A Two-Factor Uncertainty Model to Determine the Optimal Contractual Penalty for a Build-Own-Transfer Project

Public-Private Partnerships (PPP) became one of the most common types of public procurement arrangements and Build-Own-Transfer (BOT) projects, awarded through adequate bidding competitions, have been increasingly promoted by governments and other public entities. The theoretical model herein proposed is based on a contractual framework where the government grants leeway to the private entity regarding the timing for project implementation. However, the government is aware that delaying the beginning of operations will lead to the emergence of social costs, i.e., the costs that result from the corresponding loss of social welfare. This fact should motivate the government to include a contractual penalty in case the private firm does not implement the project immediately. The government also recognizes that the private entity is more efficient in constructing the project facility. Considering both the existence of social costs and the private firm's greater efficiency, the model's outcome is the optimal value for the legal penalty the government should include in the contract form. A two-factor uncertainty approach is adopted, where the facility construction costs and the cash-flows to be generated by running the subsequent activities follow geometric Brownian motions that are possibly correlated. Adkins and Paxson ((2011)) quasi-analytical approach is followed since homogeneity of degree one can not be invoked in all of the model's boundary conditions. Sensitivity analysis reveals that variations both in the correlation coefficients and in the standard deviations have a strong impact on the optimal contractual penalty. Sensitivity analysis also demonstrates that there is a level of social costs above which the inclusion of a legal penalty is necessary and, similarly, that there is a level for the comparative efficiency above which the inclusion of a legal penalty is not justifiable. The analytical solution to determine each of these values is presented. Finally, the effects of including a non-optimal penalty value in the contract form, which result from overestimating or underestimating the selected bidder's real comparative efficiency are examined, using a numerical example. Results demonstrate that overestimating (underestimating) the selected bidder's real comparative efficiency leads to the inclusion of a below-optimal (above-optimal) value for the legal penalty in the contract and produces effects that the government would prefer to prevent