The Random Quantity Mechanism: Laboratory and Field Tests of a Novel Cost-Revealing Procurement Mechanism


Information on private costs can improve the efficiency of programs that provide payment for environmental services in contexts involving information asymmetries and heterogeneous private costs. Using data from laboratory and field experiments, this paper presents and evaluates a novel private cost revealing mechanism, termed the random quantity mechanism (RQM), that can advance research in conservation contracting, payments for environmental services, and other similar settings. We examine the RQM’s performance in a laboratory setting using induced costs and report results obtained from the first field implementation of this mechanism, with smallholder farmers in Zambia. We show that the RQM is incentive-compatible, that participant decision-making maximizes expected payoffs, and that the mechanism provides non-parametric estimates of private costs. The paper contributes to economic field studies by introducing a new incentive-compatible mechanism that elicits individuals’ minimum willingness to accept across intensive margins, enabling researchers to estimate the supply of a service or commodity, and provides for exogenous variation in contract terms, which can aid in separately identifying the impacts of incentives and of participants’ willingness to accept on contract outcomes.


Procurement auctions have been receiving increased attention in recent years for their use in payments for environmental services (PES) and conservation programs that incentivize landholders to provide environmental goods and services, either through changes to management practices, land set-asides, or other activities. Procurement auctions (also known as reverse auctions) are market-like allocation mechanisms that can induce producers and land managers to reveal private information regarding their opportunity costs (Latacz-Lohmann and Van der Hamsvoort 1997). They are particularly useful in contexts involving information asymmetries and heterogeneous opportunity costs, allowing a program to allocate payments to suppliers with the lowest costs and thereby improving cost-effectiveness relative to alternative approaches such as fixed-price (take it or leave it) offers (Ferraro 2008).

However, in practice a wide range of design and contextual factors in procurement settings can affect such performance characteristics of auctions as cost-effectiveness or truthful cost revelation and incentive compatibility. In this paper, we propose and empirically evaluate a novel mechanism for revealing the private cost of providing environmental and conservation goods and services that we call the random quantity mechanism (RQM). The RQM is an extension of the Becker–DeGroot–Marschak (BDM) (1964) mechanism commonly used in economic experiments to elicit willingness to pay for a good. The RQM holds potential value for field research in procurement settings for several reasons. First, as the bids of each participant represent her minimum willingness to accept (WTA), the RQM bidding data can be used to estimate cost structures non-parametrically and construct aggregate supply across participants. Additionally, it provides experimental variation in the allocation of contracts to participants and in contract terms (quantity and total payment) for those awarded contracts. The researcher can use this exogenous variation to separately identify the impacts of the payment incentives and individuals’ WTA on contract outcomes.

The RQM is well suited for use with products or services that are not traded in markets, as is often the case with environmental services associated with land use in developing nations (e.g., conservation and climate change mitigation). Smallholder farmers, forest communities, and other private land managers in developing nations and elsewhere can make significant contributions to conservation and climate change mitigation by modifying their land use and land management practices. An RQM pilot project can estimate provision costs for such services and provide valuable information for designers of conservation and PES programs. We conduct such a pilot program in a field experiment involving smallholder farmers in Zambia using contracts for agroforestry tree planting. Using the data collected in that pilot, we then construct estimates of the available supply of agroforestry tree planting by the farmers and compare the results of the RQM to simulated results of other common auction formats using the participants’ revealed WTA. The comparison identifies tradeoffs associated with each procurement alternative. Additionally, we confirm the theoretically-derived incentive compatibility of the RQM in a controlled laboratory setting.

This research makes a practical methodological contribution by demonstrating the applicability of the RQM in agri-environmental procurement settings as a means to help resolve information asymmetries and inform program design through a better understanding of supply across intensive margins.

Background and Literature Review

Information asymmetries (adverse selection) and additionality concerns make designing incentives and contracts for goods and services challenging for conservation agents. Ferraro (2008) identifies three broad approaches that can be used to limit information asymmetry and improve the efficiency of PES designs: (1) collecting data on landholder characteristics that likely are correlated with their private costs; (2) offering screening contracts that have been tailored to the specific distribution of private costs among the potential participants so that participants with low and high cost types self-select into different contracts; and (3) allocating contracts using an auction. The RQM can help inform the design of subsequent program roll-out, providing the information needed to implement either the targeting on observables (1) or screening contract (2) approaches, and has advantages compared to the use of a single-bid auctions as it provides a more complete characterization of aggregate supply and introduces exogenous variation in contract terms, which can aid in separately identifying the impacts of incentives and of participants’ willingness to accept on contract outcomes.

Targeting on observables is relatively simple where high-quality data is readily available. However, in contexts where information on landholder characteristics is difficult to obtain, such as in developing nations, this approach can be costly. The effectiveness of an incentive based solely on observable information also depends heavily on the strength of the correlations between those observable characteristics and the true private cost of providing the good. Several studies have noted that observable characteristics can be a poor measure of the true shadow cost of compliance, particularly where market failures are present, further limiting the effectiveness of this method (Alix-Garcia and Wolff 2014; Arriagada et al. 2012). The RQM generates data that allow researchers to directly connect observable characteristics to the individual cost structures of the landholders involved and to investigate the influence of those cost structures on contract outcomes.

Screening contracts require the conservation agent to have good information on the general functional form and distribution of landholders’ opportunity costs. With that information, the conservation agent can limit adverse selection by creating an optimal menu of contracts designed specifically for the landholders with different cost types in the target population (Ferraro 2008; Arguedas and van Soest 2009, 2011). Screening contracts alleviate concerns about weak additionality; they are designed so that the landholders are never better off choosing a contract designed for a different cost type, thus improving the PES programs’ economic efficiency (Ferraro 2008). However, the limited information typically available to conservation agents reduces the potential value of screening contracts (Hanley and White 2014). An RQM pilot can be used in the design of screening contracts as it provides rich information on private costs, incentives, and outcomes.

Procurement auctions (single-bid and multi-bid) are being used increasingly in payments for environmental services in developing nations (e.g. Jack et al. 2009 in Indonesia; Jack 2013 in Malawi; Jindal et al. 2013 in Tanzania; Khalumba et al. 2014 in Kenya; and Pant 2015 in Nepal). While these auctions can be effective in allocating contracts to the lowest-cost suppliers, they also can be challenging to implement at scale and have limitations that may preference the RQM as an alternative tool for use in research or program piloting.

