Environ Resource Econ (2008) 39:397–410
DOI 10.1007/s10640-007-9132-5
Ex-post efficient permit markets: a detailed analysis
Frank C. Krysiak
Received: 11 July 2005 / Accepted: 30 April 2007 / Published online: 26 May 2007
© Springer Science+Business Media B.V. 2007
Abstract This paper provides a detailed analysis of ex-post efficient permit markets. After
a short review of institutional designs that can achieve ex-post efficiency under uncertainty
and asymmetric information, we analyze the effects of an ex-post efficient regulation on the
expected costs of the regulated firms, on their investment behavior, and on the incentives
for strategic behavior on imperfectly competitive permit markets. Also we inquire about the
budget effects of the regulation. Our results show that ex-post efficient permit markets have
considerable benefits beside the direct increase in expected social welfare.
Keywords Tradeable permits · Uncertainty · Ex-post efficiency · Investment · Imperfect
competition
JEL
Q58 · Q52 · D82 · H23
1 Introduction
As Weitzman (1974) has shown, regulating external effects under asymmetric information
and uncertainty with the commonly discussed instruments of fixed taxes, standards, or tradeable permits results only in a second-best outcome. The first-best solution, that is, an ex-post
efficient regulation, cannot be reached, so that welfare losses accrue.
Many sophisticated instruments have been advanced that can overcome this problem.
These include price-based instruments, like adaptive or progressive taxes. But mostly a permit market on which the number of permits varies with the permit price is used. Such a flexible
supply of permits can be introduced in various ways. The proposals include hybrid regulation
strategies, as in Roberts and Spence (1976), a system of rental permits, as in Collinge and
Oates (1982), options trading, as in Unold and Requate (2001), or open market operations, as
F. C. Krysiak (B)
Faculty VIII, Economics and Management Technische Universität Berlin, Straße des 17. Juni 135,
10623 Berlin, Germany
e-mail: [email protected]
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F. C. Krysiak
in Henry (1989) and Kennedy (1999).1 Given that uncertainty and asymmetric information
are present in nearly all regulation problems, these instruments are important. Many of them
allow for considerable gains in expected welfare2 while requiring only slight adjustments to
a conventional permit market.
But to assess the economic and political feasibility of introducing these instruments, information about their impact on the regulated industry is missing. The cited studies analyze these
instruments from the perspective of a social planner to show that they can achieve ex-post
efficiency. But due to this perspective, we do not have detailed information concerning the
effects of these instruments on the regulated firms, on the market structure, and on the regulating authority. We do not know how they influence the firms’ expected costs, whether they
amplify or attenuate price fluctuations on the permit market, whether they provide incentives
for strategic firm behavior, or how they influence investment decisions. In most cases, we do
not even know under which conditions they result in costs or in revenues for the regulator.
These points are often crucial in the political discussion process. Therefore, we analyze ex-post efficient permit markets in detail in this paper to provide answers to the above
questions.
We first characterize ex-post efficient permit markets in a model with uncertainty and
asymmetric information and show that they have smaller price fluctuations than conventional ex-ante efficient markets. Then we study the effects of ex-post efficient permit markets
on the firm level, that is, their effects on expected costs and on investment decisions. We also
inquire whether such markets provide new incentives for strategic firm behavior and how
they affect incentives that are present due to imperfect competition on the permit market.
Finally, we analyze the costs or revenues of the regulator.
To gain this information, we discuss these points in a modeling framework that allows
for heterogeneous production technologies and that models uncertainty at the firm level.
However, to generate easily interpretable answers and to maintain a close connection to the
above-cited studies, we conduct our analysis in the quadratic framework of Weitzman (1974).
The paper is organized as follows. In the next section, we briefly review institutional
designs for ex-post efficient permit markets. Then we advance our model setup. In Sect. 4,
we analyze ex-post efficient permit markets w.r.t. their consequences on firm behavior. Sect.
5 studies the fiscal consequences for the regulating authority. Sect. 6 concludes.
2 Institutional designs
A number of studies have advanced concepts that allow for an ex-post efficient regulation
under uncertainty and asymmetric information.
Roberts and Spence (1976) have shown that a hybrid regulation strategy, which combines
a permit market with a subsidy for abatement and an optional emission tax, can improve
upon a standard permit market in terms of expected social welfare. The subsidy provides an
incentive for abatement if permit prices are low, and the emission tax instates an upper price
limit for the permit price. In an appendix, they show that by using an increasing number of
such instruments, it is possible to achieve ex-post efficiency.
Collinge and Oates (1982) have devised a system of rental permits that can be used to the
same end. Permits are not granted but rented from the regulating authority and the permits
1 See Sect. 2 for a more detailed discussion.
2 See, e.g., Pizer (2002).
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Ex-post efficient permit markets: a detailed analysis
399
are rented at increasing prices. In this way, a price–quantity relation for emissions that equals
the marginal damage function can be implemented, which leads to ex-post efficiency.
