A Dynamic General Equilibrium Analysis Of Environmental Tax Incidence

A Dynamic General Equilibrium Analysis of Environmental Tax Incidence

Governments often choose to regulate environmental damage by taxing the negative externality. The effect of this policy is that consumers, firms, capital owners, and workers may share some of the tax burden (Garrett 2005). A partial equilibrium model cannot adequately capture this incidence, but a general equilibrium model succeeds at this task, especially when the taxes impact large segments of the economy (Kotlikoff and Summers 1987). In addition, a general equilibrium model allows government policy makers to understand more clearly the distributional effects of an environmental tax (Garrett 2005). Some previous literature uses a general equilibrium model to examine environmental tax incidence. These models often assume that input factors, such as labor and capital, are perfectly mobile across sectors (see Fullerton and Heutel 2005). Since the results of these models differ when researchers relax this assumption, some researchers, including Garrett (2005), build costs of adjustment into the model. This change allows the model to capture the short-run effects of tax incidence with imperfect factor mobility, but a drawback is that this static model does not reach a long-run equilibrium. In addition, static models abstract from intertemporal issues of capital formation, savings and investment decisions, and capitalization effects (Kotlikoff and Summers 1987). The results of these static models substantially change in a dynamic setting since a dynamic long-run model can capture some richer aspects of tax incidence by accounting for intertemporal economic decisions, as well as the full effects of adjustment costs.

2 The purpose of this paper is to examine environmental tax incidence by incorporating the framework of imperfect factor mobility of Garrett (2005) into a dynamic general equilibrium model. By constructing a dynamic model, this paper examines the short- and long-run effects and distributional burden of a one-time increase in a pollution tax. Like Garrett (2005), this paper considers the following two distinct possibilities: firms face capital adjustment costs when they alter their capital stock, whereas dismissed workers bear the labor adjustment costs when changing jobs. First, this paper models the capital adjustment costs using a framework similar to that of Goulder and Summers (1989). However, rather than using the computable general equilibrium approach of Goulder and Summers (1989), this paper derives an analytical and graphical solution to the dynamic models. The second part of this paper then allows firms to perfectly adjust all of their factors of production, but incorporates adjustment costs for a worker into a dynamic structure. By developing the framework of Garrett (2005) into a dynamic setting, this paper re-evaluates tax incidence. The contributions of this paper are that it presents a dynamic general equilibrium model for examining environmental tax incidence with imperfect factor mobility. This paper borrows certain aspects from previous static models, including the Fullerton and Heutel (2005) environmental extension of the Harberger (1962) model, and the Garrett (2005) extension of relaxing the assumption of perfect factor mobility. Like Garrett (2005), for simplification purposes this paper only considers one clean input into production in the economy, which can represent either capital or labor. Nonetheless, this paper differs from Garrett (2005), in that it presents a dynamic model to allow for the attainment of transitional dynamics and a long-run equilibrium.

3 The results for this paper show that the long-run solutions to a dynamic general equilibrium model do not noticeably differ when accounting for the nature of the adjustment costs. Garrett (2005) finds that capital adjustment costs for a firm lead to changes in the prices and rates of return relative to the no adjustment cost case. On the contrary, the result for labor adjustment costs for a worker is that prices and wages do not change, but labor supply and pollution significantly decrease. Nonetheless, by considering a dynamic model, this paper illustrates the attainment of a long-run equilibrium and its transition path for the two types of adjustment costs. This paper finds that the long-run result for both cases is the same as it is for a static model with no adjustment costs. The first section of this paper presents a dynamic general equilibrium model of firms facing adjustment costs when they alter their capital stock. Section II considers a less traditional case in the literature of displaced workers facing adjustment costs; however, in this model, firms do not face adjustment costs when hiring or firing workers. Finally, Section III concludes and ponders future research in this field. I. Model with Capital Adjustment Costs for Firms Previous examples in the literature, both theoretical and empirical, illustrate why firms may face adjustment costs when altering their clean factors of production, such as labor and capital (see Summers 1981, Goulder and Summers 1989, Hamermesh 1989, and Hamermesh and Pfann 1996). The present model uses many aspects of the Goulder and Summers (1989) model, except this model offers less generality since it does not include both capital and labor as clean factors of production. Like Garrett (2005), this model uses only one clean factor of production in order to attain an analytical solution.

