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Optimal two-part pricing in a carbon offset market: a comparison of organizational types.

1. Introduction

One of the consequences of the Kyoto Protocol (KP) and the increased focus on the role of greenhouse gases (GHGs) in climate change is that market-based approaches to reducing GHG emissions are being actively developed. In Europe, the European Union (EU) operates an emissions-trading scheme (ETS), (1) while in the United States, the Chicago Climate Exchange (CCX) trades carbon offsets among exchange members. (2) In Canada, a number of provinces have announced plans for carbon-trading schemes; four of them have joined the Western Climate Initiative (WCI), a consortium of Canadian provinces and states developing a regional cap-and-trade system (WCI 2008). The WCI and the Regional Greenhouse Gas Initiative (RGGI), a consortium of Northeastern and Mid-Atlantic states (see RGGI 2008), are part of the International Carbon Action Partnership, a group of countries and regions from around the world that has formed to actively pursue the development of mandatory cap-and-trade systems in carbon. (3)

The agriculture and forestry sectors are likely to be important suppliers of carbon offsets to the carbon market, both through carbon sequestration activities (for example, the storing of carbon in the soil or in trees) and through activities that reduce GHG emissions. For instance, estimates made by the United States at the time the KP was being negotiated projected that forest, cropland, and grazing land management sinks could have accounted for about half of U.S. reduction commitments (U.S. Department of State 2002). In Canada, the federal government estimated that agricultural and forestry sinks could provide in the order of 15--20 million metric tons of carbon offsets relative to a total of 270-300 million metric tons of reductions required under the KP (Government of Canada 2005). Food and Agricultural Organization (FAO) estimates indicate that dryland soils in the world could sequester somewhere in the range of 12,000--16,000 million metric tons of carbon (FAO 2004).

In the case of agriculture, since carbon offsets are sold in units that are much larger than most individual farmers can supply, carbon sequestration from a number of producers must be aggregated to fulfill contracts. (4) As a consequence, market intermediaries--or aggregators, as they are known--are emerging to perform this economic function. Some of these aggregators are organized as for-profit firms, while others are organized by farmers' associations. Examples of for-profit firms are as follows: C-Green Aggregators Ltd., a privately owned company that contracts with farmers in Manitoba, Saskatchewan, and Alberta for carbon sequestration (the carbon they acquire is sold on the CCX; C-Green Aggregators 2008); and AgCert International, an Irish company founded to generate emission reductions from livestock farms (AgCert International 2008). Much of AgCert's sales of carbon credits has been direct to individual emitters. On the producer association side, the Iowa Farm Bureau has created AgraGate Climate Credits Corporation to aggregate carbon credits for sale on the CCX (AgraGate Climate Credits Corporation 2008). AgraGate administers the project, transacts trades on the exchange, and distributes earnings to project participants.

The pricing structure used by the aggregators varies according to the type of project, among other things. For instance, for carbon sequestration projects, AgraGate and C-Green use only a variable fee. Farmers are paid a per metric ton [CO.sub.2] equivalent fee equal to the carbon price obtained from the CCX less a service fee; AgraGate charges a 10% service fee, while C-Green charges a 15% service fee (AgraGate 2008; C-Green 2008). (5) In contrast, AgCert takes possession of the carbon credits created through manure lagoon projects and receives the revenue from the sale of these credits. Farmers pay a portion of the cost of the lagoon; in return, they receive direct financial benefits in the form of lower manure handling costs (they also receive nonpecuniary benefits in the form of better odor management and other environmental gains). This pricing structure represents the aggregator taking a large mark-up (farmers receive no variable payment) and paying the farmer a negative fixed fee (farmers pay only a portion of the fixed cost).

The emergence of aggregators, and particularly those operated by producer associations, raises questions around the mix of variable and fixed fees that aggregators can be expected to use. There is a sizeable literature involving optimal two-part pricing by a monopolist or public agency that sells to heterogeneous consumers or buys from heterogeneous firms (for example, Oi 1971; Ng and Weisser 1974; Leland and Meyer 1976; Auerbach and Pellechio 1978; Ordover and Panzar 1982; Laffont and Tirole 1993). This literature shows that the sign and size of the optimal fixed fee and the extent to which the optimal variable fee is raised above marginal cost depend on the relative responsiveness of consumption/production and participation to the pair of pricing variables. In general, optimal pricing by a monopolist is a scaled-up version of optimal pricing by a budget-constrained public agency because inverse elasticity pricing underlies both the variable and fixed fee pricing formulas.

The purpose of this article is to examine optimal two-part pricing by an aggregator in the carbon sequestration market. In addition to constructing a framework for evaluations of the emerging carbon offset market, the article makes two contributions to the theoretical literature on efficient pricing with two-part tariffs. First, we derive specific pricing formulas for a market intermediary that simultaneously has monopsony power with respect to upstream purchases and monopoly power with respect to downstream sales. This implies that three elasticities--inframarginal supply by farmers, participation by farmers, and demand by the large final emitter (LFE)--are combined within each optimal pricing equation. With the exception of Ordover and Panzar (1982), the firm setting the two-part tariff in the standard pricing models has either monopsony or monopoly power, and so there are just two elasticities within each pricing equation. The analysis also pays particular attention to the properties of the farmer participation elasticity because the size of this elasticity largely determines whether farmers pay a fixed participation fee or receive a signing bonus when contracting with the aggregator. For instance, we show that the negative fixed fee paid by the aggregator in the manure handling case is consistent with a relatively elastic participation elasticity for farmers. In contrast, the pricing structure observed in the soil carbon sequestration case is consistent with the participation elasticity and the inframarginal supply elasticity being roughly the same.

The second theoretical contribution of the article is that it considers the pricing decisions of three different organizational types: a for-profit firm, a public agency, and a producer association. A public agency is included because this organizational form has been the focus of a great deal of the literature on optimal two-part tariffs; the public firm also serves as a useful benchmark for the pricing behavior of the other two organizational types. The for-profit firm and the producer association are examined because they are the organizational forms emerging in the carbon offset market. Consideration of the producer association is particularly important since the optimal two-part tariff for this organizational form has not been extensively examined in the literature. (6) As will be seen, the three elasticities discussed here enter the pricing decisions of the producer association in a rather counterintuitive way because the goal of the producer association is to restrict output in order to raise the price paid by the LFE while continuing to earn zero profits.

