Methodologies of project selection in the presence of constraints

Conclusions

The shadow prices associated with resource and policy constraints can be used to decentralize project selection among project managers so that their choices accord with these constraints and with the stipulated objective function. Project managers are instructed to value their inputs and outputs (suitably redefined so as to include contributions to social goals) at their shadow values, using as discount rate the tradeoff (minus 1) between consumption in year 1 and in year 0. They will choose the optimal set of independent projects if they adopt all those with positive NPVs and, among mutually exclusive projects with positive NPVs, if they select that with the highest NPV. Project managers will be indifferent toward projects whose NPVs are zero, and will require some additional directives if they are to adopt fractional projects to the extent prescribed in the optimum solution. If a new project were to become available for evaluation at this point, the same set of shadow prices can be used to determine its NPV. Only if this NPV were positive and the project sufficiently large would there be a need to recalculate the set of shadow prices.¹⁹

As was pointed out above, the dual variables corresponding to the primal constraints indicate the tradeoffs of resource amounts or target levels against the maximand which policymakers implicitly make when they select their values. The magnitudes of these tradeoffs may then lead them to revise the constrained target levels upwards or downwards until they reach more appropriate levels. Several iterations may be needed before a satisfactory or ‘best-compromise’ solution is reached.²⁰

Instead of the ‘constraint method’ outlined above, other multiobjective programming models can be used to generate as many noninferior or nondominated solutions in objective space as desired. In the ‘weighting method’ first outlined by Zadeh (1963), for example, all objectives are included in the objective function of the primal program and are assigned arbitrary positive weights. After solving the corresponding linear program, these weights are systematically varied and trace out solutions (all of which are noninferior) in objective space. Once the policymaker has chosen the best of these, the weights which have given rise to it can be used as tradeoffs for the objectives under consideration.²¹ The superiority of these multiobjective methods over the UNIDO ‘bottom-up’ approach resides in the fact that policymakers make their best-compromise choices in objective space rather than decision space. That is, they choose the most desired combination of objectives, and only as an implication of this are projects accepted or rejected. In the bottom-up approach, instead, the policymakers are assumed to choose directly in decision space, which includes the primal variables P_i, a procedure which appears to place the proverbial cart before the horse.

One of the strengths of programming models is their versatility. Models of the type discussed above can readily be adapted to allow for such factors as a greater number of time periods, the possible reinvestment of project cash flows, the assigning of distributional weights to different economic agents, the introduction of foreign trade and investment, and limitations on the availability of specialized factors. Mutual exclusiveness within broad classes of projects can also be introduced, as can other forms of project interdependence such as one-way dependence (i.e, the investment in project i being contingent on project j being also adopted), and complementarity (the simultaneous adoption of projects i and j has different net output effects than the sum of their adoption as single projects). In cases where project indivisibility is an important consideration, integer programming can be utilized. Scale economies can also be incorporated in a programming framework (see Westphal, 1975, and Hendrick and Stoutjesdijk, 1978).

This enthusiasm for programming models should be tempered by certain limitations they possess in addition to their indubitable advantages. First, in multiperiod programming models, some of the shadow prices have been found to be unstable in the sense of fluctuating sizeably from one period to the next, a feature which fails to inspire confidence in their utilization for project implementation.²² Secondly, programming models have sometimes been carelessly specified, for example by failing to bring out the implications of additional labour employment for a labour-surplus economy's consumption-investment mix, and thus attributing to labour a shadow price of zero when it is in fact positive. This would lead to the choice of unduly labour-intensive projects.²³

Thirdly, our approach assumes that a sizeable number of project proposals are ready to be evaluated at one point in time, as opposed to the UNIDO assumption that projects are considered sequentially for possible adoption. Admittedly, it is a counsel of perfection to demand of Central Planning Offices that they maintain an updated shelf full of projects ready for possible implementation. It remains, however, true that at any point in time there are limits on the amounts of financial resources and other inputs, and that deciding upon projects' merits as they roll off the drawing boards does not allow the demands on these resources to be related to the available supplies, or to the merits of alternative projects, in any systematic way. One cannot escape the conclusion that a high payoff is to be expected from devoting resources to the active recruitment of skilled project evaluators to build up a sufficiently extensive collection of potential investment projects.

A fourth objection of a technical nature has recently been raised against the use of linear programming models as approximations to nonlinear programming ones. Baumol (1982) has shown that the dual prices resulting from such approximations need bear no relation to the true shadow prices yielded by the nonlinear program. They may take on zero values in cases where the inputs are in fact scarce, and positive values when the inputs are in excess supply. This point will not be pursued further here in view of the operational difficulties of estimating nonlinear functions. It should be noted, however, that a number of applications in the water-resources area have used geometric programming techniques to deal with nonlinearities.

In spite of these and other objections that may be raised against the use of multiobjective programming models for project selection, I believe that they do serve to illuminate the rationale for shadow pricing in economies subject to market distortions, and that a cautious reliance on them can facilitate the actual process of project selection and implement the goals of efficiency and distributional equity which characterize both the UNIDO and the Little-Mirrlees methods of project appraisal.