Dynamic auction environment with subcontracting

• DOI: http://doi.org/10.1111/1756-2171.12154
• Date: 01 November 2016
• Published date: 01 November 2016
• Author: Elena Krasnokutskaya,Przemyslaw Jeziorski

RAND Journal of Economics

Vol.47, No. 4, Winter 2016

pp. 751–791

Dynamic auction environment

with subcontracting

Przemyslaw Jeziorski∗

and

Elena Krasnokutskaya∗∗

This article provides evidence on the role of subcontracting in the auction-based procurement

setting with private cost variability and capacity constraints. We demonstrate that subcontracting

allows bidders to modify their costs realizationsin a given auction as well as to control their future

costs by reducing backlogaccumulation. Restricting access to subcontracting raisesprocurement

costs for an individual projectby 12% and reduces the number of projects completed in equilibrium

by 20%. The article explains methodological and market design implications of subcontracting

availability.

1. Introduction

The classic literature on the boundaries of the firm suggests capacity constraints as one of the

reasons for outsourcing production rather than completing it in-house. In these studies, the need

for outsourcing is generated by stochastic demand and stochastic productivity shocks; however,

the analysis is frequently confined to the perfect information setting. In this article, we elaborate

on the insights from this literature in the context of governmentinfrastr ucture maintenance, a large

market in which both capacity constraints and asymmetric information about cost are significant.1

Due to the asymmetric information about costs variability, this market is organized around an

auction-based allocation mechanism, and because the capacity constraints are frequently binding,

firms are allowed to subcontract (outsource) part of their work. We analyze the impact of such

subcontracting availability on the performance of the market and inquire into its methodological

and policy-related consequences. Our analysis is based on the set of calibrated parameters, such

that the outcomes predicted by our model match the data for the California highway procurement

market. This allows us to assess realistic magnitudes of the investigated effects.

Recent developments in auctions literature are characterized by an enhanced appreciation

of the impact of capacity constraints on the performance of procurement markets. For example,

∗University of California at Berkeley; przemekj@haas.berkeley.edu.

∗∗Johns Hopkins University; ekrasno1@jhu.edu.

1About 50% of funds spent on government procurement are allocated to construction and maintenance, envi-

ronments that are traditionally associated with capacity constraints. (See, e.g., the Federal Procurement Report 2007,

available at www.fpds.gov/.)

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Jofre-Bonet and Pesendorfer (2003) and Balat (2012) estimate that an increase in capacity utiliza-

tion from one standard deviation below the average to one standard deviation above the average

results in a 24% cost increase. These studies, however, do not account for the ability of firms to

outsource part of their work in the subcontracting market. Weargue that this omission has an im-

portant impact on the quantitative findings in the literature and on their interpretation. We modify

a framework for the dynamic auction setting developed by Jofre-Bonet and Pesendorfer (2003)

in which the work in a given period is allocated through first-price auction and unfinished work

is carried over to the next period as backlog. Such backlog subsequently increases future costs in

the manner of first-order stochastic dominance. We focus on the dynamic incentives provided by

the ability to subcontract in the primary market, thus our modelling of the subcontracting market

itself is deliberately simple. Weassume that the contractors can outsource par t of the work using

the secondary market, which is composed of a large number of small firms that undertake the

amount of work comeasurable with their capacity. Each period, the contractors decide whether

to participate in the auction, and upon participation decide how much to bid and how much

to subcontract. We assume that participants have to commit to the subcontracting policy at the

time of submitting their bids. This is consistent with the rules adopted in many procurement

markets.

Computing equilibria in dynamic auction games with subcontracting is a nontrivial numerical

exercise,thus developing an algorithm that solves such class of games is one of the contributions of

this article. Our numerical approach extends that of Saini (2013), who solves the dynamic game

with capacity constraints, but without subcontracting. He shows that the equilibrium bidding

strategies can be obtained by solving a standard auction with asymmetric bidders and with

reparametrized cost distributions. This observation enables him to choose a specification where

a closed-form solution of the auction game is available. Analysis of the dynamic game with

subcontracting is more challenging because it involves deriving two interrelated policy functions

for subcontracting and bidding. In addition, in the game with subcontracting, the continuation

payoff after losing an auction depends on the current losing bid. As a result, bidding functions

are determined by a generalized version of an “all-pay” auction with asymmetric bidders, which

does not have closed-form solution for any of the known cost distributions. Instead, the bidding

strategies have to be obtained as a numerical solution to the system of differential equations

with boundary conditions. In contrast to Saini (2013), we compute equilibrium in our game

as a limit of Markov Perfect Equilibria of finite horizon games. This alleviates concerns about

the multiplicity of equilibria by providing a consistent and robust equilibrium selection rule

that enables us to compare equilibrium outcomes for different models and parameter values.