4 Goulder and Summers (1989) include more generality in their model, but they solve their model computationally. Like Fullerton and Heutel (2005) and Garrett (2005), this model examines a general equilibrium economy when the government exogenously increases its tax on pollution. Incorporating the features of Goulder and Summers (1989) that output is separable between the production and adjustment costs, and that production is less than output when firms adjust their capital stock, the production functions are the following:

X = K X − φX I X

(1.1)

Y = Y ( K Y , Z ) − φY I Y

(1.2)

where X is the clean good, Y is the dirty good, K X and K Y are capital inputs into production, Z is the pollution input into production, φ X and φY are capital adjustment cost functions for the firm, and I X and I Y are the net investments in capital. Net investment, which includes the depreciation rate of capital over time, adheres to the following transition equations: K& X = I X

(1.3)

K& Y = I Y

(1.4)

where K& X and K& Y are the rates of change in capital in each sector of the economy. The resource constraint for capital is the following: K X + KY = K

(1.5)

where K is the total fixed capital stock in the economy (Harberger 1962). Although the aggregate capital stock is fixed, capital in each sector can vary. Since this model assumes a fixed capital stock, households do not face a savings decision, and consequently, the household intertemporal elasticity of substitution does not make a difference in their

5 maximization problem. Thus, the following equation characterizes the maximization problem of a representative consumer:

αU =

d ( X / Y ) /( X / Y ) d ( pY / p X ) /( pY / p X )

(1.6)

where α U is the intratemporal elasticity of substitution between the two goods, and this elasticity depends on the two output prices, p X and pY (Harberger 1962). Since markets are perfectly competitive, households and firms take these two output prices as given when making their optimizing decisions. Also, firms face no market price for pollution except for a tax, so the price of pollution, p Z , is equal to the governmental tax on pollution, τ Z (Fullerton and Heutel 2005). In order to rule out the case when the input usage of pollution is infinite (i.e. τ Z = 0 ), I assume that the government levies a positive tax on pollution in the initial equilibrium (Fullerton and Heutel 2005). Thus, the profitmaximizing problem for the representative firm for the clean good is to choose I X in maximizing the following objective function: p X ( K X − ϕ X I X ) − rX K X

(1.7)

subject to Equation (1.3), where rX is the net return to capital owners when they invest in the clean good. This paper uses the most basic version of quadratic adjustment costs for both the clean and dirty good (Cahuc and Zylberberg 2004). As a result, the adjustment costs take the following form:

ϕ X I X = bX ⋅ ( I X ) 2

(1.8)

ϕY I Y = bY ⋅ ( I Y ) 2

(1.9)

6 where bX and bY are positive constants. Therefore, the Hamiltonian for the clean good becomes the following: H X = p X ( K X −b X ⋅( I X ) 2 ) − rX K X +λ K X [ I X ]

(1.10)

The first-order condition for the control variable of net investment is the following:

2bX I X p X = λK X

(1.11)

The interpretation of (1.11) is that the representative firm optimizes so that the marginal adjustment cost times the output price, or the marginal cost of investment, equals the shadow price on capital, or the marginal benefit of investment. Next, the costate equation for the state variable, capital, is the following:

p X + λ&K X = r

(1.12)

where r is the discount rate. Equation (1.12) says that the price of capital plus the rate of change of the shadow price of capital, equals the discount rate1. Thus, in this arbitrage condition, the marginal cost of investing in capital, or discount rate, equals the sum of two marginal benefits: the instantaneous return of capital and the change in the shadow value over time. I plot Equations (1.12) and (1.3) on a phase diagram to characterize the steady state path of capital in the clean sector, with the following result:

1

One aspect to note is that this discount rate is constant, and this feature of the model is somewhat inconsistent with the fixed capital stock assumption.

