implicit expression with

pig

:

jy e

q

q; cj ; j

being expressed in units of the numeraire output.

10

(19)

World price of energy (pD )

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Optimal price of energy input (q pig = pD + pig )

1.4823 1.5769 1.6718 1.7668 1.8622 1.9577 2.0534 2.1493 2.2454 2.3416 2.4380

Optimal price of energy consumption goods (q)

1.3359 1.4210 1.5064 1.5920 1.6777 1.7637 1.8499 1.9362 2.0226 2.1092 2.1960

Optimal tax on energy input ( D = pig )

Optimal tax on energy consumption goods (q pD )

0.4823 0.4769 0.4718 0.4668 0.4622 0.4577 0.4534 0.4493 0.4454 0.4416 0.4380

0.3359 0.3210 0.3064 0.2920 0.2777 0.2637 0.2499 0.2362 0.2226 0.2092 0.1960

Table 2: Optimal prices of, and taxes on, energy input and energy goods when

= 0:24:

To calculate these two taxes, we normalize the initial before-tax price of energy at one and set

= 0:24 which is the value used by Cremer et al. (2003) corresponding to a

Pigouvian tax of approximately 50% of the initial produces price of one. The optimal prices of energy input and energy goods are reported in columns 2 and 3, and their optimal taxes in columns 4 and 5, of Table 2 (as the before-tax price of imported energy, relative to the domestic producer price of output in the economy, increases from one to two in column 1). The optimal tax calculations reveal that while the optimal tax on energy input remains basically invariant to the world price of energy, the optimal tax on energy goods decreases as the world price of energy increases. That the tax on energy input does not change re‡ects the fact that the tax corrects for the externality caused by emissions— a cost that is basically independent of the producer price of energy. On the other hand, the decrease in the taxes on energy goods indicate that the tax should indeed be used to mitigate the impact of the increasing world energy prices on consumers. The above observations concerning the behavior of optimal energy taxes are in terms

11

of speci…c taxes. To the extent that these taxes may be levied on an ad-valorem basis, the tax rates will show a di¤erent pattern. Speci…cally, with the producer prices increasing, one would expect the tax rates to be decreasing. This is indeed the case as shown in columns 2 and 3 of Table 3. Both tax rates show sharp declines. The behavior of energy input tax rates warrant no further discussion. However, in the case of energy goods, it will be instructive to look into its di¤erent components separately. In this way, one will be able to discern how each of these components respond to the rise in the world price of energy. To this end, we …rst de…ne the concept of a “Pigouvian price” to re‡ect the “social opportunity cost” of energy. Using Cremer et pig ;

al.’s (1998) de…nition of the Pigouvian tax,

q pig = pD +

this is de…ned by pig

:

(20)

One can then divide the optimal tax on energy consumption goods into two parts: pD = q pig

q with q pig

pD =

pig

pD + q

q pig ;

denoting the Pigouvian component and q

q pig the redistributive

component. Similarly, in terms of tax rates, we have q

pD pD

=

q pig pD q q pig + : pD pD

Columns 4 and 5 in Table 3 show the calculations for these components. Interestingly, the redistributive (incentive) is negative throughout indicating that energy goods should be subsidized relative to other goods. This is in line with the previous literature on this subject and the empirical evidence that suggests energy taxes are regressive. More speci…cally, consider Table 2 again. Initially, when the world price of energy is one dollar and the external emission damage is $ 0.4823, the Pigouvian price $1.4823. Under this circumstance, redistributive concerns call for a subsidy of $0.1464 resulting in a consumer price of energy equal to $1.3359. As the price of energy doubles, the 12

World price of energy

pD 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Optimal tax rate on energy goods

Pigouvian tax component

Redistributive tax component

Redistributive component as a % of social price

pD

q pD pD

q pig pD pD

q q pig pD

q q pig q pig

48.23% 43.35% 39.32% 35.91% 33.01% 30.51% 28.34% 26.43% 24.74% 23.24% 21.90%

33.59 29.19 25.53 22.46 19.84 17.58 15.62 13.89 12.37 11.01 9.80

Optimal tax rate on energy inputs pig

% % % % % % % % % % %

48.23% 43.35% 39.32% 35.91% 33.01% 30.51% 28.34% 26.43% 24.74% 23.24% 21.90%

–14.64% –14.17% –13.78% –13.45% –13.18% –12.93% –12.72% –12.54% –12.38% –12.23% –12.10%

