Prices vs. Quantities with Banking and Borrowing

Martin L. Weitzman∗

Preliminary and confidential. November 7, 2017 version. Please do not cite.Comments Appreciated.

Abstract

It seems to be a not uncommonly held view that intertemporal banking and bor-

rowing of tradeable permits might tilt the ‘prices vs. quantities’debate towards looking

relatively more favorably upon time-flexible quantities. The present paper shows that

this view is incorrect for a natural dynamic extension of the original ‘prices vs. quan-

tities’ information structure, which here allows the firms to know and act upon the

realization of uncertain future costs ahead of the regulators. In this banking-and-

borrowing intertemporal extension, the standard original formula for the comparative

advantage of prices over quantities remains valid. The logic and implications of this

result are analyzed and discussed.

JEL Codes: Q50, Q51, Q52, Q54, Q58

Keywords: prices, quantities, prices versus quantities, regulatory instruments,

pollution, climate change

1 Introduction

Choosing the best instrument for controlling pollution has been a long-standing central

issue in environmental economics. This paper is primarily concerned with the theory of

environmental policy in terms of the basic theoretical foundations of the instrument choice

issue between prices and quantities.1

∗Department of Economics, Harvard University (mweitzman@harvard.edu). For helpful comments on anearlier draft, but without implicating them in errors or interpretations, I am indebted to:

1I sprinkle some comments on ‘practicality’throughout the paper, but this is not the main focus. Fora balanced overall discussion of both theoretical and practical issues involved in choosing between carbontaxes and cap-and-trade, see the recent review article of Goulder and Schein (2013) and the many furtherreferences that they cite.

1

Pollution is a negative externality. Pigou (1920) introduced and subsequently popu-

larized the central concept of placing a price-charge on pollution (since called a ‘Pigouvian

tax’) as an effi cient way to correct a pollution externality. This Pigouvian-tax approach

dominated economic thinking about the pollution externality problem for about the next

half-century.

Dales (1968) introduced the idea of creating property rights in the form of tradeable

pollution permits as an effi cient alternative to a Pigouvian tax. Montgomery (1972) proved

rigorously the formal equivalence between a price on pollution emissions and a dual quantity

representing the total allotment of tradeable permits, which, after the fact, is intuitive.

Henceforth, it became widely accepted that there is a fundamental isomorphism between a

Pigouvian tax on pollution and the total quantity of allotted caps in a cap-and-trade system

that ends up with all permits trading at the same competitive-equilibrium market price as

the Pigouvian tax. For every given Pigouvian tax, there is a total quantity of tradeable

permits allotted whose competitive-equilibrium market price equals the Pigouvian tax. And

for every given total quantity of tradeable permits allotted, there is a competitive-equilibrium

market price that would yield the same result if it were imposed as a Pigouvian tax.

Thus far all of the analysis took place in a deterministic context where everything was

known with certainty.

Weitzman (1974) showed that with uncertainty in cost and benefit functions there is

no longer an isomorphism between price and quantity instruments. The key distinction

under uncertainty is that setting a fixed price stabilizes marginal cost while leaving the total

quantity variable, whereas setting a fixed total quantity of tradeable permits stabilizes total

quantity while leaving price (or marginal cost) variable. The question then becomes: which

instrument is better under which circumstances?

Weitzman (1974) derived a relatively simple formula for the ‘comparative advantage of

prices over quantities,’denoted in the paper as ∆. The sign of ∆ (but, as will be explained

later, not its magnitude) depends on the relative slopes of the marginal cost curve and the

marginal benefit curve. The sign of ∆ is positive (prices are favored over quantities) when

the marginal cost curve is steeper than the marginal benefit curve. Conversely, the sign of ∆

is negative (quantities are preferred over prices) when the marginal benefit curve is steeper

than the marginal cost curve.

There subsequently developed a sizable literature, here only partially sketched, on the

optimal choice of price vs. quantity policy instruments under uncertainty.2 Adar and Griffi n

2This literature is too large to cover here all published papers concerning various aspects of prices vs.quantities. Instead, I have included here only a small subset that I subjectively judged to be most relevantto this paper.

2

(1976), Fishelson (1976), and Roberts and Spence (1976) analyzed seemingly alternative (but

ultimately similar) forms of uncertainty. Weitzman (1978), Yohe (1978), Kaplow and Shavell

(2002), and Kelly (2005) extended the basic model to cover various aspects of nonlinear

marginal benefits and nonlinear marginal costs. Yohe (1978) and Stavins (1995) analyzed

a situation where uncertain marginal costs are correlated with uncertain marginal benefits.

Chao and Wilson (1993) and Zhao (2003) incorporated investment behavior into the basic

framework of instrument choice under uncertainty. In these extensions, the results seemed

by and large to preserve the earlier insight of Weitzman (1974) that, other things being

equal, flatter marginal benefits tend to favor prices while steeper marginal benefits or flatter

marginal costs tend to favor quantities.

Extensions to cover stock externalities in a dynamic multi-period context were made by

Hoel and Karp (2002), Pizer (2002), Newell and Pizer (2003), and Fell, MacKenzie and Pizer

(2012), among others. These dynamic stock externality extensions appeared to keep much

of the “flavor”of the original ∆ story in Weitzman (1974), which was originally phrased in

terms of emission flows throughout a regulatory period (after which comes a new regulatory

period with new decision-relevant parameters). In particular, for the case of climate change

from accumulated stocks of atmospheric carbon dioxide (CO2), this stock-based literature

concludes that, throughout the relevant regulatory period, Pigouvian taxes are strongly

favored over cap-and-trade. This is because the flow of CO2 emissions throughout a

realistic regulatory period is but a tiny fraction of the total stock of atmospheric CO2 (which

actually does the damage), and therefore the corresponding marginal flow benefits of CO2abatement within, say, a five or ten year regulatory period, are very flat, implying that prices

have a strong comparative advantage over quantities.

