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## Homework Statement

"Consider the following version of the Ehrlich economic model of crime where an individual has the expected utility function:

U = p ln(I

_{u}) + (1 - p) ln (I

_{s}).

p = objective probability of being caught

I

_{u}= Criminal's income if caught

I

_{s}= Criminal's income if not caught

The criminal's initial endowment of time is T hours a day and should be divided into "time for crime" (t

_{c}) and "time for legal work" (t

_{l}). So T = t

_{c}+ t

_{l}.

If the criminal choose to work legally, he gets an income of w

_{l}> 0 per unit of time; while engaged in crime he gets an income of w

_{c}> w

_{l}per unit of time.

If the individual is caught committing crimes, he get a penalty f > w

_{c}- w

_{l}per unit of time.

So, the income of an individual who commits a crime but is not arrested is I

_{s}= w

_{l}t

_{l }+ w

_{c}t

_{c}while if he get arrested

I

_{u}= w

_{l}t

_{l }+ w

_{c}t

_{c}- f t

_{c}.

## Homework Equations

Then, the criminal divides his time between crime and work legally, an his problem is given by:

max U = p ln(I

_{u}) + (1 - p) ln (I

_{s}). with t

_{c}, t

_{l}[itex]\geq[/itex]0

s.t. I

_{s}= w

_{l}t

_{l }+ w

_{c}t

_{c}

I

_{u}= w

_{l}t

_{l }+ w

_{c}t

_{c}- f t

_{c}

T = t

_{c}+ t

_{l}

a) For solving the problem in ( I

_{u},I

_{s}), rewrite the budget constraint as a linear equation: I

_{s}= a - bI

_{u}. Find a and b.

b) Graph the problem in ( I

_{u},I

_{s}). Solve the problem in ( I

_{u},I

_{s}) when w

_{l}=1 , w

_{c}= 2, f= 1.5 , T= 3. Then find the corresponding solutions for (t

_{c},t

_{l})."

## The Attempt at a Solution

Using differentiation and the Kuhn - Tucker conditions I've concluded that

a= ((1- p) + p (I

_{s}))/(1-p)

b= p (I

_{s})/(1-p)I

_{u}

But I don't believe that's correct.

Hope you could help. Thank you for reading.