Unold and Requate (2001) have shown that an ex-post efficient price–quantity relation
can also be established by using a standard permit market together with selling call options
for additional permits and put options that allow a firm to sell permits back to the regulator.
By providing a set of options with differing strike prices, ex-post efficiency can be attained.
Newell et al. (2005) propose to use a permit market with banking and borrowing and an
adjustment scheme for the number of available permits. With a sophisticated use of such a
system, a flexible supply of permits that approximates the marginal damage function can be
implemented.
Finally, it is possible to use open market operations on a permit market to achieve a flexible
permit supply. In such a setting, the regulator buys or sells permits to implement a pre-defined
price–quantity relation on the market. Such an approach has been advanced, in Henry (1989,
Ch.2), Kennedy (1999) and in Newell et al. (2005). If the schedule according to which the
regulator buys or sells permits equals the marginal damage function, this approach leads to
ex-post efficiency. Kennedy (1999) also considers a proportional adjustment rule that can be
used to the same end.
Altogether, a considerable number of permit-based instruments exist that allow to achieve
ex-post efficiency. They all have to implement a supply function on the permit market that
equals the marginal damage function. In the following analysis, we take this as an abstract
representation of these instruments.
3 The model
Consider a partial equilibrium setting with a finite number n of firms whose emissions cause
an external damage and who are regulated by a permit market. All firms are price-taking
profit maximizers on the market for their products, on all factor markets, and on the permit
market. We depict the technology of firm i by an abatement cost function ci (ai , ηi ). This
function depends on the firm’s level of abatement ai and on the realization ηi of a firm-specific
stochastic variable θi . Following Weitzman (1974), we assume that the stochastic deviations
are sufficiently small, so that we can approximate ci by a quadratic function in which only
the linear term is uncertain.
ci (ai , ηi ) = (αi + ηi ) ai +
βi 2
a , with 0 < αi , βi < ∞, ∀i = 1, . . . , n.
2 i
(1)
The stochastic variables model unpredictable changes of the firms’ abatement costs due to
factor price fluctuations, demand side shocks, or varying climatic conditions. In each period,
the firms observe the realizations of the stochastic variables and act according to them. But
this is private information; the regulator has no knowledge concerning the ηi .
We assume that the stochastic variables are intertemporally independent, have an expected
value of zero and the same strictly positive variance σ 2 . They can be correlated among firms.
But for ease of presentation, the correlation coefficient shall be the same for all pairwise combination of firms and there shall be either a positive or no correlation, that is, we assume ≥ 0.
The emissions of firm i are given by ei = ēi − ai , where ēi denotes the firm’s emissions without abatement. For a given permit price p, the firm sets its level of abatement by
minimizing its costs
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F. C. Krysiak
min ci (ai , ηi ) + p (ēi − ai − z i ) ,
(2)
ai ≥0
where z i denotes the number of costlessly granted permits.
n
Concerning the environmental side, we assume that the total emissions E := i=1
ei
cause an external damage D(E). As in Weitzman (1974), we use a quadratic function for
D(E) albeit without the stochastic influence considered there.
δ 2
(3)
E , with γ , δ > 0.
2
To regulate the firms’ emissions, a permit market is used. If Z permits are available (each
allowing for for one unit of emissions) the market clearing price p solves E ∗ ( p, η) :=
n
∗
∗
i=1 ei ( p, ηi ) = Z , where ei ( p, ηi ) denotes the cost minimizing emissions of firm i and
where
η := (η1 , . . . , ηn ). There exists a unique market clearing price if Z ∈]0, Ē[, where
n
ēi denotes the total emissions in the absence of abatement.
Ē := i=1
Efficiency requires that for all firms i ∈ {1, . . . , n},
∂ D(E) ∂ci (ai , ηi )
=
.
(4)
n
∂ai
∂ E E= Ē−i=1
ai =Z
D(E) = γ E +
For ex-post efficiency, this has to hold for all possible realizations ηi of θi , whereas for
ex-ante efficiency, it has to hold only with the expectation operator applied to (4). Given
that (4) contains one decision variable of the regulator (the amount of permits Z ), ex-ante
efficiency can be achieved, whereas ex-post efficiency is out of reach with a fixed supply of
permits.
But with a flexible supply of permits, which can be implemented with the instruments
reviewed in the preceding section, ex-post efficiency can be attained. Such a flexible supply
has to assure that the price for permits always equals the marginal damage, that is, p = D (Z ).
Given that a cost minimizing firm sets ∂ci (ai , ηi )/∂ai = p, Eq. 4 implies that the number of
permits has to equal Z ( p) := D −1 ( p), where D −1 (·) denotes the inverse of the marginal
damage function.
As a first step of our analysis, we show that an ex-post efficient supply of permits results
in stabilizing the permit price compared to an ex-ante efficient permit market with a fixed
supply of permits. For this, we calculate the market clearing price for permits for both types
of permit markets. To simplify the exposition, we define the following parameters3
c0 := n
1
1
i=1 βi
, c1 := c0
n
αi
i=1
βi
, η̄ := c0
n
ηi
.
