The Model

The general structure of our model incorporates environmental elements into a standard Samuelson-Diamond OLG growth model. We consider an infinite-horizon economy comprised of finitely-lived individuals, perfectly competitive firms, and the government. Production creates pollution that damages environmental quality, which is treated as a renewable resource and can possibly be beneficial to both individuals’ utility and productive activities. In what follows, we in turn describe the structure of the economy.

3.2.1. Individuals

Time is discrete. A new generation (called generation t ) is born in each period

=1

t

, 2,…, and lives for two periods. There is also an initial old generation (called generation 0) that lives only in period 1. For simplicity we assume no population growth and the size of each generation is normalized to unity. All individual agents are identical except for their ages. Accordingly, the representative generation t has the following utility function:

where

c

_t^y is consumption in youth age in period t and

c

_t^o₊₁ is consumption in old age in period

t

+1;

E

_t is environmental quality in period t ; ρ∈(0,1) is the subjective discount factor; and η >0 denotes the weight in terms of the utility attached to environmental quality.

All individual agents live for two periods. In the first period (in youth age) each of the agents is endowed with one unit of labor inelastically, and it allocates its total income (the sum of wage income and government transfer payments) between savings and young-age consumption. In the second period (in old age), each of the agents is retired from the labor market and receives the return from savings and governments’ transfer payments as its old-age consumption. Therefore, the budget constraints of generation t in youth and old age are respectively given by:

t t

t y

t

s w g

c

+ = +(1−θ) , (3.2)

1 1

1 + +

+ = t t + t

o

t

R s g

c

θ , (3.3)

where

s is savings,

_t

w is labor income,

_t

R is the gross return on savings, and

_t+1

g denotes the government transfer payments. Equations (3.2) and (3.3) state that,

t

in each period, the government returns environmental tax revenues to the young and the elderly as lump-sum transfer payments according to the proportions 1−θ and θ , respectively.¹⁹

Notice that, for generation 0, there is no savings and consumption decision for each of the agents since the agent only lives in period 1. Each of the agents possesses

s as its initial asset and passively receives both transfer payments and the

₀ return from savings as its consumption in old age. Without loss of generality, we assume

s

₀ =1 in the following analysis. For generation

t

≥1, each of the agents maximizes

U in (3.1) subject to (3.2) and (3.3), and yields the following

_t

19 A more detailed discussion of θ will be presented in section 3.2.4.

consumption and saving functions:

There is a continuum of identical and perfectly competitive firms. The number of firms is normalized to unity. The representative firm produces a single final good

Y using the following production function:

t

ν where Λ is the technology level that stands for the production externalities, _t

K is

t

the aggregate physical capital,

L is the aggregate labor, and

_t

P is aggregate

_t pollution that can be regarded as a “dirty input”. Firms hire labor, capital, and dirty inputs to maximize profits taking all factor prices and the technology level as given.

The representative firm’s problem can be written as:

t

20 It should be noted that the final good serves as the numeraire in this .

b is a constant parameter.

²¹ The first-order conditions for the firm’s optimizing problem, in per-worker terms, are thus given by:

t t t

t

k p = r

Λ

^{α− β}

α

¹ _{, (3.9)}

t t

t

t

k p

¹

( 1 τ ) b

β Λ

^{α β−}

= +

^{, (3.10)}

t t t

t

k p = w

Λ

^{α β}

ν

_{, (3.11)}

where

k

_t =

K

_t/

L

_t and

p

_t =

P

_t/

L

_t. (3.9)-(3.11) indicate that the firm equates the marginal product of the capital, labor and pollution to their respective marginal cost.

We assume that there exist two kinds of positive externalities in the production sector. The first one is the capital externality suggested by the standard literature of endogenous growth theory such as Romer (1986) and Lucas (1988). The second one is the environmental production externality, which indicates that the output level can rise with a better environmental quality (see, e.g., Bovenberg and Smulders (1995), Mohtadi (1996), Fullerton and Kim (2008)).²² Given these two positive externalities, the technology level can be specified in the following form:

λ α

t t

t

= AK

⁻

E

Λ

^{1 , (3.12)}

where A is a constant, and λ(≥0) is a parameter that reflects the extent of the environmental externality.

3.2.3. Environmental quality

Following Tahvonen and Kuuluvainen (1991), Bovenberg and Smulders (1995) and Fullerton and Kim (2008), the natural environment is treated as a renewable

21 If b_t is constant over time (i.e., b_t=b), as time goes on, the aggregate pollution will become infinite and nothing will survive. Hence, in the environmental and endogenous growth literature (e.g., Bovenberg and Smulders, 1995; Nielsen et al. 1995, Oueslati, 2002; Fullerton and Kim, 2008; Pautrel, 2008) it is necessary for the price of pollution (the price could be the private price or environmental tax, or both) to evolve with another growing factor. See Smulders (1995) for an excellent discussion.

22 See also, for example, Pearce and Warford (1993) for empirical evidences suggesting that pollution can reduce productivities.

resource. We specify that environmental quality grows and declines in the following manner:

t t t

t

E E E P

E

₊₁ = +δ( − )− ,²³ (3.13)

where δ is a regeneration parameter, and E denotes the maximum level of environmental quality (i.e., environmental quality with zero pollution). We impose a condition on (δ,

E

) to assume that they are large enough to avoid negative environmental quality (

E

_t > 0 ). (3.13) indicates that environmental quality in the ∀

t

next period is specified to be positively related to the regeneration capacity of the environment δ(

E − E

_t) and negatively related to the level of pollution created in this period.²⁴

3.2.4. Government

The government is subject to a balanced-budget requirement, which levies an environmental tax on pollution and transfers the revenue to individuals. Let

g

_t be total transfer payments. In each period t , the young (generation t ) receive

g

t

) 1

( −θ while the elderly (generation

t

−1) receive θ

g

_t. Hence, the government budget constraint in period t is given by:

t t t

t

P g g

b

θ θ

τ =(1− ) + . (3.14)

The weight parameter θ plays an important role throughout the analysis. It stands for the revenue weight that the government assigns to the young and the elderly. As we will see later, θ is also a parameter that reflects the welfare conflict between different generations. It can be seen from the individual’s budget constraint reported

23 Let J(E,P)=E_t+1 -E_t denote the evolving function of the environment. The function satisfies the properties that J_P <0, J_E <0 and J(_E^~_,P^~)=0.

24 As in John et al. (1995) and Ono (1996, 2003a; 2003b), we consider a linear evolving function of environmental quality for the purpose of deriving analytical solutions. On the other hand, Tahvonen and Kuuluvainen (1991) and Bovenberg and Smulders (1995) consider a more complicated nonlinear form of evolving function.

in (3.2) and (3.3) that, when θ =0, the whole of the tax revenues are returned to the young. However, when θ =1, the elderly receive all of the tax revenues and we can treat this case as a kind of pay-as-you-go public pension system financed by environmental taxes. In particular, we refer to the case of θ =0.5 as an equal

transfer policy that indicates that tax revenues are equally distributed to each

generation.

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