The paper is structured as follows: Section 2 describes the formal model required for the analysis. Section 3 derives the optimal pricing equations for the profit-maximizing monopolist and the public agency, while section 4 gives the main pricing results for these two organizations, first under the assumption that the key elasticities are fixed, and then more generally. Section 5 compares the pricing decision by a producer association and a public agency, and section 6 presents a discussion and concluding comments.

2. The Model

The model is a standard static partial equilibrium model of a vertical economic system.

There are four agents in the system: (i) a competitive large final emitter of GHGs; (ii) a group of heterogeneous farmers that can potentially produce carbon offsets; (iii) an aggregator that serves as an intermediary in the carbon offset trading market; and (iv) an industry regulator.

The setup in this model is similar to that in Ordover and Panzar (1982), who examine optimal pricing by an upstream monopolist who sells to firms that in turn sell to a competitive downstream market. In both their model and the one in this article, the choice of pricing variables affects the aggregate volume of production, which in turn affects the final downstream price faced by competitive agents. Ordover and Panzar (1982) allow for general nonlinear pricing with heterogeneous firms and only restrict their attention to two-part pricing when firms are assumed to be homogeneous. The analysis in this article is unique in the literature because it focuses on two-part tariffs with simultaneous upstream and downstream pricing and firm heterogeneity. (7)

The analysis begins with an in-depth examination of each of the agents.


The regulator is passive in that it specifies a maximum level of emissions, [bar.x], for the LFE, but it provides this firm with the option to purchase carbon offsets for emission levels that exceed [bar.x]. This threshold amount is exogenous to the analysis. For example, [bar.x] can be viewed as being chosen by the regulator to comply with an international agreement or to meet some environmental objectives rather than being chosen to maximize a social welfare function that explicitly incorporates the welfare of agents both in and out of the model.

Large Final Emitter (LFE)

This firm produces output y and GHG emissions x as an unwanted by-product. The LFE can reduce emissions, but any reduction raises the cost of production at a given level of output. To incorporate this restriction, let d(y,x) denote the LFE's cost of producing y units of output when emissions are at level x. Assume that [d.sub.y](y, x) > 0, [d.sub.yy](y, x) > 0, [d.sub.x](y, x) < 0, and [d.sub.xx](y, x) > 0. Regulation requires x - X [less than or equal to] [bar.x], where X is the number of units of carbon offsets that are purchased by the firm at price P. The assumption that [d.sub.x](y, x) < 0 implies that x - X [less than or equal to] [bar.x] will hold as an equality when LFE profits are maximized.

Without loss of generality, assume that the selling price of y is equal to one. The profit function for the LFE can be expressed as [[pi].sup.LFE] = y - d(y, [bar.x] + X) - PX. The pair of first-order conditions for choosing y and X to maximize this function are [d.sub.y](y, [bar.x] + X) = 1 and - [d.sub.x](y, [bar.x] + X) = P. The solution to this set of equations defines the LFE's inverse demand for carbon offsets, which can be denoted [P.sub.D](X). (8) Assuming the second-order conditions hold, it is easy to show by totally differentiating the pair of first-order conditions (FOCs) that [P'.sub.D] (X) < 0 (that is, the demand curve for carbon offsets by the LFE is downward sloping). Note also that a reduction in the maximum level of emissions, [bar.x], shifts the demand curve for carbon offsets out and to the right.


There are N competitive farmers, each of whom can potentially modify their farming practices to sequester soil carbon or reduce GHG emissions and thus qualify to contract with the aggregator. (9) Farmers differ with respect to their productivity (and hence cost) of creating a carbon offset, as measured by the parameter [theta] [member of] ([theta], [[theta].bar][bar.[theta]]). The productivity heterogeneity among farmers results from, among other things, differences in management ability and differences in soil quality. The distribution of [theta] across the N firms is governed by the density and distribution functions, g([theta]) and G([theta]), respectively, which are assumed to have the standard properties. Without loss in generality, we assume N = 1 so that all results can be interpreted as the fraction of the total number of farmers rather than the number of actual farmers.

Let c(q, [theta]) denote the farmer's total cost of producing q units of carbon offsets net of the contract fixed fee/bonus. The properties of this cost function include: (i) c(q, [theta]) [greater than or equal to] [PHI] for any q > 0 (all types of farmers operate with a minimum fixed cost, [PHI]); (ii) dc/dq > 0 and [d.sup.2]c/d[q.sup.2] > 0 (variable cost is convex in output); and (iii) dc/d[theta] < 0 and [d.sup.2]c/dqd[theta] < 0.

The carbon contract offered to a farmer by the aggregator is in the form of a two-part tariff: a fixed component [alpha] and a variable component [beta]. A positive value for [alpha] implies that farmers must pay a contract signing fee, while a negative value for [alpha] implies that farmers receive a signing bonus from the aggregator; there is no fixed fee if [alpha] = 0 (that is, standard uniform pricing is practiced). (10)

The optimal production of carbon credits for a type [theta] farmer who signs a carbon contract is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where P is the price of carbon offsets. Individual farm profits prior to paying the fixed tee can be expressed as [[pi].sup.Farm](P - [beta], [theta]) = (P - [beta])q(P - [beta], [theta]) c(q[P - [beta], [theta]], [theta]). (11) Given the previous assumptions about c(q, [theta]), it follows that d[[pi].sup.Farm]/d[theta] > 0 for all [theta] [member of] ([[theta].bar], [bar.[theta]]). Assuming that farmers will only participate in the carbon market if profits are positive, the farmer who is indifferent between signing and not signing the contract has type [??], where [??] is implicitly defined by [[pi].sup.Farm](P - [beta], [??]) = [alpha]. (12) Assuming that [theta] [member of] ([[theta].bar]. [bar.[theta]]) for all relevant pricing combinations, it follows from the previous assumptions that the farmer will not sign the carbon contract if [theta] [theta] ([[theta].bar], [??]) and will sign the contract if [theta] [member of] ([??], [[theta].bar]). (13) It is assumed that it can be easily observed if farmers choose q = 0 (and thus incur no fixed costs): thus, payments are not made to nonparticipating farmers (payment for nonparticipation, of course, is only a concern if [alpha] < 0). To ensure that farmers do not have an incentive to apply for a signing bonus while choosing some arbitrarily small output, it is necessary to assume that [phi] + [alpha] > 0. For the remainder of this analysis, we assume that this restriction does not bind when [alpha] and [beta] are chosen by the aggregator.