7

Figure (1) says that if the clean sector has an initially high level of capital stock and a low value or shadow price of capital, then capital owners stop investing in the clean sector. This disinvestment means that capital in the clean sector becomes scarcer and the shadow value of capital increases, allowing a movement along the transition path to the steady state. Alternatively, if the clean sector has an initially low level of capital stock and a high value of capital, then capital owners invest more in the clean sector. This increase in investment means that capital becomes less scarce, which decreases the value of capital, and as a result, the clean sector moves along the transition path to the steady state. Figure (1) also shows that if the clean sector begins at different initial values than the ones previously mentioned, then it does not reach a steady state. This paper next derives the equations for the dirty sector in order to analyze the comparative statics of an increase in τ Z .

8 The representative firm for the dirty good faces a slightly different but analogous profit maximization problem. This firm’s objective is to choose both I Y and Z in maximizing the following objective function:

pY (Y ( K Y , Z ) − ϕ Y I Y ) − rY K Y − τ Z Z

(1.13)

subject to Equation (1.4). where rY is the net return to capital owners when they invest in capital usage in the dirty good. Adjustment costs for this problem follow the type specified in Equation (1.9). Therefore, the Hamiltonian for the dirty good becomes the following:

H Y = pY (Y ( K Y , Z ) − bY ⋅ ( I Y ) 2 ) − rY KY − τ Z Z + λKY [ I Y ]

(1.14)

The first-order conditions for the two control variables of net investment and pollution are as follows: 2bY I Y pY = λ KY

(1.15)

∂Y (⋅) =τZ ∂Z

(1.16)

pY

The interpretation of (1.15) is the same as it is for the clean good: the marginal cost of purchasing capital equals the marginal benefit of investment. The interpretation of (1.16) is that the output price times the marginal product of pollution equals the price of pollution (i.e. the marginal benefit of pollution equals the marginal cost). The co-state equation for the state variable of capital is the following: pY

∂Y (⋅) & + λKY = r ∂K Y

(1.17)

Equation (1.17) states that the sum of the two returns on capital, the instantaneous marginal benefit and the rate of change of the shadow price, equals the marginal cost of

9 investing, or the discount rate. This equation is slightly more complicated than the costate equation for the clean good, Equation (1.12), because the production function for the dirty good has two inputs. Next, I plot Equations (1.17) and (1.4) on a phase diagram. The following graph characterizes the transition dynamics and steady state for the dirty good:

The interpretation of the transition path to the steady state for Figure (2) is exactly the same as it is for Figure (1). Since Figures (1) and (2) characterize the general equilibrium of this economy, this paper now evaluates the equilibrium when the government exogenously increases an environmental tax. From Equation (1.16), when

τ Z increases, the output price increases because one of the input prices is now more expensive. Also, the marginal product of pollution increases because the dirty firm uses less pollution. When the rate of change of the shadow price of capital is zero, the discount rate equals the output price times the marginal product of capital (see Equation

10 (1.17)). Since pollution decreases, when holding the amount of capital stock constant and applying the fact that the production functions are constant returns to scale, then the marginal product of capital decreases. Whether λ&KY is positive or negative depends on the magnitudes of the elasticity of substitution for firms, α Y , between the two production inputs, and the elasticity of substitution for consumers, α U , between the two output goods. This occurrence is similar to what Garrett (2005) derives in the static general equilibrium model with no adjustment costs. This paper considers the case when α Y > α U and acknowledges that the result for the other case is merely the opposite. The fact that α Y > α U means that the percentage change of pY is larger in magnitude than the percentage change of the marginal product of capital, since consumers need a larger percentage change to switch. The result is that the line of λ&KY = 0 shifts to the right (see Figure (2)). Initially firms cannot adjust their capital stock; but since the price is larger than the marginal product of capital, capital becomes more valuable, and the shadow value of capital in the dirty sector increases. Consequently, capital owners increase investment, which causes the value of capital to decrease and the dirty sector to transition to a new steady state (see Figure (2)). Since the price of the clean good, p X , falls in the short run and the discount rate is constant, the line of λ&K X = 0 shifts to the left (see Figure (1)). As shown in Figure (1), the transition to a new steady state in the clean sector is the opposite of what occurs in the dirty sector. Figures (1) and (2) illustrate that the long-run result of the case when α Y > α U is that the capital in the dirty sector increases, while capital in the clean sector decreases. Also, the increase in the pollution tax leads to less pollution and a more ambient