The role of intertemporal banking and borrowing of emissions permits in a dynamic multi-

period context was analyzed, or at least touched upon, in a series of papers including Rubin

(1995), Newell, Pizer and Zhang (2005), Murray, Newell, and Pizer (2009), Fell, MacKenzie,

and Pizer (2012), and Pizer and Prest (2016).3 Conclusions of these papers are mixed, but

generally seem favorable to intertemporal banking and borrowing of permits.4 None of these

papers addresses directly the concept that firms can perhaps foresee future costs ahead of

regulators, which I suspect may be indirectly motivating the benign view of time-flexible

quantities that the authors of these papers generally seem to favor.

3The literature concerning intertemporal banking and borrowing is too large to cover here all publishedpapers touching upon this subject. Instead, I have included here only a small subset that I subjectivelyjudged to be most relevant to this paper.

4The Pizer and Prest (2016) working paper goes the farthest in arguing that, with dynamic updating alongwith intertemporal banking and borrowing, cap-and-trade quantities are always preferred over Pigouvianprices. I find the presumed information and game-theoretic structure in their paper to be not real, even atthe level of modeling abstraction; but interested readers can make their own judgment.

3

This paper considers and analyzes centrally the idea that firms may be able to intuit

future costs ahead of regulators. Without some such kind of intertemporal informational

asymmetry between firms and regulators, there is no good welfare rationale for intertemporal

banking and borrowing.5

Two pioneering papers (Yates and Cronshaw (2001) and, especially elegantly and com-

prehensively, Feng and Zhao (2006)) treat analytically the issue of intertemporal banking and

borrowing under uncertainty in a formal two-period setup somewhat similar to the frame-

work of this paper. Neither of these two papers, nor hardly any of the other published

papers cited in this section, consider the fundamental welfare comparison being addressed

in this paper between a fixed-price regime and a time-flexible quantity regime allowing in-

tertemporal banking and borrowing. Additionally, the simplistic structure of uncertainty

and information postulated in this paper may be found, I hope, to be somewhat more realistic

as an abstraction than some other simplistic formulations in the literature.

This paper shows that, for a natural two-period extension of the information and reward

structure under uncertainty in Weitzman (1974), intertemporal banking and borrowing does

not make a difference in comparing price and time-flexible quantity instruments. Even

when the firms know and act upon the realization of uncertain future costs ahead of the

regulators, the earlier (1974) formula for ∆ remains valid for expressing the comparative

advantage of fixed prices over time-flexible quantities. The logic and implications of this

seemingly counter-intuitive result are analyzed and discussed.

2 The Model

The emphasis in the model of this paper is on clarity of exposition and the appealing sim-

plicity of clean analytical results.

For the sake of preserving a unified familiar notation, we follow the standard convention

that goods are good. This means that rather than dealing with polluted air, we deal instead

with its negative — clean air. And instead of dealing with pollution emissions, we deal

instead with its negative —emissions reduction (or abatement). This standard convention

implies, under the usual assumptions, that the marginal benefit curve is familiarly downward

sloping and the marginal cost curve is familiarly upward sloping.

We analyze a ‘policy period’of fixed length, at the very beginning of which the regulator

chooses instrument values that the firm will respond to during the regulatory period. ‘The

regulator’is shorthand for some government regulatory agency that sets policy instruments

5This point is made forcefully and convincingly in the Yates and Cronshaw (2001) and Feng and Zhao(2006) papers, now about to be discussed.

4

to maximize overall social welfare.6 ‘The firm’refers to an aggregate of all emitting firms

in the economy, all of which are united by facing the same identical price on emissions

(whether as the outcome of a Pigouvian tax or as the outcome of the competitive market

equilibrium of a cap-and-trade system). Whatever the origin, charging a single uniform

price causes the marginal costs of all firms to be equal. But this is exactly the condition

for horizontal summation (of quantities across one parametric price) of all of the different

firms’marginal cost (or supply) curves into one overall aggregate marginal cost (or supply)

curve. This aspect justifies the rigorous aggregation of all firms into one big economy-

wide as-if firm with an aggregate cost curve as just described. Henceforth, for notational

and analytical convenience, this as-if aggregate firm will simply be referred to as the firm,

with the understanding that its marginal cost function has a rigorous aggregation-theoretic

underpinning in a situation where all firms face the same uniform price on pollution. In the

nomenclature of this paper, a ‘price’instrument is synonymous with a ‘Pigouvian tax’while

a ‘quantity’instrument is synonymous with ‘total tradeable permits’.

At the end of one policy period, a new policy period begins with new cost and benefit

functions, and the process proceeds anew, thus allowing us, at least in principle, to analyze

policy periods in isolation. Here we introduce a change in the usual one-period structure of

information, uncertainty, and decision-making.

The policy period here is divided into two subperiods of equal duration. In subperiod

zero —hereafter denoted subperiod0 —there is no uncertainty because everything is deter-

ministic common knowledge known perfectly by both the regulator and the firm. This

determinism assumption greatly simplifies the algebraic analysis.7 By contrast, in subpe-

riod one —hereafter denoted subperiod1 —costs and benefits are uncertain. The relevant

abstraction here is that, for subperiod0 the regulator has a relatively decent sense of the

firm’s immediate costs, but is relatively hazy about the firm’s future costs in subperiod1—while the firm has some premonition about the likely costs of subperiod1 already at the

beginning of subperiod0.8 This timing-information sequence on the cost side is a strategic

assumption of the paper, which, I hope, captures some relevant aspects of reality.