Aggregate supply of carbon offsets for all farmers who sign contracts can be expressed as


Recall that [P.sub.D](X) is the LFE's demand schedule for carbon offsets. The equilibrium price of carbon offsets, denoted [P.sup.*], ensures that aggregate farm supply is equal to demand by the LFE (one unit of Q is equal to one unit of X by assumption): that is, Q([[P.sup.*] - [beta]], [alpha]) = [P.sup.-1.sub.D] ([P.sup.*]). Short-hand notation for equilibrium supply, Q([P.sup.*] - [beta], [alpha]), is [Q.sup.*].


The aggregator incurs fixed cost F. which is assumed to be independent of both the number of farmers under contract and the aggregate volume of trade in carbon offsets. This cost includes the standard fixed cost items of a service-oriented firm as well as the capital cost of testing and monitoring equipment, computer networks, and so forth. The aggregator also incurs a cost m for each unit of carbon offset that it handles. This marginal cost includes the cost of testing and monitoring each unit of land that is covered by the carbon contract. (14) Since the aggregator charges a handling commission fee of size [beta] per unit of carbon offset, the equivalent of marginal cost pricing by the aggregator implies that [beta] = m.

Much of the analysis to follow examines features of the aggregator's optimally chosen fixed fee, [alpha]. Although the analysis assumes that [alpha] is an actual cash transfer between farmers and the aggregator, it should be noted that the transfer could be achieved in ways not involving cash at the time the contract is signed. For example, the aggregator may include in the contract a clause that requires the farmer to hire the services of a private accountant, or may alternatively choose to provide those services free of charge to the farmer. In general, the aggregator will have some ability to shift contract administration costs between itself and the farmers, and these shifts are equivalent to setting alternative values for [alpha].

Variable Definitions

The following variable definitions simplify the expressions for the optimal pricing equations, which are derived below.

[eta] = [partial derivative]P/[partial derivative]Q [Q.sup.*]/P

is the elasticity of inverse demand for the LFE;


is the inframarginal supply elasticity averaged across participating farmers;

[??] = q (P - [beta], [??])

is production by the marginal farmer;

[bar.q] [equivalent to] [Q.sup.*]/1 - G([theta])

is average production across participating farmers;

s = 1 - [??]/[bar.q]

is a scaling variable that measures the percentage difference in production for the "average" farmer and the marginal farmer;


is an index of local farmer heterogeneity measured at the marginal farmer; and

E = H([??], [alpha]) g([??])/1 - G([??])

is the elasticity of farm participation.

A key factor in this analysis is the local farmer heterogeneity index, H([??], [alpha]). The expression dc/d[theta] = ([partial derivative]c/[partial derivative]q)(dq/d[theta]) + [partial derivative]c/[partial derivative][theta], which makes up the denominator of H([??],[alpha]), is assumed to be positive: that is, the production cost effect (the expression [[partial derivative]c/[partial derivative]q][dq/d[theta]], which is positive) is assumed to dominate the heterogeneity cost effect (the expression [partial derivative]c/[partial derivative][theta], which is negative). (15) Nevertheless, a greater degree of cost heterogeneity across farmers (that is, a higher value for [partial derivative]c/[partial derivative][theta]), decreases the value of dc/d[theta], which in turn raises the value of the heterogeneity index, H([??], [alpha]). In general, the heterogeneity index can be interpreted as a measure of the inverse of the percentage change in farm cost for a marginal change in farm type.

The elasticity of farm participation,

E = H ([??], [alpha]) g([??])/1 - G([??]),

is the product of the heterogeneity index and the hazard rate for the distribution of farm types. Noting that

g([??]) = - d [1 - G([??])]/d[theta]

the hazard rate can be interpreted as the percentage change in the number of participating farmers. Accounting for the effects of both the heterogeneity index and the hazard rate, E can be interpreted as the percentage change in the number of participating farmers for a 1% increase in cost facing the marginal farmer; as a result, E can be viewed as a participation elasticity.

To conclude this section, it is useful to express the solution to the aggregator's pricing problem in terms of markup indexes for the marginal farmer, which can be defined respectively for [alpha] and [beta] as

[M.sub.[alpha]] = [alpha] + ([beta] - m)[??]/(P - [beta])[??]


Index [M.sub.[beta]] is the markup of the aggregator's variable lee over marginal cost relative to the revenues earned by the marginal farmer. Index [M.sub.[alpha]] is the markup of the aggregator's fixed fee over the zero entry-distortion fixed fee, -([beta] - m)[??], relative to revenues earned by the marginal farmer; as the results below show. [M.sub.[alpha]] > 0. When the index [M.sub.[alpha]] = 0 the aggregator's fixed fee [alpha] = -([beta] - m)[??] is equal to the loss in revenue occasioned by having [beta] > m; as a result, the entry-distortion effects of the variable lee are eliminated, and there is no overall fee distortion at the participation margin.

3. Optimal Pricing: Profit-Maximizing Monopoly and Public Agency

Profit-Maximizing Monopoly Case

Consider first the case where the aggregator is a profit-maximizing monopolist. The aggregator's problem can be written as follows:


On the right-hand side of Equation 1, the first term measures the variable lee revenue less the aggregator's variable cost; the second term measures the fixed fee revenue (fixed fee cost of the signing if [alpha] < 0): and F is a measure of the aggregator's fixed cost. Equation 1 is maximized when the following FOCs are satisfied:




Public Agency Case

Next consider the case where the aggregator is a public agency charged with maximizing aggregate social welfare. The aggregator must choose values for [alpha] and [beta] so as to operate on a break-even basis. 16 The objective of the aggregator is therefore to choose values for [alpha] and [beta] to maximize W([alpha], [beta]) subject to n ([alpha], [beta]) = 0, where


On the right-hand side of Equation 3, the first term measures consumer surplus, and the second and third terms measure aggregate farm profits.

