## Presumptions

We try to utilize and extend Ishii's (2008) result to our case, so that the scope of our analysis is limited to much extent by that result. In particular, we have to assume the following:

1. There are enough participants in the emission market in the trueup period, both rational and erratical participants. This can be justified by the fact that many CDM credits are available so that they can choose to enter into the market for each commitment periods which overlaps.

2. Near the final stage of the trueup period, the emission amount of each nations in the past becomes common knowledge. This is somewhat a simplification, because some exact statistics would only be available very late. But approximately, this would hold.

3. Except for the nations under consideration who have to purchase emission from the market, all the other agents who have emission at hand in the beginning of the periods analyzed are not rational or moving randomly. This is somewhat heroic assumption necessitated by the earlier analysis. And we list it again explicitly.

4. For the market participants who are not nations, they can bank the emission they hold even after the trueup period. This is an assumption which could differ from the actual rule. Thus, elimination of this assumption would be the most desirable.

### 25.3 Model

There are three types of traders in our emissions trading market: N nations, a lot of small investors, and one noise trader. They can trade at times t =1, 2. Every nation i=i,2,---, N has to buy Wi;i units of emissions over this time period. Those who have one unit of emission obtain a banked value F at time t =3. F = F""' (25.1)

where F is observed by all traders at the beginning of t=1. s follows a normal distribution that has a mean 0 and a variance aF2 at t=3.

Each nation i places a market order Si;t at time t=1,2. We require Si;i;+Si;2=Wi;i. We define Wi;2= Wi;i-Si;i. The utility of a nation i is

for constant values Xi>0, Wi.

There are infinitely many small investors. They are uniformly distributed over [0,i]. The measure of each small investor is 0. They have no position at t=0. They can borrow some money or emission credits and place limit orders at t =i,2. They face no liquidity constraint. The interest rate is 0 for simplicity. We denote the quantity possessed by a representative small investor at the end of t (t =i,2) as Bt. His utility is

Up Px , P2. f) = - exp \-p ((P2 - P,) Bx + {F - P2) 5,)] ,

for a constant value p>0. The trading volume of the representative small investor at t is MBt^Bt-^Bt-i (B_i=0)

The noise trader randomly places a market order nt at t=i,2. nt follows a normal distribution that has a mean 0 and a variance aj independently of each other and of s.

The price Pt is determined to conform the amount of all buy order's to all sell order's at every t=i,2:

At the end of period t, all traders observe the price Pt.

We define some notations. Ei [ ] and Vari [ ] are an expectation and a variance, respectively, conditional on events that the small investors can observe by the end of the time 1. Furthermore, we define