Prices of carbon dioxide and emission permits in general equilibrium

The definition of a price per equivalent ton of carbon dioxide means establishing a market price for policies to reduce greenhouse gas emissions. The definition of a price per equivalent ton of carbon dioxide is a tool or financial mechanism that must reflect the economic, social and ecological costs of climate change. There are three approaches to the corporate definition of coal prices: shadow prices, internal taxes or quotas, and implicit prices (^{UNFCCC, 2015}). The two instruments of the most used emission control policy are: the definition of carbon dioxide prices and the granting of emission permits.

In the last two years the number of instruments for the definition of carbon dioxide prices has almost doubled, since it increased from 20 to 38. Currently, forty countries and twenty cities, which together represent 25% of total emissions to worldwide, they have imposed a price on emissions. The control of emissions by this means represents 12% of global emissions (^{IBRD, 2015}). For example, this carbon dioxide price instrument covers 1 Gt CO2eq in the US, 0.5 in China and 2 in the European Union. Today, the total value of all instruments of carbon dioxide prices is 50 billion dollars, of which 70% corresponds to the systems of buying and selling of emissions and 30% to carbon taxes. The prices of emissions in equivalent units of carbon dioxide fluctuate between 1 and 130 dollars.

These policy instruments, so variable and chaotic, have not been effective to induce an effective control of the levels of greenhouse gases accumulated in the atmosphere, because the accumulated levels of C O 2 eq not only have not been reduced but have grown considerably at dangerous levels. Hence the need to define a method to quantify their efficient levels. The appropriate approach is, obviously, that of general equilibrium, since GHG emissions have effects that are not only general within each country but also have global or planetary consequences. However, global carbon markets ( CO 2 eq) are incomplete, so the general equilibrium allocations are far from efficient. Consequently, the creation of markets in which pollution permits are offered and demanded can result in optimal assignments in the Pareto sense, only if the allocations obtained there are feasible and if economic agents that produce externalities should pay the corresponding cost to the negative economic consequences of their decisions and productive activities (^{González-Estrada, 2003} and ²⁰⁰⁴).

The general equilibrium of the economy with emission permits

The creation and organization of international markets for emission permits is one of the policy measures for the control of greenhouse gas emissions that has received more attention from experts (^{Tientenberg, 1992}), due to its advantages, the most important of which is its ability to be instrumented and carried out with a minimum bureaucratic apparatus and with low transaction costs (^{Larsen and Shah, 1994}). In addition, they are the most promising instruments of environmental policy in current conditions (^{Grubb, 1989} and ¹⁹⁹⁰), since they are also flexible, effective and efficient to reduce levels of CO 2 Eq in the atmosphere.

The general equilibrium model proposed here is an adaptation of the model of a global economy with emissions of greenhouse gases and emission permits, initially built by ^{Uzawa (2010)}. Let θφ be the annual emission of greenhouse gases (GHG) measured in units of carbon dioxide equivalent, CO2eq, in the country φ=1, 2, …, n and be θ the total emissions of those gases a global level: θ= θ1 + θ2 + ⋯+ θn. For each country, its emissions: θφ, plus those of the rest of the world: θ, add the total emission: Θ. That is: Θ= θ + θφ. The set of production possibilities of each country φ is: Tφ,∀ φ= 1, 2, …, n. In each country there are L productive factors: kφ=kφ1, kφ2, …, kφL, which have the following properties: kφ≥ 0; that is: kφl≥ 0∀l= 1, 2, …, L and kφl> 0, for some l. In this economy there are k goods: j=1, 2, …, k. The minimum number of factors of production required to produce the vector of goods or products in each country is: xφ= xφ1, xφ2, …, xφL. The national income of the country φ, whose emissions are θφ, is: yφ= fφkφ, θφ. The technologies exhibit constant returns to scale and the set of production possibilities for this economy, Tφ, is closed and convex for the vectors kφ, θφ∈R1+n+c. There are k goods and the price vector is: p=p1, p2, …, pk∈R+k (^{Uzawa, 1991} and ¹⁹⁹²).

The function that represents the partial orderings for the consumption cφ and for the total global amount, at planetary level, of the annual emissions of greenhouse gases, θ, expressed in equivalent units of carbon dioxide, CO2eq, where: θ= θ1 + θ2 + ⋯+ θn, is the function uφ= ucφ,θ (^{Uzawa, 1991} and ¹⁹⁹²).

Let ψθ be the degree to which the country’s family units φ are affected by emissions θ. Obviously, ψθ can be called global warming index, identical for all countries, and, in addition: ψθ> 0, ψ 'θ< 0;ψ''θ< 0,∀ θ∈0, θ^, where θ is a critical level of emissions and means that if θ^ continues to grow beyond θ^ irreversible damage will occur to humanity. ^{Uzawa (1991} and 1992a) postulated the following index of global warming: ψθ= (θ^- θ)β,θ∈0, θ^⋏β∈0,1. The marginal rate of change of the environmental impact index due to a marginal increase in emissions of CO2 equivalent is:

This marginal rate is also known as the coefficient of impact of global warming. The vector of the tax rates on emissions is: τφ= τφ1, τφ2, …, τφk and the cost vector of the production factors is: ωφ=ωφ1, …, ωφL.

