Economic growth and economic development 521

Introduction to Modern Economic Growth
Throughout this chapter, we assume that the economy admits an infinitely-lived
representative household, with household size growing at the exponential rate n.
The preferences of the representative household at time t = 0 are given by
#
"
Z ∞
c (t)1−θ − 1
dt.
exp (− (ρ − n) t)
(11.1)
U=
1−θ
0
Labor is supplied inelastically. The flow budget constraint facing the household can
be written as
(11.2)

a˙ (t) = (r (t) − n)a (t) + w (t) − c (t) ,

where a (t) denotes assets per capita at time t, r (t) is the interest rate, w (t) is the
wage rate per capita, and n is the growth rate of population. As usual, we also need
to impose the no-Ponzi game constraint:
ẵ
Z t
áắ
[r(s) n] ds
0.
(11.3)
lim a(t) exp −
t→∞

0

The Euler equation for the representative household is the same as before and

implies the following rate of consumption growth per capita:
1
c˙ (t)
= (r (t) − ρ).
c (t)
θ

(11.4)

The other necessary condition for optimality of the consumer’s plans is the transversality condition,
(11.5)

ẵ
Z t
áắ
[r(s) n] ds
= 0.
lim a(t) exp −

t→∞

0

As before, the problem of the consumer is concave, thus any solution to these necessary conditions is in fact an optimal plan.
The production sector is similar to before, except that Assumptions 1 and 2

are not satisfied. More specifically, we adopt the following aggregate production
function:
Y (t) = AK (t) ,
with A > 0. Notice that this production function does not depend on labor, thus
wage earnings, w (t), in (11.2) will be equal to zero. This is one of the unattractive
features of the baseline AK model, but will be relaxed below (and it is also relaxed
in Exercises 11.3 and 11.4). Dividing both sides of this equation by L (t), and as
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