11 environment. The shadow values of capital return to their initial level in the long run, but these two values differ in the short run to entice capital owners to invest in K Y . The higher pollution price and the same rental rate of capital leads to less long-run output in the dirty industry. Also, since the clean representative firm uses less capital, its long-run output decreases as well. Finally, the price of the dirty good is higher than the initial equilibrium, while the price of the clean good returns to its initial level, meaning that consumers pay a higher price for the dirty good and bear some of the tax burden in the long run. II. Model with Labor Adjustment Costs for Workers2 This model differs from the one in Section I in a few ways. First, the clean factor of production is now labor instead of capital. More importantly, firms no longer directly face adjustment costs when altering the clean factor of production; instead, workers face adjustment costs when they move from one sector of the economy to the other (Garrett 2005). Jacobson, LaLonde, and Sullivan (1993) show empirically that displaced workers often deal with a number of difficulties in their reemployment period, including lower wages, less labor time, and possible transition costs. Their study also finds that some displaced workers never again attain the same level of earnings. Garrett (2005) provides a number of intuitive reasons why workers may face adjustment costs. These reasons include search costs, moving to a new area, investing in new job training or education, and dealing with the psychological issues of losing a job, which can even result in suicide (Garrett 2005).

2

The numbering of the equations in this section is not always sequential. In order to highlight the connections between equations in this section and those in Section I, the equation numbers here parallel those in Section I (e.g., Equation (2.16) in this section is analogous to Equation (1.16) in Section I).

12 The constant returns to scale production functions for labor simplify to the case of perfect factor mobility for firms:

X = LX

(2.1)

Y = Y ( LY , Z )

(2.2)

where the symbols represent the same aspects of the Section I model, except that L X and

LY are labor inputs into production. The resource constraint for labor is the following: LX + LY = L

(2.5a)

where LX = LX + ΨX N X and LY = LY + ΨY N Y

(2.5b) (2.5c).

Also, Ψ X and ΨY are labor adjustment costs for workers, N X and N Y are the net flows of labor between sectors, and L is the fixed stock of labor in the economy. Thus, workers experience the adjustment costs through a loss of labor available to supply to the market. Since workers cannot adjust to their new jobs costlessly, transition costs absorb some of their labor time. Equations (2.5b, c) show that a firm employs a dismissed worker from the other sector immediately and costlessly, but once the worker is in the new sector, he or she is not immediately productive and does not receive a wage. The representative firm hires displaced workers knowing that before the workers can be productive in the new sector, they confront moving costs and/or a job retraining period. Nonetheless, since firms hire workers immediately, the specification of Equations (2.5b, c) does not encapsulate a search cost. The net flows of labor abide by the following transition equations:

L& X = N X

(2.3)

13 L&Y = N Y

(2.4)

where L& X and L&Y are the net changes in labor supply over time in each sector of the economy. The adjustment costs for workers conform to the following basic quadratic structure (Cahuc and Zylberberg 2004): ΨX N X = d X ⋅ ( N X ) 2

(2.8)

ΨY N Y = dY ⋅ ( NY ) 2

(2.9)

where d X and dY are positive constants. Hence, the structure of the adjustment costs is the same as it is in Section I. II.A. Representative Firm’s Problem As in Section I, the prices of the clean good, dirty good, and pollution are p X ,

pY , and τ Z , respectively. The unconstrained profit maximization problem for the representative firm in the clean industry is to choose LX in maximizing the following objective function:

p X L X − wX L X

(2.7)

where wX is the net wage to workers in the clean industry. The first-order condition becomes the following:

p X = wX

(2.11)

so that the output price or marginal benefit equals the wage or marginal cost of labor. Next, the unconstrained profit maximization problem for the representative firm for the dirty good is to choose LY and Z in maximizing the following objective function:

pY Y ( LY , Z ) − wY LY − τ Z Z

(2.13)

