How Do Lagrange Multipliers Optimize Profits in Variable Marketing Models?

[GFauxPas]

Help please with a problem from my modelling and optimization class. We're doing 2 variable optimization using Lagrange Multipliers. We're also discussing shadow prices.

The first part of this problem is to maximize profit using the price and advertising budget assumptions and data.

The data and assumptions are things like:

The cost of manufacture is 600 dollars per unit, and the wholesale price is 950 dollars

Other data are that units sold per month is 10 000, maximum advertising budget allowed by management is 100 000 per month.

Here are the data I don't know how to turn into workable function or equations, and could use help please:

In a test market, lowering the price by 100 increased sales by 50%

The advertising agency claims that increasing the advertising budget by 10 000 a month would result in a sales increase of 200 units a month.

How do I use these last data, i.e, how do I write them as equations?

edit: Oh whoops, I posted this in the wrong forum, can someone please move it?

 

[GFauxPas]

Okay here's what I have.

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.

I have:

costs incurred = 700*units - 50000t - (additional money on advertising)t
revenue = 950*units + additional profit from units sold due to increase in advertising + additional profit from units sold due to decrease in price
And past that...?

 

[Ray Vickson]

Okay here's what I have.

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.

I have:

costs incurred = 700*units - 50000t - (additional money on advertising)t
revenue = 950*units + additional profit from units sold due to increase in advertising + additional profit from units sold due to decrease in price
And past that...?

I don't think it is useful to write
"revenue = 950*units + additional profit from units sold due to increase in advertising + additional profit from units sold due to decrease in price".

Instead, let U = number of units to produce (also = number to sell, if we don't hold inventory). Then revenue = s*U, where s = selling price ($/unit)---presumably s = 950 is just one of the many possibilities. What is the total cost, in terms of U and some other variable or variables? What is the relationship between U and the other variable(s)?

 

[GFauxPas]

Instead, let U = number of units to produce (also = number to sell, if we don't hold inventory). Then revenue = s*U, where s = selling price ($/unit)---presumably s = 950 is just one of the many possibilities. What is the total cost, in terms of U and some other variable or variables? What is the relationship between U and the other variable(s)?

Well I have that u starts at 10000. then I add 200a, where a is how much I'm increasing the advertising budget in 10000s.
u = 10000 + 200a + ?
where the "?" is how much more I'm selling by reducing the price.
Also, the expenses are going to be equal to:
c = 700u + (50000 + 10000a)
where the quantity in parens is the money I'm spending on advertising.
It's the relation between pricing and everything else which leaves me confused.

edit: I'm also coming up with a REGION rather than a CURVE as a constraint, and I want a curve in order to use Lagrange multipliers.

 

[Evo]

The OP is going to try creating a new thread using the template, so please do not warn him about two threads, this one will be closed after the new thread is created.

The new thread is here https://www.physicsforums.com/threads/lagrange-multipliers-and-shadow-prices.775994/