In multi-bid auctions, landholders are allowed to make more than one price-quantity bid in the form of supply schedules. This auction format holds appeal for programs that aim to procure multiple items from multiple landholders, as it allows landholders to bid across intensive margins and mitigates the presence of “lumpy” bids that occur in single-bid auctions (Tenorio 1993). All auctions impose transaction costs for the program manager and the landholders, and these costs increase with the complexity of the auction’s design, deterring participation and weakening competition (Palm-Forster et al. 2016). The transaction costs of complex designs such as multi-bid auctions will offset some of their benefits and discourage implementation in the field, particularly when the scale of the auction is large. Single-bid auctions are simpler to implement, however, those formats also tend to underestimate supply when private costs are heterogeneous and marginal costs are not constant.

Not all auctions are incentive compatible. While truthful bidding is a weakly dominant strategy in the Vickrey format and generalized Vickrey in a multi-unit setting (Ausubel 2004), the sealed-bid uniform-price and discriminatory-price multi-bid auction formats are prone to strategic bidding and supply inflation. When this strategic bidding occurs, an auction can fail to reveal landholders’ true opportunity costs, resulting in inefficient allocations (Hailu and Thoyer 2010).

Arnold et al. (2013), using a model to study adverse selection in auctions for conservation easements, find that auctions failed to promote additionality because the landholders would not have developed their properties even in the absence of the payments. Lundberg et al. (2018) also highlights the importance of ensuring additionality in PES programs. They compare the performance of auctions in a spatial simulation using various contextual factors, including the degree of correlation between private costs and provision of the environmental services and the probability of non-compliance, modeled as a function of the private costs. They find that context is an important factor in determining the relative effectiveness of the various auction formats and provide some general principles that can inform design choices.

Overall, auctions can be effective allocation mechanisms in a variety of settings but may also lead to inefficient allocations, weak additionality, and biased cost estimates due to strategic bidding. The RQM can provide practitioners and decision-makers with better information on cost structures and the relationships between private costs, incentives and outcomes, allowing them to improve the efficiency of their procurement processes when information is asymmetric. While the RQM does not necessarily allocate to lowest-cost producers, contracting producers with higher private costs in a pilot provides additional information on the performance of cost types that would otherwise remain out of sample in an auction.

Specifically, the RQM provides quasi-random allocation of contracts and terms, conditional on satisfying a participant’s minimum WTA.Footnote 1 As a result, producers who have similar cost structures are likely to receive contracts that differ in the value of incentive payments and production quantities required. This exogenous variation in allocation and terms can be important when cost and compliance are correlated. For example, Ajayi et al. (2012) suggest that when producers’ opportunity costs are uncertain, contract payments based on expected opportunity cost might lead to low contract performance, and the weak correlation between bids and compliance outcomes in a study in Indonesia may provide some evidence of this (Leimona and Carrasco 2017). Oliva et al. (2019) highlight how the dynamic nature of many land use activities further complicates programs’ efforts to use subsidies and incentives when participants’ costs are uncertain. The RQM allows one to evaluate the effect of the incentive and the effect of an established minimum WTA separately by observing producers who have similar cost structures but receive different contracts in terms of value and quantity produced (and, thus, unit prices).

The paper proceeds as follows. First, we develop theoretical predictions of the RQM and present the results of the induced-cost laboratory test of the mechanism. We then present the RQM field experiment, in which tree planting contracts were allocated to smallholder farmers, and construct landholder supply. Finally, we compare the results of the RQM to the simulated generalized Vickrey and other auction formats that have been used in similar settings and discuss potential applications for the RQM.

Evaluating the Random Quantity Mechanism

Theoretical Incentive-Compatibility

In the RQM, individual producers are offered a cash payment (hereafter we refer to this as the ‘contract value’) for supplying a good or service and are asked to respond with the maximum quantity of that good or service they are willing to supply for that payment. A random quantity is then drawn from a predetermined distribution that encompasses possible quantity bids. If the drawn quantity is less than or equal to the quantity the individual bid to supply, then the landholder receives a contract with the specified dollar value and the drawn quantity. Landholders who offer less than the drawn number of units do not receive a contract. The profit to the landholder is the dollar value of the contract less their actual opportunity cost to supply the drawn quantity, or zero if no contract is awarded. The landholders in these auctions usually are involved in agricultural or other primary production activities so making production decisions in response to price signals from the marketplace is a familiar process. Thus, the RQM bidding process mirrors their normal production decision process.

As with incentive-compatible auctions, the key principle that drives truthful cost revelation in the RQM is that the price (a function of the binding quantity drawn and the contract value offered) is set independently from the participants’ submitted bids. Participants who submit quantity bids that are smaller than the maximum they would be willing to supply reduce the probability of receiving a profitable contract and do not increase the profit for any given quantity draw so they are no better off than if they submit the true maximum quantity. Consider, for example, a landholder offered a contract value of $10 who can, given the true cost of providing the good or service, profitably produce at most four units. If the landowner bids only three units and the drawn quantity is four, the landholder has foregone profiting from a contract requiring four units. If the drawn quantity is three units, the landholder is no better off than when submitting a bid of four units. And submitting an artificially high bid of five or six units to increase the chance that the landholder will receive a contract also presents the risk of having to produce the additional units at a loss. The formal proof and further intuition for this premise are presented in Appendix 1.

As Horowitz (2006) and others have pointed out, when the independence axiom of von Neumann–Morgenstern preferences is violated, individuals may not maximize expected utility and the RQM will no longer be incentive-compatible. Participants must also believe that their responses will not affect future contract terms. If not, they might underbid, overstating their minimum WTA in an effort to increase the prices offered in future contracting. The RQM is incentive-compatible within the expected-utility framework adopted here and elicits point estimates of the private cost of production or service provision for each participant.

In order to elicit multiple WTA estimates across intensive margins, thus enabling the estimation of individual cost curves and supply, the RQM can be applied iteratively across a menu of contract values. In that case, the participant submits a quantity bid for each of the contract values listed. One of the contracts from the menu is then randomly selected for implementation and a binding quantity drawn to determine if the participant receives a contract. Since only one contract value is chosen and is selected randomly, incentive-compatibility is theoretically maintained. While Karni and Safra (1987) suggest that implementing only one randomly chosen decision can lead to some distortion of the incentives, Azrieli et al. (2018), maintain that the random-draw mechanism for payment and implementation is the only incentive-compatible mechanism (as long as state monotonicity is assumed) in auctions offering multiple possible contracts.

