ufp_exam1_12_sol

# 4 the optimal size of a city needs to balance the

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4. The optimal size of a city needs to balance the benefits of a larger city (agglomer- ation effects) and the costs (congestion effects). The costs of traffic congestion can be measured by the opportunity costs of time. Time wasted commuting could be used more productively. An economic measure of ”productive” is a person’s hourly wage. We, therefore, typically use some fraction of the hourly wage to measure the opportunity cost of time. Problem 2 1. Hannah’s maximization problem is as follows: max 2 ln( z h ) + ln( g h + g p ) s.t. z h + g h = 50 which can be written as: max 2 ln(50 - g h ) + ln( g h + g p ) The FOC with respect to g h is: - 2 50 - g h + 1 g h + g p = 0 Hence, Hannah’s reaction function is given by: g h = 1 3 (50 - 2 g p ) Philip’s maximization problem is as follows: max ln(100 - g p ) + ln( g h + g p ) 2

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The FOC with respect to g p is: - 1 100 - g p + 1 g h + g p = 0 Philip’s reaction function is: g p = 1 2 (100 - g h ) 2. If g p = 0 then g h = 50 / 3 or 16.67. If g h = 0, then g p = 50. Those two cases cover potential corner solutions.
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