βi
(5)
i=1
The total emissions can be calculated from (2) and (1). With the above definitions, they can
be compactly written as E ∗ ( p, η) = Ē + (c1 − p + η̄)/c0 .
∗
The market
clearing condition Z ( p) = E ( p, η) yields the equilibrium price of permits
as p = c0 Ē − Z ( p) + c1 + η̄. For a fixed, ex-ante efficient, and for a flexible, ex-post
efficient, supply of permits, we get the following market clearing prices.4
c0
δ + c0
c0
=
δ + c0
p ex−ante =
p ex−post
Ēδ + γ +
δ
c1 + η̄,
δ + c0
δ
δ
c1 +
η̄.
Ēδ + γ +
δ + c0
δ + c0
(6)
(7)
3 Note that for bounded β , α , we have c , c > 0 as well as E (η̄) = 0.
0 1
i i
4 These prices depend on η̄ and thus vary over time, since by (8) below, the variance of η̄ is strictly positive.
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Ex-post efficient permit markets: a detailed analysis
401
As is well-known, in the quadratic framework used here, the expected ex-ante efficient price
equals the expected ex-post efficient price. But as the following proposition shows, the ex-post
efficient price has a strictly smaller variance in this setup.5
Proposition 1 The variance of the permit price is strictly smaller on an ex-post efficient
permit market than on an ex-ante efficient permit market.
Proof By (6) and (7) both prices depend on η̄. The variance of η̄ results from (5) as
⎛
σ̄ 2 = σ 2 ⎝
n
1
i=1 β 2
+ (1 − ) n n i 1
i=1
j=1 βi β j
⎞
⎠ > 0.
(8)
It is strictly positive, because ≥ 0, all βi are strictly positive and bounded, and n is finite.
2
δ
The variance of p ex−ante is σ̄ 2 , whereas the variance of p ex−post is σ̄ 2 δ+c
. Since
0
δ, c0 > 0, the variance of p ex−post is strictly smaller than that of p ex−ante .
Proposition 1 shows that an ex-post efficient supply of permits attenuates the price fluctuations on the permit market. This is intuitive: If the aggregate demand for permits increases,
the permit price rises. On an ex-post efficient permit market, this stipulates an increase of
the supply of permits, which reduces the price increase. Decreases in the demand for permits result in a corresponding downward adjustment of the permit supply that attenuates the
price fall. An ex-ante efficient market with a fixed number of permits lacks this adjustment
mechanism and thus exhibits a higher price volatility.
4 The effects on the regulated firms
We now analyze how an ex-post efficient permit market affects the regulated firms compared
to an ex-ante efficient permit market with a fixed number of permits. We first study the effect
on the expected costs, then the effect on investment in abatement technology, and finally the
incentives for strategic firm behavior, especially on imperfectly competitive markets.
4.1 Effects on the firms’ expected costs
The introduction of an ex-post efficient permit supply increases expected social welfare.
But the effect on the expected costs of the regulated firms is not obvious. In the quadratic
setup (1) and (3), an ex-post efficient and an ex-ante efficient permit market have the same
expected price. But by Proposition 1, they differ w.r.t. the risk imposed on the firms, because
the ex-post efficient supply reduces the price fluctuations on the permit market.
To analyze the effect on a firm’s costs, we calculate the expected costs for the ex-ante
efficient regulation and the ex-post efficient regulation. With p̂ denoting the expected price,
the expected costs of firm i with a fixed, ex-ante efficient supply of permits and those under
the ex-post efficient regulation are given by
5 We are indebted to an anonymous referee for pointing out that the quadratic framework (Eqs. 1 and 3) is
essential to state this result.
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402
F. C. Krysiak
p̂ 2 + αi2
αi
σ 2 + σ̄ 2 − 2E (ηi η̄)
ĉiex−ante = p̂ ēi − z i −
−
,
(9)
−
βi
2βi
2βi
2
2 + α2
σ 2 + σ̄ 2 (c0δ+δ) − 2E (ηi η̄) c0δ+δ
p̂
α
i
ex−post
i
−
= p̂ ēi − z i −
−
.
ĉi
βi
2βi
2βi
(10)
Comparing these expressions gives rise to the following proposition.
Proposition 2 If the same amount of permits is distributed initially, the aggregated expected
abatement costs are smaller with an ex-post efficient permit supply than with a fixed, ex-ante
efficient number of permits.
Furthermore, if firm i receives the same amount of permits in both cases, then the firm’s
expected costs are smaller for the ex-post efficient than for the ex-ante efficient regulation,
whenever6
⎞
⎛
⎞
⎛
n
1
1
1
⎠−
⎝c0 (c0 + 2δ) ⎝
⎠.
(11)
>
βi
2(c0 + δ)
1−
β 2j
j=1
The firm’s expected costs are greater, if (11) is met with the reversed inequality relation.
Proof The difference between (9) and (10) is
ex−post
i := ĉiex−ante − ĉi
Calculating E (ηi η̄) from (5) yields
=
2c0 (c0 + δ)E (ηi η̄) − c0 (c0 + 2δ)σ̄ 2
.