Laboratory Experiment

The laboratory experiment provides an empirical test of the reliability and incentive-compatibility of the RQM using an induced-cost framework in a controlled laboratory setting. The experiment sessions were conducted at a large northeastern university in the United States. The subjects (N = 21) were recruited from the student population; they were 21 years old on average and 38% were women. Subjects were seated at individual cubicles, each with a computer display and keyboard. They were given written and verbal instructions and then participated in a practice round to familiarize them with the bidding mechanism and potential outcomes. Subjects were permitted to ask the experiment administrator questions but were not allowed to communicate with each other. Each session lasted approximately 45 minutes, resulting in earnings of $34 on average and a range of $24.94 to $45.26. The guaranteed minimum earning level of $5 was not binding.

The experiment used five fictional firms producing a hypothetical homogeneous good in discrete quantities between 1 and 10, with each firm described by a corresponding production cost schedule (see Table 1).

Table 1 Cost Schedule of One of the Firms

The cost schedule for each firm was generated by a power function of the general form

$$C\left( Q \right) = L + sQ^{m}$$

where C(Q) denotes the total production cost for Q units, L is the fixed cost parameter, and s and m are parameters defining the variable costs. Parameters took on different values for each firm spanning the following ranges ∈ [3.2, 5.5], ∈ [0.1, 0.35], and ∈ [1.1, 2.2]. The specific values for each firm are provided in Table 5 of Appendix 2.

Overall, each participant went through five founds of multi-bid RQM elicitation, one round for each of the hypothetical firms. Each round included three RQM bidding tasks involving contract values that were specific to the firm. In each bidding task, the screen presented participants with a contract valueFootnote 2 and prompted them to submit the maximum number of units (1–10) they would be willing to produce given the firm’s cost structure. The cost structure remained the same within each round, and was always presented on the screen during each bidding task. The order of the fictional firms (order of rounds) and of the contract values (RQM tasks within each round) were randomized to avoid potentially confounding ordering effects.

Once the five rounds were completed, the market clearing for the experiment took place. One bidding task in each round was randomly selected to be binding by drawing one of three balls numbered 1, 2, and 3. Next, the quantity threshold for that round was randomly determined by a participant volunteer drawing one ball from a bag of ten balls numbered 1 through 10. This market clearing process was repeated for each of the five rounds (firms). The draws applied to all participants within an experimental session. Participants’ bids for the contract value associated with the selected bidding tasks became binding and determined their earnings in the experiment, in addition to the $5 participation payment. Participants whose bids were equal to or greater than the drawn production requirement were awarded a contract and received its value minus the induced cost of production for the drawn quantity. Thus, if participants submitted sub-optimal bids, their cost of production could be higher than the contract value. In those cases, the difference was subtracted from the subjects’ cumulative profit in the experiment.

No feedback beyond market-clearing outcomes was provided to participants. Following Irwin et al. (1998) research on optimizing behavior and decision-making transparency in the BDM, and given the similarity between the RQM and the BDM, the experiment instructions combined with the practice round were expected to be sufficient for subjects to compute the optimal strategy.

Results of the Incentive-Compatibility Test

Optimality of Quantity Bids

We define optimal bids as bids that maximize the firm’s expected payoff. In the RQM, optimal bids are “truthful” in the sense that they reveal a producer’s actual minimum willingness to accept payment for the stated level of production. Of the 300 quantity decisions made by individuals in the experiment (five rounds each comprised of three bids by twenty participantsFootnote 3), 66.7% were strictly optimal and 18.3% deviated from optimal by no more than one unit. Figure 1 depicts the distribution of the deviations from optimal (optimal minus actual observed bid) across all subjects and production decisions. A zero difference indicates that the submitted bid was optimal.

Fig. 1

Distribution of deviations from the predicted optimal bid

Three subjects consistently submitted sub-optimal bids; bids by the other seventeen subjects were consistently optimal or nearly optimal. The subjects with relatively few optimal bids did not have uniform distributions of deviations from the optimal, suggesting they were not bidding random quantities.

There are at least two potential explanations for the bidding behavior of these three subjects. They might have viewed their goal as “winning contracts” and thus made high bids to maximize their chances. In that case, their bids would fall on the right side of the range. Alternatively, they may have calculated the net payment resulting from relatively low quantity draws for a given contract value and believed that submitting low bids would ensure a higher profit. Such bidding strategies are not optimal and might arise when the initial information and practice round had not been sufficient to ensure cognition or when reward saliency and payoff dominance were not achieved (Smith 1982).

More than 80% of the participants maximized their expected payoffs, a result that is consistent with the weakly dominant strategy of bidding truthfully. This behavior suggests that participants understood the information and the incentives of the RQM. In a field setting, where private costs and rewards are often more salient, weak payoff dominance and reward saliency may be less of a concern than in a laboratory setting. The observed bids are not statistically different from the optimal quantities in this sample (p = 0.39). For each level of the optimal bid quantity (1–9) pooled across firms, the mean observed bid differed from the optimal bid by less than one unit.

These results demonstrate that the RQM is incentive-compatible and cost-revealing. We next examine the performance of the RQM by examining the decision-making efficiency of the bids.

Efficiency of Decision-Making

We examine the aggregate and individual-level efficiency of the RQM by comparing the expected value of the payoff based on the observed bids to the maximum expected payoff value. The expected payoff in the RQM is simply the probability-weighted sum of the payoffs for all quantity values available for the draw that are less than or equal to the quantity bid, Q:

$$ E\left( {payoff} \right) = \mathop \sum \limits_{q = 1}^{Q} prob_{q} \,*\,payoff_{q} $$

The penalty for low bids is missing the chance to make a positive profit when the quantity drawn exceeds the observed bid but does not exceed the optimal bid value. The penalty for high bids, on the other hand, is the risk of a negative profit. Overall, in our parameterization, the penalty for deviating from the optimal bid is quite small.Footnote 4

Consequently, we allow for a small degree of decision-making error in identifying the quantity that results in maximal expected payoffs. Specifically, we consider observed bids for a given contract value to be optimal if they result in an expected payoff that differs from the maximum expected payoff by less than 2.5% of the rangeFootnote 5 of expected payoffs for the firm. Similarly, individual decision-making in the experiment is considered optimal if the sum of the expected payoffs for all of the observed bids by an individual is less than 2.5% below the sum of the maximum expected payoff for the contact values.

