Assume that each criminal i has the following utility function ui=bO-fO^2 where O=number of crimes committed, b=per unit benefit associated with each crime, and f=per unit fine for each crime paid by the criminal if caught. Essentially O is the demand for crime while f is the price. Finally b>0.

a) What is the demand function and hence the demand curve for crime? (Hint: From the utility function we know that the marginal benefit of a crime to the criminal is b while the corresponding marginal cost is 2fO)

b) Suppose that f=cO captures the relationship between higher crime and corresponding increases to punishment. In other words, fines (f) set by the government increases as crime (O) increases and c>0. This function is essentially the "inverse supply function" with respect to crime. Given the demand and supply curves, express O and f in terms of b and c.

c) What is the equilibrium levels of O and f if b=c=1?

d) Suppose that an increase in crime (O) also results in a per unit amount of social damage equal to d(d>0). And assume that d=1. Is the resulting socially optimal amount of crime different from the privately optimal amount you obtained in (c)?why?