The regulator sets policy just before subperiod0 and must live with the consequences

throughout the entire policy period. Initially it will not be transparent why we are taking

such a seemingly arcane two-subperiod approach, but it will become clear later when we

deal with intertemporal banking and borrowing of permits.9 We treat the no-uncertainty6The distribution side is treated separately and depends on how revenues are collected and spent.7We could deal with uncertainty of subperiod0 in the spirit of Feng and Zhou (2006), but it comes down to

a similar set of conclusions as this paper. See especially their comment immediately following their Remark2 on page 37.

8Analogously, the regulator knows subperiod0 benefits but is uncertain about subperiod1 benefits.9When subperiod0 is modeled as deterministic and known, it makes easier and more transparent the

5

perfect-knowledge subperiod0 case first. Then we will deal with the subperiod1 uncertainty

case by adding random variables that shift the marginal cost and marginal benefit curves.

Turning first to the deterministic setting of subperiod0, let q be any value of the quantity

(here pollution abatement). Consider a known deterministic quadratic specification of costs

of the form

C0(q) = f0 + γ(q − q) +c

2(q − q)2, (1)

where the known coeffi cients γ and c are positive, while the constant f0 is known and the

known value of q will be explained later. The known deterministic subperiod0 marginal cost

corresponding to (1) is the linear form

MC0(q) = γ + c(q − q). (2)

For any value of the quantity q, the quadratic specification of benefits in subperiod0 is

taken to be of the known deterministic form:

B0(q) = g0 + β(q − q)− b

2(q − q)2, (3)

where the known coeffi cients β and b are positive, while the constant g0 is known and the

known value of q will be explained later. For any value of q, the deterministic subperiod0marginal benefit corresponding to (3) is of the linear form

MB0(q) = β − b(q − q). (4)

In subperiod1 there is genuine uncertainty.10

Treating first uncertain costs, let the random variable θ represent a shift of the subperiod1marginal cost curve relative to the subperiod0 marginal cost curve. Thus, for any value of q,

MC1(q; θ) = MC0(q) + θ = γ + c(q − q) + θ. (5)

The subperiod1 uncertain total-cost function is obtained by integrating over q the ex-

pression (5), which yields the formula

treatment of time-flexible banking and borrowing between the two subperiods. This simplified two-subperiodstructure will be employed later in order to characterize banking and borrowing between subperiod0 andsubperiod1 in an especially crisp form.10The treatment of information, uncertainty, and decision-making in subperiod1 here is a parallel variant

of much of the framework of Weitzman (1974), which will be employed ubiquitously throughout this paper.Warning: some of the symbols to be used in this model sometimes differ from the symbols used in Weitzman(1974).

6

C1(q; θ) = f(θ) + (γ + θ)(q − q) +c

2(q − q)2, (6)

where the stochastic function f(θ) (serving as a ‘constant of integration’) represents a pure

vertical displacement of the total-cost curve.

Turning next to uncertain benefits, let the random variable η represent a shift of the

subperiod1 marginal benefit curve relative to the subperiod0 marginal benefit curve. Thus,

for any value of q,

MB1(q; η) = MB0(q) + η = β − b(q − q) + η. (7)

The subperiod1 uncertain total-benefit function is obtained by integrating over q the

expression (7), which yields the formula

B1(q; η) = g(η) + (β + η)(q − q)− b

2(q − q)2, (8)

where the stochastic function g(η) (serving as a ‘constant of integration’) represents a pure

vertical displacement of the total-benefit curve.

The value q is intended to represent the subperiod1 quantity where expected marginal

benefit equals expected marginal cost, or11

E[MB1(q; η)] = E[MC1(q; θ)]. (9)

Without loss of generality and to make for a neat interpretation, we normalize θ in

(5) and (6) so that E[θ] = 0. Likewise without loss of generality and to make for a neat

interpretation, we normalize η in (7) and (8) so that E[η] = 0.12 With these normalizations,

and making use of (5) and (7) for q = q, then (9) reduces to

β = γ. (10)

Henceforth we substitute γ for β in all formulas of this paper.

The above description of q allows an important elucidation of the subperiod1 cost and

benefit functions in (6) and (8). The value q is a natural focal point or point of departure

where expected marginal benefit equals expected marginal cost. The cost function C1(q; θ)

of (6) represents rigorously a quadratic approximation of costs in a relevant neighborhood

11For any random variable x, the notation E[x] represents the expected value of x.12As a consequence of E[θ] = E[η] = 0, note that (9) then implies that MB0(q) = MC0(q). This

simplifying specification, although it is not really essential for the analysis, will synchronize costs and benefitsacross the two subperiods and will bring much-needed analytical simplicity and sharpness to the formulas ofthis paper.

7

of q Analogously, the benefit function B1(q; η) of (8) represents rigorously a quadratic

approximation of benefits in a relevant neighborhood of q.

Since the unknown factors connecting q with C are likely to be quite different from those

linking q to B, it seems not implausible to assume as a base case that the random variables

θ and η are independently distributed, implying that E[θη] = 0.13 To summarize, then,

the subperiod1 formulation here has

E[θ] = E[η] = E[θη] = 0. (11)

For any random variable x, let σ2x = E [(x− E[x])2] represent the variance of x. Then

σ2θ = E[θ2] = E[(MC1(q; θ)− E[MC1(q; θ)])

2], (12)

for all q, while

σ2η = E[η2] = E[(MB1(q; η)− E[MB1(q; η)])2

], (13)

also for all q.

This concludes our description of the basic model.14 Now we are ready to apply this

framework to evaluate and analyze the expected welfare of various instrument combinations.