Applying these margins of error to the efficiency assessment, 79% of the 300 bids resulted in optimal expected payoffs given the induced costs. The percentage of optimal bids at the firm level ranged from 72 to 100%Footnote 6 when pooled across subjects by the contract value. This high proportion indicates that decision-making in the RQM results in expected payoffs that are close to optimal.

Given the heterogeneity of bids across subjects in the experiment, it is also worthwhile to evaluate the performance of individual subjects using the same efficiency metric. We find that the bids of 17 of the 20 subjects (85%) resulted in expected payoffs that were within 2.5% of the maximum expected payoff for the set of contract values randomly presented to them. The three subjects who did not meet this threshold had expected payoff values of 90%, 70%, and 64% of the maximum total expected payoff for the contract values they were assigned.

In summary, while the sample size in the laboratory experiment is small, the results offer significant evidence supporting the theoretical finding that the RQM is incentive compatible and cost revealing.

Field Experiment: Piloting with Smallholders in Zambia

Field Study Setting, Sample, and Data

The RQM was used in a tree-planting pilot with smallholder farmers in the Mambwe district of Eastern Province, Zambia, as part of an agroforestry initiative. Tree-planting initiatives have been implemented across Zambia for more than a decade,Footnote 7 but smallholder adoption and persistence with these practices have been low in many areas even though inputs and training are sometimes fully subsidized. At the time of the studyFootnote 8 agroforestry tree planting had been promoted in the area by a number of NGOs as part of a suite of conservation agriculture practices. Faidherbia albida, the tree species used in the pilot, is native to the area and commonly found growing on alluvial soils near streams and river beds but sparsely elsewhere as well. In addition to the public benefits arising from carbon sequestration during tree growth, Faidherbia can offer private benefits to smallholders when intercropped with maize as it adds nitrogen and organic matter to soils which improve yields. Almost one-third of the study participants reported the presence of this species in their fieldsFootnote 9 while a study in a neighboring district showed that only about 10% of households there had planted it in their fields (Oliva et al. 2019).

The study area is characterized by unimodal rainfall—a single wet season followed by a dry season in which there is no significant precipitation. Common crops in the area include maize, groundnuts and cotton, grown in the wet season as smallholder agriculture in Zambia is predominantly rain-fed. Successful intercropping of trees requires inputs (seeds and seedling bags) and training in propagation techniques, including scarification of the seeds and establishment of a raised nursery, which requires construction of a makeshift stand. Ongoing management recommendations include watering seedlings regularly during the first year and making firebreaks.

In the field experiment, 220 landholders were invited to participate in information sessions on tree planting in coordination with ongoing activities under a relatively new tree-planting program run by a local NGO. Of those participants, 46% were female and the average age was 39 years (SD 13.1). The average cropped area was 5.4 limas (SD 3.5), which is equivalent to 1.35 ha or 3.34 acres. The information sessions were scheduled in advance and conducted in public meeting areas in villages in groups of between 34 and 53 smallholders each of whom were invited to participate in the study. The sessions lasted approximately 3 h and included an introduction to the program, explanation of the process by which contract terms were generated (the RQM), skits that simulated the process, and training on best practices for planting and managing the trees by experienced local extension staff. The study participants represent a convenience sample conducted through the local NGO, and all of the farmers who participated satisfied the land-suitability requirements (e.g. land must not be cleared for the purpose of tree planting) that protected against forest cover loss and reductions in carbon pools.

Experiment Design

The field experiment included five total contract values offered to participant smallholders—20, 40, 70, 100, and 140 thousand Zambian Kwacha (ZMK). One U.S dollar was approximately equal to 4800 ZMK at the time of implementation. The participants were informed of the range of contract values before being called, one at a time, to begin the RQM contracting process with an enumerator. They were then asked to state the greatest number of trees they would be willing to plant, manage, and keep alive for 1 year for each of the contract values. Their quantity bids were not constrained by the field staff. After receiving their responses, the enumerator confirmed each bid with them. The enumerator then reviewed the bids with the participant one last time as final confirmation.

After confirming their bids, the landholder was asked to randomly draw one ball from each of two opaque containers. The first draw determined which of the five contract values would be implemented. The second determined the number of trees required in the contract, drawing from a pre-determined set of values [12, 25, 37, 50, 75] not known to the participants. This process randomly sets the contract value and required number of trees for each participant. Once the binding contract value was drawn, participants could not change their bids.

Farmers who bid at least the drawn quantity (number of trees required) were offered contracts, and a signed copy of the contract was provided to the participant and to the implementing NGO. Following each individual contracting session, the farmer completed a short survey that provided information about their demographic characteristics and land use. Each participant was then directed to an area separate from the waiting area for those who had not yet participated in the auction. This safeguarded the integrity of the process and prevented information from being passed to those waiting to participate.

Once the contracts were allocated, the partner NGO established and maintained seedling nurseries until the start of the wet season, at which point the seedlings, contained in plastic sleeves, were distributed according to the terms of each contract to the landholders at their homes. At that time, they were again informed of the contract terms and had an opportunity to ask questions. The NGO staff provided periodic extension support to the farmers through the course of the year.

To receive the first contract payment (50%), landholders had to have transplanted the full number of contracted seedlings to their fields, which involved preparing the land, transporting and planting the seedlings in the selected plot of land (often a significant distance from their home). This requirement was met by all of the contracted landholders by the end of the wet season when field visits were made. Approximately 1 year following RQM sessions, an extension staff member and an enumerator assessed the number of surviving trees for each farmer to determine contract performance and the amount of the final payment, which was prorated based on the number of trees that survived. The landholder was then paid after signing to confirm the tree-survival rate and resulting payment amount.


