Contracting with Private Knowledge of Production Capacity

• Date: 01 October 2015
• DOI: http://doi.org/10.1111/jems.12112
• Author: Leon Yang Chu,David E. M. Sappington
• Published date: 01 October 2015

Contracting with Private Knowledge of Production

Capacity

LEON YANG CHU

Marshall School of Business

University of Southern California

3670 TrousdaleParkway, Los Angeles, CA 90089

leonyzhu@usc.edu

DAVID E. M. SAPPINGTON

Department of Economics

Matherly Hall, University of Florida

Gainesville, FL32611-7140

sapping@ufl.edu

We analyze the design of procurement contracts when the supplier is privately informed about

both his innate production capacity (K ) and his innate unit cost of production. We identify

conditions under which the supplier will strategically employ an inefficient production technology

to expand output above K . We also show that when the buyer employs the simple fixed-price

cost-reimbursement (FPCR) contracts in the setting examined by Rogerson (2003), the supplier

has no incentive to exaggerate K . Furthermore, the buyer can secure with FPCR contracts at

least 75% of the surplus she secures with fully optimal contracts.

1. Introduction

In Laffont and Tirole’s (1986) seminal model of procurement and in many extensions of

this model, the supplier is privately informed about his innate unit cost of production

(i.e., his operating cost if he does not deliver any cost-reducing effort). In contrast, these

models typically presume the supplier’s production capacity to be common knowledge.

In particular, it is generally assumed to be common knowledge that the supplier faces

no capacity constraints.

Although this assumption helps to simplify the analysis of theoretical models, ca-

pacity constraints are common in practice. Furthermore, suppliers typically are better in-

formed than buyers about prevailing capacity constraints for several reasons, including

the following two. First, producers often have superior information about the available

supply of critical factors of production. For example, electricity suppliers typically are

privately informed about whether their generating units are fully functional or must

be taken offline for maintenance or repair (Joskow and Kahn, 2002). Second, suppliers

often have privileged knowledge of the contracts they have signed with other buyers,

and consequently have a better understanding of the maximum output they can deliver

to any particular buyer.

We seek to determine whether a buyer’s procurement problem is complicated

by a supplier’s private knowledge of his production capacity. We address this issue

both under unrestricted procurement, where the set of procurementcontracts the buyer

We thank the coeditor and refereesfor very helpful comments and suggestions.

C2015 Wiley Periodicals, Inc.

Journal of Economics & Management Strategy, Volume24, Number 4, Winter 2015, 752–785

Private Knowledge of Production Capacity 753

can employ is not restricted, as in Laffont and Tirole (1986), and when the buyer em-

ploys fixed-price cost-reimbursement (FPCR) contracts, as analyzed by Rogerson (2003).

FPCR contracts are attractive in part because of their simplicity. FPCR contracts offer the

supplier a choice between: (i) a fixed-price (PR) contract, which sets a specific payment—

independent of the supplier’s realized production costs—at which the buyer will pur-

chase a specified amount of output from the supplier; and (ii) a cost-reimbursement (CR)

contract, under which the buyer simply reimburses the supplier fully for his observed

costs of producing the specified output. As Rogerson (2003) observes, FPCR contracts

are effectively employed, for example, when a defense contractor is afforded the op-

portunity to provide a specified deliverable in return for a fixed payment rather than

operating under the CR policy that the Department of Defense typically employs.1

Rogerson (2003) analyzes a setting in which a buyer procures a single unit of

output from a supplier who is privately informed about his innate cost of produc-

tion. Somewhat remarkably, Rogerson shows that in a fairly broad and arguably plau-

sible set of environments, the buyer can secure with optimally designed FPCR con-

tracts at least 75% of the surplus that she can secure with fully optimal contracts. We

seek to determine whether, in the corresponding setting in which the buyer is not

restricted to procure a single unit of output and in which the supplier is privately

informed about both his innate unit cost of production and his innate capacity, the

buyer can continue to achieve this impressive performance when she employs FPCR

contracts.2

To analyze these issues, we extend Laffont and Tirole (1986)’s classic procurement

model to consider a setting in which the supplier has access to two technologies: a “stan-

dard technology” and a higher cost “alternative technology.” The standard technology,

which has a limited capacity, might be viewed as encompassing the supplier’s nor-

mal operating procedures. The alternative technology might entail the subcontracting

of production to a less efficient supplier, for example, or the unavoidable substitution

of more expensive inputs for the less expensive inputs that the supplier customarily

employs when they are readily available. The alternative technology is presumed to be

inefficient in the sense that the unit cost of production using this technology (βa) exceeds

the buyer’s unit valuation of output (v).