The main novelty of this paper will be to focus especially on the possibility of intertemporal

banking and borrowing between subperiod0 and subperiod1 in a situation where the regu-

lator knows just before subperiod0 (when regulations must be issued) only the distribution

of the random variable θ in subperiod1, whereas the firm knows already at the very start of

subperiod0 the realization of the random variable θ in subperiod1 and can bank or borrow

accordingly beginning in subperiod0. Each of the next three brief sections of the paper

constitutes a kind of necessary ‘warm-up exercise’that will be indispensable for understand-

ing the remaining sections of the paper. These remaining sections of the paper (after the

three brief ‘warm-up exercises’) will all be concerned with the main theme of evaluating the

welfare implications of time-flexible quantities that respond to the possibility of unlimited

banking and borrowing by the firm.

13The condition E[θη] = 0 seems like an especially relevant abstraction for the case of climate change. Thesituation where θ and η have non-zero correlation is considered in the footnote on page 485 of Weitzman(1974) and explicated in more detail in Yohe (1978) and Stavins (1995).14It goes without saying that we are imposing a tremendous amount of structure on the basic model (linear-

quadratic costs and benefits, additive shocks, two-period analysis, a simplified information and decisionstructure, a purely static flow analysis, and so forth and so on). Absent such kind of structure it is diffi cult,or even impossible, to address simply and analytically the basic issues of this paper.

8

3 Expected Welfare of an Optimally-Fixed Quantity

Assume that the regulator imposes the same fixed quantity q on output in both subperiod0and subperiod1.15 We now seek to know the regulator’s expected welfare in this two-

subperiod fixed-quantity setting, which will be indispensable for addressing and understand-

ing the main issue of prices vs. time-flexible quantities with banking and borrowing later in

this paper.

The regulator seeks to maximize over q the welfare expression

B0(q)− C0(q) + E [B1 (q; η)]− E [C1 (q; θ)] , (14)

implying that the solution q∗ must satisfy the first-order condition

MB0(q∗)−MC0(q

∗) + E[MB1(q∗; η)]− E[MC1(q

∗; θ)] = 0. (15)

Plug (4), (2), (7), (5), and (10) into (15), take expected values where indicated, and make

use of (11) to infer (after eliminating and consolidating terms) that (15) then implies

q∗ = q. (16)

Let EW ∗q stand for expected welfare in the optimally-fixed-quantity setup of this section

where the basic equation (16) holds. Then EW ∗q equals benefits minus costs in subperiod0

plus expected benefits minus expected costs in subperiod1. More formally, making use of

(16) in (14), we have

EW ∗q = B0(q)− C0(q) + E [B1 (q; η)]− E [C1 (q; θ)] . (17)

Substitute the four basic terms of the right-hand side of (17) into the corresponding

equations (3) for B0(q), (1) for C0(q), (8) for B1 (q; η), and (6) for C1 (q; θ) (paying attention

to (10) whenever applicable). Take expected values in the substituted version of (17) where

indicated. Use (11) to eliminate terms. After some cancelling and combining, we obtain

the basic result

EW ∗q = k, (18)

where the constant k is defined by the equation

k = g0 − f0 + E[g(η)]− E[f(θ)]. (19)

15It makes no difference if the regulator is allowed to set different quantities in the two different subperiods,since in an optimal quantity solution here they will both end up being the same anyway.

9

4 Expected Welfare of an Optimally-Fixed Price

Assume that the regulator imposes the same fixed price on output in subperiod0 and subperiod1.16

We now seek to know the regulator’s expected welfare in this two-subperiod fixed-price set-

ting, which will be indispensable for addressing and understanding, later in this paper, the

main issue of prices vs. time-flexible quantities with banking and borrowing.

If the regulator imposes the price p in subperiod0, the firm’s quantity response is then to

react with quantity q̃0(p) satisfying MC0(q̃0(p)) = p. Substituting from (2) and inverting

to solve for q̃0(p) yields the subperiod0 quantity-reaction or supply function

q̃0(p) = q +p− γc

. (20)

If the regulator imposes the price p in subperiod1, the firm’s quantity response is then

to react with quantity q̃1(p; θ) satisfying MC1(q̃1(p; θ); θ) = p. Substituting from (5) and

inverting to solve for q̃1(p; θ) yields the subperiod1 quantity-reaction or supply function

q̃1(p; θ) = q +p− γ − θ

c. (21)

Suppose next that the regulator chooses p to maximize benefits minus costs in subperiod0plus expected benefits minus expected costs in subperiod1.17 The solution p∗ must satisfy

the first-order condition

MB0(q̃0(p∗))−MC0(q̃0(p

∗)) + E[MB1(q̃1(p∗; θ); η)]− E[MC1(q̃1(p

∗; θ); θ)] = 0. (22)

Plug (20), (21), (4), (2), (7), (5), and (10) into (22), take expected values where indicated

and make use of (11) to infer (after eliminating and consolidating terms) that (22) then

implies

p∗ = γ, (23)

in which case (20) and (21) reduce to

q̃0(p∗) = q (24)

and, more interestingly,

16It makes no difference if the regulator is allowed to set different prices in the two different subperiods,since in an optimal price solution here they will both end up being the same anyway.17For simplicity, we assume throughout this paper that there is no discounting (r = θ = 0 in the notation

of Feng and Zhou (2006)).