Contracting Under the RQM

Of the 220 landholders who participated in the RQM sessions, 131 received tree planting contracts. Table 2 provides the mean quantity bid for each contract value for the full sample and for those who did and did not receive a contract. To maintain incentive-compatibility, the contract allocation rule of the RQM is that participants whose bids are equal to or greater than the drawn quantity are awarded contracts. Hence, no farmers were required, via a contract or otherwise, to produce a greater quantity than they were willing to produce for the price offered.

Table 2 Means of the quantity bids for the five contract values

While the drawn quantity and contract value were randomly assigned, the farmers’ probability of receiving a contract increased with the quantities they bid. Moreover, to enable direct comparability of quantity assignments across contracts, we used the same five pre-determined binding quantities (selected by recipients drawing one of five balls) for each contract value. These two features of the RQM can be seen clearly in Table 2. First, comparing the mean bids across rows, it is evident that participants with relatively low private costs (higher quantity bids) are more likely to receive contracts. The allocation rule involves a stochastic process (quantity draws) that is a function of the landholders’ private costs and parameters of the experiment design (the distribution of possible quantities) so the resulting allocation is different from one under an incentive-compatible auction. Second, the proportion of landholders who are granted contracts increases with the contract value. This is a direct result of the experiment using a draw from a fixed distribution of pre-determined quantities for all of the contract values. Because the quantity bids are generally nondecreasing with contract value, the probability of drawing a quantity that is less than the bid increases as the contract value increases.

Figure 2 plots the submitted quantity bids. The positive relationship between the bid quantity and the contract value is highly statistically significant. Including a range of control variables minimally changes the scale of the price-specific coefficients in a linear regression but does not affect the level of significance.

Fig. 2

Notes: The respondents’ bids include random noise to improve visibility. One respondent’s bids are omitted from this figure as outliers

Distribution of tree quantity bids.

Aggregate Supply

The RQM directly measures revealed WTA, allowing us to non-parametrically estimate individual and aggregate supply. Figure 3 presents the aggregate supply curve for the participants constructed using the marginal cost of production. To calculate the marginal costs of production for each participant, we determine the quantity bid increment between each contract and the next-smallest-valued contract for which (1) there was a non-zero bid and (2) the bid increment was positive. We then divide the increment in contract value by the increment in the quantity bid. We exclude bids for which the calculated marginal cost would result in a negative profit,Footnote 10 and re-calculate the increment using the next qualified bid (skipping over excluded contract bids). In constructing aggregate supply, we also exclude marginal cost and bid pairs that are dominated by another bid from the same participant—one with a larger bid and lower or equal marginal cost

Fig. 3

Note: The prices are restricted to a maximum of 10,000 ZMK per tree in this figure for ease of viewing

Revealed aggregate supply.

The dashed lines in Fig. 3 are constructed from the bids and the implied marginal costs for each of the contract values offered, while the continuous line pools all RQM bidding data. The marginal cost (WTA per tree) in this sample ranges from 400 to 50,000 ZMK (approximately $0.08 to $10.42), and maximum supply is slightly more than 11,000 trees, equivalent to an average of 51 trees per respondent. This apparent maximum threshold is partly a consequence of the experiment design but may also be related to land constraints. Evidence for the latter is seen in patterns of quantity bids for some respondents that appear to flatten toward the upper range of values. Twenty-five percent of the sample reported total land available of one hectare or less and the recommended planting spacing is 100 trees per hectare.

Figure 3 also provides a useful demonstration of one of the key advantages of a multi-bid (MB) RQM format, in which participants bid for each of a menu of contract values, over a single-bid (SB) format. When implemented using the MB format, the RQM provides WTA estimates for multiple points on an individual’s cost structure and allows for non-parametric modeling of supply that incorporates adjustments along both the extensive and intensive margins. Supply curves constructed from an SB RQM and from uniform-price procurement auctions, as we show in the next section, tend to underestimate aggregate supply because they do not incorporate changes in production along intensive margins (individual supply responses). A buyer using data from a SB RQM to inform a posted-offer price could set the price too high, overshooting the budget or quantity target, or could face an allocation challenge associated with the presence of excess supply in which the quantity available from sellers at a given price is greater than demand as determined by the budget. This scenario is described in more detail in the next section.

Comparing the RQM to Procurement Auctions

The RQM shares some features with procurement auctions in applications such as payments for environmental services (PES) when information asymmetries between the buyer and seller are present. Procurement auctions create temporary markets in which competitive forces provide incentives for landholders to reveal private information, allowing a PES program to supply contracts to those who have the lowest costs of provision.

We compare the MB RQM format to a generalized Vickrey format, where participants can submit multiple price-quantity bids, and to a uniform-price single-bid auction in two configurations—landholders bidding a single price-quantity pair and landholders bidding a price for a fixed-quantity contract. We chose these three mechanisms because theoretically they are incentive-compatible and because they all have closed-form solutions for bidding equilibria, allowing us to compare budgets or allocative efficiency (Hailu and Thoyer 2010). The uniform-price auction format is relatively intuitive, simple to implement, and commonly used in field applications. In these auctions contract prices are set according to the lowest rejected price-quantity bid, and so the weakly dominant strategy is to reveal one’s true cost of production (Latacz-Lohmann and Schilizzi 2005). The generalized Vickrey auction can reveal WTA at multiple points along cost curves, like the RQM.

Since all three mechanisms are incentive-compatible, in principle they should reveal the same cost estimatesFootnote 11 when used to allocate contracts (the allocations, however, would be different). Therefore, it is possible to simulate the bidding and allocations under these auctions using the revealed-cost data from the RQM bidsFootnote 12 and directly examine their relative efficiency in this setting.

A general caveat worth noting is that theoretically incentive-compatible mechanisms applied in a procurement setting are not necessarily cost-revealing in practice depending on the nature of the contract’s design and enforcement provisions (Spulber 1990). Political acceptability and practical considerations influence important elements of the contract’s design and implementation, such as the timing (and/or conditions) of incentive payments and how the incentives are tied to performance (for example, a fixed payment for reaching a threshold or a proportional payment). When incentive payments are made without conditions (irrespective of performance or in advance of implementation without recourse for default), landholders have an incentive to bid a lower price to increase their chances of winning a contract since they do not expect to have to comply (a moral hazard problem). It is not uncommon, however, for procurement contracts to include upfront payment of all or part of the incentive, particularly in programs in developing countries (Jack et al. 2009 in Indonesia; Jindal et al. 2013 in Tanzania; Khalumba et al. 2014 in Kenya).

