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Let's say I would like to create some composite score for multiple of goods...

EDIT: More concise version based on @BrsG comments... I would come up with the following scenario. I have a consumer with preferences $U(x_1,x_2,x_3,x_4)$ and let's say I would like to combine $x_2, x_3, x_4$ into one good $CG$ such that $U(x_1, CG)$ esentially describes the same preferences. What would be the price of 1 $CG$ then? How to construct $CG$? Is there any problem?

I want to construct a metric for measuring units of food and its relevant price but right now I only know what milk, bread, and sweets are... Nothing else. I do not know what $1$ food is.

Let's assume the prices are following:

  • milk: $P_{Milk} = 3$
  • bread: $P_{Bread} = 5$
  • sweets: $P_{Sweets} = 10$

I think the food should be a function of the amount of purchased goods: $Food = f(Milk; Bread; Sweets)$

Is it arbitrary how I create the food score (or does it have any rules)? For example by some linear function (coefficients chosen randomly) such as

$$Food = 0.1*Milk + 0.3*Bread + 0.2*Sweets$$

Prices would then follow the same weights?

$$P_{Food} = 0.1*P_{Milk} + 0.3*P_{Bread} + 0.2*P_{Sweets}$$

Could I say that $1$ Food is any combination that gives me $Food = 1$?

Is there problem with preferences? What if the consumer likes to consume Milk, Bread, and Sweets together (Cobb-Douglas on all three)? Can I just assume the consumer would always choose the best combination of goods or is there some hidden problem?

Should the function for composite good reflect the optimization problem instead?

Following questions:

  • Is this intuition correct?
  • Can I choose nonlinear function to create it?
  • Are there additional rules that have to be followed?
  • What if prices are dependent on the quantity of goods to be purchased?
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  • $\begingroup$ It would be helpful if you stated what you really want to do - the underlying problem you want to solve by constructing the metric. Without that context it's difficult to get a good answer. $\endgroup$
    – BrsG
    Commented Feb 21, 2023 at 11:01
  • $\begingroup$ I just want to find a way how I could construct a metric for "food" which in turn could allow me to work only with it and not with Milk, Bread, and Sweets. Since goods in shop are usually discrete I want to find a way how to construct a good which is as good as if it was continuous. $\endgroup$
    – Athaeneus
    Commented Feb 21, 2023 at 11:49
  • $\begingroup$ But what is the background? What you need may likely depend on what problem you are trying to solve, ie why would you want to work only with food and not the components. The answer may differ depending on whether the metric is used in an optimisation problem (you mention preferences) or statistical work related to volumes or prices etc. $\endgroup$
    – BrsG
    Commented Feb 21, 2023 at 12:00
  • $\begingroup$ Well, to be honest, there is no special background... What I try to solve is to create for myself some better (internally consistent) understanding of economics. The basic model of consumer choice works with continuous goods and usually a statement is given that we assume infinite divisability of goods... Which I call bullshit... Goods are not infinitely divisible... But here is the solution: Composite goods... So I try to think of the variables in consumer choice as of composite goods ("food" instead of "milk"). So, optimization and statistical work could be both prefered answers) $\endgroup$
    – Athaeneus
    Commented Feb 21, 2023 at 12:07
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    $\begingroup$ Then you might want to look into how GDP components are constructed, in particular households consumption. You will get a price index (the deflator) and a volume series. And nominal consumption (food) and volume consumption will be the same, e.g. equal to one in the base period. $\endgroup$
    – BrsG
    Commented Feb 21, 2023 at 12:29

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