2016 fall Kennedy: Environmental Economics 2
2. THE EQUILIBRIUM PRICE EFFECTS OF POLLUTION POLICY
2.1 THE MODEL
Recall our simple model of a single price-taking firm from section 1.7 in Chapter 1. There are two key elements of that model. The first is the relationship between emissions and output:
The second is the production-cost function:
(2.2) γ (1 2 ) 2 c( y, x) x y + =
where γ ≥ 0 is a production-cost parameter. We now extend that model to one with multiple firms, where γ is idiosyncratic.
Output is homogeneous and aggregate demand is (2.3) Q( p) = a .bp where a > 0 and b > 0 .
We begin by deriving the market equilibrium in the absence of regulation. This will serve as a template for solving the more complicated problem when emissions are constrained. Moreover, the solution that we later derive for aggregate abatement cost can be usefully expressed in terms of the unregulated equilibrium values, so it is helpful to derive those first.
2.2 MARKET EQUILIBRIUM WITHOUT REGULATION
Equilibrium Market Surplus
Market surplus is the sum of aggregate profit and consumer surplus. Aggregate profit is simply equal to the sum of individual profits:
A Complication: Free Entry and the Number of Firms
The size of β in this example captures the extent to which firms differ.
In the limit as β →∞ , all firms are identical and Ω →α .
2.3 MARKET EQUILIBRIUM WITH AN EMISSIONS CONSTRAINT
p6
We can equate this aggregate output to aggregate demand from (2.3) and solve for p to yield the equilibrium market price as a function of E:
Any aggregate emissions constraint lower than 0 E must cause the equilibrium price to rise, since ∂p(E) / ∂E < 0 R . This price rise has implications for both producers and consumers.
The Impact on Consumer Surplus
This is increasing in E at any E < E0 .
Thus, a binding constraint on aggregate emissions reduces consumer surplus via its impact on the market price.
The Impact on Profits
2.4 THE LEAST-COST ALLOCATION OF INDIVIDUAL EMISSIONS QUOTAS
The least-cost allocation of the aggregate target is that which maximizes market surplus subject to the target being met. This generalizes our criterion from section 1.8 in Chapter 1 to a setting with price effects
Given our assumptions, profit is still positive for firm i at any 0 ˆi i e ≤ e so there is no change in the number of firms, and no change in ϕ . Thus, consumer surplus is independent of individual allocations in our model. This means that maximizing social surplus is equivalent to maximizing aggregate profit.
Maximization of Aggregate Profit
and solve for ρ as a function of E, denoted ρ *(E) . Making this substitution for ρ in (2.23) then yields the optimal quota for firm i:
Thus, the quota for firm i is calculated as a share of the aggregate target, where that share is based on its production cost relative to that for the industry as a whole.
The Envelope Theorem (see the Appendix to Chapter 1) tells us that the impact of E on the maximized aggregate profit is equal to ρ *(E) plus the direct effect of E on the aggregate profit.
2.5 AGGREGATE ABATEMENT COST AND ITS INCIDENCE
Our first goal in this section is to derive the marginal aggregate abatement cost function. To do so we first need to calculate market surplus at the surplus-maximizing solution:
The abatement cost incurred from meeting an aggregate emissions target is the associated loss of market surplus:
Thus, marginal aggregate abatement cost in this simple setting with linear demand and quadratic costs takes the same familiar, linear form that we derived in section 1.8 from Chapter 1 when price did not change – see equation (1.70) from that chapter – but where the slope term now depends on key parameters from both supply and demand.
The Incidence of Abatement Cost
The loss of market surplus due to the aggregate emissions constraint is borne partly by consumers and partly by firms, where the relative burden – the incidence of abatement cost – depends on the key parameters underlying demand and supply in the market, and critically, on the strictness of the emissions target. Moreover, the impact on firms can actually be positive in some cases; aggregate profit can be higher than in the unregulated equilibrium.
To investigate these issues further, let us decompose the change in market surplus into the change in aggregate profit and the change in consumer surplus, beginning with the later.