–9.88% –9.88% –9.89% –9.90% –9.90% –9.91% –9.91% –9.92% –9.92% –9.92% –9.93%

Table 3: Optimal tax rate on energy inputs and energy consumption goods with its Pigouvian and redistributive components when = 0:24: Pigouvian tax changes only slightly (from $0.4823 to $ 0.4380). The redistributive subsidy, on the other hand, changes substantially (from $0.1464 to $0.2420 or an increase of 65%). Translating these changes into relative terms in Table 3, one observes that it is the Pigouvian tax rate which changes dramatically (from 48.23% of the energy price to 21.90% in colum 4). On the other hand, the redistributive subsidy does not change much (decreasing from 14.64% to 12.10% in column 5). To summarize these …ndings, the rise in the world price of energy does not a¤ect the Pigouvian tax but increases the required redistributive subsidy measured in dollars. In percentage terms, on the other hand, the rise in the world price of energy lowers the Pigouvian tax drastically but does not a¤ect the subsidy rate by much. Of these latter two changes, that the Pigouvian tax rate decreases is basically an arithmetic artifact: a constant marginal social damage is divided by a higher world price.13 13

If the social damage function is convex, marginal social damage decreases as world price of energy increases (because consumption of energy decreases). This will reinforce the reduction in the Pigouvian

13

Figure 1: Prices and subsidies when

= 0:24

In contrast, the behavior of the redistributive subsidy re‡ects a fundamental point that the subsidy rates, as a percentage of the “social price”, are constant. This …nding is borne out by the calculations reported in the last column in Table 4. They show the subsidy rate on energy goods as a percentage of the Pigouvian price or the social price— re‡ecting both the producer price of energy as well as the social damage of emissions— as the producer price of energy doubles (increases from one to two). Similarly, Figure 1 depicts the Pigouvian price, the optimal price, and the implicit subsidy q pig

q in

dollars as the world price of energy doubles. Finally, it is interesting to note that the constancy of the tax/subsidy rate as a percentage of the social price of energy holds regardless of the emission costs and even if there are no such costs. Table 4 and Figure 2 depict the results in the absence of externality ( = 0). Under this circumstance, the world price of energy, pD ; re‡ects the social cost of a unit of energy and the consumer price, q, di¤ers from the energy price tax rate.

14

Figure 2: Prices and subsidies when by the redistributive subsidy (q

5

=0

pD ). The subsidy rate is then equal to (q

pD ) =pD :

Conclusion

This paper examines if an energy price shock should be compensated by a reduction in energy taxes to mitigate its impact on consumer prices. Such an adjustment is often debated and advocated for redistributive reasons. Our investigation is based on a modi…ed version of optimal emission taxation developed by Cremer et al. (1998, 2003 and 2010). This model has allowed us to characterizes second-best optimal taxes in the presence of an externality generated by energy consumption. Energy is used by households as a consumption good and by the productive sector as an input. We have calibrated this model on US data and proceed with simulations of this empirical model. The model shows that optimal energy taxes on consumption goods, as opposed to inputs, are a¤ected by redistributive consideration and that optimal energy tax is less

15

pD 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

q 0.8993 0.9892 1.0791 1.1690 1.2589 1.3488 1.4387 1.5287 1.6186 1.7085 1.7984

pD q 0.1007 0.1108 0.1209 0.1310 0.1411 0.1512 0.1613 0.1713 0.1814 0.1915 0.2016

pD q pD

10.07 10.07 10.07 10.07 10.08 10.08 10.08 10.08 10.08 10.08 10.08

% % % % % % % % % % %

Table 4: Subsidy rates on energy consumption goods as a percentage of their social price when = 0: than the Pigouvian tax (marginal social damage). The di¤erence is a subsidy representing roughly 10% of the Pigouvian price of energy (its true social cost). Assuming that energy prices are subject to an exogenous shock, we have calculated the optimal tax mix, including income, commodity and energy taxes, for di¤erent levels of this shock. Simulations show that variation in the energy price has an almost negligible e¤ect on the subsidy rate as a percentage of the Pigouvian price. On the other hand, the Pigouvian tax rate decreases substantially as the price of energy increases. This latter e¤ect is simply a purely arithmetic adjustment due to the fact that the marginal social damage does not change. Nevertheless, it is also true that, in dollars, the subsidy to the consumer price of energy increases— albeit by a small amount— so that this price does not increase by as much as world energy prices.

