The government would like to introduce a subsidy to help citizens who need their car for work and cannot afford to buy a ULEZ-compliant car. It also wants to ensure that only citizens who need to drive their car regularly use the subsidy. Instead of the two groups of citizens discussed above, we focus on one particular citizen who does not drive very often, only once a year. As a result, this citizen does not emit excessive pollution even with a car that is not ULEZ-compliant. The government would therefore prefer citizens like her not to buy a new car to avoid having to pay the subsidy unnecessarily. Once a year, the citizen derives a benefit of driving through the ULEZ area of B = 20. She can choose to either buy a new ULEZ-compliant car at a cost of C = 10, 000, denoted x = new, or stick with her old car at no cost, denoted x = old.If she sticks with her old car, she has to pay a ULEZ fee of τ = 12.50 on the day that she drives. Her net benefit of driving on that day is therefore B(old) = 20 −12.50 = 7.50. If she buys a new ULEZ-compliant car, she doesn’t have to pay the fee. Her net benefit of driving on that day is therefore B(new) = 20. However, if she buys a new car she also has to pay the cost of the car, C, but can deduct the government subsidy, S from that price. The citizen decides whether to buy the new car by comparing her net utility of having a ULEZ-compliant car, U(x = new) = B(new) − (C − S), to her net utility of keeping her old car, U(x = old) = B(old). 1. Suppose first that there is no scrappage scheme, so the subsidy is equal to S = 0. If the citizen buys the new ULEZ-compliant car, she has to pay the full cost: C = 10, 000. What is her utility of buying the new car? What is her utility of buying the old car? Does the citizen prefer to buy the new car?
The government would like to introduce a subsidy to help citizens who need their car for work and cannot afford to buy a ULEZ-compliant car. It also wants to ensure that only citizens who need to drive their car regularly use the subsidy. Instead of the two groups of citizens discussed above, we focus on one particular citizen who does not drive very often, only once a year. As a result, this citizen does not emit excessive pollution even with a car that is not ULEZ-compliant. The government would therefore prefer citizens like her not to buy a new car to avoid having to pay the subsidy unnecessarily. Once a year, the citizen derives a benefit of driving through the ULEZ area of B = 20. She can choose to either buy a new ULEZ-compliant car at a cost of C = 10, 000, denoted x = new, or stick with her old car at no cost, denoted x = old.If she sticks with her old car, she has to pay a ULEZ fee of τ = 12.50 on the day that she drives. Her net benefit of driving on that day is therefore B(old) = 20 −12.50 = 7.50. If she buys a new ULEZ-compliant car, she doesn’t have to pay the fee. Her net benefit of driving on that day is therefore B(new) = 20. However, if she buys a new car she also has to pay the cost of the car, C, but can deduct the government subsidy, S from that price. The citizen decides whether to buy the new car by comparing her net utility of having a ULEZ-compliant car, U(x = new) = B(new) − (C − S), to her net utility of keeping her old car, U(x = old) = B(old).
1. Suppose first that there is no scrappage scheme, so the subsidy is equal to S = 0. If the citizen buys the new ULEZ-compliant car, she has to pay the full cost: C = 10, 000. What is her utility of buying the new car? What is her utility of buying the old car? Does the citizen prefer to buy the new car?
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