Let [lambda] denote the Lagrange multiplier for this constrained optimization problem. (17) Using the expressions for d[pi]/d[alpha], and d[pi]/d[beta] given by Equation 2, the FOCs for choosing values for and [beta] that maximize W([alpha], [beta]), subject to [pi]([beta], [beta]) = 0, can be rearranged and written as




Nested Monopoly and Public Agency Cases

A comparison of Equations 2 and 4 reveals that with [lambda] [right arrow] [infinity] the pricing equations for the public agency aggregator are the same as the pricing equations for the profit-maximizing monopolist aggregator. Hence, Equation 4 captures both aggregator scenarios. This outcome is expected given the standard pricing comparison of monopoly and a public agency (see, for example, Atkinson and Stiglitz 1987).

If the various elasticity expressions defined already are substituted into Equation 4, the resulting expressions can be written as




Expressions for d(P - [beta])/d[beta], dP/d[alpha], d[??]d[beta], d[??]/d[alpha], and d[??]/d[alpha] can be derived by totally differentiating the market equilibrium conditions,


and [[pi].sup.Farm](P - [beta], [??]) = [alpha] (see Appendix). The expressions for these differentials can be substituted into Equation 5 to give




After solving Equation 6 for [M.sub.[alpha]] and [M.sub.[beta]], and then substituting in [M.sub.[beta]] = ([beta] - m)[??]/(P - [beta])[??] = [beta] - m/P - [beta]


[M.sub.[alpha]] = [alpha] + ([beta] - m)[??]/(P - [beta])[??],

the optimal pricing equations can be written as




For the special case of [eta] = 0 (that is, perfectly elastic demand by the LEE), Equations 7a and 7b are the same as the pair of pricing formulas derived by Laffont and Tirole (1993, ch. 2), and they are similar to those derived in related studies of optimal two-part pricing with heterogeneous consumers. Equation 7 extends the standard pricing equations by assuming that the firm in question has both upstream and downstream market power (this is the reason [eta] appears in the optimal pricing equations).

For the case of a monopoly aggregator, [lambda] [right arrow] [infinity] and thus ([lambda] - 1)/[lambda] = 1, in which case Equation 7 can be solved to give




For the case of the public agency aggregator, Equation 7 must be solved with the budget constraint ([beta] - m)Q + (1 - G[[??]]) [alpha] = F, which can be rewritten as ([beta] - m)[bar.q] = [bar.f] - [alpha], where [bar.f] is a measure of average aggregator fixed costs per participating farmer and is equal to F/(1 - G[[??]]). The public agency solution is




4. Results: Monopoly and Public Agency

General comparative static analysis of Equations 8 and 9 is not possible because the elasticity and share variables are all endogenous. Nevertheless, formal results (hereafter termed fixed elasticity results) can be derived for a set of specific functional forms for the distribution of [theta], the farm-level cost function, and LFE demand, which ensure that [eta], [epsilon], E, and s in Equations 8 and 9 take on constant values. The fixed elasticity results will reflect the general features of the pricing solution. With nonconstant values for [eta], [epsilon], E, and s, it is possible that the fixed elasticity results no longer hold. Later in the analysis, the constant hazard rate assumption is relaxed, and an example is presented where the fixed elasticity results are reversed when the equilibrium switches from public agency pricing to monopoly pricing.

Fixed Elasticity, Results

If we assume that the inverse demand for carbon offsets by the LFE is given by the power function, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [a.sub.0] and [a.sub.1] are positive-valued constants, and we further assume that farm cost is governed by the combined power-exponential function, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [b.sub.0], [b.sub.1]. and [b.sub.2] are positive-valued constants, it is easy to establish that the elasticity of inverse demand, q, and the supply elasticity averaged over participating farmers, s, both take on constant values. Also, with [alpha] = 0. the farmer heterogeneity index,


is a constant. Thus, by further assuming that the distribution of farmer types follows an exponential distribution, which has a constant hazard rate, it follows that the participation elasticity,

E = g([??])/1 - g([??]) H,

is a constant when [alpha] = 0. The second-order variation in E during the comparative static analysis due to [alpha] [not equal to] 0 is ignored. Finally. these assumptions ensure that individual farm production, q(P - [beta], [theta]), the traction of participating farmers, 1 - G([??]), and aggregate industry production are all exponential functions of [theta]. Thus, s = 1 - [??]/[bar.q] also takes on a constant value.

RESULT 1. Fixed elasticity: With one exception, both the monopolist and the public agency aggregator set a lower variable fee and a higher fixed fee at larger inframarginal production elasticity, [epsilon], and smaller participation elasticity, E. The converse is also true. The one exception is that the monopolist's variable lee is independent of E.

PROOF OF RESULT: Recognizing that [eta], [epsilon], E, and s take on constant values, Result 1 can easily be established by first differentiating the right-hand sides of Equations 8 and 9 with respect to s and E, and then signing the resulting expressions. QED.

Result 1 reflects the well-known public finance inverse elasticity rule, where the implicit tax is set relatively low at the more elastic margin. (Baumol and Bradford [1970] present a general formulation of the inverse elasticity rule: they also provide a detailed history and overview of the early literature on the rule.) At the one extreme where inframarginal supply is highly elastic and participation is highly inelastic, the aggregator will primarily use fixed fees to generate revenue. At the opposite extreme, where participation is highly elastic and inframarginal supply is highly inelastic, the aggregator will primarily use variable fees to generate revenue.

Result 1 indicates that the profit-maximizing monopolist decreases the variable fee and increases the fixed fee in response to an increase in the inframarginal supply elasticity, but only the fixed fee is adjusted in response to an increase in the participation elasticity. An increase in [epsilon] induces the monopolist to charge a lower [beta]; the lower [beta] combined with the higher [epsilon] results in farmers supplying more carbon offsets. The greater output and lower [beta] generate greater profits for farmers, which increases the marginal benefit of raising [alpha]. On the other hand, the impact of an increase in E plays out differently. As participation becomes more elastic, the marginal benefit of raising [alpha] falls; as a result, the monopolist lowers [alpha]. Because the fixed fee has no impact on the inframarginal supply elasticity and thus the marginal benefit and marginal cost of changing [beta], no change in [beta] occurs with the increase in E.