16

Appendix A The optimal general income plus linear commodity taxes The Lagrangian for the second-best problem is (where p is set equal to 1),

$=

4 X

1

Ij j4 V q; cj ; ; wnj

1 j=1 8 < O (K; L; D) :

XX j

jk

2

4 X

j

x q; cj ;

4 X

j

j=1

j

rK

j=1

V q; cj ;

k6=j

j

y q; cj ;

2 4 X pD 4

31

D5

j

j

y q; cj ;

j=1

Ij ; wnj

j

V q; ck ;

Ik ; wnj

j

+ [w

j

+ 3

+ D5

R

9 = ;

+

OL (K; D; L)] :

(A1)

where ; jk and are the multipliers associated respectively with the resource constraints, the incentive constraint and the endogenous wage condition. The …rst-order conditions are, for j = 1; 2; 3; 4, 2 3 4 4 j X X @$ I j4 j = ; j V q; cj ; y q; cj ; j D5 @q wnj j=1 j=1 2 3 4 4 j X X j j j 5 j 4Vq q; cj ; I ; j yq q; c ; xq q; cj ; j + pD yq q; cj ; wnj j=1

4 X X j=1 k6=j

jk

Vq q; cj ;

Ij ; wnj

j

+

j=1

j

Vq q; ck ;

17

Ik ; wnj

j

= 0;

(A2)

@$ @cj

=

2

Ij j4 V q; cj ; ; wnj j

yc q; cj ;

j

4 X

j

j

j=1

j

xc q; cj ;

j

4 X

j

y q; cj ;

3

D5

j

j=1

2

4V q; cj ;

+ pD yc q; cj ;

Ij wnj

j

4 X

j

;

j

kj

Vc q; cj ;

k6=j

@$ @I j

=

j

2

4V q; cj ;

@$ @D

wnk

Ij

+

jk

+

@$ @w

4 X

jk

k6=j

j

3

D5

(A3)

j

y q; cj ;

3

j

1 Ij VL q; cj ; ; j wn wnj

j 1 j I V q; c ; ; L wnk wnk k6=j 2 4 X Ij j4 = ; j V q; cj ; wnj

D5

j 1 j I q; c ; V ; L wnj wnj

j

+

j

j

k

OLL (L; K; D) = 0;

w 4 X

j=1

pD ]

Ij ; wnj

j

= 0;

kj

[OD (L; K; D)

@$ @K

X

j

Vc q; cj ;

j=1

j

w

k

;

j

; wnj

OL (L; K; D) X

Ij

y q; cj ;

Ij ; wnj

j=1

X k6=j

X

Vc q; cj ;

j

y q; cj ;

j=1

OLD (L; K; D) = 0;

[OK (L; K; D) r] OLK (L; K; D) = 0; 2 4 4 X X Ij j4 j j = V q; cj ; ; y q; cj ; wnj

j

(A4)

3

D5

+ (A5)

=

j=1

4

j=1 k6=j

j

j=1

OL (L; K; D) 4 X X

(A6)

jk

1X

w2 Ik

nj w 2

j=1

j j

I +

4 X X

jk

j=1 k6=j

VL q; ck ;

Ik wnj

;

j

Ij nj w 2 2

3

D5 VL q; cj ;

4 1 X 4 + 1+ 2 w j=1

Ij nj w 2 Ij ; wnj

VL q; cj ; j

3

I OLL (L; K; D)5 = 0:

j j

j

+

+

We now show that whereas the optimal tax on the polluting good is non-Pigouvian (Proposition A1), the optimal tax on polluting input is Pigouvian (Proposition A2). Consider …rst the polluting good tax. We have: 18

Ij ; wnj

(A7)