As in other studies (e.g. Jack 2013 in Malawi), we prorated the contract payment based on the degree of compliance with the contract, measured by the number of contracted trees that survived, which could affect the optimal bidding strategy in the RQM. However, the contract in this study stipulated an initial payment of 50% of the contract’s value that was conditional on the farmer transplanting the full number of trees required by the contract, thus mitigating the potential effect of a partial payment on bidding.Footnote 13 Nevertheless, we cannot rule out the possibility that the prorated payment may have influenced bidders’ behavior.

Linking part of the incentive payment to necessary activities (transplanting seedlings and related activities such as land preparation) and the remainder to (uncertain) future outcomes (tree survival) provides an incentive for farmers to invest effort while balancing risk-sharing between the program manager and the landholders. Intermediate levels of incentive for uncertain outcomes have been found in laboratory experiments to maximize the environmental benefit provided when landholders are risk-averse at the expense of cost-effectiveness through a tradeoff between effort and participation (Schilizzi and Latacz-Lohmann 2016).

We simulate participants’ bidding behavior in a hypothetical uniform-price procurement auction by selecting from each participant the contract value and quantity bid pair that represents the participant’s lowest marginal cost of production.Footnote 14 Figure 4 presents the resulting auction supply curve and the MB RQM-derived supply curve previously presented. In principle, bidding in an incentive-compatible auction such as the RQM is truthful in that participants are incentivized to reveal their true private costs. In the following analysis, we rely on this principle and proceed on the assumption that landholders’ private costs (minimum WTA) are revealed by both mechanisms and, therefore, that the RQM bids can be used directly to simulate bids in the other formats.

Fig. 4

Supply curves for the multi-bid RQM and uniform-price auction. Note: The solid black line includes the bids for all of the contract values while the dashed red line is constructed using the single-quantity bid and contract value from the bids that minimized the marginal cost of production of each participant. (Color figure online)

An auction with uniform pricing in which participants submit a single price-quantity bid is incentive-compatible and theoretically cost-revealing but does not necessarily provide the most cost-effective allocation of contracts. It might be possible to produce the contracted quantity at a lower cost than the allocation achieved in the auction, and this efficiency loss increases with the heterogeneity of the private costs. This SB format does not allow flexibility for adjustments along intensive margins, as can be clearly seen in Fig. 4. The derived supply curve based on simulated bids (dashed red line) is to the left of the MB RQM supply curve (solid black line), which incorporates adjustments along intensive margins.

We find that the constructed supply curves from the uniform-price single-bid auction show a generally smaller supply of trees than the aggregate supply revealed by the MB RQM. The relative position of the supply curve derived from the SB RQM compared to the supply generated from a uniform price auction varies based on the contract value offered in the former. When prices are sufficiently low, the supply estimated using the single-bid auction data will tend to be equal to or greater than the supply estimated from the SB RQM. This occurs because the single contract value adds a constraint that either (1) excludes landholders who would require larger contract values to reach their minimum costs of production (e.g. due to high fixed costs and relatively low marginal costs) or (2) limits the production of landholders who could produce more at a given price under a higher contract value. At higher prices, the reverse can be true. The supply derived from contract values of 100,000 ZMK and 140,000 ZMK in the MB RQM, for example, is greater than the supply derived from the single-bid auction when prices are sufficiently high.

This analysis highlights the tradeoffs associated with these mechanisms and the potential value of the MB RQM format in estimating aggregate supply. While the single-bid uniform price auction provides an incentive for truthfully reporting private opportunity costs, it does not necessarily result in least-cost allocations when private costs are heterogeneous because it does not allow for production responses across intensive margins. Consequently, it can introduce the potential for excess supply when used to inform price-setting in a fixed-price posted-offer market. The SB RQM also does not capture production responses along intensive margins and underestimates aggregate supply relative to the MB RQM. In contrast, the MB RQM provides multiple point estimates along individual cost curves and thus accommodates potential production responses along intensive margins, providing a more-accurate reflection of the true aggregate supply from a given sample.

The generalized Vickrey auction format, in which suppliers provide price-quantity schedules that provide WTA estimates across intensive margins, is incentive-compatible and efficiently allocates supply contracts. In principle, it obtains the same individual and aggregate supply curves as the MB RQM format since both formats involve incentives to truthfully reveal WTA for a range of quantities. The MB RQM aggregate supply in Fig. 4 (solid black line) therefore corresponds directly with the aggregate supply resulting from a generalized Vickrey auction implemented with these landholders. While the aggregate supply curves are equivalent the expected contract outcomes are different; the relative efficiency, allocations, expenditures, incentive levels and contracted quantities depend on design choices (e.g. the set of available contract values and quantities in the RQM) and the targets established by the program. The contract-allocating agency may want to set a market-clearing price to meet a specific quantity or budget target. In principle, such targets can be achieved using a MB RQM by enlisting additional participants until the awarded contracts collectively reach the desired target.

To highlight differences in efficiency and allocations across mechanisms and formats, with a view to evaluating their respective attributes for piloting and research use, we conduct two sets of comparisons. We first describe the supply outcomes in our setting resulting from a posted price (take it or leave it offer) that has been selected using a single bid RQM supply curve, and compare this to the supply outcomes predicted by the MD RQM and uniform price auction. The price is set to clear the market and meet an illustrative $3000 budget constraint. We then directly compare the contracting results of the MB RQM to the simulated results of the auction formats discussed above.

Consider setting the price using a supply curve derived from SB RQM data from the 140,000 ZMK contract in a hypothetical market with an eligible population identical to the participants in the pilot. The supply curve constructed using bids for the 140,000 ZMK contract indicates that, to remain within the $3000 illustrative budget and clear the market,Footnote 15 the price should be set at 2667 ZMK ($0.56), which would result in 5281 trees supplied by 53 landholders. By comparison, the (simulated) single-bid auction supply suggests this price would result in slightly fewer trees (5047) from a far larger number of eligible landholders (197), and also meets the budget constraint. However, supply derived from MB RQM data shows that 101 landholders would accept a 2667 price and supply 9368 trees at a total cost of $5204, representing excess supply and an allocation problem. To satisfy the $3000 budget constraint and clear the market, the MB RQM data indicates that the price should be set at 1875 ZMK ($0.39) resulting in 6238 trees supplied at a total cost of $2436 (18% more trees than predicted by the SO RQM curve, and 19% under budget).Footnote 16 The degree of underestimation resulting from the SB RQM and from the single-bid auctions will increase with the heterogeneity of the landholders’ opportunity costs and elasticity of their supplies. This analysis underscores the potential for efficiency losses resulting from single-bid procurement mechanisms and the limitations of using these tools to set posted prices.