How should the general equilibrium of this economy be defined with emissions of greenhouse gases and taxes? What are the necessary and sufficient conditions for the existence of a general equilibrium with efficient assignments of pollution permits?

In order to answer these questions, it is assumed that there is an international market for emission permits, in which countries buy and sell permits to issue CO2Eq. Let θφ be the amount of CO2Eq issued by the country φ= 1, 2, …, n and let bφ, the amount of the emission permits for that country. If θφ> bφ, the country would have to buy θφ - bφ emission permits, but if θφ >bφ, then it could sell bφ - θφ permits. From an institutional point of view, the central problem is how to reach an agreement regarding the definition of the initial quantity of permits offered (^{Rose and Stevens, 1993}; ^{Bertram, 1992}; Larson and Shah, 1992; 1994). It is assumed that both θ and b were already determined and that, as already said: b= ∑φ=1nbφ,θ=∑φ=1nθφ. The condition of the balance of payments for the country φ is: pcφ=pxφ-qθφ-bφ,where p=p1, p2, …, pk is the price vector of the k goods produced and q is the price of the emission permits in the international market. In addition, cφ is the consumption vector and xφ is the vector of goods produced in the country φ=1, 2, …, n.[/p]

The general equilibrium of the world economy with emission permits is the matrix of production, consumption and price allocations: p*, q*, ωφ*, pem*; xφ*, cφ*, kφ*, θφ*, such that: a) in each country, companies maximize profits πφ= p.fφkφ, θφ- wφkφ - qθφ; b) solve Max’s problems uφcφ,θ=ψθuφcφ, subject to p.cφ= pxφ - qθφ - bφ; c) globally, goods markets are balanced: ∑φ=1 ncφ=∑φ=1n xφ, and finally, d) the sum of the emissions of CO2Eq of all countries is equal to the authorized amount of emission permits: Σ ∑φ=1n θφ= θ=∑φ=1n bφ= b. The general equilibrium conditions, necessary and sufficient, given concavity conditions, are:

If they are multiplied (1) and (2) respectively by k and θφ if they are summed and if Eüler’s theorem is applied, we get:

That is, yφ= wk+qθφ= p·cφ. This means that the national income or the total value of production is equal to the sum of the payments to the factors of production: capital and labor force. Since qθφ is the value of the emission allowance assignment at market prices, then: y yφ= wk+qbφ= pxφ - q θφ + qbφ= pxφ - qθφ - bφ≡ p·cφ. Clearly: p·xφ - q θφ≡ wk. Note also, that if both sides of condition (4) are multiplied by cφ and the Eüler Theorem is applied, we get:

The general equilibrium requires that: a) the vectors p*,q*meet the conditions (1)-(8); b) in each country the taxes on the emissions are equal to the market price of the same, q*, and c) the balance of payments is balanced, defined as: pcφ= pxφ - qθφ - bφ. The general equilibrium of the country /φ for φ = 1, 2, ..., n, characterized by conditions (1) - (8) will be efficient in the Pareto sense, if θ= b and, of course, if: θφ=bφ,∀ φ=1, 2, …, n. Condition (4) implies that:

where: αφ= 1/λφ> 0, ∀φ y λφ is the opportunity cost of money or the marginal utility of income. Finally, from the general equilibrium conditions, from the definition equation: y= p·c and from Eüler’s theorem, we obtain : rφθ=τφθ/yφ, so that:

Therefore, the tax rate that the country φ must pay for its greenhouse gas emissions is: τφ*θ= rφθyφ, where τφ*θ is the tax on emissions of CO2eq in the country φ,rφθ is the coefficient of environmental impact and yφis the national income of the country φ. Given that p·c= p·x= y, then the optimal tax corresponding to sector j in country φ is given by the following expression: τφj*θ=rφjθyφj, where rφjθ is the tax rate that sector j of the country φ must pay for its emissions of greenhouse gases and yφj is the national income produced by sector j in the country φ.