2βi (c0 + δ)2
E (ηi η̄) = σ 2 + (1 − )
c0
βi
(12)
,
(13)
n
σ̄ 2
i = 2(cc0+δ)
with (13) and (8), Eq. 12 implies that i=1
2 is strictly positive, proving the
0
first part of the proposition. Substituting (13) and (8) into (12) leads directly to (11).
Proposition 2 shows that in the aggregate, an ex-post efficient regulation benefits the regulated industry. The reason is that, as depicted by (9) and (10), the expected costs of a firm
decrease with the variance of the firm’s costs and with the variance of the permit price, but
they increase with the covariance between a firm’s costs and the aggregate price fluctuations. These relations follow from the strict convexity of the cost function: A firm can adjust
to changes of the permit price and of its costs and thereby reduce its expected costs.7 But
this adjustment is smaller on average if the permit price can be expected to be high (low),
whenever the firm’s abatement costs are high (low).
A switch from an ex-ante to an ex-post efficient regulation reduces both the variance of
the permit price, as indicated by the factor δ 2 /(c0 + δ)2 < 1 of σ̄ 2 in (10), and the covariance
between the permit price and the firm’s cost changes, which results in the factor δ/(c0 +δ) < 1
6 Note that the right-hand side of (11) depends on β via the sum and via c , so that, depending on the
0
i
parameters, there can be disconnected intervals of βi for which (11) is met.
7 Both stochastic variables enter the cost function linearly. Thus without adjustments, the expected costs
would not be influenced by the variance of these variables. By adjusting optimally, a firm can do better than
not adjusting at all and thus reduce its expected costs.
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Ex-post efficient permit markets: a detailed analysis
403
of E (ηi η̄) in (10).8 The variance effect increases the expected costs, whereas the covariance
effect decreases them. Our result shows that in the aggregate, the covariance effect is more
important and thus aggregate expected costs are reduced by the switch.
But this does not necessarily hold for each individual firm. Whereas the variance effect
is the same for all firms, the covariance effect depends on a firm’s technology and can thus
differ among firms. This is intuitive: A firm with an “inflexible” technology (high βi ) can
adjust its emissions only slightly to stochastic cost changes and has therefore nearly constant
emissions. So if a firm has a less flexible technology than the other firms, its cost changes will
contribute less to the variations of the total permit demand and thus the covariance between
the firm’s costs and the permit price is comparatively small. Since the covariance effect is
proportional to the covariance between a firm’s costs and the price fluctuations, this beneficial effect is small for a firm with an inflexible technology. In contrast, the negative variance
effect is the same for all firms. So, as (11) shows, a firm with an “inflexible” technology (high
βi ) tends to lose from the ex-post efficient regulation.
However, in addition to a firm’s technology, we also have to consider the correlation
between the cost changes of different firms, that is, . A high implies that the costs of
different firms tend to move in the same direction, which implies that, irrespective of a firm’s
technology, its demand for permits is correlated with the aggregate demand for permits and
thus with the permit price. Therefore, we have the term −/(1 − ) in (11), which shows
that the higher the correlation between the firms’ cost changes is, the less flexible a firm’s
technology has to be for the firm to benefit from the ex-post efficient regulation.9
It is interesting to note that the question whether a firm benefits or loses from an ex-post
efficient regulation depends only on the technology of the firm and not on its market position,
that is, on its initial endowment of permits. The reason is that the expected prices are the
same for both types of markets considered here and that the initial endowments enter the
costs linearly, so that the variance of the permit price does not influence the expected revenue
of selling the initially granted permits.
Finally, note that since the ex-post efficient regulation benefits the industry in the aggregate (compared to the ex-ante efficient regulation), this gain can be distributed, so that the
expected costs of all firms are reduced. This is possible by combining the switch to an ex-post
efficient regulation with a reallocation of permits: Firms that lose according to Proposition 2
are compensated by initially receiving more permits at the expense of firms that gain. Since
the aggregate expected costs are smaller under the ex-post than under the ex-ante efficient
regulation, there exists a reallocation of permits that assures that all firms benefit from the
combined effect of the reallocation and of the introduction of the ex-post efficient regulation.
In conclusion, an ex-post efficient supply of permits does not only increase expected social
welfare but is also beneficial for the regulated industry and, when combined with a suitable
reallocation of permits, even for all individual firms.
4.2 Investment behavior
The choice of a regulatory instrument also affects investment into abatement technology.
Thus it is interesting to inquire which consequences ex-post efficient permit markets have
for investment and thus for long-run efficiency.
8 The flexible supply of permits counteracts the effect of a firm’s cost changes on the permit market. Thus it
reduces the covariance between individual cost changes and changes of the permit price.
9 Indeed, for → 1, all firms benefit, regardless of their technology. In this case, no firm can adjust its
emissions to its cost changes on an ex-ante efficient permit market. The ex-post efficient regulation allows for
some adjustment and thus reduces the expected costs of all firms.
