Carbon Trading with Blockchain

Abstract

Blockchain has the potential to accelerate the worldwide deployment of an emissions trading system (ETS) and improve the efficiency of existing systems. In this paper, we present a model for a permissioned blockchain implementation based on the successful European Union (EU) ETS and discuss its potential advantages over existing technology. The proposed ETS model is both backward compatible and future-proof, characterised by interconnectedness, transparency, tamper-resistance and continuous liquidity. Further, we identify key challenges to implementation of blockchain in ETS, as well as areas of future work required to enable a fully decentralised blockchain-based ETS.

Appendix

1.1 Bancor Algorithm for Token Exchange

As demonstrated with Algorithms 7 and 8, the Bancor exchange protocol  [25, 39] ensures constant tradability of a token, as it prices a token algorithmically, as opposed through matching a buyer and a seller. We use the notation listed in Table 3 to explain the protocol.

For demonstration purposes, we assume that the medium of exchange is a stablecoin, measured in €, that circulates on the same blockchain as the permit tokens.

Table 3 Mathematical notation for token exchange

It holds that, the stablecoin reserve C (in €), always equals a fraction, preset as $F\in (0,1)$, of the product of token price P (in €/token) and outstanding token supply s (in tokens). That is, the following equation is always true:

$$\begin{aligned} C(s) \equiv F\, s\, P(s) \end{aligned}$$

(1)

Taking the derivative with respect to s on both sides:

$$\begin{aligned} \frac{\mathrm {d} C(s)}{\mathrm {d} s} \equiv F\left[ P(s)+s\frac{\mathrm {d} P(s)}{\mathrm {d} s}\right] \end{aligned}$$

(2)

There exists another relationship between C, P and s: if one buys from the exchange an infinitesimal amount of tokens, $\mathrm {d}s$, when the outstanding token supply is s, then the unit token price at purchase would be P(s). The exchange receives stablecoins and thus its reserve increases according to:

$$\begin{aligned} \mathrm {d} C(s) = P(s)\, \mathrm {d} s \end{aligned}$$

Rearranging:

$$\begin{aligned} P(s) = \frac{\mathrm {d} C(s)}{\mathrm {d} s} \end{aligned}$$

(3)

Combining (2) and (3),

$$\begin{aligned} P(s)&=F\,\left[ P(s)+s\frac{\mathrm {d} P(s)}{\mathrm {d} s}\right] \nonumber \\ \frac{\mathrm {d} P(s)}{P(s)}&=\left( \frac{1}{F}-1\right) \frac{\mathrm {d} s}{s} \nonumber \\ \end{aligned}$$

Integrating over $s \in (s_0, s)$:

$$\begin{aligned} \int _{x=P(s_0)}^{P(s)}\frac{\mathrm {d} x}{x}&=\left( \frac{1}{F}-1\right) \int _{y=s_0}^s\frac{\mathrm {d} y}{y} \nonumber \\ \ln P(s) - \ln \frac{c_0}{F\,s_0}&=\left( \frac{1}{F}-1\right) (\ln s - \ln s_0)\\ \end{aligned}$$

Now we can express token price P(.) as a function of s:

$$\begin{aligned} P(s)&= \frac{c_0}{F\,s} \root F \of {\frac{s}{s_0}} \end{aligned}$$

(4)

Plugging (4) into (1), we can derive the exchange’s stablecoin reserve C(.) as a function of s:

$$\begin{aligned} C(s)=F\, s\, \frac{c_0}{F\,s} \root F \of {\frac{s}{s_0}} = c_0\, \root F \of {\frac{s}{s_0}} \end{aligned}$$

(5)

Assume one spends t amount of stablecoins in exchange for e amount of tokens when the outstanding token supply equal $s_0$. After the purchase, the outstanding token supply becomes $s_0+e$, while the stablecoin reserve increases by t, i.e.,

$$\begin{aligned} t + c_0 = C(s_0+e) \overset{\text {according to } (5)}{=} c_0\, \root F \of {\frac{s_0+e}{s_0}} = c_0\, \root F \of {1+\frac{e}{s_0}} \end{aligned}$$

(6)

Rearranging (6), we get:

• the amount of stablecoins to be paid (or received when negative), t, based on the amount of tokens to be bought (or sold when negative), e, and the outstanding token supply $s_0$ (Algorithm 7),

  $$\begin{aligned} t = c_0 \Bigg (\root F \of {1+\frac{e}{s_0}}-1\Bigg ) \end{aligned}$$

  (7)

• the amount of tokens to be bought (or sold when negative), e, based on the amount of stablecoins to be paid (or received when negative), t, and the outstanding token supply $s_0$ (Algorithm 8).

  $$\begin{aligned} e = s_0\,\left[ \left( \frac{t}{c_0} + 1\right) ^F - 1\right] \end{aligned}$$

  (8)