Lagrangian utility maximization with a ''complex'' summation

[Boardish]

Hello there! It's my first time posting here, I hope you guys will be good to me :).

I took a one year break to study a language abroad, and now it seems like I forgot everything math-wise. I'm preparing for a test and I'm having a really hard time doing the following problem.
I need to maximize the utility of a consumer with obviously a constraint.

https://www.physicsforums.com/attachments/3396._xfImport

https://www.physicsforums.com/attachments/3397._xfImport

https://www.physicsforums.com/attachments/3395._xfImport

Where C is the aggregate consumption, P the aggregate price and Z the level of expenditure.

I need to maximize the utility function (the consumption) C for the level of expenditure Z.

I also need to show with the first order conditions that
https://www.physicsforums.com/attachments/3398._xfImport

I found a somewhat similar question (http://mathhelpboards.com/calculus-10/lagrange-multipliers-summation-function-constraint-3821.html) but I have a hard time working with the exponents and the summation.

Thanks a lot!

EDIT : Or if someone can only explain to me how I actually derive the summation
https://www.physicsforums.com/attachments/3396._xfImport
I'll try to figure everything else :)

 

[jvicens]

Hello there! It's great to see you reaching out for help with your math problem. Don't worry, forgetting things after taking a break is completely normal and it's great that you're taking the time to study and prepare for your test.

To start, let's break down the problem step by step. You are trying to maximize the utility function for a consumer, which is represented by the variable C. This function is subject to a constraint, which is represented by the variable Z. In order to maximize C, we need to find the optimal level of expenditure Z.

To solve this problem, we can use the method of Lagrange multipliers. This method involves creating a new function, called the Lagrangian, by combining the original utility function and the constraint function. In this case, the Lagrangian would be:

L = C - λ(Z)

Where λ is the Lagrange multiplier. We then take the partial derivatives of the Lagrangian with respect to C and Z, and set them equal to 0.

∂L/∂C = 1 - λ = 0
∂L/∂Z = -λ = 0

Solving for λ, we get λ = 1. This means that the optimal level of expenditure Z is equal to 1. To show this using the first order conditions, we need to plug in the value of λ into the original constraint function:

Z = 1 = ∑P^(1/α) * C^(β/α)

To solve for C, we can take the natural log of both sides and rearrange the equation to get:

C = (Z/∑P^(1/α))^(α/β)

This is the equation for the optimal level of consumption C, given the constraint Z.

Now, let's move on to the summation function you mentioned. In order to derive this summation, we need to use the chain rule. First, let's take the natural log of both sides of the original constraint function:

ln(Z) = ln(∑P^(1/α) * C^(β/α))

Next, we can apply the chain rule to the right side of the equation:

d(ln(Z))/dC = β/α * C^((β/α)-1)

We can then rearrange the equation to get:

C^((β/α)-1) = d(ln(Z))/dC * α/