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GFauxPas

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This is a homework in mathematical modeling and optimization; we're up to Lagrange multipliers and shadow prices.

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.

A main part of the exercise is to set up the relevant equations.

Time is in months, money in dollars.

Let

I have that

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:

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?

1. Homework Statement1. Homework Statement

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.

## Homework Equations

A main part of the exercise is to set up the relevant equations.

## 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 200*a*, where*a*is how much I'm increasing the advertising budget in 10000's of dollars.*u*= 10000 + 200*a*+ ?where the "?" is

Also, the expenses are going to be equal to:

*c*= 700*u*+ (50000 + 10000*a*)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?

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