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

Abstract

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.

Appendices

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.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;

f():

the quantity distribution from which R is drawn;

Y _o :

initial income;

C(Q):

the private opportunity cost of producing Q units;

U(Y):

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 ($c = L + sx^{m} )$