Carbon Tax with Individuals’ Heterogeneous Environmental Concerns

A1

The supply of unskilled and skilled labor is the same as in Dinopoulos and Segerstrom (1999). Each individual chooses to train and become skilled at the beginning of life; the duration of the training period – when the individual cannot work – is exogenously fixed at T > 0. Hence an individual with ability θ decides to train if and only if the following arbitrage condition is satisfied:

with 0 < γ < 1/2. Note that an individual with ability θ > γ is postulated able to accumulate human capital θ − γ after training, whereas an individual with an ability lower than γ (i.e. θ ≤ γ) never gets any skill from schooling. Therefore, a skilled worker with ability θ > γ > 0 earns a wage θ − γ w H after training from time t + T > 0 onwards and does not earn any wage during her period of training. Since the stock market is assumed to be perfectly efficient, the returns of each financial asset must be equal to the riskless rate of return r, and therefore r t = ρ holds at all dates t. From equation (6) this implies a constant value of the consumption c _θ,η , quantity of goods q q , q a , and unskilled and skilled wage w _L , w _H respectively. Considering eq. (A1.1) with equality, the ability threshold θ ₀ is obtained which renders an individual indifferent to becoming skilled or to remaining unskilled for all her life. Hence, the individual will train if and only if her ability is higher than

where σ ≡ 1 / e − ρ T . An individual with ability θ > θ ₀ will decide to train and will accumulate quantity θ − γ of human capital.

The supply of unskilled labor at each time t, L, equals the number of individuals in the population who decide to remain unskilled, i.e. L = θ ₀ N = θ ₀, and the skilled labor force at each time t is H = 1 − θ 0 N = 1 − θ 0 . The average skill level of workers who have finished training is 1 − γ / 2 + θ 0 − γ / 2 = θ 0 + 1 − 2 γ / 2 . Therefore the supply of skilled labor at each time t, measured in efficiency units, is given by H = θ 0 + 1 − 2 γ 1 − θ 0 1 2 . Q.E.D.

A2

In the main text, a quality ladder model is employed, as exemplified by Grossman and Helpman (1991). Specifically, in the context of non-drastic innovations and Bertrand competition, producers of top-quality services and those offering the highest environmental quality services adopt a limit pricing strategy. With free entry into the production of the second-best quality, profits are expected to be zero in equilibrium.

Considering the aforementioned context, if the subsequent innovation introduces a superior quality services product j + 1 , considering the top environmental quality services 1 a i + 1 , free entry into producing the second-best environmental quality services 1 a i is viable. Therefore, the price-quality ratio of the top quality leader producing quality services j + 1 is p n , ω q λ j + 1 / a i , where λ is the exogenous and constant quality services jump in each variety, whereas the price-quality ratio of the follower producing the second best quality services j of the same variety is m c λ j / a i . The quality services leader has the lowest price-quality ratio whenever p n q λ j + 1 ≤ m c λ j , i.e. whenever p n q ≤ λ m c , which implies p n q = λ m c . Similarly, if the next innovation introduces a better environmental quality product and, given the top quality services λ ^j+1, free entry in producing the second best quality services λ ^j is viable. The price-quality ratio of the top environmental quality leader producing with the i + 1 t h version of a variety is p n a λ j / a i + 1 , whereas the price-quality ratio of the follower producing with the i t h second best environmental quality version of the same variety is m c λ j / a i . The top environmental quality leader has the lowest price-quality ratio whenever p n a λ j / a i + 1 ≤ m c λ j / a i , i.e. whenever p n a ≤ m c a , which implies p n a = m c a . Therefore, the top quality services leader has the lowest price-quality ratio p n q = λ m c , and the top environmental quality leader has the lowest price-quality ratio p n a = m c a . Q.E.D.

A3

It is assumed that individuals differ in their willingness to pay η for the environmental quality services of the product distributed in 0,1 according to any continuous cumulative distribution function (cdf) F η with usual properties F ′ η > 0 , F 0 = 0 , F 1 = 1 . It is assumed that the individual’s type η and personal ability θ are independently distributed. The individual type η is private information, and its cumulative distribution function across individuals is assumed to be common knowledge. To ensure the generation of strictly positive profit flows, the leading provider of environmental quality should set a price that satisfies the inequality p n a = η m c a > m c , indicating η > η ̄ = a . Consequently, in equilibrium, the demand for all individuals with type η ≤ η ̄ = a is zero. Since an individual’s type η is private information and only the top environmental quality product can be sold at a price higher than the marginal cost mc, all individuals with η > η ̄ = a would be willing to pay at most p n a = m c a . Accordingly, the top environmental quality leader should opt for a uniform price p n a = m c a for all individuals with η > η ̄ = a to maximize profits. Therefore, 1 − F η ̄ , with η ̄ = a , represents the population share with a strictly positive demand for state-of-the-art environmental quality service products, and F η ̄ indicates the proportion of individuals who demand the state-of-the-art quality service product. Q.E.D.