10

q̃1(p∗; θ) = q − θ

c. (25)

Let EW ∗p stand for expected welfare in the optimally-fixed-price setup of this section

where the basic equations (24) and (25) both hold. Then EW ∗p equals benefits minus costs

in subperiod0 plus expected benefits minus expected costs in subperiod1 More formally,

making use of (24) and (25), we have

EW ∗p = B0(q)− C0(q) + E[B1(q −

θ

c; η)]− E[C1(q −

θ

c; θ)]. (26)

Substitute carefully the four basic terms of the right-hand side of (26) into the corre-

sponding equations (3) for B0(q), (1) for C0(q), (8) for B1(q − θ

c; η), and (6) for C1

(q − θ

c; θ)

(paying attention to (10) whenever applicable). Carefully take expected values in the substi-

tuted version of (26) where indicated. Then use (11) and (12) to consolidate and eliminate

terms. After much algebra and rearrangement we obtain the basic result

EW ∗p = k +

σ2θ2c2

(c− b), (27)

where k is given by equation (19).

5 Comparative Advantage of Prices over Quantities

In the two-subperiod situation of this paper it is natural to define the comparative advantage

of (fixed) prices over (fixed) quantities, here denoted ∆pq, as

∆pq = EW ∗

p − EW ∗q . (28)

Substituting from (27) and (18) into (28), we then have the fundamental two-subperiod

result that

∆pq =

σ2θ2c2

(c− b). (29)

The two-subperiod formula (29) for ∆pq is identical to the original uni-period formula for

∆ (the comparative advantage of prices over quantities) that first appeared in Weitzman

(1974). This identity of formulas may be seen, after the fact, as a natural two-subperiod

extension of the uni-period case. The properties of ∆ (=∆pq) have been much discussed

throughout the literature. For example, σ2η defined by (13) does not appear in the formula

(29) because it affects price and quantity instruments equally adversely. Here I wish to

emphasize a point that has been largely overlooked in the literature.

11

The literature subsequent to Weitzman (1974) has, almost exclusively, emphasized the

purely qualitative interpretation that the sign of ∆ (=∆pq) in formula (29) depends only on

the sign of (c−b). Thus, in this standard literature interpretation, quantities are preferred toprices when the slope of the marginal benefit curve is steeper than the slope of the marginal

cost curve (b > c), while prices are preferred to quantities when the slope of the marginal

cost curve is steeper than the slope of the marginal benefit curve (c > b). This qualitative

inference about the role of the relative slopes of the marginal benefit and marginal cost curves

in determining the sign of ∆ (=∆pq), while it is of course true and can be very useful as a

reduced-form summary of prices vs. quantities, overlooks the possibly-important quantitative

side of formula (29).

The coeffi cient ∆pq (=∆) in (29) is intended to be a purely economic measure of the

relative advantage of prices over quantities. It goes without saying that making a decision

to use price or quantity regulatory instruments in a specific instance is more complicated

than just consulting the formula for ∆pq (=∆). There are also going to be a host of prac-

tical considerations outside the scope of this model, including political, ideological, legal,

social, historical, administrative, motivational, informational, timing, monitoring, enforcing,

or other behavioral issues. Having a purely-economic quantitative measure of comparative

advantage, like the full formula for ∆pq (=∆) in (29), is important because it can serve as a

benchmark against which the magnitude of practical ingredients might be reckoned in reach-

ing a final judgment about whether it would be worthwhile or not for an adviser to fight for

the verdict of pure economic analysis given by the sign of ∆pq (=∆) in (29) (which is exactly

the sign of (c− b)).On this account, attention only to the sign of (c− b) may not be fully adequate.18 For

example, the magnitude of σ2θ in (29) is significant because it magnifies (or shrinks) the

expected loss from using the ‘wrong’instrument with comparative disadvantage. Another

important factor in determining the quantitative magnitude of formula (29) is the term c2

in the denominator of the right hand side, ignored when only looking at the term (c − b).The basic issue is that a ceteris paribus discussion of the role of c in the term (c − b) is

different from a ceteris paribus discussion of the role of c in the term (c − b)/c2. As

one extreme example of how it can be misleading to ignore the c2 divisor in (29), consider

a situation where b > 0 but c is huge, approaching infinity in the limit. Then merely

inspecting the term (c − b) might be interpreted as indicating that price has an enormouscomparative advantage over quantity. Actually, as c→∞ in (29), then ∆p

q → 0, indicating

that neither instrument has a comparative advantage in such a limiting situation. This is

18The comments that are presented here in this paragraph have widespread applicability and will applythroughout the rest of this paper where similar formulas to (29) will be derived and need to be analyzed.

12

because if marginal costs are very steeply rising around the optimum, as would be the case

with fixed capacity and low variable costs corresponding to c→∞, then there is not muchdifference between regulating by price or quantity instruments because the resulting output

response will be almost the same with either mode. In such a situation, as with the case

σ2θ → 0, non-economic ‘practical’factors should play the decisive role in determining which

instrument to regulate.

Despite the above warning to pay attention to the quantitative magnitude of the right

hand side of (29), I disregard my own advice throughout the rest of this paper. Instead I

focus, like virtually all of the rest of the literature, on the qualitative sign of the comparative

advantage formulas to be derived in the remainder of the paper. This focus on the purely-

economic side allows neat sharp results. The important task of rigorously extending the

analysis to cover the quantitative side of comparative advantage formulas is left for further

research.

6 Expected Welfare of Optimal Time-Flexible Quanti-

ties

In this section begins the main novelty of this paper, to be additionally spelled out in subse-

quent sections. By time-flexible quantities, we mean the ability of the firm to intertemporally

bank and borrow freely on the basis of some premonition of future costs. Beginning with

this section, the rationale behind the two-subperiod framework of this paper really comes

into its own as a useful workhorse formulation.

As usual, for simplicity the specification of subperiod0 is entirely deterministic and is

known completely by firm and regulator alike. The regulator sets its control instruments

just before the beginning of the policy period, which here means just before subperiod0.