14 where wY is the net wage in the dirty industry. Maximizing this equation yields the following two first-order conditions: pY

∂Y (⋅) = wY ∂LY

(2.15)

pY

∂Y (⋅) =τZ ∂Z

(2.16)

The interpretation of Equation (2.15) is that the output price times the marginal product of labor, or marginal benefit of labor, equals the wage or marginal cost of labor in the dirty industry. The first-order condition for pollution is the same as it is in Section I (see Equation (1.16)). Rewriting these two first-order conditions yields the following result:

∂Y (⋅) ∂LY w = Y ∂Y (⋅) τ Z ∂Z

(2.18).

This equation shows that the representative firm in the dirty industry optimizes so that the marginal rate of transformation equals the price ratio.

II.B. Consumer’s Optimization Problem Representative consumers maximize the following utility function over an infinite time horizon: ∞

U = ∫ e ρ ( t − s )C s ( X , Y )ds

(2.6)

t

where ρ is the rate of time preference, and Cs ( X , Y ) is the overall consumption of the two goods. As long as the function Cs ( X , Y ) is HD(1), the utility functional form specifies the special case when the intertemporal elasticity of substitution, σ U , is

15 infinity. Also, like in Section I, the intratemporal elasticity of substitution is α U . Thus, consumers maximize Equation (2.6) subject to the transition equations, Equations (2.3) and (2.4), and the following static budget constraint:

p X X + pY Y = wX LX + wY LY + Π X + Π Y

(2.19)

where Π X and Π Y are the shares of firm profits in each industry, which the representative consumer receives exogenously. The solutions to the Lagrangian problem of choosing how much of each good to buy are the following: ∂C s (⋅) = λp X ∂X

(2.20)

∂C s (⋅) = λpY ∂Y

(2.21)

where λ is the Lagrange multiplier on Equation (2.19). Since λ binds, rearranging Equations (2.20) and (2.21) yields the following familiar result:

∂Cs (⋅) ∂X = p X ∂Cs (⋅) pY ∂Y

(2.22).

This equation shows that the representative consumer optimizes so that the marginal rate of substitution between the two goods equals the price ratio. The consumer uses the necessary condition of Equation (2.22) to obtain an indirect utility function. The following indirect utility function is a function of prices and income:

V ( p X , pY , wX LX + wY LY + Π X + Π Y )

(2.23).

Thus, the Hamiltonian, or intertemporal optimization problem, for the representative consumer becomes the following:

H = V (⋅) + λLX [ N X ] + λLY [− N X ]

(2.24)

16 where − N X = N Y from Equation (2.5a). Thus, the first-order condition for choosing the net flow of labor is the following:

λ L − 2 d X N X wX X

where

∂V (⋅) ∂V (⋅) = λLY − 2dY NY wY ∂I ∂I

(2.25)

∂V (⋅) is the marginal value of income. This necessary condition shows that the ∂I

consumer intertemporal optimization problem takes into account the shadow price of labor supply in each industry, the marginal value of income, and the product of the marginal adjustment cost and the wage in each industry. Thus, consumers maximize their indirect utility so that the marginal benefit of working in the clean sector is equal to the marginal benefit of working in the dirty sector, net of the marginal adjustment costs in each industry.

II.C. The Equilibrium The co-state equations for the two state variables, L X and LY , are the following:

∂V (⋅) wX λ& L ∂I + X = λ LX

(2.26)

∂V (⋅) wY λ& ∂I + LY = λLY

(2.27).

ρ

ρ

ρ

ρ

Thus, for each industry, the present discounted value of the sum of the instantaneous return on working and the rate of change of the value of labor equals the shadow value of labor. Next, I plot the equations characterizing the equilibrium on two graphs (see Equations (2.3) and (2.26) in Figure (3) and Equations (2.4) and (2.27) in Figure (4) on the following page).

