Collusion in public procurement: the role of subcontracting

Abstract

This paper investigates how subcontracting affects collusion in public procurement. In a model in which a public buyer runs simultaneous or sequential competitive procedures we show that the stability of collusive agreements depends on the level of subcontracting share and it is not necessarily increasing in this share. In a repeated procurement in which contractors and subcontractors are involved in collusive agreements enforced by slit award and bid rotation strategies we find that simultaneous procedures induce less collusion than sequential procedures, with split award strategies allowing the less stable collusive scheme. We also find that allowing a further increase in the subcontracting share strengthens collusion when the share is low but it mitigates collusion when the share is high. Thus, the competitive format and the allowed subcontracting share must be carefully managed by the public buyer in order to prevent collusion.

Appendix

Proof of Proposition 1

We simply find the lowest discount factor such that collusion is stable in any format for any collusive strategy.

(1) Simultaneous format with bid rotation. Assume firm 1 gets the collusive profit at $t=0$, then when sticking to the collusive path of $\sigma _{R}$ its discounted profit is:

$$\begin{aligned} V_{1}^{C}=\sum _{t=0}^{\infty }2v\left( 1-\alpha \right) \delta ^{2t}=2v\left( 1-\alpha \right) \frac{1}{1-\delta ^{2}} \end{aligned}$$

(1)

whereas when deviating the discounted profits starting from the current period of deviation are:

$$\begin{aligned} V_{1}^{D}=2v \end{aligned}$$

Note that in this case the deviation at the bidding and the execution stage give the same deviation profit, then:

$$\begin{aligned} V_{1}^{C}\ge V_{1}^{D}\text { if }\delta \ge \delta _{1,R}=\sqrt{\alpha } \end{aligned}$$

Is is easy to see that we have the same discount factor for firm 2. The collusive and deviation profits for firm 3 are respectively:

$$\begin{aligned} V_{3}^{C}=\sum _{t=0}^{\infty }2\alpha v_{3}\delta ^{t}=2\alpha v_{3}\frac{1}{ 1-\delta } \end{aligned}$$

(2)

and

$$\begin{aligned} V_{3}^{D}=2v_{3} \end{aligned}$$

(3)

then

$$\begin{aligned} V_{3}^{C}\ge V_{3}^{D}\text { if }\delta _{3,R}\ge 1-\alpha \end{aligned}$$

(2) Simultaneous format with split award. The collusive profit for each firm, when sticking to the collusive path in $\sigma _{S}$, is:

$$\begin{aligned} V_{1}^{C}=\frac{1}{1-\delta }v\left( 1-\alpha \right) \end{aligned}$$

whereas when deviation at time $t=0$, the discounted profits are:

$$\begin{aligned} V_{1}^{D}=2v \end{aligned}$$

where $V^{C}\ge V^{D}$ if:

$$\begin{aligned} \delta \ge \delta _{1,S}=\frac{1}{2}\left( 1+\alpha \right) \end{aligned}$$

The discounted cooperative profits for firm 3 are the same in (2), its deviation profit is the same as in (3), then:

$$\begin{aligned} V_{3}^{C}\ge V_{3}^{D}\text { if }\delta _{3,R}\ge 1-\alpha \end{aligned}$$

(3) Sequential format with bid rotation. The cooperative discounted profits are the same in (1), the best deviation is now that one at the execution stage by the firm awarded the contract. Thus, when deviating at the first lot, the deviation profits are:

$$\begin{aligned} V_{1}^{D}=\left( 1-\alpha \right) v+v \end{aligned}$$

Then:

$$\begin{aligned} V_{1}^{C}\ge V_{1}^{D}\text { if }\delta \ge \delta _{1}=\frac{\sqrt{\alpha \left( 2-\alpha \right) }}{2-\alpha } \end{aligned}$$

For the firm 3 the cooperative profit is the same in (2). Its deviating profits are:

$$\begin{aligned}&V_{3}^{D}=\alpha v_{3}+v_{3}\\&V_{3}^{C}\ge V_{3}^{D}\text { if }\delta \ge \delta _{3,c}=\frac{1-\alpha }{1+\alpha } \end{aligned}$$

(4) Sequential format with split award. Cooperatives profits are:

$$\begin{aligned} V_{1}^{C}=\sum _{t=0}^{\infty }v\left( 1-\alpha \right) \delta ^{t}=v\left( 1-\alpha \right) \frac{1}{1-\delta } \end{aligned}$$

the highest incentive from deviation is gained by the firm awarded the auction at the first stage (first lot), in fact, it is better off by winning the first lot, as the agreement entails, subcontracting without destroying the agreement and getting $v\left( 1-\alpha \right)$, but then deviating at the second lot, where it obtains v.

$$\begin{aligned} V_{1}^{D}=v\left( 1-\alpha \right) +v \end{aligned}$$

then:

$$\begin{aligned} V_{1}^{C}\ge V_{1}^{D}\text { if }\delta _{1}\ge \frac{1}{2-\alpha } \end{aligned}$$

For firm 3 we have the same results of the case 3. $\square$

Extension to the case of an efficient subcontractor

Consider $c>k$. In this case firm 3 is more efficient, this changes its profit in the asymmetric Bertrand equilibrium of the punishment path, that now becomes $c-k$. The discount factors for firm $i=1,2$ do not change. Thus in what follows we can only consider firm 3. The proof follows the steps of the proof of the Proposition 1.

(1) Simultaneous format with bid rotation. For firm 3, the collusive profit does not change but the new deviation profits are respectively:

$$\begin{aligned} V_{3}^{C}=\sum _{t=0}^{\infty }2\alpha v_{3}\delta ^{t}=2\alpha v_{3}\frac{1}{ 1-\delta } \end{aligned}$$

(4)

and:

$$\begin{aligned} V_{3}^{D}=2v_{3}+\frac{\delta }{1-\delta }2\left( c-k\right) \end{aligned}$$

(5)

Note that in order to let the participation constraint to hold we need the assumption $\alpha >\frac{c-k}{r-k},$ ensuring that the collusive profit is higher than the profit of the Nash equilibrium of the Bertrand equilibrium (during the punishment). Then:

$$\begin{aligned} V_{3}^{C}\ge V_{3}^{D}\text { if }\delta _{3}\ge \left( 1-\alpha \right) \frac{r-k}{r-c}=\left( 1-\alpha \right) \beta \end{aligned}$$

(6)

with $\beta =\frac{r-k}{r-c}>1$.

(2) Simultaneous format with split award. The discounted cooperative profits for firm 3 are the same in (2), it deviation profits are the same in (5), then the discount factor is the same of (6).

(3) Sequential format with bid rotation. For the firm 3 the cooperative profit is the same in (2). Its deviating profits are:

$$\begin{aligned} V_{3}^{D}= & {} \alpha v_{3}+v_{3}+\frac{\delta }{1-\delta }2\left( c-k\right) \\ V_{3}^{C}\ge & {} V_{3}^{D}\text { if }\delta \ge \delta _{3,c}=\frac{1-\alpha }{ 1+\alpha -2\frac{\left( c-k\right) }{\left( r-k\right) }} \end{aligned}$$

Note that $\delta _{3,c}=\frac{1-\alpha }{1+\alpha -2\frac{\left( c-k\right) }{\left( r-k\right) }}>\delta _{3,c}=\frac{1-\alpha }{1+\alpha }$.

(4) Sequential format with split award. For firm 3,  is the same as in the case3.

It possible to see that the new critical discount factor for firm 3 shifts up (since $\beta >1$). $\square$