At that initial instant, the regulator comprehends θ as a probability distribution that will

make its realization known only during subperiod1, well after regulatory policy has already

been set. The firm, however, is allowed to have perfect foresight throughout by knowing

already at the beginning of subperiod0 the realization of the random variable θ in subperiod1.

Furthermore, the firm can freely bank or borrow intertemporally between subperiod0 and

subperiod1 as it sees fit on the basis of its prescient knowledge of θ. If anything, I think

this extreme formulation errs on the side of making intertemporal flexibility more attractive

to the firm than it actually is in practice.19

19For example, in reality firms have at best imperfect foresight and, while they are able to bank permitsin many regulatory situations, they rarely can borrow.

13

At the very beginning of the regulatory period, when θ in subperiod1 is conceptualized

by the regulator only as a probability distribution, let the regulator here set a total quantity

of permits Q covering both subperiods. With unlimited banking and borrowing by the firm,

the form in which Q is assigned is inessential. For example, the regulator might specify

an assigned number of permits qa0 for subperiod0 and an assigned number of permits qa1 for

subperiod1. With the ability to bank or borrow unlimited amounts, the firm cares only about

the total quantity of permits Q = qa0 + qa1 . Henceforth we deal only with Q.

Given Q and the firm’s perfect-foresight prescient knowledge of the subperiod1 realization

of the random variable θ already at the very beginning of subperiod0, the firm (postulated as

having the unlimited ability to bank or borrow) faces the problem of optimally determining

q̂0(Q; θ) in subperiod0 and q̂1(Q; θ) = Q− q̂0(Q; θ) in subperiod1. The firm therefore seeks

to minimize the total costs of both periods, which means that it minimizes over q̂0 the

expression

C0(q̂0) + C1(Q− q̂0; θ). (30)

Let the solution of the firm minimizing over q̂0 the total-cost expression (30) be denoted

q̂∗0(Q; θ). Then q̂∗0(Q; θ) must satisfy the first order condition

MC0(q̂∗0(Q; θ)) = MC1(Q− q̂∗0(Q; θ); θ). (31)

Substituting q̂∗0(Q; θ) into the MC0 formula (2) yields

MC0(q̂∗0(Q; θ)) = γ + c(q̂∗0(Q; θ)− q). (32)

Substituting Q− q̂∗0(Q; θ) into the MC1 formula (5) yields

MC1(Q− q̂∗0(Q; θ); θ) = γ + c(Q− q̂∗0(Q; θ)− q) + θ. (33)

Next plug (32) into the left hand side of (31) and plug (33) into the right hand side of

equation (31). Then, after carefully cancelling, consolidating, and rearranging terms, the

substituted version of (31) reduces to

q̂∗0(Q; θ) =Q

2+

θ

2c, (34)

which implies

q̂∗1(Q; θ) =Q

2− θ

2c. (35)

Thus far the regulator imposes a value of Q and the super-prescient firm reacts with (34)

14

and (35). Suppose next that the regulator sets Q at the instant just before subperiod0,

when it knows θ only as a probability distribution. The regulator seeks to maximize over

Q the expression

E[B0(q̂∗0(Q; θ))]− E[C0(q̂

∗0(Q; θ))] + E [B1 (q̂∗1(Q; θ); η)]− E [C1 (q̂∗1(Q; θ); θ)] . (36)

Let the value of Q that maximizes (36) be denoted Q∗. Then Q∗ must satisfy the first-

order condition

(E[MB0(q̂∗0(Q

∗; θ))]− E[MC0(q̂∗0(Q

∗; θ))])

(∂q̂∗0∂Q

)

+(E [MB1 (q̂∗1(Q∗; θ); η)]− E [MC1 (q̂∗1(Q

∗; θ); θ)])

(∂q̂∗1∂Q

)= 0. (37)

Note from (34) and (35) that∂q̂∗0∂Q

=∂q̂∗1∂Q

, (38)

and therefore these two terms of (38) cancel out of equation (37).

Next, use the result (34) that q̂∗0(Q; θ) = Q2

+ θ2cand the result (35) that q̂∗1(Q; θ) = Q

2− θ2c.

Plug (34) into formulas (4) and (2) and plug (35) into formulas (7) and (5) (paying attention

to condition (10) when applicable), and thereafter plug the resulting four pre-expectation

formulas into (37). Take expected values, remembering that E[θ] = E[η] = 0. Rearrange

terms to derive the basic consequence that

Q∗ = 2q. (39)

With (39) holding, (34) becomes

q̂∗0(Q∗; θ) = q +

θ

2c, (40)

while with (39) holding, (35) becomes

q̂∗1(Q∗; θ) = q − θ

2c. (41)

Let the expected welfare of optimized time-flexible quantities with banking and borrowing

here be denoted EW ∗q̂ . Making use of formulas (40) and (41), we then have

EW ∗q̂ = E[B0(q +

θ

2c)]− E[C0(q +

θ

2c)] + E[B1(q −

θ

2c; η)]− E[C1(q −

θ

2c; θ)]. (42)

15

Substitute carefully the four basic pre-expectation terms of the right-hand side of (42)

into the corresponding equations (3) for B0(q+ θ2c

), (1) for C0(q+ θ2c

), (8) for B1(q − θ

2c; η),

and (6) for C1(q − θ

2c; θ)(paying attention to (10) when applicable). Then carefully take

expected values of the resulting substituted-for expression. Use (11) and (12) to consolidate

and eliminate terms. After much careful algebra, rearrangement, and cancellation, we obtain

finally the basic result

EW ∗q̂ = k +

σ2θ4c2

(c− b). (43)

where k is given by (26).

