Environment and Statecraft: The Strategy of Environmental Treaty-Making

By Scott Barrett | Go to book overview

Appendix 12.1 The Trade Model

Let inverse demand in country i be linear and given by p(xi) = 1 − xi, where xi is consumption of the good in country i.

Firm j's costs are also linear and are given by C(xj, qj) = φqjxj, where xj is total output by firm j and qj is the abatement standard for firm j. Abatement is binary; that is, qj ∈ {0, 1}. Firm j takes qj as given and so takes unit costs as given and constant. Transport costs are zero. Emissions by firm j (and thus country j) are xj(1 − qj). If country j plays qj = 1, then emissions by j will equal zero; if j plays qj = 0, then emissions by j will equal firm j's output. 12

Firms are assumed to choose their outputs for each market simultaneously. Specifically, firm j takes its own abatement standard, the standards imposed on other firms, and the segmented outputs of other firms as given and chooses a quantity to produce and ship to market i,

, so as to maximize(12A.1)

The first order conditions (for an interior solution) require that every firm increase output so long as the gain in revenue exceeds the corresponding increase in costs. Formally, this requires(12A.2)

It is easy to confirm that eqn (12A.2) is also sufficient and that the solution is unique. I shall restrict choice of parameter values such that firms produce positive quantities in equilibrium.

Since the players are symmetric, all firms required to abate will produce the same output in equilibrium, as will all firms allowed to pollute. Let xA(xP) denote the output of a firm that abates (pollutes), and let z represent the number of other countries that play Abate. Country j takes z as given but can anticipate how the market for CFCs will “respond” to its own choice of whether to play Abate or Pollute. If j plays Pollute, eqn (12A.2) becomes(12A.3a)

If j plays Abate, eqn (12A.2) becomes(12A.3b)

Consider now the decision facing country j.j's net benefits are the sum of firm j's profits and the consumer surplus realized by the citizens of j, less the environmental damage suffered by country j. Pollution is assumed to be a pure public bad and aggregate emissions are given by

. Marginal environmental damage for each country is a constant, b. Given the demand specification, consumer surplus for country j is (xj)2/2. j's payoff is thus(12A.4)

-328-

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