This tells us that ΔCS(E) < 0 at any 0 E < E but the rate of loss diminishes as E is reduced, as depicted in Figures 2-2 and 2-3. These figures plot − ΔCS(E) under two different parameter scenarios relating to the size of b.
A more surprising relationship exists between E and aggregate profit.
∂ΔΠ(E) / ∂E < 0 at 0 E = E ; a reduction in E from the unregulated level initially causes aggregate profit to rise. This increase in aggregate profit continues as E is reduced further until E reaches a critical threshold E+ > 0 at which aggregate profit is at its highest for any value of E.
Any further reduction in E below E+ causes aggregate profit to decline but it will not necessarily fall below 0 Π at any E > 0 .
b 小的话，消费者弹性小。消费者承担大部分份额，厂商受损小。
that if $b < \bar b$ ,
That is, a binding emissions constraint at any nonnegative value causes aggregate profit to rise above its unregulated level. This is the case depicted in Figure 2-2
Conversely, if $b > \bar b$ then ΔΠ(E) eventually becomes negative
2.6 THE OPTIMAL LEVEL OF AGGREGATE EMISSIONS
Our next task is to identify the optimal level of aggregate emissions. This is straightforward because we have already derived the marginal aggregate abatement cost function, in (2.45) above. Setting this equal to marginal damage from (1.58) in Chapter 1 yields
2.7 EQUILIBRIUM PRICE EFFECTS OF A POLLUTION TAX
这回，收税不同于quota。
对厂商效果，不同。
对消费者，影响也不一样。
in Chapter 1 that a Pigouvian tax on emissions is in some respects equivalent to a set of individual emissions quotas that are chosen to implement the optimal aggregate target at least-cost. In particular, the tax brings into equality marginal abatement costs across sources, and since it is set equal to marginal damage, it achieves the optimal level of aggregate emissions. However, because the tax involves a payment from the regulated firms, its impact on profits is quite different from the impact of a set of emissions quotas. Moreover, at least part of the tax is passed on to consumers via a higher product price, so consumers are also affected differently under the two regulatory approaches. We examine these incidence issues in this section, and begin by characterizing the market equilibrium under the tax using our simple quadratic model from section 2.1
2.7-1 Market Equilibrium with a Pollution Tax
These results tell us that the tax induces the firm to reduce emissions but it also reduces its output. Why? The tax on emissions induces the firm to adopt a cleaner production process (a higher value of x) but this raises its marginal cost of production. It therefore cuts production at any given output price; its supply curve shifts.
- b 代表了需求方的弹性。
- ϕ 代表了供给方的弹性。
Note the role of b and ϕ as determinants of this rate of increase. A high value of b means that demand is highly responsive to price so a smaller fraction of the tax is passed on to consumers as a price increase. Conversely, a high value of ϕ means that supply is very responsive to price (because marginal cost of production is relatively flat) so a larger fraction of the tax is passed on to consumers via a price increase.
2.7-2 The Incidence of a Pollution Tax
The Impact on Consumers
The Impact on Firms
t0 is a critical value of the tax rate at which profits are just unchanged by the application of the tax
收税，也有意外的效果，能起到减少产量，类似于垄断到效果。企业firm的利润，反而可能增加。
这很大程度上取决于b。
The same joint-monopoly effect arises under the tax policy but it is at least partly offset by the tax payment itself in terms of the overall impact on aggregate net profits. The size of b is again a critical determinant of that impact. We consider two distinct cases.
The Pigouvian Tax Rate Let us now compare 0 t with the Pigouvian tax rate. We know from (2.53) that the socially-optimal level of aggregate emissions in this setting is
Figure 2-6 depicts the case where b < b and δ <δ . In this case, ΔΠ(t* ) < 0 ; aggregate net profits are lower under the Pigouvian tax than in the unregulated equilibrium. Conversely, Figure 2-7 depicts the case where b < b and δ >δ . In this case, Π(t* ) > 0 ; aggregate net profits are higher under the Pigouvian tax than in the unregulated equilibrium.