# Lagraingian constrained optimization problem

1. Dec 15, 2005

Im not sure if this is the right place, but I have an optimization problem where I assume we are supposed to use the Lagraingian method:

Consider the labour supply problem for an individual over an entire year. Suppose the individuals utility is described by the function U = (C^0.5) x (H^0.5). Further, suppose that the individuals combined time/income constraint is given by the equation C + wh = Tw, where T = 8760 is the number of hours in a standard year. Suppose the initial wage rate is $10/hour. Suppose that the government imposes a progressive income tax of 10% on all income above$ 25,000. That is, the individual pays no tax on the first $25,000 they earn. However, any income above$ 25,000 per year is taxed at a rate of 10%. Given this tax, what are the individuals optimum choices of consumption (C) and leisure (H).

I know how to do the problem without the tax, but i have no idea how to deal with the tax. Any help is greatly appreciated (its gonna be on a final i have tomorow, so the faster the better. Thanks.

2. Dec 15, 2005

any ideas? anyone?

3. Dec 16, 2005

### HallsofIvy

Staff Emeritus
It might help if you would tell us what your variables, C, H, wh (or is that c*h?) mean! I can see no relationship between U and the tax!

4. Dec 16, 2005

### uart

I agree with Hall's, the variables involved are not very clearly explained and the mixed case (H vs h etc) is confusing.

Anyway, I assume the consumption contraint, c = (T-h)w, is just for the non-taxed case. That is where w is a constant of \$10.00 per hour.

This case is very easily solved, no need to use Lagrange multiplers (though you can if you wish). You just need to subst the constraint into the functional to make it a simple function of one variable "h" and the find the maxima.

If you do this it will tell you that you have to work 12hrs per day, 7 days per week and 365 days per year just to be maximally happy, Rats! Thank God for taxation however (j/k) because when you repeat the problem with taxation included then at least it tells you to work a little bit less.

To handle the case with taxation it's best to split the problem into two regions (case 1 for annual income <= 25000 and case two for annual income > 25000). Remember that if a local maxima for a given case does not fall within the required region for that case then the actual maximum will occur at the boundary.

For case 2 (income>25000) use the modified constraint of c = 2500 + 9(T-h) ok.