Optimal provision for a Public Good

[JoshMaths]

So a homework question so just hints would be appreciated.

we have each persons utility function as U_i = A^a_iB_i

Solve for optimum level of the public good, both goods cost £1/unit

Ok do you use Lagrangian?
Do you need to express it in the set function way as in u_i[A^a₁+A^a₂,(y_i - A^a_i)/p]

or do you already have the expression you need?

Thanks,

Josh

 

[JoshMaths]

Wow, viewed nearly 500 times and no answers or help?

Maybe this is why we need an Economics forum...

 

[D H]

Wow, viewed nearly 500 times and no answers or help?

A good number of those views were by web crawlers.

Maybe this is why we need an Economics forum...

You didn't receive help because there's not much to go on. You didn't describe your terms. What does that subscript i on the left hand side represent? How about on the right? Do those subscripts represent the same thing? What are A and B? Why the superscript on A? What motivated your attempt at a solution?

We can't read your mind. You did not give us a well worded problem with which we we can offer help you with.

 

[JoshMaths]

Hi D H thanks for the reply, i am sure you have helped me in the past, i remember the name.

I assumed that since it was in the Economics section people would be familiar with utility functions and the like, same way when you see an integral or power series you know it without someone having to go through it.

A = A public good B = a private good the 'i's are for each persons differing utility functions, normally constrained at two people so U1 and U2 and all remaining subscripts following that pattern.

A has a superscript same way anything does, because U is a function and A and B are variables.

It was a homework attempt so that is my motivation.

Thanks for your help.

 

[LudusRex]

As a scientist, my response would be that the optimal provision for a public good can be determined using economic principles and mathematical models. In this case, the utility function for each person, Ui, is given as AaiBi, where A and B represent individual preferences for the public good. To determine the optimal level of the public good, we can use the Lagrangian method to find the maximum value of the utility function subject to the budget constraint of the goods costing £1/unit. This will give us the optimal values for A and B, which can then be used to calculate the optimal provision for the public good.

Alternatively, we can express the utility function in the set function form as ui[Aa1+Aa2,(yi - Aai)/p], where p is the price of the public good and yi is the individual's income. Using this form, we can again use mathematical methods to find the maximum value of the utility function and determine the optimal provision for the public good.

In conclusion, the optimal provision for a public good can be determined using various economic and mathematical techniques, such as the Lagrangian method and set function form. These methods allow us to find the optimal level of the public good that maximizes overall utility for individuals while taking into account budget constraints.