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F. C. Krysiak
There exist many studies that analyze investment behavior under environmental regulation. Since the price on a permit market will change due to investment into abatement
technology and since the firms will anticipate this change, investment is lower if the firms are
regulated by a permit market than if they are subject to an emission tax, if both instruments
are designed to be ex-ante efficient before investment has occurred.10 This effect can change
if the instruments are adjusted after investment has taken place. Furthermore, with investment, the welfare effects of the regulation depend strongly on modeling assumptions and
the details of the instrument design. A robust, welfare-based ranking of policy instruments
does not seem to exist, see Requate and Unold (2003) or Fischer et al. (2003). Additional
complications arise, if technological spillovers or uncertainty are accounted for, as in Fischer
et al. (2003) and Zhao (2003).
We use a simple extension of our model to analyze a firm’s investment decision. Let the
cost function of firm i (i = 1, . . . , n) depend not only on the permit price and the stochastic
variable, but also on a stock of capital ki , which incurs the costs r ki . So the total costs of firm
i are ci (ai , ki , ηi ) + r ki + p(ēi − ai − z i ).
We assume that the expected long-run costs E ci (ai∗ (ki , θ ), ki , θi ) , where ai∗ (ki , η)
denotes the firm’s optimal abatement, are a continuously
differentiable, strictly decreas
ing and strictly convex function of ki , with
limki →0 E ∂ci (ai∗ (ki , θ ), ki , θi )/∂ki → −∞
and limki →∞ E ∂ci (ai∗ (ki , θ ), ki , θi )/∂ki → 0. Albeit being stated rather directly, these
assumptions mainly imply that investment into abatement technology reduces the expected
abatement costs but at a decreasing rate. In (1), we let αi and βi depend on ki and assume
that these parameters are strictly monotonic decreasing in ki .
To maintain our static model setup, we analyze only the long-run effects. We assume that
all firms base their investment decisions on the above defined expected long-run costs,11 and
that they all are small compared to the market, so that they do not take into account the effect
of their investment on the permit price. Also, we abstract from all sources of inefficiencies
that are not directly related to the question of ex-post efficient permit markets.
In this context, the introduction of an ex-post efficient regulation has two effects: it changes
the volatility of the permit price and, via investment, it can change the expected price. The
investment effects of differing permit prices have already been widely discussed. Therefore we
focus on the effects of the changes of the price risk. For this, we compare the investment induced
by an ex-post efficient permit market and by a permit market with a fixed supply of permits for
the case in which both markets have the same expected price after investment has occurred.
There is a considerable literature on the effects of price uncertainty on investment. For the
special case of permit markets with volatile prices, Chao and Wilson (1993), Insley (2003),
and Zhao (2003) have studied the effects of the uncertainty on a firm’s investment behavior.
They all show that uncertainty will either reduce the incentives for investment or result in
investment delays. Closest to our setting is Zhao (2003), who shows that investment becomes
less probable with an increasing firm-specific uncertainty but that the effects of an increase
in the variance of a common shock can be ambiguous.
Although our model is considerably less sophisticated than those of the cited studies, it
can provide some insight into the investment effects of stabilizing the permit price,12 as the
following proposition shows.
10 See, e.g., Milliman and Prince (1989) or Requate and Unold (2003).
11 This will, e.g., be the case if investment involves sunk costs and if the investment good is used for a
sufficiently large number of periods.
12 Note that in our model, the expected permit price is the same in all periods, that is, the passing of time does
not reveal new information. Thus some of the effects present in the above studies, like investment delays and
the option value of permits, do not arise our setting.
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Ex-post efficient permit markets: a detailed analysis
405
Fig. 1 The relation between abatement capital ki and the expected marginal productivity of abatement capital for the ex-post and for the ex-ante efficient market (dashed curve). At ki× , (11) is met with equality for
βi× := βi (ki× )
Proposition 3 Firm i invests more with an ex-post efficient permit supply than with a fixed
supply of permits that results in the same expected price after investment, if condition (11)
of Proposition 2 is met at the optimal level of ki .13 It invests less if this condition is met with
the reversed inequality relation at the optimal level of ki .
Proof In our modeling context, a firm’s optimal levelof investment is given
by E ∂ci (ai∗ (ki , θ ),
ki , θi )/∂ki ) = −r . Due to the strict convexity of E ci (ai∗ (ki , θ ), ki , θi ) , the left-hand side
of this expression is strictly monotone in ki . Thus for a given r , there is a unique optimal
level of ki , which can, however,
differ for both types of permit markets, see the points A, B
and C, D in Fig. 1. By E ∂ci (ai∗ (ki , θ ), ki , θi )/∂ki being strictly
monotonic increasing in
ki (approaching zero from below), a larger absolute value of E ∂ci (ai∗ (ki , θ ), ki , θi )/∂ki
corresponds to a lower ki .