Proposition A1 The optimal tax on the polluting good is non-Pigouvian. Proof. Multiply equation (A3) by y q; cj ; j ; sum over j, and add the resulting equation to (A2). Simplifying, using Roy’s identity, results in 4 X

j

j=1

2 4 X 4

j

2

4V q; cj ;

yq q; cj ;

Ij wnj

j

4 X

j

;

j

y q; cj ;

j=1

+ y q; cj ;

j

yc q; cj ;

j

xq q; cj ;

j

+ y q; cj ;

j

5

j

xc q; cj ;

D5

j

3

j=1

4 X

3

j

j=1

pD

4 X

j

yq q; cj ;

j

+ y q; cj ;

j

yc q; cj ;

j

j=1

4 X X

kj

y q; cj ;

j

Vc q; cj ;

j=1 k6=j

Ij ; wnk

k

jk

+

Vq q; ck ;

Ik ; wnj

j

= 0: (A8)

To simplify equation (A8), partially di¤erentiate the j-type individual’s budget constraint, x q; cj ; j + qy q; cj ; j = cj ; once with respect to cj and once with respect to q: This yields xc q; cj ; j

xq q; c ;

j j

+ qyc q; cj ; j

+ qyq q; c ; j

Multiply equation (A9) by y q; cj ; We get xq q; cj ;

j

j

q yq q; c ;

(A9) j

=

j

y q; c ;

:

(A10)

and add the resulting equation to equation (A10).

+ y q; cj ; j

= 1;

j

j

j

xc q; cj ; j

+ y q; c ;

19

j

j

= j

yc q; c ;

j

:

(A11)

Substituting from equation (A11) into (A8), the latter equation is rewritten as 4 X

j

yq q; cj ;

j

+ y q; cj ;

j=1

8 < :

(q

4 X

pD )

2

4V q; cj ;

j

j=1

4 X X

kj

j

y q; cj ;

j

yc q; cj ; Ij wnj

Vc q; cj ;

j=1 k6=j

;

j

4 X

j

j

y q; cj ;

D5

j

j=1

Ij ; wnk

k

jk

+

Vq q; ck ;

9 =

3

Ik ; wnj

j

;

= 0: (A12)

Next, rewrite the last term on the left-hand side of equation (A12) as 4 X X

jk

Vq q; ck ;

j=1 k6=j

Ik ; wnj

j

=

4 X X

kj

Vq q; cj ;

Ij ; wnk

k

kj

Vc q; cj ;

Ij ; wnk

k

j=1 k6=j

=

4 X X j=1 k6=j

y q; cj ;

k

; (A13)

where in going from the second to the last expression, we have made use of Roy’s identity. Now substituting from (A13) into (A12) results in 4 X

j

yq q; cj ;

j

+ y q; cj ;

j=1

8 < :

(q

pD )

4 X j=1

4 X X j=1 k6=j

kj

j

yc q; cj ;

2

Ij j4 V q; cj ; ; wnj

Vc q; cj ;

Ij ; wnk

k

j

4 X

j

h y q; cj ;

j

y q; cj ;

j

j=1

j

y q; cj ;

k

i

3

D5 = 0:

9 = ;

e q; cj ; j : Substituting y eq q; cj ; j Denote the compensated demand function for y by y j j j j j j for yq q; c ; + y q; c ; yc q; c ; in above, dividing the resulting equation by

20

4 P

jy e

j=1

q

q

q; cj ;

j

; and rearranging yields 4 X

pD =

j=1

4 P P

2

Ij j4 V q; cj ; ; wnj kj

j=1 k6=j

4 X

j

j

y q; cj ;

3

j

j=1

n Ij Vc q; cj ; wn k; 4 P

k

jy e

j=1

q

y q; cj ;

j

y q; cj ;

D5 k

+ o

:

