Transitions General Equilibrium Model

(HTGEM)

The carbon-economy equilibrium

Our analytical approach follows a general equilibrium model in which society faces the choice of either consume goods and services, invest in productive capital or address climate change.1 Following Nordhaus (1992), the choice is represented by an aggregate utility function based on consumption so that:

(1) where U is the utility derived from consumption, c(t) represents consumption at time t, N is population at time t, and τ is the social preference rate. Thus, the equation depicts the time-discounted value of the sum of utility functions. The maximization is then subject to two restrictions in terms of economic growth and in terms of emissions.

The set of economic constraints, in particular to the decreasing returns to capital model in Ramsey (1928). Thus, the maximized utility equation (1), is based on the utility definition:

(2) where α is the rate of inequality aversion ranging from 0 to 1, so that the larger α becomes, the more aversion to inequality a given society displays and a preference for more egalitarian systems.

Labor, skills, and technology for carbon-neutrality

Consumers then obtain personal or household incomes according to their marginal productivity expressed in wages. The greater the skill the greater their contribution to productivity and reflected in higher wages. The economy wide income derived from labor is obtained by the sum of all workers’ remunerations according to their individual skill:

(3) where W stands for wages, L for number of workers (labor) and u and s refer to unskilled and skilled workers correspondingly. While the EGD’s human transitions will require skilled workers for the production of capital and consumption goods and services, the technological progress required to meet the challenge would also require an emphasis on research workers (r), so that the total skilled workers’ wage bill will comprise research ( r ) and non-research (nr) workers:

(4)

It follows that labor is made of unskilled and skilled (including research) workers subject to skill-specific shares reflected in ω (1 for skilled and 2 for unskilled), and the elasticity of substitution between types of workers ():

(5)

To allow for labor mobility across countries as it is the case in the EU, foreign workers (f) are assumed to be able substitutes for local workers (l) with an elasticity of substitution among them (ρ2):

(6)

(7) where ω continues to be the share of worker type (3 for skilled local, 4 for skilled foreign, 4 for unskilled local, and 5 for unskilled foreign), and elasticity of substitution between types of workers () according to 2 local to foreign among skilled workers, and 3 for local substitution for foreigners among unskilled workers.

Assuming there are no consumption externalities of choosing one location over another, workers will move to locations chiefly based on relative factor prices, specifically relative wages. However, locations with higher wages will attract further workers, until the receiving and sending locations’ wage gap narrows. The narrowing of the wage gap will be the result of rising wages in locations losing labor, and declining wages in the region that receives the inflow of workers. The same phenomenon will occur by segment of the labor market (i.e. unskilled and skilled) subject to the elasticity of substitution (ρ). Each location’s resulting labor share (λ) of unskilled workers will therefore be:

(8) and correspondingly for skilled workers:

(9) where location i is different from location j

For simplicity, the production process could occur under a constant-returns-to-scale Cobb-Douglas production function in which total factor productivity is represented by A, capital (K), effective labor (L) and natural resources (E):

(10) subject to β + γ ≤ 1

Aggregate demand will therefore be the results of all factors of production at their marginal product (assuming payments to the factors of production reflect their true contribution to productivity as in the Arrow-Debreu model), so that the linear equation for (10) is:2

(11)

(12) where r refers to the payments to capital, and p is prices under the carbon (c) economy.

However, the EGD implies an investment of physical and human capital to face policy shocks related to the objectives in the new green economy. The demand therefore for innovation will be significant. Technological progress will then accelerate. As a consequence, some firms may be able to acquire or develop newer technologies; however, some others would likely not. Therefore, assuming constant returns to scale as in equation 10, at least in some of the sectors would not be realistic. Instead, technology and innovation will determine market structure for firms. Innovation in this Schumpeterian view leads to a creative destruction (Schumpeter, 1934), which is only possible if some protection is provided to new ideas in the form of patents and copy rights. The likely unintended consequence is, some degree market concentration. Monopolist and oligopolistic competition would then require firms’ profit-maximizing supply to be:

(13) where represents the cost of renting or acquiring capital, and is the share of revenues that accrue to workers. Following Aghion and Howitt (1999), A becomes the average productivity parameter and (1−θ) represents profits. It is therefore implied that the sum of intermediate outputs equals the intensity in the use of capital (i.e. capital-labor ratio) so that:

(14) and the associated cost of acquiring capital will decline with past accumulations expressed in the intensity of capital:

(15)

With this market structure favoring innovation in firms exerting market power, innovation (V) in a steady-state growth path would become:

(16) where is the maximum A which in turn indicates productivity gains stemming from the leader’s technology (or the leading-edge technology); represents as before, firms’ profits and k continues to be capital intensity at steady-state level. Similarly, r is the discount rate applied to future consumption and φ is the innovation probability flow and ϑ represents the level of research.