For the case of public agency pricing, the goal of the aggregator is to efficiently maintain a balanced budget. To do so, the public agency equates the marginal deadweight loss associated with [alpha] and [beta] while ensuring that revenues equal costs. An increase in [epsilon] results in a higher marginal deadweight loss associated with [beta]; decreasing [beta] allows this marginal deadweight loss to be lowered somewhat. To maintain a balanced budget, [alpha] is increased; increasing a also brings the marginal deadweight loss associated with [alpha] in line with the higher marginal deadweight loss associated with [beta]. In a similar fashion, an increase in E results in a higher variable fee and a lower fixed fee for the public agency aggregator.

Which economic factors give rise to more elastic participation by farmers? Recall that the participation elasticity, E, is defined as the percentage change in the number of participating farmers given a 1% increase in the profitability of the marginal farmer. Further recall that

E = H([??], [alpha]) g([??])/1 - G([??])

is the product of the farmer heterogeneity index and the hazard rate for the distribution of farm types. Thus, participation is more elastic for greater degrees of local cost heterogeneity across the pool of marginal farmers and greater concentrations of farmers with costs similar to the marginal farmer, as measured by the hazard rate.

Finally, a word about the parameter s. As Equations 8 and 9 indicate, the impact of the inframarginal supply elasticity [epsilon] is modified by the parameter s. To understand the impact of s, suppose [??] is small relative to [bar.q], thus making s relatively large. Under these conditions, the inframarginal elasticity [epsilon] overstates the marginal elasticity of supply; in effect, supply is less elastic than is suggested by [epsilon]. (The adjustment of average values to reflect marginal conditions is also found in Ng and Weisser [1974] and Laffont and Tirole [1993].) The implication of this less elastic supply is that [beta] can be increased further than is implied by [epsilon]. Notice also from Equations 8 and 9 that with [eta] = 0, the scaling variable s is a determinant of [beta] but not [alpha]. This is because [beta] affects both inframarginal supply and participation; whereas, [alpha] affects only participation.

RESULT 2. Fixed elasticity: Both the monopolist and public agency aggregator set a higher variable fee and a lower fixed fee with a less elastic demand by the LFE.

PROOF OF RESULT: Result 2 follows directly if the right-hand sides of Equations 8 and 9 are differentiated with respect to [eta] and signs for the resulting equations are established (recall that a less elastic demand implies a larger absolute value for [eta]).

First consider Result 2 in the context of the profit-maximizing monopolist. A less elastic demand by the LFE induces the aggregator to increase the variable fee in order to reduce production and raise the equilibrium price, P. Increasing [beta] reduces the profits earned by farmers and, in turn, the marginal benefit of raising [alpha]; as a result, the monopolist lowers the fixed fee [alpha].

Next consider the case of a public agency aggregator. A less elastic demand by the LEE lowers the marginal deadweight loss associated with [beta]. Increasing [beta] serves to increase this cost in order to bring it back into line with the marginal deadweight loss associated with [alpha]. To maintain a balanced budget, [alpha] is lowered; decreasing [alpha] also brings the marginal deadweight loss associated with this variable in line with the lower marginal deadweight loss associated with [beta].

RESULT 3. Fixed elasticity: For both the monopolist and public agency aggregator, farmers will be charged a fixed lee if 1/E > s/[epsilon]; farmers will be provided a signing bonus if 1/E < s/[epsilon]; and they will face pure variable fee pricing if 1/E = s/[epsilon].

PROOF OF RESULT: Result 3 follows directly from Equations 8b and 9b.

Result 3 specifically identifies the optimal balance between [[alpha].sup.*] > 0 (a fixed fee) and [[alpha].sup.*] < 0 (a signing bonus). Result 3 is a natural extension of result 1, which indicates that the size of the fixed fee is a decreasing function of the participation elasticity. A sufficiently large participation elasticity, which occurs if cost heterogeneity across farmers is relatively high and the concentration of farmers at the participation margin is relatively high, implies that both the monopolist aggregator and the public agency aggregator should offer farmers a signing bonus rather than charging them a fixed fee. For the public agency that maintains a balanced budget, the signing bonus must be financed exclusively through excess variable fee revenue.

Result 3 provides an interesting contrast with the results of Ordover and Panzar (1982), who show that the optimal fixed fee is zero when the firms purchasing from the monopolist are homogeneous and the technology is fixed proportions. Although fixed proportions are assumed in this model, the introduction of firm heterogeneity results in fixed fees that are not zero. (Ng and Weisser [1974], Leland and Meyer [1976], and Auerbach and Pellechio [1978] similarly find that the fixed lee is generally not zero.)

RESULT 4. Fixed elasticity: A monopolist aggregator sets a higher variable fee than a public agency aggregator, lf 1/E > s/[epsilon], the monopolist sets a higher fixed fee than the public agency, and if 1/E < s/[epsilon], the monopolist offers a higher signing bonus than the public agency.

PROOF OF RESULT: Substitute [bar.f] + [bar.[pi]] for [bar.f] in Equation 9, where [bar.[pi]] is a measure of the average profits earned by the aggregator per participating farmer. Result 4 holds if the absolute value of Equation 9 is an increasing function of [bar.[pi]]. This is clearly the case because the left-hand side of Equation 9 is proportional to [bar.f] + [bar.[pi]]. QED.

The result that a profit-seeking monopolist aggregator sets a higher variable fee and a higher fixed fee than a zero-profit public agency aggregator is intuitive. It is less intuitive that the monopolist offers farmers a higher signing bonus than the public agency when 1/E - s/[epsilon] < 0. When 1/E - s/[epsilon] < 0, participation is relatively elastic compared to inframarginal supply. From result 1, both firms will wish to raise [beta] and lower [alpha]. The monopolist finds it optimal to offer a large signing bonus, which, given the relatively large value of E. entices a large number of farmers to sign the contract. The profits earned by the farmers are then captured by the monopolist through a higher [beta]. For the public aggregator, there is less incentive to raise [beta], since doing so raises the price to the LFE and lowers welfare. Given the lower value of [beta] and the requirement to balance its budget, the public aggregator then chooses a smaller signing bonus.

