How to pick from several products/services to maximize utility considering the cost of each option?

Question

What is the proper way to consider the utility of a product/service versus its cost in order to make the best rational decision to get the "most for my dollar" if there are several choices for me to choose from?

Let me explain with a made-up example. I am looking at some widgets. x represents the utility I would get from each widget if I were to buy it and y represents the cost of each of those widgets.

x = {1,2,3,4,5,6,7,8}

y = {1,2,3,4,100,150,200,300}

Note here that I am only considering widgets that I can afford. So I can buy any of them. But I only want to buy the one that is "worth it" the most. Here my intuition says, buy the fourth widget....so obvious. The cost of the other four is not worth their price. It is not worth it to pay $96 to get only one more unit of utility where I can pay only $4 for the first units of utility. I need help formalizing this "intuition" and quantifying it so that it can be applied to more complex decisions.

Now some real-world examples. And yes, these are real decisions I am looking at right now. But I am looking for a general method which I can apply to any such decision. All of the data presented here from now on is real.

I am looking to buy a new TV. Assume that my utility is directly correlated to the size of the TV. The bigger, the better; the happier I am. But bigger TV's cost more (mostly) and I don't have an unlimited supply of money. So my question is, given a bunch of TV models and their prices, how do I decide which model to purchase so that I get maximum amount of utility considering how much money I would have to pay? The prices look something like this.

The curve is not monotonically increasing because of sales on some models. Assuming, y representing the vertical axis (the prices) and x representing the horizontal axis (size of the TV);

1. Should I be looking at the average cost, which would be the cost per inch for each TV, and pick the TV with the smallest average cost? This is equivalent to minimizing y/x.

2. Should I be looking at the marginal cost of the next inch? For all of the TV models, what would be the cost for the next inch? This would be the quantity dy/dx (the first derivative of y with respect to x). But then do I want to minimize this quantity or do I want this quantity to be as close to zero as possible (minimize abs(dy/dx))?

3. Or do I look at the marginal cost of the next inch...per inch. This would be the quantity (dy/dx)/x. If yes, is this to be minimized or what?

4. Or is there another quantity to look at?

Some more examples!

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This second plot shows the rental price of a car versus the utility I would get out of renting this car. The utility is assigned (entirely subjectively, of course) by me considering how much space it has, how easy/hard it is drive, the cost of gasoline and its mileage, etc. Which car should I rent?

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This third example shows the premium I would pay for auto insurance versus how much total liability coverage the plans gives me. Of course, the insurance provider, the car, other coverage options, etc. all remain fixed. The only variable I changed was the liability. More liability coverage is better. Which plan should I buy?

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This example is different. This example shows the premiums versus the comprehensive/collision deductible I would have to pay if something happened. In this case the x-axis is not increasing but rather decreasing with my utility. I want the deductible to be as small as possible. What kind of a relationship between the deductible and my utility makes sense? Should I consider

utility = 1/deductible

or something like

utility = -1*deductible?

The relationship is inverted but how? And of course, how can I decide which coverage to get? Note that the fourth data point (to the far right) is outside the range of the plot. That point corresponds to no collision/comprehensive coverage. So I set the "deductible" to the value of the car itself because in the worst case scenario, my out-of-pocket cost would be the complete value of the car if the car was completely destroyed. The premium for that option is $474. So the data point is at coordinates (12000,474).

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Additional Question 1: This question is more basic than the one I asked above. To even just visually look at such data, how to decide whether it should be y vs. x or x vs. y? Should it be price vs. utility or the reverse? Similarly, looking at quantities like mileage for a car, should I look at miles/gallon or gallon/miles? They are both the same mathematically. Maximizing the first is the same as minimizing the second. But they get distorted when their reciprocals are taken so plots can be very different. Which one to pick?

Additional Question 2: This question is a multivariate version of my post. How to make a decision when more than one variable is involved? For example, for my insurance quotes, let's say I start varying liability and underinsured/uninsured, or even comprehensive/collision deductibles (with their inverse relationship)? Which function of my utility should I be playing with here so that I get the "best" combination?

Answer 1

I think this is a great question. There is no simple answer for all real-world cases, but that is usually because in real-world situations you cannot accurately assign simple utility values to individual items.

That said, my basic answer to your basic question (separated into your questions 1-4 about the TV) is that you should try to maximize utils (utility units) per dollar, which is equivalent to minimizing dollars per util. This is, crucially, assuming that you have accurately quantified the utilities (which, as I'll mention below, I don't think you have), and that you have already restricted focus to cases which meet your minimum need in terms of utility. (For instance, in the insurance case, you would begin by filtering out any options that don't offer the minimum amount of coverage required by law in your jurisdiction.)

