- #1
Dev06
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Well, I've been working on this problem, but I can't get the right path to the solution.
"Consider the following version of the Ehrlich economic model of crime where an individual has the expected utility function:
U = p ln(Iu) + (1 - p) ln (Is).
p = objective probability of being caught
Iu= Criminal's income if caught
Is= 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" (tc) and "time for legal work" (tl). So T = tc + tl.
If the criminal choose to work legally, he gets an income of wl> 0 per unit of time; while engaged in crime he gets an income of wc> wl per unit of time.
If the individual is caught committing crimes, he get a penalty f > wc - wl per unit of time.
So, the income of an individual who commits a crime but is not arrested is Is = wltl + wctc while if he get arrested
Iu = wltl + wctc - f tc .
Then, the criminal divides his time between crime and work legally, an his problem is given by:
max U = p ln(Iu) + (1 - p) ln (Is). with tc, tl[itex]\geq[/itex]0
s.t. Is= wltl + wctc
Iu = wltl + wctc - f tc
T = tc + tl
a) For solving the problem in ( Iu,Is), rewrite the budget constraint as a linear equation: Is = a - bIu . Find a and b.
b) Graph the problem in ( Iu,Is). Solve the problem in ( Iu,Is) when wl=1 , wc= 2, f= 1.5 , T= 3. Then find the corresponding solutions for (tc,tl)."
Using differentiation and the Kuhn - Tucker conditions I've concluded that
a= ((1- p) + p (Is))/(1-p)
b= p (Is)/(1-p)Iu
But I don't believe that's correct.
Hope you could help. Thank you for reading.
Homework Statement
"Consider the following version of the Ehrlich economic model of crime where an individual has the expected utility function:
U = p ln(Iu) + (1 - p) ln (Is).
p = objective probability of being caught
Iu= Criminal's income if caught
Is= 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" (tc) and "time for legal work" (tl). So T = tc + tl.
If the criminal choose to work legally, he gets an income of wl> 0 per unit of time; while engaged in crime he gets an income of wc> wl per unit of time.
If the individual is caught committing crimes, he get a penalty f > wc - wl per unit of time.
So, the income of an individual who commits a crime but is not arrested is Is = wltl + wctc while if he get arrested
Iu = wltl + wctc - f tc .
Homework Equations
Then, the criminal divides his time between crime and work legally, an his problem is given by:
max U = p ln(Iu) + (1 - p) ln (Is). with tc, tl[itex]\geq[/itex]0
s.t. Is= wltl + wctc
Iu = wltl + wctc - f tc
T = tc + tl
a) For solving the problem in ( Iu,Is), rewrite the budget constraint as a linear equation: Is = a - bIu . Find a and b.
b) Graph the problem in ( Iu,Is). Solve the problem in ( Iu,Is) when wl=1 , wc= 2, f= 1.5 , T= 3. Then find the corresponding solutions for (tc,tl)."
The Attempt at a Solution
Using differentiation and the Kuhn - Tucker conditions I've concluded that
a= ((1- p) + p (Is))/(1-p)
b= p (Is)/(1-p)Iu
But I don't believe that's correct.
Hope you could help. Thank you for reading.