Table 3 summarizes the key metrics of the RQM field study and of three auction types discussed above and simulated using the RQM bid data—a generalized Vickrey auction and two versions of a single-bid uniform price auction, one in which landholders choose both the contract price and the quantity to supply (column 3) and another that involves price bids for contracts requiring a fixed number (50) of trees (column 4). For comparison purposes, we apply a minimum quantity target based on the quantity contracted for in the field experiment of 3677 trees to determine the market-clearing prices in the auctions.

Table 3 Bidding and allocation metrics for the RQM and alternative auction mechanisms with a target constraint of producing a minimum of 3677 trees

For any desired quantity target, the generalized Vickrey mechanism will, in principle, allocate contracts to individuals whose quantity bids will result in the lowest production costs. In the RQM, on the other hand, each participant has a positive probability of receiving a contract. Both of the single-bid auctions result in intermediate outcomes.

The mean private cost of production for farmers who receive contracts under each mechanism illustrates this point. Among farmers who receive contracts, the mean cost per tree is 705 ZMK in the generalized Vickrey mechanism and 1783 ZMK in the MB RQM, which also allocates contracts to landholders who vary more in their private opportunity costs (standard deviation of 786 compared to 98 in Vickrey). The mean contract payment is higher under the RQM (94,198) than under the Vickrey auction (72,097). The RQM leads to a much smaller number of trees per person—approximately 28 compared to 90 in the Vickrey auction. Similarly, the price paid per tree as a proportion of the private cost of production is far higher under the RQM at 2.65 compared to the Vickrey auction value of 1.17 (i.e. 17% above private costs). Consequently, the degree of overcompensation (payment exceeding their minimum WTA) also is greater in the RQM as a fundamental consequence of the mechanism design.

Similarly, uniform-price auctions necessarily result in overcompensation when landholders’ opportunity costs are heterogeneous; and fall in between the results of the Vickrey and MB RQM. The total value of awarded contracts and mean price per tree under the RQM are 12,340,000 ZMK ($2571) and 4338 ZMK ($0.90) respectively, or approximately four and five times greater than the same values under the generalized Vickrey auction for a similar supply. These results indicate that obtaining the additional information provided by the RQM comes at cost. However, the RQM also allocates contracts to more than three times as many farmers as the simulated Vickrey auction, allowing for predictions that would otherwise be out of sample, and provides exogenous variation in contract terms resulting in varying degrees of overcompensation relative to a participant’s minimum WTA.

The MB RQM provides accurate non-parametric estimates of private opportunity costs that can be used to construct supply and inform pricing in programs that establish posted-offer markets, unlike the single-bid procurement auctions that tend to underestimate aggregate supply when private costs are heterogeneous. The RQM also allocates contracts to each farmer with a positive probability, unlike the generalized Vickrey mechanism, allowing for out-of-sample predictions. Finally, the RQM varies the level of overcompensation across private opportunity cost types, which allows researchers to independently identify the relationship between incentives, private costs and outcomes, information that can be used to guide efforts to scale-up in future programs. The relationships between private costs, incentives, and compliance in a particular setting might motivate the choice of allocation mechanism and other aspects of program design. For example, a program manager might be inclined to use discriminatory pricing because it may reduce overcompensation and improve budgetary efficiency in contracting.Footnote 17


This paper presents a new cost-revealing procurement mechanism, the Random Quantity Mechanism, that is well suited for use as a research tool in conservation and PES settings. The RQM is shown to be incentive-compatible in an expected utility framework and the results of a laboratory experiment with induced-costs support this finding. We report on the first field implementation of the RQM and demonstrate the mechanism’s feasibility in field applications designed to examine producers’ opportunity costs for supplying an environmental good. The multi-bid RQM allows for non-parametric estimation of private costs that can be used by policymakers to examine the effect of a subsidy and other similar policy instruments on provision of non-market goods and services. The RQM also can be used by conservation agents to predict the resulting supply of the good under a given budget and evaluate the program’s cost-effectiveness relative to provision of other environmental goods that could achieve similar goals (e.g. pollution abatement through different mechanisms).

The RQM offers some advantages over procurement auctions in the presence of information asymmetries and/or uncertainty regarding the bidders’ actual opportunity costs, and in settings in which the landholders’ investment of effort affects the outcomes. It allocates contracts to each participant with positive probability, thereby allowing for predictions that would otherwise be out of sample. It also provides experimental variation in contract allocation, precise estimates of individual participants’ minimum WTA (which they base on their true costs of production), random variation in drawn binding quantity and contract values (conditional on satisfying the minimum WTA), and enables construction of aggregate supply. While these features come at a cost—greater overcompensation relative to procurement auctions, they help to isolate the impacts of incentives and producers’ private costs on contract performance. When used as a research tool or in a pilot program to estimate the relationships between effort, private costs, incentives, and environmental outcomes, the RQM can provide valuable input for program designers supporting efforts to improve the efficiency and effectiveness of conservation investments.


  1. 1.

    The mechanics of this are described in more detail in the next section.

  2. 2.

    Four of the five firms each had nine possible contract values, specific to each firm, such that the nine values corresponded to the nine possible (optimal) bids 1–9. In the design phase, these nine contract values were blocked into three sets of three values, with each block corresponding to a single bidding task. The value presented in a bidding task was randomly selected from the corresponding block, so that not all participants responded to the same values for a given firm. This design ensured submitted bids (if optimal) would cover the full range of possible quantities on the production schedule. The fifth firm had one value in each of the three blocks, and thus these three values were presented to all participants. The fifth firm served as a simple comparison point of a more limited range and variation in WTA observations. The contract values and blocking system (indicated by a contract number) are provided in Appendix 2, Table 4.

  3. 3.

    Twenty-one subjects participated in the experiment, but one subject was dropped from the analysis, as 13 of their bids were 10 units and the remaining two bids were 7 and 9 out of possible 10 units.