Relaxing the Fixed Elasticity Assumption

In the previous section, the comparative static analysis was simplified by assuming functional forms that gave rise to constant values for the various elasticity and share variables. These results are also expected to hold in the more general case where the elasticity and share variables are not constants, provided that their values do not change excessively when the pricing variables shift from one equilibrium outcome to another.

Probably the most restrictive assumption made in the previous section concerns the distribution of [theta]. To achieve a constant hazard rate, which was necessary to achieve a constant elasticity of participation, [theta] was assumed to follow an exponential distribution. Most common distributions, such as the uniform, normal, and gamma, have an increasing hazard rate. An increasing hazard rate implies that the rate of exclusion of participating farmers grows as the pool of participating farmers shrinks due to higher values for [alpha] and [beta].

Recall that

E = H([??], [alpha]) g([??])/1 - G([??])

and that [??] takes on a higher value with monopoly pricing versus public agency pricing. An increasing hazard rate therefore implies that participation will be more elastic with monopoly pricing versus public agency pricing. Equation 9b shows that more elastic participation with monopoly pricing will shrink the differential between the fixed fee set by the monopolist versus the public agency when 1/E - s/[epsilon] > 0 and will increase the differential between the signing bonuses set by the monopolists versus the public agency when 1/E - s/[epsilon] < 0. In fact, simulation results demonstrate that the fixed fee for the monopolist can be lower than the fixed fee for the public agency, and in some cases, a signing bonus is optimal for the monopolist; whereas, a fixed fee is optimal for the public agency aggregator. (18)

5. Producer Association

Now consider the pricing decisions of an aggregator that is operated by a producer association. The problem facing the producer association is similar to the public agency in that both attempt to extract just enough revenue from the various market participants to cover the costs of operation. However, the two organizations differ in that the producer association is concerned only with the welfare of its members, while the public firm considers the welfare of both producers and the LFE.

To compare the pricing decision of the producer association and public agency, we use a two-stage process. In stage one, the aggregator chooses optimal values for [alpha] and [beta] conditioned on industry output, Q, and in stage two, the optimal value for Q is chosen. (19) Note that in stage one, for the same value of Q, the public agency and the producer association will choose the same values for [alpha] and [beta].

In stage two, when Q is the choice variable, the comparison of a producer association and public agency is equivalent to a standard comparison of monopoly and pure competition. Specifically, to maximize profits for its members, the producer association chooses Q to ensure that marginal revenue, [P.sub.D](Q) + Q[P'.sub.D], is equal to marginal cost, where marginal cost includes the marginal cost of aggregation plus the marginal cost of carbon offset production by farmers. The public agency, on the other hand, chooses Q to ensure that [P.sub.D](Q) is equal to marginal cost. Because marginal cost is the same for both the producer association and the public agency, and because [P.sub.'D] < 0, it follows immediately that the producer association's choice of Q is smaller than that of the public agency.

In stage one, when optimal values for [alpha] and [beta] are conditioned on predetermined stage-two output, [Q.sup.0], the aggregator faces two constraints: Aggregate market supply must equal demand by the LFE, and profits for the aggregator must equal zero. These two constraints can be written as



[pi]([alpha], [beta]) = ([beta] - m)Q(P - [beta], [alpha]) + (1 -G([??])[alpha] - F = 0)(10b)

The values for at and 13 that solve Equation 10, which are denoted [alpha]*([Q.sup.0]) and [beta]*([Q.sup.0]), respectively, are the stage-one solution values for the aggregator's pricing problem.

Given that the optimal stage-two choice of Q is smaller for the producer association than for the public agency, it is possible to determine how [alpha]* and [beta]* differ for the two organizational types in the limit as [eta] [right arrow] 0. Specifically, with [eta] negative but marginally close to zero, the producer association will choose a value for Q that is marginally below the value of Q chosen by the public agency, which is denoted Q**. Differences in pricing strategies for the two organizational forms in this limiting case are revealed when Equation l0 is totally differentiating with respect to [alpha], [beta], and [Q.sup.0], and then the signs of the resulting equations are evaluated at [Q.sup.0] = Q**. Conclusions about differences in pricing strategies for more general values of q can be inferred from the limit case results.

RESULT 5. Suppose inverse demand by the LFE has a "small" negative slope. If 1/E - s/[epsilon] > 0, the producer association should decrease [alpha] and increase [beta] relative to the pricing solution of the public agency. The opposite results hold if 1/E - s/[epsilon] < 0.

PROOF OF RESULT: Equations 10a and 10b are totally differentiated with respect to [alpha], [beta], and [Q.sup.0], and then Cramer's rule is used to obtain expressions for d[alpha]*/[Q.sup.0] and d[beta]*/d[Q.sup.0], evaluated at Q = Q**. After simplifying the resulting equations by substituting in the various price differential expressions contained in the Appendix and the elasticity definitions, the following results emerge:

d[alpha]*/d[Q.sup.0] = 1 - [eta][epsilon] - (1 - s)[eta]W/(1 - s) (1/E - s/[epsilon]) E[epsilon], (P - [beta])[??]/[Q.sup.0] (11a)


d[beta]*/d[Q.sup.0] = -1/(1/E - s/[epsilon])[epsilon]E (P - [beta])/[Q.sup.0]. (11b)

Equation 11 shows how the producer association can marginally reduce [Q.sup.0] below Q** to raise profits for farmers relative to the public agency case. Notice that (i) if 1/E > s/[epsilon], then the association must decrease [alpha] (because d[alpha]*/d[Q.sup.0] > 0) and increase [beta] (because d[beta]*/d[Q.sup.0] < 0); and (ii) if 1/E > s/[epsilon], then the association must do the opposite. QED.

