more relevant. In this model, firms’ decisions regarding permit trading is an _ex ante_ choice in the sense that optimal emissions and permit banking decisions depend not only on current but also on expectation of future output and input prices. Second, I empirically test the theoretical prediction in a real trading program. To the best of my knowledge, this is the first attempt to quantitatively estimate the effects of uncertainty on emission trading based on actual market data. Although empirical analysis is conducted in the context of the U.S. SO2 allowance trading program, the conclusion is general enough to be extended to the analysis of global carbon trading program.

# **III. Modeling Framework**

This section contains the basic theory of intertemporal permit trading under uncertainty. I begin by setting up a firm’s dynamic optimization problem, and then state and prove Proposition 1 and Lemma 1 on the relationship between uncertainty, and banking/emissions. I then go on to show how conclusions can be affected by imperfect competition in the permit and output markets and returns to scale of production technology.

## **III.(i) A Firm Model of Intertemporal Permit Trading Under Uncertainty**

Consider a risk neutral firm that uses adjustable levels of low- and high-carbon fuel to produce electricity. In each time period, the firm decides the electricity output ( _gt_ ) chooses the mix of low- and high-carbon fuel ( _lt_ and _ht_ ), and the amount of permits to buy ( _xt_ > 0) or sell ( _xt_ < 0) to maximize its discounted present profits for a constrained level of emissions. Uncertainty exists in the supply and demand for electricity. Suppose this uncertainty is characterized by electricity prices _Pe_ , which is a random variable that follows a Markov process. The probability law of _Pe_ is known to all firms. At the start of each period _t_ , the firm observes electricity price ( _Pet_ ), permit price ( _Pat_ ), the price for low- and high-carbon fuel ( _Plt_ and _Pht)_ , and the initial endowment of permits which is the sum of permits issued by the government at current period ( _At_ ) and the amount of banked permits carried from the previous period ( _Bt_ ). Refer to Table I for a thorough description of all model parameters used in the section.

Firms are facing a dynamic optimization problem because choosing how many permits to save for the future has to be made before uncertainties over future prices are resolved. Assuming that firms are price takers in all markets, I model individual firm behavior as an intra-firm game. Taking the strategies of other firms as given, each firm picks a strategy in each time period that is optimal from the firm’s perspective in that period. The firm’s strategy is thus a map from the Markov state Λ _t_ = { _Pt_ , _At_ , _Bt_ } to choose variables { _lt_ , _ht_ , _xt_ }, where _Pt_ is a price vector, i.e. _Pt_ = { _Pet_ , _Pat_ , _Plt_ , _Pht_ }.

I assume firms employ three compliance strategies: abating emissions through blending or switching to low-carbon fuel, purchasing allowances in addition to initial allocation, and adjusting output levels. Other capital intensive strategies such as scrubbing, re-powering or permanently retiring a facility are not considered because regulatory, financial and other uncertainties in today’s volatile energy markets provide firms incentive to avoid capital intensive investment as long as possible.

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