A4

Here, we explore a scenario where the government provides a fiscal incentive to companies producing environmentally superior products. In this context, the fiscal impact on polluting emissions is determined by τ a ( i max − m ) a i max , where m ≥ 0 is a non-negative integer. When m = 0, the fiscal impact on polluting emissions simplifies to τ a i max a i max = τ . For m ≥ 1, the fiscal impact on polluting emissions for the top environmental quality product becomes τ a ( i max − m ) a i max = τ a m < τ . Consequently, as ‘m’ increases, the environmental tax on state-of-the-art environmental quality products decreases, emphasizing an inverse relationship between m and the tax rate. Therefore, the profit flows of the patent holder solves the following maximization problem:

where instantaneous profit flows net of the tax burden charged on consumers are considered, l = q , a , e = 1 is used. The maximization problem (A4.1) reduces to:

As above, the innovation’s target is quality services improvement, and the solution to the maximization problem as in equation (A4.2) implies p n q = λ m c . When the innovation’s target is pollution abatement improvement, the solution to the maximization problem as in equation (A4.2) implies p n a = m c a . Q.E.D.

A5

There is a set Λ = Q , A comprising R&D firms dedicated to patenting improvements in quality services (Q) and environmental quality services (A), respectively, with Q ∩ A = ∅. The Lebesgue measure is μ q Q ∈ 0,1 and μ a A ∈ 0,1 , respectively. Consequently, the combined rate of innovation arrival at time t is represented by I t = ∫ Q ∪ A I f t d f , signifying the collective Poisson arrival rate of innovations generated by all R&D firms. As usual in quality ladder models à la Grossman and Helpman (1991) and Aghion and Howitt (1992), Arrow’s effect is at work. Cozzi (2007) has proved that the standard Schumpeterian growth models are compatible with positive and finite R&D investment by the incumbent monopolist. All the analysis in this paper is compatible with Cozzi’s (2007) findings. Therefore, this model allows for positive, yet non-strategic sighted, R&D investment by the incumbent monopolist.

A6

Let us analyze the incumbent leader producer both of top quality services and top environmental quality services. Let us v l t , with l = q , a , denote the expected discounted profit flows of a successful quality leader at time t producing the best quality services (q) and the best environmental quality services (a), respectively. Since each incumbent firm is targeted by R&D firms that try to discover the next best quality services and the next best environmental services, the shareholder suffers a loss v l t with probability I l t d t = ∫ l I f t d f d t , which represents the aggregate Poisson arrival rate of innovation produced by all R&D firms that aim to producing the best quality services (l = q) and the best environmental quality services (l = a), respectively. Thus, the event of no innovation occurs with probability 1 − I l t d t . Over a time interval dt, the shareholder of a stock issued by a successful R&D firm receives a dividend π l t d t , and the value of the firm appreciates by d v l t = v ̇ l t d t . Since the stock market is assumed to be perfectly efficient, the expected rate of return of a stock issued by a successful R&D firm must be equal to the riskless rate of return r:

Dividing by dt, and taking the limits as dt → 0, the following condition for the expected discounted value of the firm producing either the top quality services or pollution abatement services respectively is obtained:

where v . l t v l t = π ̇ l t π l t = 0 . Hence, the expected discounted value (A6.2) boils down to

where r = ρ in equilibrium, and ρ > 0. Q.E.D.

B1

Substituting (18) in (10) and (9), we can write the quantity of each variety targeted by quality service innovations and environmental quality service innovations respectively as:

and

The stream of monopoly profit flows accruing to the firm that manufactures the state-of-the-art of quality services and environmental quality services respectively are therefore:

and

Considering (B1.3) and (B1.4), the no-arbitrage condition for the state-of-the-art quality services and the state-of-the-art environmental quality services as in equation (16) can be written respectively as:

and

where w H = σ θ 0 − γ has been used.

Solving equations (B1.5) and (B1.6) for I ^q and I ^a respectively, the following Poisson arrival rate of innovations are obtained:

and

Summing equations (B1.7) and (B1.8), the aggregate supply of skilled workers H = b ^q I ^q  + b ^a I ^a is obtained

where Δ = λ m c + τ m c a + τ a m F η ̄ m c a + τ a m + 1 − F η ̄ λ m c + τ has been used. Considering equation (B1.9), the skilled labor market clearing condition (19) can be written as:

where Ω ≡ m c σ 1 − s m c w L Λ > 0 , with

Λ ≡ F η ̄ λ − 1 m c a + τ a m + 1 − F η ̄ 1 a − 1 λ m c + τ F η ̄ m c a + τ a m + 1 − F η ̄ λ m c + τ .

