# Optimization, Lagrange multipier

A Parent has two children living in tow cities A & B. Cost of living in B is 3 times cost of living in A. The child in city A has income $\$3000$and child in city B has income of$\$9000$.The parent has $\$4000$to give. He will give in way that will maximize the function U= Ca* Cb where Ca is consumption of child in city A and Cb is the consmption of child in city B.How will she divide the money? I have assumed that Child A gets X and child B gets Y. Setting up the Lagrangian as $$l(m,x,y)=(3000+x) [(9000+y)/3] - m (x+y-4000)$$$x+y=4000, x>0,y>0$. I am not sure how the conditions$x>0$and$y>0$will affect the solution, but when I solved without these restrictions I got$x=5000$which is obviously inadmissible. • There is something odd in this problem. The answer is totally standing alone: the relative cost of living doesn't play any role in the result. Anyway your answer is right. According this rules x = 4000, y=0 is the optimal solution in the range of your model and this is obvious because the formulation is insensitive to the costs. – Tetis Jan 7 '14 at 20:37 ## 1 Answer Lagrange multipliers is overkill when one variable is a function of the other, as here where$y=4000-x$. Then it's just a matter of maximizing $$(3000+x)(9000+4000-x)\frac 13 =$$ The factor of$1/3$is a constant and irrelevant for the optimization (so it doesn't matter which city is the expensive one, or by how much), and what is left is just the product of two linear expressions whose sum is constant. It is clear even without calculating anything that such a function has its maximum when the two factors are equal -- its graph is a parabola, so its apex is halfway between the roots. So ideally the mother should split her gift such that the two children's total is as close to each other as possible. The$x=5000$solution you've already found corresponds to giving$5000$to child A and$-1000$to child B which make both children end at 8000 -- but since you can't give negative gifts, the best thing the mother can do is giving all 4000 to A, which will push the product-of-consumptions as high up the parabola as she can. However, it really doesn't make much sense to add an income (which is per some-unit-of-time) to a one-time gift. So it is tempting to understand the problem differently: The children's current incomes are red herrings, just like the cost-of-living factor already is, and what we're being asked to maximize is the product of the additional consumption each child can make from the gift. Then we're maximizing $$x(4000-x)\frac13$$ and just as before the$\frac13$factor is irrelevant, so the way to maximize the product is to give each child the same dollar amount, that is, \$2000.

Which kind of parent gets the crazy idea to maximize the product of their children's purchase power, though? It does lead to seeking equality between the children, which is a noble and fair goal, but why on earth phrase it as maximizing a product?

• Hi Henning..thanks for the detailed explanation..$2000 each is among the list of options,so i believe you are right..but if am still going to maximization of of Ca*Cb, then$2000 each will make this 14,666,666 . If i give \$4000 to A only,then it becomes 7000*3000=21,000,000 which is more.. i am not able reconcile these two facts... the parent is very crazy – kangkan Jan 8 '14 at 10:17