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

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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.

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Fig. 1
Fig. 2

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

Fig. 3

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

Fig. 4


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

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

  7. See, for example, ICRAF:

  8. October/November of 2010.

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

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

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

  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. Setting aside the problem of bid-shading.

  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

1.1 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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