(A14)

q; cj ; j

This proves that q pD is non-Pigouvian unless the polluting good demand depends only on one’s income but not on his taste so that the second expression on the right-hand side of (A14) will be zero. Second, we prove that the input tax is Pigouvian regardless of individuals’ tastes. The proof is facilitated through the following lemma. Lemma A1 In the optimal income tax problem (A1), and characterized by the …rstorder conditions (A2)–(A7), the Lagrange multiplier associated with the constraint w = OL (K; D; L), ; is equal to zero. Proof. Multiply equation (A4) through by I j =w, sum over j; and simplify to get 2 3 4 4 j j X 1 X jIj 4 I j j j j j 5 V q; cj ; I ; j V q; c ; ; y q; c ; D w2 nj wnj wnj j=1

j=1

1 X X Ij L+ 2 ( j) w n j

k6=j

Ij jk V q; cj ; ; wnj

L

Ij ( k) n

j

kj

V q; cj ;

Ij ; wnk

k

1 OLL (L; K; D)(wL) = 0: w2

(A15)

Substituting (A15) into (A7) and simplifying, we get X X Ij ( k) n j

kj

VL q; cj ;

k6=j

Ij ; wnk

k

= w2 +

X X Ik ( j) n j

jk

VL q; ck ;

k6=j

Ik ; wnj

j

: (A16)

Then rewrite the left-hand side of (A16) as X X Ij ( k) n j

k6=j

kj

VL q; cj ;

Ij ; wnk

+

k

=

X X Ik ( j) n j

21

k6=j

jk

VL q; ck ;

Ik ; wnj

j

:

(A17)

Substituting from (A17) into (A16) implies = 0:

Observe that Lemma A1 is in fact an application of the production e¢ ciency result as it tells us that w = OL (K; D; L) imposes no constraint on our second-best problem. Using this lemma, we can easily show: Proposition A2 The optimal tax on energy input is Pigouvian. Proof. Using the result that = 0 in the …rst-order conditions (A4)–(A7), simpli…es them to 2 3 4 j j X 1 @$ I j4 j j j j j j I 5 = V q; c ; V q; c ; j + ; y q; c ; D ; L @I j wnj wnj wnj j=1

j

+

X k6=j

@$ = @D

4 X j=1

jk

j 1 j I V q; c ; ; L wnj wnj

2

Ij j4 ; V q; cj ; wnj

[OD (L; K; D)

j=1 k6=j

jk

j

y q; cj ;

pD ] = 0;

1

w2

4 X j=1

Ik nj w 2

j j

I +

j 1 j I V q; c ; ; L wnk wnk

k

= 0;

(A18)

j=1

j

3

D5

+ (A19) (A20)

4 X

j=1

4 X X

kj

k6=j

4 X

j

@$ = [OK (L; K; D) r] = 0; @K 2 4 @$ X j 4 Ij = V q; cj ; ; j @w wnj OL (L; K; D)

X

j

j

y q; cj ;

j

j=1

4 X X

Ij nj w 2

jk

j=1 k6=j

VL q; ck ;

Ik ; wnj

j

3

D5 VL q; cj ;

Ij nj w 2 Ij ; wnj

VL q; cj ; j

:

That the input tax is Pigouvian follows immediately from equation (A19).

22

Ij ; wnj

+

(A21)

j

+

Appendix B B1

Calibration of the parameters of the nested CES production function

The data used from EUKLEMS include: gross value added, capital compensation, labor compensation, intermediate energy compensation, gross value added at current basic prices, capital services, volume indices, 1995 = 100, total hours worked by persons engaged (millions), number of persons engaged (thousands), intermediate energy inputs, volume indices, 1995 = 100, gross value added, volume indices, 1995 = 100, capital-Labor elasticity (from the literature). From these data, we K, D, L, r, w, pE = pD (1 + D ) can easily be derived. These data also allow to build price indexes for O and , respectively pO and p . To do this de…ne the following notation,

KD

pK K + pD D p = C C = elasticity of substitution between K and D;

KL

= elasticity of substitution between K and L;

LD

= elasticity of substitution between L and D:

S

Using the properties of our nested CES function it follows that, KL KD

= =

LD

1 S

= + 1

23

1 S

(B1)

Econometric estimations found in the literature show that is close to 0.8 14 . There is less consensus about the value of KD but it is generally shown to be weakly complement in the short run and substitutable in the long run. We take the value KD = 0:2 and derive the value of the parameter from equation (B1). The parameters , , A and B are then the solution to the following system of equations15 , h O = B (1 h = A K

)L 1

1

+ (1

1

+ )D

O L (L; K; D) = w;

i

1

i

1

1

; ;

O K (L; K; D) = r; O D (L; K; D) = pD (1 +

D ):

where their analytical expressions as derived from the CES forms combining O and are substituted for O L (L; K; D), O K (L; K; D) = r, O D (L; K; D) = pD (1 + D ).