Now assume that the firm will invest the same amount for both types
of regulation. For
this to be feasible, the difference between E ∂ci (ai∗ (ki , θ ), ki , θi )/∂ki for the fixed supply
of permits and for the ex-post efficient supply of permits has to be zero. This difference is
given by
ex−post
(ai∗ (ki , θ ), ki , θi )
∂cifixed (ai∗ (ki , θ ), ki , θi )
∂ci
E
−E
(14)
∂ki
∂ki
c0 2(c0 + δ)E (ηi η̄) − (c0 + 2δ)σ̄ 2 ∂βi (ki )
.
=−
2βi (ki )(c0 + δ)2
∂ki
Thus we need 2(c0 + δ)E (ηi η̄) − (c0 + 2δ)σ̄ 2 = 0, for (14) to be zero. Using (13) and (8)
shows that this is the case, if (11) is met with equality. Furthermore, the right-hand side of
(14) is strictly positive if (11) is met, and strictly negative
if (11) is met with the reversed
inequality relation. By the strict monotonicity of E ∂ci (ai∗ (ki , θ ), ki , θi )/∂ki and by βi
being a strictly monotonic decreasing function of ki , this implies that the firm’s optimal capital stock is greater for the ex-post efficient regulation, whenever (11) is met at the optimal
ki (the points C, D in Fig. 1), and smaller, whenever (11) is met (at the optimal ki ) with the
reversed inequality relation (the points A, B in Fig. 1). Note that the situation does not change
13 Both the optimal level of k for the ex-post and for the ex-ante efficient permit market lead to the same
i
conclusion.
123
406
F. C. Krysiak
if there are multiple values of βi at
Furthermore, due to the
which (11) is met with equality.
monotonicity and continuity of E ∂ci (ai∗ (ki , θ ), ki , θi )/∂ki , it is irrelevant whether we use
the optimal ki of the ex-post or of the ex-ante efficient permit market.
Proposition 3 shows that the effect of the ex-post efficient regulation on investment can
differ in its direction among the regulated firms. This resembles the ambiguous effect of a
change in the variance of the common stochastic shock in Zhao (2003).
The result is intuitive: A firm gains from the introduction of the ex-post efficient regulation, because the correlation between its cost changes and the permit price is reduced
(correlation effect), and it loses, because the variance of the permit price is reduced (volatility effect). Both effects are amplified by investment, since a lower value of βi implies that
the firm can adjust its output more strongly in response to cost and price changes and thus
gains more from volatile prices and loses more from the cost/price correlation (cf. (9) and
(10)).
If, in the long-run optimum, condition (11) is met, the correlation effect outweighs the
volatility effect. Reducing βi increases the firm’s benefit from the ex-post efficient regulation, so that a smaller βi is more valuable under the ex-post than under the ex-ante efficient
regulation. If (11) is met with the reverse inequality sign, the volatility effect outweighs
the correlation effect. Reducing βi increases the detrimental effect of the ex-post efficient
regulation on the firm and therefore a smaller βi is less valuable under the ex-post than under
the ex-ante efficient regulation.
Note that due to the firm heterogeneity, it is not possible to infer the effect on aggregate
investment without using more specific assumptions on the dependency of the short-run costs
on the capital stock. However, if we assume that all firms have the same technological possibilities, that is, if αi (ki ) and βi (ki ) have identical functional forms for all firms, then they
will all use the same capital stock in the long-run. In this case, we can sum over (14) and, as
shown in the proof of Proposition 2, the resulting expression is strictly positive. Applying the
proof of Proposition 3 shows that in this case, aggregate investment is strictly greater under
the ex-post than under the ex-ante efficient regulation.
Proposition 3 addresses the positive question whether a firm invests more or less under an
ex-post efficient regulation. It does not answer the normative question, whether the induced
level of investment is socially desirable.
In the context of this section, the answer to the latter question is simple: Under the assumptions outlined above, an ex-post efficient regulation sets efficient investment incentives. The
ex-post efficient regulation assures a perfect internalization of the external damage. Thus
despite uncertainty and asymmetric information, each firm faces the same optimization problem as the social planner. Consequently, each firm acts socially optimally. This is a general
characteristic of an ex-post efficient regulation; it is not constrained to the Weitzman setting
characterized by (1) and (3).
So in addition to maximizing instantaneous welfare, an ex-post efficient permit market
also assures efficient behavior in the long run.
This feature of ex-post efficient permit markets has been established in a slightly different
setting, by Kennedy (1999), who analyzes the investment decisions induced by permit markets in a two-period model with an initial uncertainty about the external damage and where
the number of permits can be adjusted after this uncertainty is resolved. Kennedy shows
that if ex-post efficiency in the second period is realized by open market operations or by
a proportional adjustment rule, then this regulation will also induce the efficient amount of
investment. Our setting is complementary to his analysis, since we consider demand-side
uncertainty instead of supply-side uncertainty on the permit market.
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Ex-post efficient permit markets: a detailed analysis
407
4.3 Strategic firm behavior
As a final step of our analysis of firm behavior, we study whether an ex-post efficient permit
market provides an incentive for strategic firm behavior, and whether existing incentives are
changed.
