URGENT!

1. (30 points) California is designing its statewide high-speed rail system. The system runs a number of high speed trains N to accommodate a number of travelers, P. The value of the system is a function of N and P:F(N,P). On the other hand, operating the system incurs a cost, taken to be the number of trains N times the average cost per train c. 1.1. What is the sign of the first-order derivative of F(N,P) with respect to N and P ? (5 points) 1.2. If the objective is to maximize the net value of the system, express the objective function. In addition, each train in the system has a seat capacity of K seats. Write out the seat capacity constraint. (8 points) 1.3. Write Kuhn-Tucker conditions of the above constrained maximization problem. You may use fN and fP to denote the first-order partial derivatives of F with respect to N and P. (7 points) 1.4. Do you think the Lagrange multiplier involved in the Kuhn-Tucker conditions will be zero? If not, express fP as a function of fN(5 points) 1.5. Economic theory indicates that the optimal train fare, t, should be set equal to the marginal value to train travelers, t=fP. In order for the system to operate break-even, would the state subsidy be needed? If yes, what would be the amount? (5 points)