The carbon-neutral economy

An EGD framework requires to account for the coupled relationship between production—and its growth—and carbon emissions, as well as natural resource consumption. The framework assumes that a short-term fixed rate, but long-run variable ratio, of carbon is an externality of production. Similarly, the model also incorporates a rate with similar intertemporal properties for production and natural resource consumption. The depletion of natural resources is then given by:

(17) where D is the net depletion of natural resources, n ranges from 0 to 1 and represents the ratio of natural resources consumption to gross production (Q). D is net of any offsetting elements provided by reforestation or any action that leads to replenishment (R) of natural resources (E).

Greenhouse gas (GHG) emissions are an externality of production and thereby a ratio that changes over time with efficiency (ε), and a ratio to production represented by σ:

(18)

It follows that overall impact (I) of production will be:

(19)

Following Hallegate et. al. (2012), a vector of policies that aim at controlling or reducing can be considered under the vector P(e), which may have the effect of inducing efficiency (such as those related to the European Green Deal):

(20) where ψ measures production efficiency. P(e) will be assumed to lead to a reduction in GHG through the ratio μ. Thus P(e) in equation 20 will transform equation 18 so that:

(21)

In order to better understand how environmental-upgrading policies may entail costs to different economic agents, but at the same time lead to efficiencies and positive externalities, economy-wide effects need to be modeled from the basic production equation using the output approach:

(22) where aggregate demand is represented by Y, P are prices and Q production for i industry in t time. Prices in equation 22 are t=0 where carbon industries are still largely unaffected by ay elements in P(e). Since Y can also be accounted for using the expenditure approach:

(23)

All output in the economy (equation 22) is equal to all expenditures (equation 23) and to the sum of all payments to the factors of production (the simplified form of equation 12), so that:

(24)

The EGD entails regulations that increasingly put pressure on economic agents to change behaviors and production methods. These regulations under P(e) may have two main effects. On the one hand, these policies may foster innovation and technology adoption in firms that leads to efficiency. On the other hand, they also represent an additional cost for firms to operate. The former is already considered under equation 20. The latter would alter relative prices in the economy resulting in:

(25)

Payments to the factors of production will change accordingly, by changing r and w in equation 24, increasing k in equation 14 and changing relative wages in λ (skilled and unskilled in equation 9). Changes in consumption will follow so that: (i) carbon-neutral production ( Qn ) at carbon-neutral prices (Pn) would be favored following EGD policies; (ii) investment moves towards carbon-neutral technologies that will trigger a change in labor demand patterns; and (iii) relative wages will increase in favor of skilled workers.

The short-run equilibrium

Assuming firms have no particular market power so that n firms operate in i location, labor market equilibrium condition would imply that labor demand (Ld) and supply (Lt as in equation 5) are effectively matched:

(26)

Firms will then invest in capital according to the market structure, in those markets in which technologies are widely available (e.g. solar panels), the industries market structure can be assumed to reach a monopolistic competition equilibrium and thus, face prices that include an elasticity of substitution among varieties of the same product so that:

(27) where are prices in location i under the new conditions imposed by carbon neutrality (n). These prices are explained as individual prices in location i and subject to the elasticity of substitution for other varieties (ε > 1). In contrast, for industries in which technologies are the result of innovation, market structure can be assumed to take the form of an oligopoly that will appear at the speed dictated by V in equation 16 and with an associated cost of as in equation 15. Finally, we recall that profit maximization is as in equation 13, with an elasticity of substitution of 0. For consumers, welfare implications of the new carbon-neutral economy start considering that they seek the maximization of their utility function expressed in equation 1. However, relative prices and wages will imply:

(28) where the inequality aversion term continues to determine utility as in equation 1. However, equation 28 includes changes in income due to carbonto-carbon-neutral relative prices (P*), and relative wages triggered by P(e) restrictions stemming from the EGD.

The long-run equilibrium

In the long-run, firms have the capacity to evaluate their choice regarding the mix of factors of production. Given the transition to a carbon-neutral economy, technological requirements most likely will lead to both, an increase in the capital-labor ratio and to a shift in skills in the latter. Equation 26 would therefore transform to:

(29)

Equation 29 proposes that total labor demand is the sum of individual firms in location i proportional to their individual capital-labor ratio (k as in equation 14). Thus, the EGD’s requirement to increase efficiency will over time, likely raise individual firms’ capital-labor ratio (k) and alter the demand for labor. However, since labor is in this model, not homogeneous, but instead allows for differences in skills (research, skilled and unskilled), as well as for labor mobility (by allowing local and foreign labor to substitute each other), labor demand will also be subject to relative wages, relative labor forces and the elasticity of substitution as per equation 9. As a result, labor demand will be explained by long-run adjustments related to workers’ migration choices based on relative wages and the elasticity of substitution in other locations in Europe:

(30)

In the long-run, k will continue to change, and firms will face market structure determined by technological progress and innovation. Welfare implications in the long run are similar to equation 28. However, the long-run utility function adds a term that reflect positive externalities from the new carbon-neutral economy in terms of the cleaner air and environmental amenities (Z(t)) that can be enjoyed including a health improvement term (H(t)).

(31)