We embed the numerical algorithm into a routine which calibrates parameters of our model,

so that model outcomes match those in the data from the California procurement market. We

subsequently use these parameters to study procurement outcomes and policy effects associated

with subcontracting availability.

The central feature of our setting is that bidders’ (effective) costs, which underlie the prices,

are endogenously determined. Specifically, subcontracting reduces current costs by allowing to

modify unfavorable within-period draws and lowers future costs by mitigating the accumula-

tion of backlog. Beyond reducing costs, the availability of subcontracting has consequences for

equilibrium pricing. In particular, it induces ex ante symmetrization of cost distributions and

ex post symmetrization of specific cost draws, which intensifies competition and lowers bidders’

markups. The within-project cost modification also reduces the importance of private information

and lowers informational rents. Moreover, an impact of backlog on future costs is reduced, there-

fore the dynamic considerations become less important, which further lowers equilibrium prices.

These effects decrease the cost of procurement for an individual project. Additionally, lower

prices allow for the allocation and completion of a greater number of projects because lower

winning bids meet the reserve price more frequently. The higher rate of allocation also enables

and is in part facilitated by the higher rate of participation. We estimate that in the California

market, the availability of subcontracting leads to a 12% decrease in the average procurement

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costs for an individual project and in a 20% increase in the number of projects completed relative

to the case without subcontracting.

Subcontracting availability has methodological ramifications. Specifically, in the markets

with substantial subcontracting activity, omitting such subcontracting when estimating the dis-

tribution of private costs and the parameters associated with the impact of capacity constraints

results in a downward bias. The bias from using a misspecified model without subcontracting

is caused by an incorrect attribution of low equilibrium prices to low baseline costs and low

importance of capacity constraints. For the California data, these effects are substantial: the mean

of the cost distribution is biased downward by 8%, 23%, and 33%, and standard deviation by

29%, 50%, and 67%, under various representative levels of backlog. Similarly, the parameters

capturing the effect of capacity utilization on the mean and the standard deviation of private costs

are biased by on average 100%. Interestingly, the cost distributions recovered using the model

without subcontracting correspond neither to the distribution of modified (effective) costs nor

to the static component of modified costs. This is because the biases are mostly driven by the

incorrect option value component imposed by the model without subcontracting in estimation.

The presence of subcontracting has important implications for the market design. We find

that in an environment without subcontracting, the equilibrium outcomes differ along several

dimensions depending on using first- or the second-price auction. Specifically, the second-price

auction delivers 6% greater allocative efficiency and results in 10% higher number of projects

allocated. However, it is also characterized by 14.6% higher procurement costs per individual

project, which is caused by higher cost resulting from greater backlog accumulation and byhigher

markups charged in equilibrium. Formally, the difference in procurement costs across auction

formats arises because of the cost asymmetry inherent in the setting with capacity constraints,

and because of the interdependence in bidders’ effective costs generated by the continuation

value. The latter effect is similar to that documented in the auction models with resale by Haile

(2001) and Bikhchandani and Huang (1989). The availability of subcontracting allows bidders

to endogenously modify both the cost asymmetries and the interdependence. We find that in the

setting with subcontracting, the difference in the procurement cost across two formats are reduced

to 1%. At the same time, the differences in allocative efficiency and in the number of allocated

projects between formats remain important and amount to 4.8% and 6.3%, respectively. Thus, the

choice of the auction format in the setting without subcontracting involves important trade-offs

whereas in the setting with subcontracting, this choice is less ambiguous.

Tosummarize, the article makes four contributions. First, we analyze the mechanism through

which subcontracting works in the markets similar to the California procurement market and

measure the impact of subcontracting on procurement outcomes. Second, we study the implica-

tions of subcontracting availability for the choice between first-price and second-price allocative

mechanisms. Third, we demonstrate methodological consequences of subcontracting availability

and measure biases that would arise under a misspecified model. Finally, we develop a nu-

merical algorithm that enables computing equilibria of the class dynamic auction games with

subcontracting.

The rest of the article is organized as follows. In Section 2, we summarize the related

literature. Section 3 describes the model. In Section 4, we characterize the equilibrium with

subcontracting. The calibration exercise is summarized in Section 5. We analyze the properties of

computed equilibrium in Section 6, study the implication of subcontracting for the choice of an

auction format in Section 7, and discuss the consequences of using the misspecified model without

subcontracting in estimation in Section 8. Section 9 discusses empirically relevant extensions.

Section 10 concludes.

2. Related literature

Our article is related to the literature on boundary of a firm which is represented by Coase

(1937), Coase (1988), Williamson (1975), Jensen and Mechling (1976), Alchian and Demsetz

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The RAND Corporation 2016.