B2

Calibration of the parameters of the nested CES utility functions

We combine data from the US Bureau of Labor Statistics (BLS) and the Panel Study of Income Dynamics (PSID). We need those two sources because the BLS does not give any information on labor supply. The data from the BLS are given for di¤erent categories of households while the data from the PSID are individual data about 7407 households. Consequently we must allocate the 7407 households of the PSID to the di¤erent categories of households found in the BLS nomenclature. This is performed by considering the PSID information about the main occupation of the family head. We consider 4 categories of households, managers and professionals, technical sales and clerical workers, service workers, operators, fabricators and laborers, construction workers and mechanics. 14

See for instance, Diewert and Wales (1987). Let us note that the same values for , , A and B can be obtained by using the duality properties of CES cost and production functions. 15

24

The data used from the PSID are: family interview ID number, labor income of the head, labor income of the wife, main occupation of the head, main occupation of the wife, work weeks by the head, work hours by the head, wage rate by the head, work weeks by the wife, work hours by the wife, wage rate by the wife, The data used from the BLS are: number of consumer units, income before and after taxes, wages and salaries, personal taxes (Federal income taxes, State and local income taxes, other taxes), average number of persons, of child under 18, of persons 65 and over, of earners, of vehicle in the consumer unit (sex of reference, person, age,...), average annual expenditures. The last item is given in a very detailed nomenclature of goods and services. We use those data to build the small data set we need to calibrate the parameters of the utility functions of the di¤erent categories of households listed above. This data set includes the following variables: labor supply (L), 25

before and after tax wage (respectively w and wn ), non energy consumption and energy consumption (respectively x and y). The j’s consumer budget constraint is linearized and thus written, pxj + qy j = Gj + M j + wnj Lj ; where p and q are the consumer prices of x and y, Gj is the income adjustment term needed for linearizing the budget constraint (virtual income that can be estimated by using the data above together with the o¢ cial US tax Schedule) and M j is the individual’s exogenous income. The …rst-order conditions for a j-type’s optimization problem are given by 1

aj xj aj yj (1

aj bj

1 !

q = ; p

bj ) xj =(1

h aj + (1

Lj )

aj )(xj =y j )

1

1 ! !

i

! (1 !)

wnj = : p

As the energy and non-energy taxes are almost the same in US, the relative price of energy is not distorted by the tax system, we therefore set p = q = 1. We have 3 equations and 5 unknown parameters, aj ; bj , !, and M j to estimate. As shown in the above equations, the parameters of the elasticities of substitution are the same for all categories of households. The value of these parameters has been chosen by the help of some very simple econometric estimations and values found in the existing literature. The sensitivity of the results to the choice of ! is very small; the sensitivity to the choice is larger but the results are not qualitatively a¤ected by the choice of .

26

References [1] Bourguignon, F and A. Spadaro, Social preferences revealed through e¤ective marginal tax rates, Mimeo, 2000. [2] Cremer, H., F. Gahvari and N. Ladoux (1998), “Externalities and optimal taxation”, Journal of Public Economics, 70, 1998, 343–364. [3] Cremer, H., F. Gahvari and N. Ladoux (2003), “Environmental taxes with heterogeneous consumers: an application to energy consumption in France”, Journal of Public Economics, 87, 2003, 2791–2815. [4] Cremer, H., F. Gahvari and N. Ladoux (2010), “Environmental tax design with endogenous earning abilities (with applications to France)”, Journal of Environmental Economics and Management, 59, 2010, 82–93. [5] Diewert, W.E. and Wales, T.J. (1987), “Flexible functional forms and global curvature conditions”, Econometrica, vol. 55, 2010, 43–68. [6] Goulder, L., 1994: Energy Taxes: Traditional E¢ ciency E¤ects and Environmental Implications, in James M.Poterba, ed., Tax Policy and the Economy 8, Cambridge: MIT Press.

27