If the marginal external effect is adverse as in the heat example, the tax is negative (each marginal product MP i k 1 k is negative) the producer price p i must exceed the consumer supply price q i. The only difference from the consumer externality is that the tax equals the value of the marginal external effects rather than the negative of this value, simply because the firm is paying the tax. The divergence between the marginal rate of substitution and the marginal external effects, MRS i, 1 − ∑ k = 1 J MP i, k 1 k from Eqn (7.21), is independent of j. (7.24) MRS i, 1 = MP j i, j 1 j + ∑ k = 1 j MP i, k 1 k j = 1, …, JĪ single Pigovian tax is sufficient because the marginal damage to any firm depends only on the aggregate use of factor i. In terms of the general model, this simply means that MP i,k 1 k = 0 for most k in the summation of the external effects. ![]() Realistically, any source of pollution affects only a small subset of firms in the economy. Note, finally, that the production model has been written in its most general form. With the single tax then, the consumer’s marginal rate of substitution is correctly equated to the full social marginal product of factor i in the production of good 1. Conversely, for external economies, each firm is subsidized by an amount equal to the aggregate marginal external benefit of the activity. If the marginal external effect is adverse, as in the heat example, the tax is negative (each marginal product MP i,k 1 k is negative) the producer price p i must exceed the consumer supply price q i. ![]() The only difference from the consumer externality is that the tax equals the value of the marginal external effects rather than the negative of this value simply because the firm is paying the tax. ![]() The divergence between the marginal rate of substitution and the marginal external effects, MRS i, 1 undefined − ∑ k = l J MP i, k 1 k from Eq. (7.24) MRS i, 1 = MP ji, j 1 j + ∑ k = 1 J MP i, k 1 k j = 1, … ,JĪ single Pigovian tax is sufficient because the marginal damage to any firm depends only on the aggregate use of factor i.
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