Result 5 is counterintuitive because it does not correspond to the general inverse elasticity rule. Standard efficient pricing results would require the association to raise 13 with a more inelastic inframarginal supply, and to raise the fixed fee with a more inelastic participation elasticity. Instead, for the producer association, a smaller c leads to a smaller [beta] compared to the public agency. While counterintuitive, the result is logical. Suppose [epsilon] is quite small and E is quite large, which implies that relative to the public agency, the producer association will set a higher value for [alpha] and a lower value for [beta]. Because E is large, increasing [alpha] will drive out a sizeable number of marginal farmers and will thus achieve a relatively large reduction in output. Lowering [beta] will transfer the surplus revenue to the inframarginal farmers to ensure the aggregator's break-even constraint is maintained. However, because [epsilon] is small, this transfer of surplus invokes a comparatively small supply response. So the producer association has achieved its objective of lowering output while not violating the two constraints. Of course, the opposite also holds. When E is small relative to [epsilon], the association raises [beta], which has a comparatively large negative impact on inframarginal production, and lowers [alpha], which has a comparatively small impact on production at the participation margin.

Result 5 thus indicates that the standard inverse elasticity rule must be modified for the producer association. To the extent that the producer association wishes to make efficient use of resources, it will follow a version of the inverse elasticity rule. However, the necessity of transferring revenue to its members, all the while trying to reduce overall output, means that the inverse elasticity rule has to be modified. In short, the counterintuitive result for the producer association occurs because the association trades off economic efficiency in return for the ability to redistribute revenue and income.

6. Discussion and Conclusion

The development of carbon markets and the emergence of aggregators to supply carbon offsets leads to an interesting pricing problem that to date has not been examined in the literature. The model developed in this article shows that three elasticities--the inframarginal elasticity of supply, the participation elasticity, and the demand elasticity--are critical to the setting of a two-part price when the monopoly and the monopsony powers of the aggregator are considered.

The results of the article show that optimal pricing patterns are distributed throughout a wide range depending on the relative values of the elasticities. For instance, the use of a variable fee with no fixed fee (as is observed in the soil carbon sequestration market) is optimal when the adjusted inframarglnal elasticity of supply ([epsilon/s]) equals the elasticity of participation (E), while the use of negative fixed fees is optimal when participation is relatively elastic compared to inframarginal supply (which is likely to be the case in manure lagoon projects, where this type of pricing pattern is in fact observed; see Introduction).

The impact of these elasticities, however, differs with the type of aggregator that is considered. When the various elasticities considered in the article are assumed to be constant, the profit-maximizing aggregator is likely to set a higher variable fee than a public agency aggregator. If 1/E > s/[epsilon], the monopolist sets a higher fixed fee than the public agency, which in turn sets a higher [alpha] than the producer association (the producer association correspondingly has a higher variable fee than the public agency). If 1/E < s/[epsilon], the monopolist sets a higher bonus than the public agency, which in turn has a larger bonus than the producer association (the producer association has a lower variable fee than the public agency).

Section 5 showed that the fees charged by the producer association deviate from those charged by the public agency in a direction that does not accord with the intuition provided by the inverse elasticity rule. This deviation can be understood, however, as the result of the open membership policy of the association and its need to redistribute surplus revenues to its members.

These two characteristics are important in another regard. Helmberger (1964) observes that open membership cooperatives--which share the same characteristics as the producer associations examined in this article--generally lead to smaller departures from the competitive equilibrium than those associated with a monopoly for-profit firm. Given the emergence of producer associations in the carbon offset market, more work is required to determine the effectiveness of open membership producer associations in achieving social welfare objectives, particularly in situations where they are competing with for-profit firms in a mixed oligopoly setting.


In this appendix, expressions for (d[P - [beta])/d[beta], dP/d[alpha], d[??]/d[beta], and d[??]/d[alpha] are derived. The market equilibrium conditions are as follows:


If these two equations are totally differentiated and converted to elasticities (note that an envelope condition implies that

D[[pi].sup.Farm](P - [beta], [theta])/d[theta] = dc[q(P - [beta], [theta]), [theta]]/d[theta]),

we find


Using Cramer's rule, the solution to this system of equations can be written as


Received January 2008; accepted January 2009.


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Murray Fulton * and James Vercammen [dagger]

* Johnson-Shoyama Graduate School of Public Policy, Diefenbaker Building, University of Saskatchewan, Saskatoon, SK S7N 5B8, Canada; Phone (306) 966-8507; E-mail; corresponding author.

[dagger] Food and Resource Economics and Sauder School of Business, 2053 Main Mall, University of British Columbia, Vancouver, BC V6T 1Z2, Canada; Phone (604) 822-8475; E-mail

The authors would like to thank Tom Ross and Ralph Winter for valuable comments on an early version of this article, as well as the Southern Economic Journal editor and two anonymous reviewers. The usual caveats apply.

(1) The ETS began operation in 2005. The scheme limits emissions from 12,000 plants in specified industries, and member countries allocated legally binding allowances to emitters. The scheme allows them to trade emission credits with other EU firms. The ETS will not trade credits generated by Land Use, Land Use Change, and Forestry (LULUCF) activities, reflecting the EU position when negotiating the use of sinks in the KP and continued opposition on the part of EU environmental groups and other stakeholders to the use of sinks (Young et al. 2007).

(2) The CCX became operational in 2003 with the stated purpose of developing the institutions and skills needed to facilitate trade in GHG credits. The CCX members have pledged to reduce their GHG emissions by 6% by 2010, from a baseline of 1998-2001; this reduction commitment serves as the demand in the market (Chicago Climate Exchange 2008).

(3) Founding members include the European Commission and selected countries from the EU, the WCI, and the RGGI, New Zealand, and Norway (International Carbon Action Partnership 2008).

(4) For example, carbon offsets are sold in units of 100,000 metric tons carbon equivalents (MTCEs) on the CCX. In comparison, a farmer might be expected to sequester 0.4 metric tons of carbon per acre in a year; over 10 years, a 10,000 acre farm would be in a position to supply 40,000 MTCEs. A similar situation would emerge if small-scale landowners were to supply carbon credits by planting trees; while the remainder of the article focuses on the agriculture case, the analysis is also applicable in this scenario.

