Composite good and preferences

Usually in economics, we could see some versions of multiplicative utility:

$$U(\boldsymbol{x}) = x*y$$

The thing is that most of the time an additional statement is given that $$y$$ is some composite good/commodity with some price $$P_y$$.

If I have preferences like this:

$$U(\boldsymbol{x}) = x_1*(x_2*x_3 + x_2-x_3)$$

Can I say that composite good $$y = x_2*x_3 + x_2-x_3$$ and related price $$P_y = P_2 x_2 + P_3 x_3$$ ?

Does this mean I do not have to know the final preferences of consumer in order to create a composite good, if we know that price of composite good is just a linear function of prices and quantities bought?

Or does this mean I do have to know the preferences of consumer? Because I know what one $$y$$ is equal to, therefore, I have to know what would be the optimum value of $$x_2$$ ; $$x_3$$ in order to determine its price?

Both approaches actually co-exist.

I do have to know the preferences of consumer

Historically, aggregate goods were defined as those occurring in the utility function (whenever this happens). The main drawback of this approach was soon highlighted by Pu (1946, p.299): "If this criterion is strictly adhered to, one will find in most cases that either no aggregates can be found to fulfil this criterion, or, in the case where this criterion is satisfied by the manipulation of the construction of aggregates, the aggregates will become such monsters that they are completely void of any economic significance".

I do have to know the preferences of consumer

In contrast, other economists take for granted the given economic or statistical definition of the aggregate good or price, and construct aggregate utility (or production) functions as $$V(x_1,\textbf{x}_2) = \max_{x_2} \{ U(x_1,x_{21},x_{22},..,x_{2K}) : a(x_2)=\mathbf{x}_2 \},$$ where the expression of the aggregator $$a: \mathbb{R}^K \rightarrow \mathbb{R}$$ is given, as well as its value $$\mathbf{x}_2$$. For a contribution with this approach, see Blackorby and Schworm (1988). (Note that in general the optimal solution $$x_2^*(\mathbf{x}_2)$$ is not unique, but is a multivalued relationship.) With the aggregate quantity is $$\mathbf{x}_2$$, the corresponding price could be $$\mathbf{p}_2 = \sum_k p_{2k}x_{2k} /\mathbf{x}_2$$. (This is not fully compatible with your $$P_y$$ unless you divide it by $$y$$ to ensure that $$P_yy=P_2x_2+P_3x_3.$$ )

A third approach in this literature explicitly considers aggregation errors, and replaces all the elementary goods (or prices) by their aggregate counterparts and an error term: $$x_{2k}=\textbf{x}_2\rho_{2k}$$, so that the utility (production) function becomes: $$W(x_1,\textbf{x}_2;\rho) = U(x_1,x_{21},x_{22},\ldots,x_{2K}) = U(x_1,\textbf{x}_2\rho_{21},\ldots,\textbf{x}_2\rho_{2K}),$$ where the vector $$\rho$$ denotes the share of each elementary good in the aggregate: $$\rho = (x_{21},x_{22},\ldots,x_{2K}) / \textbf{x}_2.$$ Restrictions on the distributions of the unobserved $$\rho$$ are useful to achieve some interesting properties on the aggregate demand and supply functions. See for instance Lewbel (1996) for a contribution with this approach.

Blackorby, Charles, and William Schworm. “The Existence of Input and Output Aggregates in Aggregate Production Functions.” Econometrica, vol. 56, no. 3, 1988, pp. 613–43.

Lewbel, Arthur. “Aggregation Without Separability: A Generalized Composite Commodity Theorem.” The American Economic Review, vol. 86, no. 3, 1996, pp. 524–43.

Pu, Shou Shan. “A Note on Macroeconomics.” Econometrica, vol. 14, no. 4, 1946, pp. 299–302.

• Thank you very much! May I ask further for the Pu's argument? If I understand it correctly, the main idea is that the utility function should depend only on the magnitudes of composites not on their distribution? Would the perfect knowledge of someone's utility function solve this problem, allowing us to work with dependence on distribution? The core idea would be the following: I would assume that the consumer always pays the least required amount for $1 y$. Then, I would know all the possible combinations that would give me this $1 y$ and could deduct the price of it (thru expansion path)? Feb 27 at 8:53
• My interpretation of Pu's statement, is that aggregates like $y = x_2x_3 + x_2-x_3$ or (why not) $y = x_2^{x_3 + \sqrt x_2}F(x_3)$, where $F$ denotes any increasing function, are "monsters", because no statistical institute will ever compute such individual specific aggregates, and utility functions cannot be observed anyway. It seems to me that the approach you describe is close to the one behind $V(x_1,\mathbf{x_2})$. Feb 27 at 22:32