I don't think that attempting to optimize the marginal utility is a good way to go about it. The reason is that each potential purchase is effectively a world unto itself, which you are considering in relation to ALL other potential purchases, not just the next-most-expensive or next-most-useful one. If your utility numbers are accurate, then you want to consider each item's price and utility independently of all others.

Another reason this is so is that, if you look at these slopes, your assessment of the relative value of items might change depending on the existence or nonexistence of intermediate options. Measuring the slope from A to B and comparing that to the slope from B to C might give a different result from just looking at the slope from A to C. But it would be very strange to say, for instance, that option A is better than option C when option B is not present, but that adding option B somehow makes option C better than option A (due to a change in the intermediate slopes). This sort of absurd result is avoided if you consider each item's utility and cost individually.

The other fundamental component to a thoroughgoing understanding of utility is the concept of opportunity cost. That is, when considering the purchase of a TV, you have to consider not only the utility of the TV, but the utility you could obtain by spending less on the TV and doing something else with the money. Accurate computation of utility values for a dense web of such comparisons can be nearly impossible, because it requires you to answer a large of number of difficult questions like how many scoops of ice cream you would be willing to give up for an extra 1 megapixel of resolution on your cellphone camera.

Nonetheless, attempting such comparisons can be a valuable way to draw your attention to opportunity costs and help you to consider them. In the TV example, for instance, supposing you could spend an extra $100 to get an extra 5 inches of TV size, you may want to do a rough calculation along these lines: "I can go out to a nice dinner for $20. Would I rather have the 55 inch TV or go out to dinner 5 times?" Thus, to modify my advice above about what to maximize, what you're trying to maximize here is not the utility per dollar of the TV, but the utility per dollar of your life as a whole.

My guess is that in many cases, honest answers to these kinds of questions will make you realize that, despite what you said, your utility calculations were inaccurate. I am highly skeptical that anyone would really rather increase their TV size from, say, 100 to 110 inches rather than go out to their favorite restaurant five times. For almost any good, there simply comes a point at which your utility diminishes. Thus you are probably wrong if you assume your utility is directly correlated to the size of the TV; in fact, it is almost surely the case that your actual enjoyment of ever larger TVs will decrease the larger they get.

This is perhaps most clear with your example of car liability coverage. If I had liability coverage for, say, a billion dollars, I would certainly not pay an extra $100 a month to get two billion dollars worth of coverage, even if doing so would result in a better ratio of coverage to dollars. There is simply no conceivable case in which I could make use of the extra billion dollars of coverage, so it doesn't have any actual utility for me. Similarly, there is no use buying, say, a metric ton of ice cream, even if the price per kilogram is very low and you really like ice cream, because you just can't actually extract enough utility from it.

Assigning utility is a subjective matter, and thus a difficult problem to solve, but what it comes down to is that if you really want to maximize your utility, you need to do the difficult but fascinating and rewarding introspective work of accurately assessing what utility you assign to things. In your comment, you say: "I assign utility to each product based on how much satisfaction I think I would gain by owning that product." But in your examples (except perhaps the rental car one) you haven't actually done that; you've simply equated utility with some easily measurable feature of the good (inches in the TV, dollars of liability coverage, etc.). If you're able to actually convert such objective measures into accurate measures of your subjective utility, then I think you will indeed be able to apply objective reasoning to those utility numbers. The results won't be ironclad, but that's nothing to worry about, because even applying objective reasoning to objective data doesn't generally produce ironclad results.

To address your "additional questions" at the end, in reverse order

Question 2: The beauty of doing your computations in utils is that things like liability versus collision are all shoved under the rug. As I suggested above, the hard part is converting objective numbers like dollars of liability coverage into accurate measurements of subjective utility. There is no way to objectively find the "best" combination when considering only things like dollars of coverage. You have to convert them to utils, and once you do, everything is monovariate.

Question 1: For personal decisions like the ones you show here, I believe it makes the most sense to put price on the X axis and utility on the Y axis. The reason is that price is the independent variable because you are choosing how much money to spend; utility is the dependent variable because the utility you get depends on what you buy. In a typical graph of a function, x is what you put in and y is what you get out. In making decisions about what products to purchase, dollars are what you put in and utility is what you get out.

Answer 2

how do I decide which model to purchase so that I get maximum amount of utility considering how much money I would have to pay?"

You should have a fixed budget for buying a TV, based on a small portion of your available cash, and not spend more.