Introduction to Modern Economic Growth sequences Let X ≡ Q i∈H X i be the Cartesian product of these consumption sets, which can be thought of as the aggregate consumption set of the economy We also use the notation x ≡ {xi }i∈H and ω ≡ {ω i }i∈H to describe the entire consumption allocation and endowments in the economy Feasibility of a consumption allocation requires that x ∈ X Each household in H has a well defined preference ordering over consumption bundles At the most general level, this preference ordering can be represented by a relationship %i for household i, such that x0 %i x implies that household i weakly prefers x0 to x When these preferences satisfy some relatively weak properties (completeness, reflexivity and transitivity), they can equivalently be represented by a real-valued utility function ui : X i → R, such that whenever x0 %i x, we have ui (x0 ) ≥ ui (x) The domain of this function is X i ⊂ R∞ Let u ≡ {ui }i∈H be the set of utility functions Let us next describe the production side As already noted before, everything in this book can be done in terms of aggregate production sets However, to keep in the spirit of general equilibrium theory, let us assume that there is a finite number of firms represented by the set F and that each firm f ∈ F is characterized by a production set Y f , which specifies what levelsn of ooutput firm f can produce from ∞ is a feasible production plan specified levels of inputs In other words, y f ≡ yjf j=0 f f for firm f if y ∈ Y For example, if there were only two commodities, labor and a final good, Y f would include pairs (−l, y) such that with labor input l (hence Q a negative sign), the firm can produce at most as much as y Let Y ≡ f ∈F Y f â ê represent the aggregate production set in this economy and y ≡ y f f ∈F such that y f ∈ Y f for all f , or equivalently, y ∈ Y The final object that needs to be described is the ownership structure of firms In particular, if firms make profits, they should be distributed to some agents in the economy We capture this by assuming that there exists a sequence of numbers â ê (profit shares) represented by θ ≡ θif f ∈F,i∈H such that θif ≥ for all f and i, and P i i i∈H θ f = for all f ∈ F The number θ f is the share of profits of firm f that will accrue to household i 234