  4. 4.

    For example, if the firm described in Table 1 bids eight units (instead of the profit-maximizing seven) in response to a contract payment of $11.64, the firm’s cost of production is $11.76. Taking the probability of eight units being drawn, bidding eight units decreases the cumulative expected payoff by less than $0.01 relative to the maximum possible expected payoff.

  5. 5.

    This range is the appropriate reference value since it accurately measures the difference between the best and worst potential outcome for a given firm in which the worst potential outcome can be negative.

  6. 6.

    Since no feedback was given to participants after bidding commenced, learning was unlikely to occur and is not examined in this paper.

  7. 7.

    See, for example, ICRAF:

  8. 8.

    October/November of 2010.

  9. 9.

    The survey question made no distinction between naturally occurring and farmer planted trees, but the field officers reported limited intercrop planting of Faidherbia. Moreover, visual inspection of satellite imagery of the fields that participants selected for planting showed few pre-existing trees present.

  10. 10.

    Calculated as price (marginal cost) times the bid less the total cost of production (equivalent to the contract value).

  11. 11.

    Procedural invariance does not always hold in field applications; for examples, see Jack (2013) and Berry et al. (2015).

  12. 12.

    This depends on the range of contract-value/quantity-bid pairs encompassing the production level for each individual that results in the minimum cost, a reasonable assumption given the setting.

  13. 13.

    Under certain circumstances, partial payments may result in higher bids than would be made when the payment “all or nothing” since the bids establish the lower bound of the unit price for a given contract value and, given pro-rated payments, participants may then determine their bids based on their marginal costs such that the unit price is greater than or equal to the marginal cost at some level of production below the bid and is not based on the participant’s total cost.

  14. 14.

    Discrete contract values result in discrete point estimates of WTA. While the true production level that results in the minimum marginal cost may be between two quantity bids under a contract value, the potential error is bounded by quantity bids and so is not expected to be large. Furthermore, the errors generated in the auction and RQM formats would be consistent and thus would not weaken the comparison of the supply curves. In this analysis, the marginal costs are calculated using the same procedure as was used for the RQM aggregate supply.

  15. 15.

    Based on the 140,000-ZMK supply curve, expenditures would be $2933.89.

  16. 16.

    While a 1875 ZMK price significantly underspends the budget, setting the price at the next higher marginal costs (2000 ZMK) results in excess supply.

  17. 17.

    Setting aside the problem of bid-shading.

  18. 18.

    When total payment equals total private cost for a specific quantity, q, the optimal solution is not unique since q and q − 1 result in the same expected payoff.


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We thank Kelsey Jack, Bill Schulze, Jennifer Alix-Garcia and CBEAR conference participants for helpful comments and suggestions. We also thank the editors and two anonymous reviewers for their helpful comments. We are grateful for financial support from Kenneth L. Robinson endowment. Specific thanks to Mwape Chibale for excellent field work assistance.

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Appendix 1

RQM Theory

Under the maintained assumption that opportunity costs are weakly increasing in quantity, the intuition behind the incentive-compatibility of this mechanism is straightforward. The dominant strategy for the individual, faced with uncertainty in the quantity required under the contract, is to state the maximum quantity for which the individual’s opportunity costs are no greater than the contract value.Footnote 18 Or equivalently, when costs are nondecreasing, to state the maximum quantity that would result in a non-negative profit should that quantity be drawn. This strategy maximizes the chances of receiving a contract that would meet or exceed the individual’s total opportunity cost and excludes the possibility of receiving a contract in which the contract value is smaller than the cost.

More formally, let:

T :

the contract value (total payment) offered;

Q :

the bid (quantity offered) for the contract value;

R :

the random quantity draw, required by the contract;


the quantity distribution from which R is drawn;

Y o :

initial income;


the private opportunity cost of producing Q units;


utility, a function of money income including the net value of the contract;

If R ≤ Q, the individual receives a contract at (R, T) and the expected utility is

$$U\left( {Y_{o} + T{-}C\left( R \right)} \right) \, \ge U\left( {Y_{o} + T{-}C\left( Q \right)} \right)$$

under the assumption that Cʹ(Q) ≥ 0 and Uʹ(Y) > 0. In an expected utility framework, the optimal quantity bid solves

$$\mathop {\hbox{max} }\limits_{Q} \mathop \smallint \limits_{0}^{Q} [U\left( {Y_{0} + T - C\left( R \right)} \right)p\left( R \right) + U(Y_{0} )\left( {1 - p\left( R \right)} \right)]dR.$$

The first term describes expected payoff for random quantities R less than or equal to the bid Q, and the second term describes the expected payoff for a randomly drawn quantity greater than the bid (Q). In our laboratory experiment setting, a predetermined value will limit the maximum bid that participants may submit.

First-order conditions define the optimal bid Q*, which satisfies

$$U\left( {Y_{0} + T - C\left( {Q^{*} } \right)} \right) - U\left( Y \right) = 0$$
$$T = C\left( {Q^{ *} } \right).$$

The optimal bid occurs when the total cost equals the contract value.

In a discrete production setting in which units of the good are not infinitely divisible, individuals maximize expected utility by bidding Q* such that C(Q*) ≤ T and C(Q* + 1) > T. In the continuous output scenario, the RQM provides accurate point estimates of private costs (the minimum WTA payment for producing the quantity bid), while in the discrete case it provides a bounded estimate of private costs. It is worth noting, then, that the RQM in a discrete setting only definitively indicates that the cost of production of the quantity bid is less than the transfer and that the cost of production of Q + 1 units is more than the contract value.

Appendix 2

See Tables 4 and 5.

Table 4 Summary of available contract values, per bidding task, per firm
Table 5 Parameter values for firms’ induced cost power functions (\(c = L + sx^{m} )\)

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Bell, S.D., Streletskaya, N.A. The Random Quantity Mechanism: Laboratory and Field Tests of a Novel Cost-Revealing Procurement Mechanism. Environ Resource Econ 73, 899–921 (2019).

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  • Random quantity mechanism
  • Conservation auction
  • Procurement auction
  • Information asymmetry
  • Additionality
  • Payments for environmental services
  • BDM
  • Reverse auction
  • RQM
  • Zambia
  • Africa