(5) Farmers are paid on 80% of the carbon that is sequestered; the remaining 20% is held in reserve to cover losses that may occur because of noncompliance or nonparticipation (AgraGate 2008; C-Green 2008). Farmers are also responsible for offset registration fees, offset credit-trading fees, and offset verification costs (these costs are deducted from the carbon sale proceeds before the farmer is paid); all of these fees are also on a per ton basis.

(6) There is a small body of literature that examines nonlinear pricing by cooperative firms (see Sexton 1986; Vercammen, Fulton, and Hyde 1996; and Bontems and Fulton, in press). None of these papers explicitly considers the details of two-part tariffs along with an examination of farmer participation.

(7) Relative to the two-part tariff, nonlinear pricing would be more profitable for the aggregator and more efficient for society. While general nonlinear pricing is not expected in real-world carbon offset markets due to the inherent complexity of the pricing schedules, other types of simpler pricing discrimination schemes that are still more efficient than a two-part tariff (for example, volume discounts or fixed fees based on farm size) cannot be ruled out. One advantage of restricting the analysis to two-part tariffs is that the specific contributions of the various elasticities and share variables can be readily analyzed, which is not the case for nonlinear pricing. See Wilson (1993) for a general comparison of two-part tariffs and nonlinear pricing in an applied setting.

(8) Graphically, the demand curve [P.sub.D](X) is given by the portion of the - [d.sub.x] curve that lies to the right of a vertical line at [bar.x]. If the LFE can sell any shortfall in emissions below [bar.x] to the carbon market, then [P.sub.D](X) extends into the upper left-hand quadrant, and X can take on both negative and positive values (that is, the [P.sub.D][X] curve becomes the entire--[d.sub.x] curve). For the purposes of this article, it is assumed that [bar.x] is such that X > 0 for all feasible values of P.

(9) For an example of the types of projects eligible as offsets in agriculture, see Climate Change Central (2008).

(10) The conditions in (iii) in the previous paragraph mean that higher [theta] results in lower variable cost and lowermarginal cost, which in turn mean that the marginal cost curves of farmers with different [theta]s do not cross. Following Oi (1971) and Cassou and Hause (1999), noncrossing marginal cost curves imply that the optimal variable fee [beta] will be greater than or equal to marginal cost.

(11) The model assumes that carbon credit creation activities do not affect the profits from farming. As a first approximation, this assumption is reasonable. For instance, the construction of a new lagoon to manage manure in livestock operations has little impact on the overall farming operation. For soil carbon sequestration, most farmers adopt carbon sequestering practices for the direct benefits (for example, better water retention) that they provide. In making a decision to participate in the carbon market, farmers examine the benefits and costs of signing a contract and not of actually changing their production practices.

(12) This assumption is consistent with a situation where the next best alternative for all farmers generates profits equal to zero, or a situation where [pi]([P - [beta]], [theta]) implicitly reflects the profits over the next best alternative. A more explicit characterization of the next best alternative would complicate the model without materially changing the results.

(13) Fixed and variable payments, along with farmer heterogeneity, have been shown to affect whether farmers remain in agriculture and. in turn, aggregate agricultural production (see, for instance. Chau and de Gorter 2000: Goodwin and Mishra 2006).

(14) The aggregator may also incur a per farm cost (for example, the cost associated with opening a new account), and this cost is strictly neither a fixed cost nor a marginal cost. The following analysis assumes this cost is either zero or is effectively captured in the other costs.

(15) To see this more formally, note from the farmer's FOC that dq/d[theta] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 0, given the assumed properties of c(q, [theta]). Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The first term in the brackets reflects the increase m cost due to higher production, while the second term in the brackets reflects the decrease in cost with a marginal change in farm type for the same level of production. The former expression is assumed to dominate the latter, so that dc/d[theta] > 0.

(16) For the public agency case, this assumption implies that public funds are not transferred in or out of the agency (that is. public firms operate with predefined budgets). For the producer association case, this assumption implies that association bylaws prevent the organization from operating with either a deficit or a surplus.

(17) The analysis and results would be the same if [lambda] were specified as an exogenous variable and equal to the shadow cost of transferring taxpayer funding to cover losses incurred by the public agency aggregator (or the shadow benefit of transferring surplus earned by the aggregator to taxpayers). In this situation, [beta] - m and [alpha] = 0 become the optimal pricing outcome as [lambda], [right arrow] 1 (that is, the standard marginal cost linear pricing rule emerges as optimal).

(18) The simulation results are based on the following assumptions. The farmer's cost function is given by c(q, [theta]) 5[q.sup.3]/[theta] + 10, which implies that [epsilon] = 0.5. The LFE has perfectly elastic demand at P = 40. The marginal cost for the aggregator is m = 1, and fixed costs are given by F = 2. The distribution of farmer types is given by the Weibull distribution, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for which the hazard rate is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. With k = 1, the Weibull distribution takes the form of an exponential distribution with a constant hazard rate. With k > l, the hazard rate of the Weibull distribution is an increasing function of [theta]. Different combinations of values of the two Weibull parameters, [gamma] and k, are assumed for the simulation. The simulation results show that when the hazard rate is increasing at a relatively rapid rate because k/[gamma] takes on a comparatively large value, then a signing bonus is optimal for the monopolist, while a fixed fee is optimal for the public agency. The simulation results are available from the authors on request.

(19) The two-stage process could have been used earlier in the article to derive the optimal pricing conditions for the public agency. This approach, however, does not work for the monopoly, since given the optimal Q, there exists an infinite number of [alpha] and [beta] combinations that generate this Q. (The problem is that there is only one constraint, namely, that aggregate demand has to equal aggregate supply.) The two-stage approach can be used in the public agency case (and the producer association case) because an added constraint is present, namely, the need to balance revenues with costs. While the direct approach that was used in the article could have been used for the producer association case, the lack of the consumer surplus component in the objective function means that the producer association outcome is not nested within the other two outcomes, which in turn makes a comparison difficult. The use of the two-stage process for the producer association, however, allows for an easy comparison of the pricing outcome for the public agency and producer association.
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Comment:Optimal two-part pricing in a carbon offset market: a comparison of organizational types.
Author:Fulton, Murray; Vercammen, James
Publication:Southern Economic Journal
Geographic Code:1CANA
Date:Oct 1, 2009
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