The cost of nitrogenous fertilization emissions in Mexican agriculture

Climate change is an inexorable reality, whose effects, at best, can only be mitigated. According to ^{Sachs (2008)}, the most severe and catastrophic impacts could be avoided if the global temperature of the atmosphere increases less than 2 degrees Celsius in relation to the average temperature prevailing in the reference period: the beginnings of the industrial revolution to mid eighteenth century. To achieve this, it is necessary to reduce the 52.76 giga- tons of CO2eq produced in 2012 to only 32 by the year 2050. At present, the global costs associated with mitigation and adaptation to climate change add up to an amount between 385 billion dollars per year; the investments that are expected to be made by the year 2030 should be of 400 billion dollars per year and by 2050, of 2 billion dollars per year (^{IBRD, 2015}). Today, coal prices are fragmented by countries and, within them, by sectors of the economy (^{IBRD, 2015}). In addition, they are fixed in a very wide range: from 1 to 130 dollars. It is obvious that those prices are far from being efficient, socially speaking. This dispersion reflects the fact that the richest countries transfer the most polluting activities to the poorest countries, who, in order to attract investments, lower environmental standards. It also explains why the ten largest developing economies pollute more than the ten most developed and richest countries.

^{Uzawa (2010)} shows that if b*= θ; that is, if emission permits equal the total amount of emissions, then the general equilibrium solution with emission permits is equal to the general equilibrium with emission taxes, so the total planetary cost of emissions of CO2eq is: q*=rθy. Consequently, the cost of a ton of CO2eq is:

This estimator is consistent with the results obtained by ^{Golosov et al. (2014)}, who, with a stochastic general equilibrium dynamic model, estimated an efficient cost of emissions of 56.9 dollars per ton if the discount rate were 0.5% and 496 dollars if that rate was 1.5%. If you want to avoid that the global temperature of the atmosphere increases less than 2 degrees Celsius in relation to the average temperature prevailing in the mid-eighteenth century and that the economic, social and environmental effects of climate change reach catastrophic and unavoidable levels by the year 2050 , the price that each emission permit must have is $100.00 dollars per ton of CO2eq. Something similar is reported by the International Bank for Reconstruction and Development (^{IBRD, 2015}). This is the cost to maintain the production of humanity within the set of non-catastrophic feasibility. Consequently, the costs of the emissions produced by chemical fertilization in Mexican agriculture are the following.

Efficient emission control policies

The nitrous oxide emissions produced by the application of chemical-nitrogen fertilizers represent 50.4% of the emissions of the agricultural sector (^{SEMARNAT, 2013}). According to Table 2, it is estimated that in 2014 emissions from this sector were 11 617 million tons of CO 2 eq, which represent a cost of 19 982.2 million pesos in 2017. In that year, the Agricultural GDP was 280 billion pesos, so the ad valorem tax on emissions of greenhouse gases should equal 7.14%. In order to better understand this result as part of a whole, ^{González-Estrada et al. (2011)} cite that in 2004 the total costs for depletion and environmental degradation (CTADA) in Mexico represented 9.2% of GDP; that is, 712 344 million pesos, an amount that reflects the amount of natural capital appropriated as surplus or extraordinary profits by the economic agents benefited without paying anything in return.

^{Gonzalez-Estrada et al. (2011)} estimated the efficient environmental tax for each economic sector in Mexico. The agricultural, forestry and fishing sector produced an environmental degradation equivalent to 49 226 million pesos, so the efficient environmental tax that should be paid is 13.15%. This ad valorem tax of 7.14%, obtained here, although efficient in terms of Pareto, would be ineffective due to the high levels of tax evasion and, in addition, would have high transaction costs, since its implementation would be onerous.

With the help of models of general dynamic equilibrium of the world economy it was estimated that the efficient cost of a permit to issue a ton of C2Oeq should be $ 100.0 dollars (IBRD, 2015), and that this is the efficient price, in Pareto terms, of an emission permit for a ton of C2O eq. According to ^{Uzawa (2010)}, if b= θ, the general equilibrium with emission permits corresponds to the differential equilibrium with taxes on emissions.

Therefore, an equivalent and equally efficient policy would be to establish a Pigouvian tax on chemical fertilizers, according to the relative nitrogen content, using the same method followed to obtain the total cost of emissions in the Table 3 and in accordance with the procedure proposed by ^{González-Estrada (2003} and ²⁰⁰⁴).

Irrationality of the use of fertilizers in Mexican agriculture

Chemical fertilization in Mexican agriculture is inefficient economically and socially, in the sense that they define efficiency ^{MasColell, et al. (1995)}; ^{Varian (1992)}; ^{Jehle and Reny (2011)}, because neither the producers nor the authorities nor society take into account the costs of nitrous oxide emissions that fertilizer applications have. Moreover, in the calculation of the optimal-economic doses of fertilization, emissions costs have not been taken into account and, therefore, fertilization practices in Mexico are not economically or socially efficient either.

The current policy of fertilizer subsidies, which promotes greater use than optimal-economic doses and also does not take into account the costs of nitrogen-containing chemical fertilizer emissions, is also inefficient, firstly, because subsidies are inefficient in their own right and, secondly, because they encourage excessive use of nitrogen fertilizers and, therefore, stimulate emissions of nitrous oxide, which has a heating potential 310 times greater than that of carbon dioxide, as it is reported by ^{SEMARNAT (2013)}.