Economic growth and economic development 240

Introduction to Modern Economic Growth
The hypothesis that it is a solution to the first problem also implies that
X
X
X
X
∗∗ i
αi ai +
αi b (p∗∗
≥
αi ai +
αi b (p∗ ) (y ∗ )i
α ) (yα )
i∈H

(5.8)

i∈H

X

i∈H

α

i

∗∗ i
b (p∗∗
α ) (yα )

i∈H

≥

X

i∈H

α b (p ) (y ∗ )i .
i

∗

i∈H

∗∗

Then, it can be seen that the solution (p , y ∗∗ ) to the Pareto optimal allocation
/ HM . In view of this and the choice of (p∗ , y ∗ ) in
problem satisfies y i = 0 for any i ∈
(5.7), equation (5.8) implies

αM b (p∗∗
α )

X

(yα∗∗ )i ≥ αM b (p∗ )

i∈H

∗∗
b (p∗∗
α ) (yα )

∗

∗

X

(y ∗ )i

i∈H

≥ b (p ) (y ) ,

which contradicts equation (5.8), and establishes that, under the stated assumptions,
any Pareto optimal allocation maximizes the utility of the representative household.
Ô
5.3. Infinite Planning Horizon
Another important microfoundation for the standard preferences used in growth
theory and macroeconomics concerns the planning horizon of individuals. While,
as we will see in Chapter 9, some growth models are formulated with finitely-lived
individuals, most growth and macro models assume that individuals have an infiniteplanning horizon as in equation (5.2) or equation (5.21) below. A natural question
to ask is whether this is a good approximation to reality. After all, most individuals
we know are not infinitely-lived.
There are two reasonable microfoundations for this assumption. The first comes
from the “Poisson death model” or the perpetual youth model, which will be discussed
in greater detail in Chapter 9. The general justification for this approach is that,
while individuals are finitely-lived, they are not aware of when they will die. Even

somebody who is 95 years old will recognize that he cannot consume all his assets,
since there is a fair chance that he will live for another 5 or 10 years. At the simplest
level, we can consider a discrete-time model and assume that each individual faces
a constant probability of death equal to ν. This is a strong simplifying assumption,
since the likelihood of survival to the next age in reality is not a constant, but a
226