This paper is concerned with the mathematical analysis of emissions markets. We review the existing quantitative analyses on the subject and introduce some of the mathematical challenges posed by the implementation of the new phase of the European Union Emissions Trading Scheme as well as the cap-and-trade schemes touted by the U.S., Canada, Australia, and Japan. From a practical point of view, the main thrust of the paper is the design and numerical analysis of new cap-and-trade schemes for the control and reduction of atmospheric pollution. We develop tools intended to help policy makers and regulators understand the pros and cons of the emissions markets. We propose a model for an economy where risk neutral firms produce goods to satisfy an inelastic demand and are endowed with permits in order to offset their pollution at compliance time and avoid having to pay a penalty. Firms that can easily reduce emissions do so, while those for which it is harder buy permits from firms that anticipate they will not need all their permits, creating a financial market for pollution credits. Our equilibrium model elucidates the joint price formation for goods and pollution allowances, capturing most of the features of the first phase of the European Union Emissions Trading Scheme. We show existence of an equilibrium and uniqueness of emissions credit prices. We also characterize the equilibrium prices of goods and the optimal production and trading strategies of the firms. We use the electricity market in Texas to numerically illustrate the qualitative properties of these cap-and-trade schemes. Comparing the numerical implications of cap-and-trade schemes to the business-as-usual benchmark, we show that our numerical results match those observed during the implementation of the first phase of the European Union cap-and-trade CO${}_2$ emissions scheme. In particular, we confirm the presence of windfall profits criticized by the opponents of these markets. We also demonstrate the shortcomings of tax and subsidy alternatives. Finally we introduce a relative allocation scheme which, while easy to implement, leads to smaller windfall profits than the standard scheme.

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