17

18 The interpretations of the transition paths to the steady state in Figures (3) and (4) are very similar to that of Figures (1) and (2). The only difference from the graphs in Section I is that the λ& = 0 lines are downward sloping because a higher labor stock

implies a lower wage and lower shadow value of labor (see Equation (2.15)). Since Equations (2.16) and (1.16) are the same, previous analysis shows that an increase in the pollution tax leads to an increase in both the price of the dirty good and the marginal product of pollution. The decrease in pollution causes the marginal product of labor to decrease since the production function is constant returns to scale. Therefore, from Equation (2.15), the sign of the wage in the dirty industry is ambiguous, and similar to analysis in Section I, this sign depends on the magnitudes of α Y and α U . In order to compare the results from the Section I and Section II models, again this paper considers the case when α Y > α U . Since the magnitude of the percentage change of pY is larger than the percentage change of the marginal product of labor, the wage in the dirty sector increases. From Equation (2.27), the increase in the wage causes the line of λ&LY = 0 to shift to the right (see Figure (4)). The shadow value of labor in the dirty industry increases initially, and since workers receive a higher wage in that industry, over time they shift their labor supply from the clean to the dirty industry. This increase in labor supply causes the wage to decrease and the dirty sector returns to a steady state (see Figure (4)). The increase in the pollution tax causes the wage in the clean sector to decrease when α Y > α U , and from Equation (2.26), the line of λ&LX = 0 shifts to the left (see Figure (3)). Again, like in Section I, the transition to a new steady state in the clean sector moves in the opposite direction as it does in the dirty sector.

19 Figures (3) and (4) illustrate that in the long run, the labor supply in the clean sector decreases, whereas the labor supply in the dirty sector increases when α Y > α U . While the shadow values of labor and wages change in the short run to attract workers to move to the dirty sector, they eventually return to the initial equilibrium level. Also, like in Section I, the increase in the environmental tax accomplishes the policy goal of decreasing pollution. By similar reasoning as in Section I, the long-run output in both industries decreases when α Y > α U . The fact that wages return to their initial level means that the price of the clean good also returns to its initial level, but the price of the dirty good is higher in the long run than initially because the input price of pollution is higher. Thus, like in Section I, consumers pay a higher price for the dirty good and bear some of the burden of the tax in the long run. Finally, an interesting but expected result is that both capital owners in Section I and workers in this section only experience tax incidence in the short run as they face the adjustment costs. III. Conclusion This paper examines tax incidence when the government increases an environmental tax; it accomplishes this goal using two dynamic models with imperfect factor mobility. This paper finds that the long-run equilibrium result is independent of the nature of the adjustment costs, and that this equilibrium is the same as it is in a static model with no adjustment costs. Nonetheless, due to the adjustment costs, the models in this paper derive a different result in the short run than the static model with no adjustment costs. In both the capital adjustment cost case for a firm, and the labor adjustment cost case for a worker, the shadow values change in the short run. This shortrun change induces capital owners to change their investment and workers to move to a

20 different sector. Over time the shadow values in each sector and the price of the clean sector return to their initial level, but the increase in the price of the dirty good in the long run means that consumers share some of the tax burden. Importantly, the pollution tax accomplishes the policy goal of reducing the negative externality. Finally, these results are independent of whether firms face adjustment costs via their production functions or workers face adjustment costs via the labor resource constraint. Researchers can find a number of ways to extend this paper in future research. The first way is to allow for more generality in the model by including both capital and labor as clean inputs into production (see Goulder and Summers 1989 and Fullerton and Heutel 2005). Another way to extend the model is to use different functional forms for the adjustment costs other than quadratic adjustment costs. This research can examine if the results of this paper depend on the structure of the adjustment costs. Finally, researchers can relax the assumption of a fixed stock of labor or capital to allow for savings decisions in the model. The results of this paper launch certain policy implications. Clearly, taxing a negative externality like pollution leads to a lower amount of the externality. The problem with this taxation is that certain economic agents bear the burden of the tax. Using a dynamic model, this paper shows that regardless of the nature of the adjustment costs, the long-run incidence only affects consumers who want to buy the dirty good. Therefore, in the long run, the externality is internalized, and governments can view direct taxation as an effective and economically costless way of reducing negative externalities. (Look at last paragraphs of first two sections, especially at the end of Section II.)

21

22 References

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