Economic growth and economic development 256

Introduction to Modern Economic Growth
is to consider the Arrow securities already discussed in Chapter 2. Arrow securities
are an economical means of transferring resources across different dates and different
states of nature. Instead of completing all trades at a single point in time, say at
time t = 0, households can trade Arrow securities and then use these securities
to purchase goods at different dates or after different states of nature have been
revealed. While Arrow securities are most useful when there is uncertainty as well
as a temporal dimension, for our purposes it is sufficient to focus on the transfer of
resources across different dates.
The reason why sequential trading with Arrow securities achieves the same result
as trading at a single point in time is simple: by the definition of a competitive
equilibrium, households correctly anticipate all the prices that they will be facing at
different dates (and under different states of nature) and purchase sufficient Arrow
securities to cover the expenses that they will incur once the time to trade comes.
In other words, instead of buying claims at time t = 0 for xhi,t0 units of commodity
i = 1, ..., N at date t0 at prices (p1,t0 , ..., pN,t ), it is sufficient for household h to have
P
h
an income of N
i=1 pi,t0 xi,t0 and know that it can purchase as many units of each
commodity as it wishes at time t0 at the price vector (p1,t0 , ..., pN,t0 ).

This result can be stated in a slightly more formal manner. Let us consider
a dynamic exchange economy running across periods t = 0, 1, ..., T , possibly with
T = ∞.3 Nothing here depends on the assumption that we are focusing on an
exchange economy, but suppressing production simplifies notation. Imagine that

there are N goods at each date, denoted by (x1,t , ..., xN,t ), and let the consumption
of good i by household h at time t be denoted by xhi,t . Suppose that these goods
are perishable, so that they are indeed consumed at time t. Denote the set of
households by H and suppose that each household h ∈ H has a vector of endowment

¢
¡ h
ω 1,t , ..., ω hN,t at time t, and preferences given by the time separable function of the
form

T
X
t=0

3When

¡
¢
β th uh xh1,t , ..., xhN,t ,

T = ∞, we assume that all the summations take a finite value.

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