Given that such permit markets involve a sophisticated regulatory setup, they might be
vulnerable to strategic exploitation. Moledina et al.(2003) have shown that in a setting in
which the number of permits in one period depends on the market price of the previous
period, firms can have an incentive for under-abatement, since this allows them to get more
permits in the following period.
The difference to our setting is that Moledina et al. analyze a permit market in which the
additional permits are costlessly granted. In contrast, the instruments discussed in Sect. 2
imply that additional emissions incur costs for the receiving firms that at least equal the
current market price of permits. Thus the incentives present in Moledina et al. (2003) cannot
exist in our setting. It is not rational for a firm to under-abate only to buy additional permits
at an increased price.
So exploiting the flexible supply of permits is not a strategic option for the regulated
firms. But the flexible supply might change existing incentives for strategic behavior. As an
example, we consider the strategic exploitation of market power on a permit market.
We use the model of Hahn (1984), in which one firm on the permit market acts strategically.14 Assume that, as in the last sections, there are n price-taking firms (the competitive fringe) but that an additional firm (firm n + 1) can influence the price on the permit
market.
Following Hahn’s model, the market clearing price for permits can be written for a fixed
permit supply15 Z̄ and for the flexible supply that assures p = D (Z ) as16
p fixed = c0 Ē + ēn+1 − an+1 − Z̄ + c1 + η̄,
(15)
c
δ
δ
0
( Ē + ēn+1 − an+1 )δ + γ +
c1 +
η̄.
(16)
p flexible =
δ + c0
δ + c0
δ + c0
The strategic player sets its abatement according to
min cn+1 (an+1 , ηn+1 ) − p(an+1 , η)(ēn+1 − an+1 − z n+1 ),
(17)
an+1 ≥0
where p(an+1 , η) denotes either p fixed or p flexible as given above, depending on the analyzed
market setting. The necessary condition for the optimum is
∂cn+1 (an+1 , ηn+1 )
∂ p(an+1 , η)
= p(an+1 , η) +
(ēn+1 − z n+1 − an+1 ).
∂an+1
∂an+1
(18)
n+1 ,η)
Compared to the case of perfect competition, (18) contains the additional term ∂ p(a
∂an+1
(ēn+1 − z n+1 − an+1 ), which implies a deviation from the efficient solution if z n+1 does not
always equal the efficient level of the strategic player’s emissions. Due to the stochastic cost
changes, z n+1 will not always equal this efficient level, so that the regulation is inefficient.
14 We could use the more general model of Westskog (1996), which allows for more than one strategically
acting firm. But this would only result in a more complex notation without adding insight.
15 We could specialize to the supply of permits that is ex-ante efficient under perfect competition, which
would render the Eqs. 15 and 16 more symmetric, but this would unnecessarily reduce the scope of our result.
16 The definitions of the aggregate parameters c , c , and of the aggregate stochastic variable η̄ remain
0 1
unchanged from those stated in Sect. 3.
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408
F. C. Krysiak
The question is how the flexible supply of permits influences this strategic behavior. The
following proposition answers this question.
Proposition 4 The flexible supply of permits Z ( p) = D −1 ( p) induces a deviation of the
allocation of emissions from the competitive solution that is strictly smaller than that resulting
from a fixed supply of permits, if both markets are compared for the same initial allocation
of permits.
Proof Since both markets are compared for the same initial allocation of permits, the term
∂ p(an+1 ,η)
determines the deviation from the efficient solution. With a fixed supply of permits,
∂an+1
this term equals −c0 , whereas for the flexible supply it equals −c0 δ/(c0 + δ), confirm (15)
and (16). Because c0 , δ > 0, the assertion follows.
Thus a price stabilizing regulation reduces the effects of market power. The intuition behind
this result is straightforward: the flexible supply increases the price elasticity of the aggregate
demand/supply of permits that the strategic player faces and thus diminishes the incentives
for over- or under-abatement. If, for example, the strategic player is a seller of permits for
some realization of θ and thus tries to raise the permit price by selling less than the efficient
amount of permits, the flexible supply will counteract this behavior by making more permits
available due to the observed price increase. The strategic player anticipates this and thus
sells more permits (although not the efficient amount), because the marginal price increase
achieved by withholding permits is smaller than it would be without the flexible supply.
So introducing a flexible supply of permits along the lines discussed in Sect. 2 does not
result in new possibilities for strategic firm behavior. In contrast, it even reduces possibly
pre-existing incentives for a strategic exploitation of market power.
5 The regulator’s perspective
As a final step of our analysis, we consider the fiscal consequences of the ex-post efficient
regulation. We specialize to a setting in which the initial allocation of permits is costless for
the firms and in which the regulator’s marginal revenue of permits equals the market clearing
price for permits.17
Given the costless initial allocation of permits, a standard permit market will be budgetneutral. This is different for ex-post efficient permit markets, because these require permanent
activity by the regulating authority, which can result in a revenue or in costs for the regulator.