The left hand side of the equation (B1.10) is a strictly concave quadratic polynomial in θ ₀ with roots 2 γ − 1 < 0 (recall γ ∈ 0 , 1 2 ) and 1. The right hand side of the same equation is a strictly convex quadratic polynomial in θ ₀ with two real roots, one negative and one positive, where the positive root is:

if the stated parameter restrictions are satisfied. Therefore, one and only one real and positive steady state solution θ 0 * ∈ γ , 1 exists. Q.E.D.

Therefore, equations (19) and (B1.11) imply a constant value of the threshold ability θ ₀, that implies a constant value of the aggregate quantities (B1.1) and (B1.2), profits (B1.3) and (B1.4), Poisson arrival rate of innovations (B1.7), (B1.8), and (B1.9), no-arbitrage condition (16), the per capita consumption (18). In this way, the Euler equation is satisfied for r t = ρ . Therefore, the economy is in a steady-state. This also implies that the per capita average instantaneous utility function grows at the same pace as the aggregate innovation, i.e. u ̇ u = g = I q ⁡ ln ⁡ λ + I a ⁡ ln 1 / a . As usual, the individual utility growth rate is also interpreted as the measure of the log-run economic growth rate of the GDP per capita. Q.E.D.

D1

This appendix establishes the impact of more stringent ERTs on incentives for human capital accumulation and, consequently, on wage inequality and per capita growth rate. In equation (B1.11), the ERTs τ are expressed in the variable Ω ≡ m c σ 1 − s m c w L Λ . To analyze the effects of a marginal change in the ERTs τ on the threshold ability parameter θ 0 * , we can focus on the term Λ ≡ F η ̄ λ − 1 m c a + τ a m + 1 − F η ̄ 1 a − 1 λ m c + τ F η ̄ m c a + τ a m + 1 − F η ̄ λ m c + τ . Using calculus, the following relationships are established:

In the first row of condition (D1.1), when 1 / a > λ is satisfied, more stringent ERTs τ lead to ∂ Λ ∂ τ > 0 . Consequently, increased ERTs τ result in a higher value for Ω, a lower positive root in equation (B1.11), a reduced threshold ability parameter θ 0 * , and an elevated aggregate Poisson arrival rate of innovation (B1.9) along the new equilibrium. The second row of condition (D1.1) determines the same qualitative effects as before when λ > 1 / a and λ > 1 / a m + 1 .

When λ > 1 / a and λ < 1 / a m + 1 is true, based on the third row of condition (D1.1), more stringent ERTs τ lead to ∂ Λ ∂ τ < 0 . Consequently, increased ERTs τ result in a lower value for Ω, a higher positive root in equation (B1.11), an elevated threshold ability parameter θ 0 * , and a reduced aggregate Poisson arrival rate of innovation (B1.9) along the new equilibrium. Q.E.D.

D2

This appendix establishes the impact of a change in the proportion of environmentally conscious individuals on incentives for human capital accumulation and, consequently, on wage inequality and per capita growth rate. In equation (B1.11), the ERTs τ are expressed in the variable Ω ≡ m c σ 1 − s m c w L Λ . To analyze the effects of a marginal change in the proportion of environmental conscious individuals on the threshold ability parameter θ 0 * , we can focus on the term Λ ≡ F η ̄ λ − 1 m c a + τ a m + 1 − F η ̄ 1 a − 1 λ m c + τ F η ̄ m c a + τ a m + 1 − F η ̄ λ m c + τ . Using calculus, the following relationships are established:

In the first row of condition (D2.1), when λ > 1 / a is satisfied, a larger proportion of environmentally conscious individuals, i.e. a lower F η ̄ , leads to a lower Λ, and therefore to a lower value for Ω. Consequently, when λ > 1 / a is satisfied, a larger proportion of environmentally conscious individuals leads to higher positive root in equation (B1.11), an increased threshold ability parameter θ 0 * , and a reduced aggregate Poisson arrival rate of innovation (B1.9) along the new equilibrium.

When 1 / a > λ is true, based on the second row of condition (D2.1), a larger proportion of environmentally conscious individuals, i.e. a lower F η ̄ , leads to a higher Λ, and therefore to a higher value for Ω. Consequently, when 1 / a > λ is satisfied, a larger proportion of environmentally conscious individuals leads to lower positive root in equation (B1.11), a reduced threshold ability parameter θ 0 * , and an elevated aggregate Poisson arrival rate of innovation (B1.9) along the new equilibrium. Q.E.D.

Q.E.D.