Productivity in Procurement Auctions of Pavement Contracts in Mexico

Abstract

When it comes to allocating contracts, governments must weigh the decision of whether to exercise discretion in hiring or to allow for greater competition without firm selection. It is not always clear which allocation format will lead to better outcomes. This trade-off is influenced by the government’s ability to select the best firms when competition is restricted, as well as the likelihood that this practice will lead to corruption. In this paper, I examine the allocation of street pavement contracts in Mexico. By combining auction methods with a productivity analysis, I am able to indirectly analyze whether local governments select firms with low excess costs when competition is restricted. This indirect approach allows for monitoring contract allocation in situations where there is limited information available on firm reputation. I find that firms selected for settings with less competition have lower costs for complex pavement contracts, but higher costs for simple ones. These results suggest that the government would benefit from using public auctions for simple pavement contracts, which is the opposite of current practices.

Appendix

1.1 Estimation Details of the Procurement Auction Model

Evidence of asymmetry

Figure 2 shows the conditional bid cdfs evaluated at different project sizes, approximated by the total amount of concrete z. When evaluating the cdfs at z’s 50th percentile, the distributions are quite similar. Nevertheless, this changes when evaluating the cdfs for small and larger projects. Especially for smaller projects, class 0 firms dominate class 1 firms.

Fig. 2

Conditional bid CDFs evaluated at different project sizes, z. Class 1: firms that, in addition to participating in public auctions, have participated in an auction by invitation or received a project by direct allocation. Class 0: firms that only participate in public auctions

Evidence to support the model - public auction

A key assumption of the structural procurement auction model is that the firm’s strategy functions are strictly increasing in its cost. Figure 3 shows the estimated strategy functions and the estimated costs. As expected, the functions are increasing for both classes of firms, with a few exceptions on the tails.

Fig. 3

ξ⁻¹( ⋅ ∣z) evaluated at the median of the project size. Class 1: firms that, in addition to participating in public auctions, have participated in an auction by invitation or received a project by direct allocation. Class 0: firms that only participate in public auctions

1.2 Conditional Cost Density, Evaluated at Different Project Characteristics

Figure 4 displays the estimated cost distributions for both classes of firms, conditional on different levels of the total amount of concrete z, and whether the project includes sewage work.

Fig. 4

Conditional cost density, evaluated at different project characteristics (z). Class 1: firms that, in addition to participating in public auctions, have participated in an auction by invitation or received a project by direct allocation. Class 0: firms that only participate in public auctions

Conditioning the cost distributions on the amount of concrete allows me to control for the size of the project, and conditioning the cost distribution for whether the project includes sewage work allows me to control for the contracts’ complexity. The different project sizes I condition for correspond to the 25th, 50th and 75th percentiles of the total amount of concrete used in a project.

Bid data of auctions by invitation (I3P)

Find in Table 7 the bid data of auctions by invitation and, for easy of comparison, the bid data of public auctions from Table 1.

Table 7 Summary statistics, first-price sealed bid auctions of auctions by invitation and public auctions: street pavement contracts

Overall, the median bid per cubic meter is lower under I3P. The projects under I3P are half the size of projects allocated by public auctions, but take on average only five percent less time to complete. A lower proportion of I3P projects include sewage work, 22 percent against 37 percent of public auctions, and an equal proportion of projects are allocated by the municipality (compared to the state government) in both auction formats.

1.3 Estimation Details of the Stochastic Frontier Analysis

Relationship between the cost frontier and the actual cost

Using Shephard’s lemma, we notice:

$$\begin{array}{lll}\qquad\,\frac{\partial {C}^{* }}{\partial {w}_{k}}\,=\,{x}_{k}{e}^{-\eta },\\ \Rightarrow \frac{\partial \ln {C}^{* }}{\partial \ln {w}_{k}}\,=\,\frac{{w}_{k}{x}_{k}{e}^{-\eta }}{{C}^{* }}=\frac{{w}_{k}{x}_{k}}{{w}^{{\prime} }x}={S}_{k},\end{array}$$

where S_k denotes the input k’s share of the total cost. By rearrenging terms, we then have that ${x}_{k}=\frac{{C}^{* }{S}_{k}}{{e}^{-\eta }{w}_{k}}$, therefore, when calculating the actual cost:

$$\begin{array}{lll}\qquad\,\,{C}^{a}\,=\,\mathop{\sum}\limits_{k}{w}_{k}{x}_{k}=\mathop{\sum}\limits_{k}\frac{{w}_{k}{C}^{* }{S}_{k}}{{e}^{-\eta }{w}_{k}}=\frac{{C}^{* }}{{e}^{-\eta }}\mathop{\sum}\limits_{k}{S}_{k}={C}^{* }\exp (\eta ),\\ \Rightarrow \ln {C}^{a}\,=\,\ln {C}^{* }+\eta .\end{array}$$

Estimation of the efficiency index and marginal effects on E [ η ]

Note that the conditional distribution of η is known, so we can derive moments of the continuous function of η∣ϵ, where ϵ = η + ν. Following Battese and Coelli (1988), we can show:

$$\begin{array}{lll}\qquad\quad E[{\eta }_{i}| {\epsilon }_{i}]\,=\,\frac{{\sigma }_{* }\phi \left(\frac{{\mu }_{* i}}{{\sigma }_{* }}\right)}{\Phi \left(\frac{{\mu }_{* i}}{{\sigma }_{* }}\right)}+{\mu }_{* i},\\ E[\exp (-{\eta }_{i})| {\epsilon }_{i}]\,=\,\exp \left(-{\mu }_{* i}+\frac{1}{2}{\sigma }_{* }^{2}\right)\frac{\Phi \left(\frac{{\mu }_{* i}}{{\sigma }_{* }}-{\sigma }_{* }\right)}{\Phi \left(\frac{{\mu }_{* i}}{{\sigma }_{* }}\right)}.\end{array}$$