The necessary amount of regulatory interference on the permit
n market depends on the
amount of permits that has been initially allocated. Let Z 0 := i=1
z i be the number of initially allocated permits. There are two appealing choices for Z 0 . It can be chosen to minimize
the interference of the regulator on the market, that is, to minimize the expected number of
additional permits issued plus the expected number of permits taken from the market. Or it can
be chosen so that the regulation is budget-neutral in the long-run. The following proposition
characterizes both choices.
17 The latter assumption is met for the concept of open market operations, for the system of rental permits,
and for an ex-post efficient hybrid regulation. It is not met for the flexible banking and borrowing concept of
Newell et al. (2005), which does not induce financial flows between the firms and the regulator at all. In the
case of options trading, it depends on the details of the regulation whether this assumption is met or not.
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Ex-post efficient permit markets: a detailed analysis
409
Proposition 5
(a) The total number of permits issued additionally or taken from the market is minimized, if
the number of initially distributed permits equals the ex-ante efficient level of emissions
1 −γ
Ẑ = c0 Ē+c
.
δ+c0
(b) Let p̂ denote the expected price for permits. Then the expected revenue per period
of the regulation is zero, if the number of initially distributed permits equals Z 0R =
δ
σ̄ 2
.
p̂ (δ+c0 )2
have Z 0R >
Ẑ +
(c) We
Ẑ . Furthermore, the expected revenue per period of the regulation is
strictly positive if Z 0 < Z 0R permits are initially distributed.
Proof The expected number of permits issued additionally or taken from the market to
achieve ex-post efficiency is given by E (|Z ( p) − Z 0 |), where Z ( p) is the ex-post efficient
permit supply. Standard statistical theory shows that this expression is minimized by choosing Z 0 = E (Z ( p)), which, under (1) and (3), equals the ex-ante efficient number of permits
Ẑ . The expression for Ẑ follows directly from (4) and (6). This proves
(a).
The expected revenue of the regulation is given by E (R(Z 0 )) := E p ex−post (Z ( p)− Z 0 ) ,
where the ex-post efficient supply Z ( p) can be calculated from Z = D −1 ( p) together with
η̄
1 −γ
(7) as Z ( p) = c0 Ē+c
+ δ+c
. Setting E (R(Z 0 )) = 0 and substituting (7) proves (b). Part
δ+c0
0
(c) follows from (b), because c0 , δ, p̂ > 0, and because E (R(Z 0 )) is a strictly monotonic
decreasing function of Z 0 .
There are several interesting implications of Proposition 5. Issuing the ex-ante efficient number of permits minimizes the regulatory interference on the permit market, but generates a
profit for the regulator in the long-run. Thus this kind of regulation draws money from the
permit market.
However, a money transfer between the regulator and the firms is not unavoidable; a regulation that is budget-neutral on average is possible. But the number of initially issued permits
needed for a budget-neutral regulation is strictly greater than the ex-ante efficient number of
permits.
The intuition behind these results is simple. Due to D (E) being an increasing function of
E, the supply of permits is increasing in the permit price. But this implies that the regulator
will sell permits if the price is high and will buy them if the price is low. If the initial supply of
permits is the ex-ante efficient amount, both cases are equally probable, so that the regulation
generates a revenue for the regulator. To avoid such a revenue, the regulator has to reduce
the probability of selling permits, which is done by initially granting more than the ex-ante
efficient amount of permits.
6 Conclusions
In this paper, we have provided a detailed analysis of the effects of regulating an external
damage by ex-post efficient permit markets. We have first reviewed several implementation
possibilities. Using an abstract representation of these instruments, we have inquired about
their effects on the price fluctuations, on the expected costs of the regulated firms, on the
incentives for investment and for strategic firm behavior, and on the costs or revenues of the
regulating authority.
This analysis complements the existing studies on first-best regulation under uncertainty
and asymmetric information, which mostly employ a normative perspective and an aggregated
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F. C. Krysiak
modeling framework. In contrast, we have used a disaggregated model with firm-specific
uncertainty and have considered both a normative and a positive perspective. This has enabled
us to gain detailed insights into firm-specific cost changes and thus into the question who
gains and who loses by the implementation of an ex-post efficient regulation. Furthermore,
we have considered several adjacent problems, like the long-run effects on investment and
the question of strategic behavior, which are usually not covered in the literature on ex-post
efficient regulation under uncertainty.
Our results suggest that using ex-post efficient permit markets has benefits beside the
direct increase of expected social welfare. These markets reduce the consequences of imperfect competition and they provide efficient incentives for investment. Both points can lead to
an indirect increase of expected social welfare. Furthermore, the shift from an ex-ante to an
ex-post efficient permit market reduces the aggregate expected costs of the regulated industry
and can simultaneously generate a revenue for the regulating authority. If such a revenue is
not desired, the ex-post efficient regulation can also be designed to be budget-neutral, which
will imply an even greater reduction of the industry’s expected costs.
So an ex-post efficient permit market seems to be both a socially desirable as well as a
politically feasible regulation.
Acknowledgement We are indebted to two anonymous referees whose comments helped considerably to
improve the paper.
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