Kuhn-Tucker Optimization Problem

[Dev06]

Well, I've been working on this problem, but I can't get the right path to the solution.

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_lt_l + w_ct_c while if he get arrested
I_u = w_lt_l + w_ct_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$\geq$0

s.t. I_s= w_lt_l + w_ct_c
I_u = w_lt_l + w_ct_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.

 

[Ray Vickson]

Well, I've been working on this problem, but I can't get the right path to the solution.

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_lt_l + w_ct_c while if he get arrested
I_u = w_lt_l + w_ct_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$\geq$0

s.t. I_s= w_lt_l + w_ct_c
I_u = w_lt_l + w_ct_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.

Your expressions for a and b are incorrect.

I hate this question's notation, so I re-cast it as
max p*log(x1) + (1-p)*log(x2),
s.t.
x1 = w1*t1+w2*t2-f*t2
x2 = w1*t1 + w2*t2
T = t1+t2,
all vars >= 0.
You want to use the three constraints to eliminate x1, and so express x2, t1 and t2 in terms of x1. That is a simple linear-equation-solving exercise.

RGV

 

[Dev06]

Thank you for your reply. It was very helpful .