7 Comparative Advantage of Time-Flexible Quantities

over Fixed Quantities

In the two-subperiod framework of this paper, denote the comparative advantage of time-

flexible banked-and-borrowed quantities over fixed quantities as ∆q̂q. Then it follows that

∆q̂q = EW ∗

q̂ − EW ∗q . (44)

Substituting from formulas (43) and (18) into (44), we then have the fundamental result

that

∆q̂q =

σ2θ4c2

(c− b). (45)

Equation (45) signifies that the q̂ regulatory system is preferred to the q regulatory system

when the slope of the marginal cost curve is steeper than the slope of the marginal benefit

curve (c > b), while the q regulatory system is preferred to the q̂ regulatory system when

the slope of the marginal benefit curve is steeper than the slope of the marginal cost curve

(b > c). Thus, the sign of ∆q̂q in equation (45) is negative when b > c.

The prime question, which is begging to be answered, is why the sign of ∆q̂q in equation

(45) is not strictly positive, but rather depends on the relative magnitudes of b and c in

exactly the same way as the formula (29) for the traditional comparative advantage of prices

over quantities ∆pq (=∆). After all, with the same total output across both subperiods

(= 2q), the time-flexible q̂ system is intertemporally cost-effi cient (MC0 = MC1) whereas

the time-fixed q system is intertemporally cost-ineffi cient (MC0 6= MC1).20 This cost-

effi ciency aspect is, perhaps naturally, what most economists have instinctively focused upon

in intuitively preferring q̂ to q. Why, then, is the sign of ∆q̂q in (45) not unambiguously

20More precisely, MC0 6= MC1 except on a set of measure zero.

16

positive?

The answer is that in the choice of control instruments we must pay attention to the

benefit side (as well as the cost side). While the time-flexible q̂ system is intertemporally

cost-effective, it increases the variability of q̂0 in (40) and q̂1 in (41) relative to the fixed

quantity assignments of q0 = q1 = q. The risk-aversion to quantity variability here is

captured by the b coeffi cient in the quadratic total-welfare formulas (3) and (8) when q0is given by (40) and q1 is given by (41). As b increases in formula (45), the more are the

expected drawbacks of quantity variability diminishing the expected benefits minus expected

costs (summed over the two subperiods) of time-flexible cost-effective quantities relative to

fixed quantities.

The sign of formula (45) for ∆q̂q is the same as the sign of the traditional formula (29) for

the comparative advantage of prices over quantities∆pq because the underlying considerations

are similar —namely the stability of fixed quantities against the cost-effectiveness of setting

marginal costs to be equal across both subperiods. Time-flexible quantities with banking

and borrowing are preferred to fixed quantities if and only if the slope of the marginal cost

curve is steeper than the slope of the marginal benefit curve. Thus, on purely economic

grounds, flexible quantities are favored relative to fixed quantities if and only if fixed prices

are favored relative to fixed quantities.21

8 Comparative Advantage of Fixed Prices over Time-

Flexible Quantities

This section represents the culmination of the paper, being the comparison of fixed prices

vs. time-flexible quantities with banking and borrowing.

In the two-subperiod framework of this paper, denote the comparative advantage of fixed

prices over time-flexible banked-and-borrowed quantities as ∆pq̂. Then it follows that ∆p

q̂

can be expressed in a readily-analyzable form as22

∆pq̂ = ∆p

q −∆q̂q. (46)

21That the multiplicative factor 14 in formula (45) is different from the multiplicative factor 1

2 in formula(29) would be explained by a full quantitative analysis of the difference between the two regulatory controlinstruments, which is not undertaken in this paper. Basically, the conclusion that ∆q̂

q = 12∆p

q comes fromthe fact that q̂ and q in formula (45) are more similar in their effects (because both fix the same total outputacross the two subperiods) than p and q in formula (29).22An equivalent form is ∆p

q̂ = EW ∗p − EW ∗

q̂ , which is not nearly as useful for analysis here as the form(46).

17

Substituting from formulas (29) and (45) into (46), we then derive the fundamental

sought-after result of this paper:

∆pq̂ =

σ2θ4c2

(c− b). (47)

Once again, here, the sign of ∆pq̂ in expression (47) is exactly equal to the sign of (c− b).

Fixed prices are preferred to time-flexible quantities with banking and borrowing when the

slope of the marginal cost curve is steeper than the slope of the marginal benefit curve

(c > b). Conversely, time-flexible quantities with banking and borrowing are preferred to

fixed prices when the slope of the marginal benefit curve is steeper than the slope of the

marginal cost curve (b > c).

The sign of formula (47) for ∆pq̂ is exactly the same as the sign of the traditional formula

(29) for the comparative advantage of prices over quantities ∆ (=∆pq). Both the fixed-

quantity regulatory system q and the time-flexible quantity system q̂ set the same total

output across the two subperiods equal to 2q. Formulas (47) for ∆pq̂ and (29) for ∆p

q are

indicating that this feature of the same total output across the two subperiods is apparently

enough to make the economic advice given to the regulator for choosing between fixed prices

and time-flexible quantities be exactly identical to the economic advice given to the regulator

for choosing between fixed prices and fixed quantities.23 Everything seems to come back

to the same crucial issue of the sign of (c− b).What this means for policy is that in principle the regulator need not at all consult sepa-

rately formula (47) for ∆pq̂. All information required for the regulator to decide on economic

grounds between fixed p and time-flexible q̂ is already fully contained in the traditional for-

mula (29) for the comparative advantage of fixed prices over fixed quantities ∆ (=∆pq). In

this sense fixed prices vs. time-flexible quantities with banking and borrowing presents no

new issues relative to the standard original analysis offered by considering fixed prices vs.

fixed quantities. Nothing is gained or added to the economic analysis, nor is any information

lost, by conceptualizing banking and borrowing as somehow different from fixed quantities.

