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Q3 (20 points) Consider an individual who has a choice between drive alone (DA), carpool (CP), bus (B), and light rail (LR). Let the deterministic component of each utility function be V =0.8-T -0.25CDM Vop =0.2-Top -0.25Ccp V. =-0.2-T -0.25CB VER = -TLA -0.25CLR where 7 and C are the travel time in hours and traveling cost in dollars. Let PDA, Pop, Pp and PLR denote the choice probabilities predicted by the logit model for a given set of values, T and C, for all four travel modes. (1) Suppose that the traveling cost by light rail is increased by a certain amount (all other variables keep unchanged), resulting increases in the choice probabilities of the three other modes by APp,, APp, AP;, show that: APp/ PDA = AP;/P: = APcp/ Pop . (2) If original choice probabilities are: Poy = 0.458 , Pop = 0.251, P; =0.121, PLA =0.170. An increase in the traveling cost by light rail (all other variables keep unchanged) has resulted in an decrease in its choice probability by AP =-0.016 (or a new choice probability of PR = 0.154), determine the new choice probability Pox , Pap and P