The supplier produces output at constant unit cost under both technologies. The

supplier is privately informed about: (i) the capacity (K) of the standard technology;

(ii) the innate unit cost of production (β) for output below capacity using the standard

technology; (iii) the effort he devotes to reducing his unit cost of production under

the standard technology; and (iv) the amount of output he produces using each tech-

nology. We examine whether, in addition to his normal incentive to exaggerate β,the

supplier may find it profitable to misrepresent K. The supplier can understate Kwith

1. Rogerson (2003) also notes that a regulator can be viewed as employing FPCR contracts when she

offers the regulated firm a choice between rate of return regulation and price cap regulation. Sappington

(2002) describes a related policy that the U.S. Federal Communications Commission has implemented. The

policy affords suppliers of telecommunications services a choice between prices that are not explicitly linked

to realized production costs and prices that can vary as realized earnings increase above or decline below

specified thresholds.

2. Rogerson (2003) derives his key finding in a setting where the supplier’s innate cost has a uniform

distribution, the supplier’s personal cost of delivering cost-reducing effort eis a quadratic function of e,and

the buyer effectively values only one unit of the product that she procures.We focus on corresponding settings

in which the buyer derives value v>0 from each unit of output that she procures. Alternative settings are

discussed in the concluding section.

754 Journal of Economics & Management Strategy

impunity.He can also exaggerate Kby employing the alternative production technology

to generate output in excess of K.3

We find that the supplier generally has no incentive to understate K.4However,

under unrestricted procurement, the supplier may sometimes find it profitable to exag-

gerate Kby employing the high-cost technology to deliver output in excess of K.Thisis

the case because the larger is the supplier’s output, the more rapidly his total production

cost declines as his increased effort reduces his unit cost of production. Consequently,

the more output the buyer procures, the more cost-reducing effort she typically induces

the supplier to deliver.The supplier secures increased rent from the associated expanded

opportunity to substitute a low innate cost (β) for cost-reducing effort.

In contrast, the supplier does not find it optimal to employ the alternative tech-

nology to increase output above his innate capacity when the buyer employs FPCR

contracts. This conclusion reflects, in part, the fact that under the optimal set of FPCR

contracts, the fixed payment increases with delivered output at a rate that is less than v,

the buyer’s marginal valuation of output. Consequently,because the supplier ’s marginal

cost of production using the alternative production technology (βa) exceeds v,thein-

cremental revenue the supplier can secure by employing the alternative technology to

expand output is less than the corresponding incremental cost.

In addition to mitigating the supplier’s incentive to exaggerate his production

capacity, optimally designed FPCR contracts continue to secure for the buyer at least

75% of the surplus that she can secure with fully optimal contracts. Therefore, Rogerson’s

important observation about the performance of FPCR contracts extends to settings in

which the supplier is privately informed about both βand Kand in which the buyer

procures multiple units of the supplier’s product.5

Our derivation and presentation of these findings proceeds as follows. Section 2

describes our model. Section 3 provides a formal statement of the buyer’s problem under

unrestricted procurement. Section 4 identifies the outcomes that arise in two benchmark

settings: one in which the buyer is fully informed about both the supplier’s innate

production capacity (K) and his innate unit cost (β) and one in which the buyer is fully

informed only about K. Section 5 specifies conditions under which the buyer’s limited

knowledge of Kis constraining under unrestricted procurement. Section 6 demonstrates

that the supplier is not constrained by her limited knowledge of Kwhen she employs

FPCR contracts. Section 6 also proves that when she employs FPCR contracts in this

setting, the buyer can always secure at least 75% of the surplus that she can secure

under unrestricted procurement. Section 7 concludes and suggests directions for future

research. The Appendix presents the proofs of all formal conclusions that do not appear

in the text.

3. Thus, as in Lacker and Weinberg(1989) and Kartik (2009), the supplier can misrepresent the prevailing

environment by undertaking personally costly “falsification” activities.

4. Incentives to understate capacity can arise in other settings where, for example, the suppliers of a

commodity (e.g., electricity) are paid a unit price that equates the realizeddemand and supply of a commodity.

See Borenstein et al. (2002), Joskow and Kahn (2002), and Tellidouand Bakirtzis (2007).

5. In contrast to other analyses of multidimensional adverse selection problems (e.g., McAfee and McMil-

lan, 1988; Rochet and Chon´

e, 1998; Armstrong and Rochet, 1999), we do not characterize the optimal contract

for the buyer under unrestricted procurement.We explain in Section 6 why this characterization is not required

to demonstrate our primary conclusion, namely, that the buyer can secure with FPCR contracts at least 75%

of the surplus that she can secure under unrestricted contracting.