Economic growth and economic development 248

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
© ª
(profit shares) represented by θ ≡ θif f ∈F,i∈H such that θif ≥ 0 for all f and i, and
P
i
i
i∈H θ f = 1 for all f ∈ F. The number θ f is the share of profits of firm f that will

accrue to household i.

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