Enunciation/notation in utility maximisation model

[vabm]

Hi Everyone. I am working on a model that I think can be defined as a utility optimisation problem but I'm struggling with the enunciation and notation.

The model should describe how the utilities of a set of agents A={1,2,...,n} increase with the availability of a larger set of product types P={1,2,...n}, while the utilities of a supplier S decrease with the size of P. The ideal state for A is P_n and for S is P₁ (one product type for each agent v/s a single product type for all). This assuming that each product type can potentially give a certain amount of utilities to each agent, so for A₁ utilities might be u(P₁) > u(P₂), for A₂ utilities might be u(P₁) < u(P₂), and for A₃ utilities might be u(P₁) = u(P₂), etc. The utilities perceived are different for each agent and they are not indifferent to any product, so if P needs to be reduced to e.g. 5 types, the problem of choosing element types is not irrelevant.

Then, the (optimisation-like) problem is to maximise utilities for A_n finding the best combination of products among the whole 'virtual' set P_n if the 'actual' set P has to have no more than x number of elements.

Is this more or less ok or am I completely lost? How can I express it as a utility maximisation equation? Any help/directions refining this would be great help.

 

[jvicens]

Hello,

Your model does seem to fall under the category of utility optimization. To express it as a utility maximization equation, you can use the following notation:

Let U(A) represent the total utility of the set of agents A. Similarly, let U(S) represent the total utility of the supplier S.

Now, let U(Pi) represent the utility of product type i for the agents in set A. This can be further broken down as follows:

U(Pi) = u(Pi) - c(Pi)

Where u(Pi) represents the utility perceived by the agents for product type i and c(Pi) represents the cost of that product type for the supplier.

Using this notation, your objective function can be expressed as:

Maximize U(A) - U(S)

Subject to the constraint:

Summation of c(Pi) <= x

This constraint ensures that the actual set P has no more than x number of elements.

I hope this helps. Let me know if you have any further questions or need any clarification. Good luck with your model!