Maximize Happiness: Find Values of Expenses for Maximum Present Value

[beamthegreat]

Hi all. I was bored so I created the following spreadsheet: https://docs.google.com/spreadsheet...ei9iMjSFwupyYWjfOgzbbK-EBM/edit#gid=911477072

I'm currently having some trouble with this. I want to find the values of expenditures for each year such that the happiness present value is maximized (explained below).

Basically, change all the values of the orange cells to maximize the green cell. (Note: Your wealth cannot go below zero.) I cannot find a consistent way to do this.

Assumptions I've made:

1. Happiness is a direct function of expenditure.

2. The same expenditure will have a lower impact on happiness as you age
3. Increasing expenditures will have diminishing returns on happiness.

Using assumptions 2 and 3, I've created a function to modulate that, shown in Tables 1 and 2.

4. Happiness now is worth more than the identical value in the future.

Thus, the present value of happiness value can found by discounting the future "streams" of happiness given a happiness discount rate. (Exactly identical to finding present value in finance.) This is the value I want to maximize.

5. You start with a wealth of 100 units. You consume X units immediately at the beginning of the year and make a 7% return on investment. Meaning, if you consume 4 units, you'll have 102.72 unit the next year [(100-4)*1.07].

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How it works:

So basically, your happiness for the current year can be found by inputing your expenditure for that year into the diminishing return function, followed by the age function.

Example.)

Diminishing return function:

Expenditures/Base expenditures = X
Happiness(X)=(0.7232*LN(X) + 0.9265)/0.93

So if your base expenditure is $4 and you spend $8, you'll have a happiness of 1.54 units
X = 8/4 = 2
Happiness(2)=(0.7232*LN(2) + 0.9265)/0.93
or simply look at Table 1

Next, this value is multiplied with the happiness multi (age function), so if you're 41 years old, you multiply that with 0.90 (Table 2)

Thanks for reading all that! Any guidance on how I can solve this will be much appreciated.

 

[mfb]

You certainly want to end up with 0 wealth in this scenario (but probably not in real life). The final wealth can be expressed as function of the initial wealth and all the expenditures in a nice handy way. I don't think you ever run into the problem where negative expenditures (-> negative wealth) would be interesting, a requirement to end up with 0 is probably sufficient.
Generally speaking, the 7% return of investment makes earlier investments more expensive. You can multiply this factor with column E (happiness multiplier) to get rid of all effects of interest. Afterwards this should be a fitting problem that various tools should be able to handle.

I would expect the largest expenses somewhere around age 70. Cheap in terms of investment but not too much discounted from the happiness factor.

 

[beamthegreat]

You can multiply this factor with column E (happiness multiplier) to get rid of all effects of interest. Afterwards this should be a fitting problem that various tools should be able to handle.

Right. I could probably simplify that and also the discount rate to make a single happiness multiplier.

However, it's still not obvious to me what the function I should use to fit the expenditure. I'd imagine the optimal function would be non-linear.

 

[mfb]

Sure, it is a non-linear problem, but one that should be well-behaved. You can even set up more equations by comparing two different years only and finding the optimal distribution given a fixed spending for them. I would expect that the corresponding system of equations doesn't have an analytic solution, but it will certainly have a numeric solution.

 

[beamthegreat]

I would expect that the corresponding system of equations doesn't have an analytic solution, but it will certainly have a numeric solution.

I've finally found three solutions using quadratic, power, and linear equations to fit the expenditures. The former seems to provide the best results. But how do I know whether I really have the best solution? (How do I know using a cubic or quartic equation to fit the expenditure would not yield better results?)

Is there really no better way than to perform a numerical trial and error approach to these types of problems?

 

[mfb]

But how do I know whether I really have the best solution?

See if the derivative of total happiness is zero if you shift money from one year to another. For the optimum this should be zero for every pair of years.

 

[beamthegreat]

See if the derivative of total happiness is zero if you shift money from one year to another. For the optimum this should be zero for every pair of years.

Hmm, not sure what you mean. Wouldn't that provide the global maximum, but only within each fit?

So for example if I fit my expenditure to a linear function y=mx+b, I can vary b, solve for m (so that ending wealth = 0), and plot the amount of happiness with b. I used numerical methods to find the maximum, but finding the derivative of the total happiness to find the max would work too (if I'm understanding that correctly).

However, that only guarantees that the solution is optimal under linear fit. It turns out that quadratic equations yield higher happiness, so I'm wondering if there's any better way to do this than to arbitrarily use a random equation to force a fit.

 

[mfb]

Wouldn't that provide the global maximum, but only within each fit?

What do you mean by "within each fit"?
It is a condition for the global maximum, sure. I thought that's what you want to find.

If you only want to look at specific classes of spending profiles, then you need the derivative with respect to the free parameters of these classes.