Quantity Choice in Unit Price Contract Procurements

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Svante Mandell

We study the optimal quantity to procure when unit price contracts (UPC) are used and there is uncertainty surrounding the actualrequired quantity will be. The model shows that the optimal quantity, i.e., the one that minimizes the procurer’s expected total cost,is determined by a fundamental trade-off between (1) the risk of having to pay for more units than actually necessary and (2) therisk of having to conduct costly renegotiations. In optimum, the procured quantity will increase in costs associated with a possiblerenegotiation. It will decrease in the expected per unit price. Typically, if the renegotiation does not result in too large mark-up inper unit price, the procured quantity decreases in the uncertainty surrounding the actual quantity required.When the procured quantity is low compared to the expected amount required, the risk that the final amount exceeds the procuredone is obviously large. This implies that the actual total cost, with high probability, will exceed the total sum agreed upon in thecontract. If we allow ourselves to define this as being a measure of cost-overrun, we may conclude that not only is it rational andoptimal to allow for cost-overruns. It is actually more likely, in optimum, to see cost-overruns in projects that are expected to runsmoothly in the sense that the costs of renegotiations are expected to be low.Previous work dealing with UPC more or less implicitly assumes that the principal will procure the estimated amount of eachactivity. We show that this notion is not correct. Rather, there are cases in which the principal should – in order to minimize herexpected total costs – procure a quantity exceeding the estimated or expected one and other cases in which the procured quantityshould be lower. More importantly, the model will provide us with an intuitive understanding for the mechanisms at work.This paper is akin to a literature on strategic bidding in UPC procurements due to agents having superior information, e.g., Ateyand Levin (2001), Bajari et.al. (2007), and Ewerhart and Fieseler (2003). This literature typically models the bidding agents’behavior in UPC auctions while the present paper models the procurer’s behavior. Thus, the present paper is a step towards aunified model which allows for strategic behavior of both procurer and bidders.Furthermore, our model adds to the literature on cost overruns. That cost overruns are frequently occurring in infrastructureprojects seems to be an established fact in the literature; Flyvbjerg et.al. (2003) and Odeck (2004), and Priemus et al (2008). Thelatter provides a couple of plausible explanations for systematic miscalculation leading to cost underestimation. Our modelsuggests one additional reason. Namely that the procurer, in some situations, contracts on a low quantity knowing that the requiredquantity with high probability will be larger than the contracted one and, thus, that total costs ex post most probably will exceed thecontracted sum. Papers that, as in the present study, formally models procurements in which cost-overruns are likely to occur inoptimum include Ganuza (2007) and Gaspar and Leite (1989/90). However, these studies focus on different aspects of procurementthan our model (impact on competiveness and a selection bias effect, respectively).

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