To analyze prices vs. quantities by the traditional comparative-advantage formula for ∆ cov-

ers both the case where q is fixed and the newly-analyzed case here where q is the outcome

of time-flexible banking and borrowing.

23That the multiplicative factor 14 in formula (47) is different from the multiplicative factor 1

2 in formula(29) would be explained by a full quantitative analysis of the difference between the two regulatory controlinstruments, which is not undertaken in this paper. Basically, the conclusion that ∆p

q̂= 1

2∆pq comes from

the fact that p and q̂ in formula (47) are more similar in their effects (because both cost-effectively fix equalprices across the two subperiods) than p and q in formula (29).

18

9 Concluding Remarks

There is no question but that the model of this paper is highly stylized with a great many

restrictive assumptions (including linear-quadratic costs and benefits, additive shocks, two-

period analysis, independence of costs and benefits, a simplified information and decision

structure, a purely static flow analysis, and so forth and so on). The problem confronting

a modeler here is that the issues addressed in the model of this paper become analytically

intractable without making strong structural assumptions. As a consequence, the results

of this paper are applicable only for providing insight and rough guidance at a high level of

abstraction. The model of this paper might be appended and extended, but largely at the

detriment of adding complexity.

This paper shows that the addition of time-flexible banking and borrowing does not alter

the fundamental insights gained from the traditional framework for analyzing fixed prices

vs. fixed quantities. The customary formula for the comparative advantage of prices over

quantities is suffi cient for the regulators to make the correct economic decision even when

banking and borrowing are allowed in place of fixed quantities. I believe this insight is

useful, and parts of it might even survive under more realistic assumptions.

References

[1] Adar, Zvi, and James M. Griffi n (1976). “Uncertainty and the Choice of Pollution

Control Instruments.” Journal of Environmental Economics and Management, 3: 178-

188.

[2] Chao, Hung-Po, and Robert Wilson (1993). “Option Value of Emission Allowances.”

Journal of Regulatory Economics, 5: 233-249.

[3] Dales, John H. (1968). Pollution, Property, and Prices: An Essay on Policy-making

and Economics. Toronto: University of Toronto Press.

[4] Fell, Harrison, Ian A. MacKenzie, andWilliamA. Pizer (2012). “Prices versus Quantities

versus Bankable Quantities.” Resource and Energy Economics, 34(4): 607-623.

[5] Feng, Hongli, and Jinhua Zhao (2006). “Alternative intertemporal trading regimes with

stochastic abatement costs.” Resource and Energy Economics, 28(1): 24-40.

[6] Fishelson, Gideon (1976). “Emission Control Policies under Uncertainty.” Journal

of Environmental Economics and Management, 3: 189—197.

19

[7] Goulder, Lawrence H., and Andrew R. Schein (2013). “Carbon Taxes vs. Cap and

Trade: A Critical Review.” Climate Change Economics, 4(3): 1-28.

[8] Hoel, Michael, and Larry Karp (2002). “Taxes versus Quotas for a Stock Pollutant.”

Resource and Energy Economics, 24(4): 367-384.

[9] Kaplow, Louis, and Steven Shavell (2002). “On the superiority of corrective taxes to

quantity regulation.” American Law and Economics Review, 4(1): 1-17.

[10] Kelly, David L. (2005). “Price and quantity regulation in general equilibrium.” Journal

of Economic Theory, 125: 36-60.

[11] Montgomery, W. David (1972). “Markets in Licenses and Effi cient Pollution Control

Programs.” Journal of Economic Theory, 5(3): 395-418.

[12] Murray, Brian C., Richard G. Newell, and William A. Pizer (2009). “Balancing cost and

emission certainty: An allowance reserve for cap-and-trade.” Review of Environmental

Economics and Policy, 3: 84-103.

[13] Newell, Richard, and William Pizer (2003). “Regulating stock externalities under un-

certainty.” Journal of Environmental Economics and Management, 45: 416-432.

[14] Newell, Richard, William Pizer, and Jiangfeng Zhang (2005). “Managing permit mar-

kets to stabilize prices.” Environment and Resource Economics, 31: 133-157.

[15] Pigou, Arthur C. (1920). The Economics of Welfare. London: Macmillan.

[16] Pizer, William A. (2002). “Combining price and quantity controls to mitigate global

climate change.” Journal of Public Economics, 85:409-434.

[17] Pizer, William A. and Brian Prest (2016). “Prices versus Quantities with Policy Updat-

ing.”NBER Working Paper No. 22379.

[18] Roberts, Mark J., and A Michael Spence (1976). “Effl uent charges and licenses under

uncertainty.” Journal of Public Economics, 5: 193-208.

[19] Rubin, Jonathan D. (1995). “A Model of Intertemporal Emission Trading, Banking,

and Borrowing.” Journal of Environmental Economics and Management, 31: 269-286.

[20] Weitzman, Martin L. (1974). “Prices vs. Quantities.” Review of Economic Studies,

41(4): 477-491.

20

[21] Weitzman, Martin L. (1978). “Optimal rewards for economic regulation.” American

Economic Review, 68: 683-691.

[22] Yates, Andrew J., and Mark B. Cronshaw (2001). “Pollution permit markets with

intertemporal trading and asymmetric information.” Journal of Environmental Eco-

nomics and Management, 42: 104-118.

[23] Yohe, Gary W. (1978). “Towards a general comparison of price controls and quantity

controls under uncertainty.” Review of Economic Studies, 45: 229-238.

[24] Zhao, Jinhua (2003). “Irreversible abatement investment under cost uncertainties: trad-

able emission permits and emissions charges.” Journal of Public Economics, 87: 2765-

2789.

21