Based on budgetary restrictions, the city employs a police force of 20 officers. Let C denote the number of crimes and P the number of police in the neighborhood. The relationships between crime and police for each of the two neighborhoods are given by: Cpoor =110−2Ppoor and Crich =100−4Prich
a) Graph the crime-curves (or crime-lines, in this case) for each of the neighborhoods with the x-axis representing police (P) and y-axis representing Crime (C). Explain intuitively the differences between the slopes and intercepts of these crime-curves.
b) Do you think the difference in "productivity" of an extra police officer between rich and poor neighborhoods is realistic? Provide a general explanation based on of your socio-economic understanding of the issue of crime.
c) Derive and graph the city's transformation curve between Cpoor and Crich , subject to the constraint that Ppoor and Prich sum-up to 20. For the graph, represent Cpoor on the x-axis and Crich on the y-axis. Note that because of the linear relationship between C and P for each group, the transformation curve is a straight line. (Hint: you should be able to find the transformation curve solely by locating its endpoints and then connecting them.)
d) Find and graph the crime level if the city divided the police officers equally between the two neighborhoods.
e) Find and graph the crime level that would result if the city divided the police officers to equalize crime across the two neighborhoods.
f) Finally, consider the case where the city's goal is to minimize overall crime, which equals (Cpoor +Crich )/2. How should the city divide its police officers across the two neighborhoods? Find the answer by using the iso-crime line approach described in the chapter. Provide an intuitive explanation for the answer that you have calculated. [Hint: Given the linear lines, tangency occurs at an end-point of the transformation curve; the question is at which end-point will total crime in minimized?]
g) Next consider if the coefficient of Ppoor in the above formula were equal to 4 and the coefficient of Prich were equal to 2. Without drawing any diagrams or doing any computations, use your economic intuition on police productivity (developed in part (b) above) to tell how the city would allocate the police force if its goal were to minimize total crime. What allocation would be chosen? Provide an intuitive explanation.
