Economic growth and economic development 248

Let X≡Q i∈HXi 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 ≡ {

Introduction to Modern Economic Growth sequences Let X≡Q

i∈HXi 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 ≡ {ωi

}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 : Xi → R, such that whenever x0 %i x, we have

ui(x0)≥ ui(x) The domain of this function is Xi

⊂ 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 Yf, which specifies what levels of output firm f can produce from specified levels of inputs In other words, yf≡n

yfjo∞ j=0 is a feasible production plan for firm f if yf ∈ Yf For example, if there were only two commodities, labor and

a final good, Yf would include pairs (−l, y) such that with labor input l (hence

a negative sign), the firm can produce at most as much as y Let Y≡Q

f ∈FYf

represent the aggregate production set in this economy and y≡©

yfª

f ∈F such that

yf

∈ Yf 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 (profit shares) represented by θ≡©

θifª

f ∈F,i∈H such that θif ≥ 0 for all f and i, and P

i∈Hθif = 1 for all f ∈ F The number θif is the share of profits of firm f that will accrue to household i

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