One of the reasons for public confusion regarding climate change is a general failure to communicate several rather simple concepts clearly and carefully. I will leave the question of whether these are errors of omission or commission to others.
A carbon "cap" is simply a maximum limit on annual carbon emissions by some entity, whether it be a country, an industry or a specific company. Rationally, the "cap" would be set slightly above current annual emissions to allow for economic growth between the time when the "cap" is established and the time when it takes effect. A "cap" is not a market-based mechanism. A "cap" is a command and control mechanism, imposed by legislative or executive fiat.
A "declining cap" is a "cap" which is imposed at some initial emissions level and then declines or decreases by a pre-determined amount each year until the pre-determined reduced emissions level is achieved. A "declining cap" is also not a market-based mechanism, since the initial "cap", the rate of decline and the final "cap" are determined by legislative or executive fiat.
A "trade" is an action taken by a willing (if compelled) buyer and a willing seller in which the seller, who has reduced carbon emissions more rapidly than would have been required by the declining "cap", "trades" unneeded emissions allowances to a buyer, who has not reduced carbon emissions to the extent required by the declining "cap". A "trade" is a market-based mechanism, in that the seller and the buyer agree on a value for their transaction which is a function of the supply of emission allowances available from those who have reduced emissions to a greater extent than required and the demand for emissions allowances from those who have not reduced their emissions to the extent required.
The market in which emissions allowances are "traded" establishes a "price" for carbon which is a function of the relative cost of carbon emissions reductions by those participating in the market. This carbon "price" would not produce a federal revenue stream, to any significant extent, since the revenue from the sale would be taxable to the seller, while the purchase price of the allowances would be a tax deduction for the buyer.
The government could also impose a "price" for carbon by selling the carbon emissions allowances available under the current "cap" to carbon emitters and others who desire to purchase the allowances. There are a variety of criteria which could be used to set this "price" for carbon. The "price" would incent those for whom carbon emissions reductions could be achieved rapidly and for less than the established "price" to do so, rather than pay the higher "price" for the emissions allowances. A government-set "price" for carbon emissions allowances is not a market-based mechanism, since the price is set by legislative or executive fiat. However, if the government-set "price" were set below the average cost of achieving carbon emissions reductions, some entities might choose to purchase allowances in excess of their need in anticipation of the opportunity to sell those allowances for a profit to another entity with higher emissions reduction costs or a longer emissions reduction timeframe. The government could also "price" carbon by simply imposing a direct tax on carbon emissions.
Another mechanism which also establishes a "price" for carbon is the investment required on the part of the carbon emitters to actually reduce carbon emissions to comply with the current emissions "cap". This "price" is a function of the magnitude of the required investment in low/no carbon emissions facilities and equipment and the timeframe required to place the low/no carbon emissions facilities and equipment into service. In some cases, this "price" may also be a function of the commercial availability of the necessary facilities and equipment. This "price" for carbon emissions is in addition to any "price" established in any other way. This "price" is a market-based mechanism.
Any "price" imposed on carbon emissions by government through any means would increase the total cost of achieving carbon emissions reductions and thus increase the price of energy in the economy. Thus, any "price" imposed on carbon emissions by government would be "regressive" with regard to achieving emissions reductions. However, the distribution of the revenue received by government as the result of imposing the "price" would likely be "progressive" to some degree, offsetting all or a portion of the imposed "price" for certain groups.

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