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?