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Equation - Utility Generated from Successfully Igniting an Arson Wildfire

The expected value of the criminal's utility function U subscript i for a given number of offenses committed O subscript i, equals the sum of two quantities. The first quantity is the probability of getting caught pi subscript i times the utility function U subscript i, determined by the the criminal's psychic and income benefits from the illegal activity for example fire setting, quantity g subscript i, minus the production cost for the activity c subscript i minus a function representing the loss from being caught and convicted of the crime f subscript i in turn determined by employment status upper case W subscript i and the wage rate lower case w subscript i. The second quantity is a second utility function U subscript i determined by the difference between g subscript i and c subscript i, with the utility function multiplied by the probability of not getting caught, 1 minus pi subscript i