Knowledge that Transforms

In this paper, a continuous time model for a monopoly firm of the Evans' type, encompassing operations, investments, and output prices, is formulated as an optimal control problem. In the model the objective of the firm is to maximize, subject to various constraints, the integral of production profits less interest and investment costs over a finite decision-making interval, plus the value of the capacity at the end of the period. The state variables are capacity, debt, and output price; the controls are the scale of operation, rate of purchase of new capacity, and rate of change of the output price. Final capacity, price, and debt are control parameters. There are several inequality constraints. Using results in control theory, the optimal controls are characterized for a model basically linear in structure. It shows that the one case suggested by Evans for further analysis is a trivial problem. These results are interpreted using the properties of the value equation. In addition, the control model is formulated alternatively as a mathematical programming problem. Solutions may then be computed by published algorithms.

IF EACH AUTOMOBILE PAYS AVERAGE RATHER THAN MARGINAL SOCIAL COST OF A HIGHWAY TRIP, THERE WILL BE TOO MUCH AUTO TRAVEL DURING PERIODS OF CONGESTION. BY USING ONLY INPUTS TAXES, ADJUSTMENTS IN FARES ON ALTERNATIVE TRANSIT MODES, AND INCOME REDISTRIBUTION, RESOLVE THIS PROBLEM IN TWO WAYS: COMPLETE (FIRST-BEST) OPTIMALITY IN PEAK PERIODS AND SECOND-BEST OPTIMALITY IN OFF-PEAK PERIODS; OR FIRST-BEST OPTIMALITY IN OFF-PEAK PERIODS AND SECOND-BEST OPTIMALITY AT THE PEAK. IT IS SHOWN THAT WITH CONGESTION INTERDEPENDENCE, AS WHEN AUTOMOBILES AND BUSES CONTRIBUTE TO ONE ANOTHERS CONGESTION, THE SECOND-BEST PEAK SOLUTION CAN WARRANT AN URBAN BUS TRANSIT FARE BELOW AVERAGE COST CALLING FOR A SUBSIDY. AND UNDER A FIRST-BEST PEAK SOLUTION, INVOLVING BOTH AN INPUTS TAX AND A TRANSIT FARE ADJUSTMENT, A COMPANION SECOND-BEST OFF-PEAK TRANSIT FARE IS SHOWN THAT WILL MITIGATE THE (THEN INAPPROPRIATE BUT STILL EFFECTIVE) INPUTS TAX. /AUTHOR/

[The paper discusses the problem of pooling cross-section and time-series data in the framework of likelihood functions and the associated posterior distributions and shows how this analysis can be useful in deciding whether or not to pool.]