For the marginal effects of η’s mean determinants, I follow Wang (2002). If $E[{\eta }_{i}]={{{{\bf{w}}}}}_{i}^{{\prime} }{{{\boldsymbol{\delta }}}}$, we have:

$$\begin{array}{l}\frac{\partial E({\eta }_{i})}{\partial {{{\mbox{w}}}}_{r}}\,=\,{\delta }_{r}\left[1-{\Lambda }_{i}\left[\frac{\phi ({\Lambda }_{i})}{\Phi ({\Lambda }_{i})}\right]-{\left[\frac{\phi ({\Lambda }_{i})}{\Phi ({\Lambda }_{i})}\right]}^{2}\right],\\ \frac{\partial V({\eta }_{i})}{\partial {{{\mbox{w}}}}_{r}}\,=\,\frac{{\delta }_{r}}{{\sigma }_{\eta }}\left[\frac{\phi ({\Lambda }_{i})}{\Phi ({\Lambda }_{i})}\right](E{({\eta }_{i})}^{2}-V(\eta )),\end{array}$$

where Λ_i = μ_i/σ_η.

Summary statistics: SFA input data

Table 8 displays the summary statistics of the input data. The price of concrete is in MX pesos per seven cubic meters of concrete, a common measure in which the concrete is transported. The price of capital is the per day rent of a flattening roller of one ton. This is the most expensive piece of equipment needed for paving a street, and the size of the roller is common to pave small streets. Finally, the wages reported are the average monthly wage in MX pesos in the construction sector.

Table 8 Summary statistics: data on inputs of Stochastic Frontier Analysis

Distribution of the efficiency index by firm class and allocation procedure

See in Fig. 5 the unweighted distribution of the efficiency index by firm class, using the public auctions. In the estimation, three inputs are used assuming that the inefficiency term η follows a Truncated-Normal, with its mean being influenced by three observable characteristics: the firm’s class, whether the project takes place during the year before elections, and a dummy for party alignment between the mayor and governor. We see that class 0 firms have a higher efficiency index. This is mainly driven by the fact that they are more efficient in projects without sewage work, which are more common than projects with sewage work.

Fig. 5

Densities - Efficiency index. Class 1: firms that, in addition to participating in public auctions, have participated in an auction by invitation or received a project by direct allocation. Class 0: firms that only participate in public auctions. The efficiency index is estimated by $E[\exp (-{\eta }_{i})| {\epsilon }_{i}]$ and ranges from 0 to 1. For interpretation purposes, a value of 0.6 denotes that the minimum cost is 60 percent of the actual cost

Robustness analysis: model specification with scaling property

A comparison of the Truncated-Normal model with model specifications under the scaling property of Wang and Schmidt (2002) deserves further discussion. So far, under the Truncated-Normal specification, I have parameterized the mean of the inefficiency term as a function of external variables w. Alternatively, Wang and Schmidt (2002) propose that the inefficiency term η may follow the form, η_i ~ h(w_i, δ)η*. With some abuse of notation, the function h(⋅) ≥ 0 does not represent the production function as above, but rather any observation specific non-stochastic function of the exogenous determinants of the inefficiency term, and η* ≥ 0 is a random variable, common to all observations. The authors call h(⋅) the scaling function, and η* the basic distribution. Models with the scaling property are attractive because the shape of the inefficiency term η_i is the same for all firms, and h(⋅) scales the distribution. In comparison, for the Truncated Normal specification, where the mean of η is parameterized, each observation has a different truncation point.

As a robustness analysis, I fit compare the results with a Half-Normal Model and with a Truncated-Normal with the scaling property, as specified in Wang and Schmidt (2002). Nevertheless, in order to achieve convergence, the translog specification of the frontier needs to be restricted.

Table 9 presents the estimates for various restrictions to the translog function. The results displayed are the marginal effects of the exogenous determinants of inefficiency, the main estimates sought. For comparison purposes, the first row displays the Truncated-Normal estimates presented in Section 4, and the second row displays the Truncated-Normal with no interactions included in the cost frontier specification. As can be seen from the analysis, when comparing the results with no interactions, the marginal effects vary little irrespective of the specification. Hence, the preferred model specification is the Truncated-Normal for two reasons: first, it does not require restrictions in the translog function, and second, the marginal effects are similar to the estimates from models with the scaling property.

Table 9 Robustness analysis of marginal effects on the expected inefficiency E[η]: comparing the Truncated-Normal with specifications with the scaling property

Distribution of the efficiency index by state

In Fig. 6 I aggregate the efficiency results estimated using public auctions. Following Sickles and Zelenyuk (2019), each firm’s efficiency is weighted by their relative cost in their respective state. Overall, we can see that the northern states are more efficient, along with the gulf states. The state that fares the worst is Chiapas, in the south, consistent with the fact that it is a region that is hard to access due to the difficult terrain.

Fig. 6

Weighted average efficiency index per state. States in black have no or less than 10 projects in the sample. The efficiency index is estimated by $E[\exp (-{\eta }_{i})| {\epsilon }_{i}]$ and ranges from 0 to 1. For interpretation purposes, a value of 0.6 denotes that the minimum cost is 60 percent of the actual cost

Political influence on the mechanism choice

Table 10 displays the results probit estimates of the choice of mechanisms, but separately for municipality and state contracts. It is interesting to observe that the increase in the use of mechanisms with hiring discretion is more prominent in state contracts. Nevertheless, this represent a minority of observations in the sample.

Table 10 Probit: dependent variable, dummy = 1 if Direct or Auction by Invitation (I3P), dummy = 0 if by Public Auction