This is a homework in mathematical modeling and optimization; we're up to Lagrange multipliers and shadow prices. 1. The problem statement, all variables and given/known data A manufacturer of PCs currently sells 10,000 units per month of a basic model. The cost of manufacture is 700$/unit, and the wholesale price is $950. The cost of manufacture is $700/unit, and the wholesale price is $950. During the last quarter the manufacturer lowered the price $100 dollars in a few test markets, and the result was a 50% increase in sales. The company has been advertising its product nationwide at a cost of $50000 per month. The advertising agency claims that increasing the advertising budget by $10000 a month would result in a sales increase of 200 units a month. Managemeny has agreed to consider an increase in the advertising budget to no more than $100000 a month. I have to find the price and advertising budget that will maximize profits, among other analyses. 2. Relevant equations A main part of the exercise is to set up the relevant equations. 3. The attempt at a solution Time is in months, money in dollars. Let u be the units sold and produced. I have that u starts at 10000. then I add 200a, where a is how much I'm increasing the advertising budget in 10000's of dollars. u = 10000 + 200a + ? where the "?" is how much more revenue edit: how many more units sold I'm getting by reducing the price. Also, the expenses are going to be equal to: c = 700u + (50000 + 10000a) where the quantity in parenthesis is the budget put to advertising. I don't know how to deal with the price variable and everything connected to it; what is it I'm increasing by 50%? The relation between pricing and everything else has me confused. Also, I have that the constraint is [itex]a \in [0..5][/itex], but that's an interval, not a curve, and don't I need the constraint to be a curve to use Lagrange multipliers?