# Equity Principles method

The first method used for the climate calculator is based on Bretschger (2013). The paper discusses four different equity principles that are important for the distribution of a global carbon budget.

• Ability to pay: The larger the economic capacity of country is, the more it should contribute to global climate policy,
• Cost sharing: The higher the costs of climate policy are for a country, the less it should contribute to global climate policy
• Technical contribution: The more a country has contributed to advance carbon-efficient technologies in the past, the less it should contribute to global climate policy
• Technical development: Earlier emissions are weighted less than later emissions, because carbon-efficient technologies become increasingly available over time.

To operationalize these principles, we have to find proxies for the four different principles. For this, we use population, Li, Gross Domestic Product (GDP), Yi, and Emissions, Ei. We measure the four measures according to the following table.

Ability to pay Cost sharing Technical contribution Technical development
Measure
\frac{L_i}{Y_i}
\frac{E_i}{L_i}
\frac{Y_i}{E_i}
\frac{E_i}{L_i}
Weight α β γ δ

Having defined how to measure the four principles, we have to combine them into a fairness index, Fi, to measure the performance of countries. We use a specification that combines the principles in a multiplicative way and relies on a Cobb Douglas functional form. The different exponents define the weight of each principle.

F_i = \left(\frac{L_i}{Y_i}\right)^\alpha\left(\frac{E_i}{L_i}\right)^\beta\left(\frac{Y_i}{E_i}\right)^\gamma\left(\frac{E_i}{L_i}\right)^\delta

With the additional assumption of α + β + γ + δ = 1, you can think of the exponents as percentages of the importance of each principles.

Example: If you think that “Ability to pay” is twice as important as “Cost sharing” which you consider equally important as “Technical contribution” and “Technical development” you pick the numbers:
α = 0.4; β = γ = δ = 0.2

You can also type α = 40; β = γ = δ = 20 and due to the normalization only the relative size of the parameters matters. To get the budget, Zi, of each country, we weigh the fairness index by population and multiply it with the chosen budget, Z including previous emissions from 1990 to 2000, E_{1990-2000} (around 230 GtCO2).

Z_i = \frac{L_i F_i}{\sum L_i F_i} \left(Z + E_{1990-2000}\right)

To account for historic responsibility, we deduct the historical country emissions from the country budget discounted by θ, the responsibility factor. So if θ = 0.5 we take away 50% of the previous emissions.

Z_i(\theta) = Z_i - \theta \cdot E_{\text{hist}, i}

This budget has to be scaled by the ratio of the total budget to the responsibility budget. If we only deducted θ = 0.5, the total budget would be larger than the original budget, since the amount has been already emitted.

To get the remaining fixed budget, Z_i^\text{final}, we multiply the total budget by the ratio of the reduced budget over the responsibility budget.

Z_i^\text{final} = \frac{Z^\text{rem}_\text{scaled}}{\sum Z_i(\theta)} Z_i(\theta),

where Z^\text{rem}_\text{scaled} is the remaining re-scaled budget as described in General settings.