Economic growth and economic development 260

Introduction to Modern Economic Growth
This maximization problem can be solved in a number of different ways, for
example, by setting up an infinite dimensional Lagrangian. But the most convenient
and common way of approaching it is by using dynamic programming.
It is also useful to note that even if we wished to bypass the Second Welfare
Theorem and directly solve for competitive equilibria, we would have to solve a
problem similar to the maximization of (5.18) subject to (5.19). In particular, to
characterize the equilibrium, we would need to start with the maximizing behavior
of households. Since the economy admits a representative household, we only need
to look at the maximization problem of this consumer. Assuming that the representative household has one unit of labor supplied inelastically, this problem can be
written as:
max

{c(t),k(t)}∞
t=0

subject to some given a (0) and
(5.20)

∞
X

β t u (c (t))

t=0

a (t + 1) = r (t) [a (t) − c (t) + w (t)] ,

where a (t) denotes the assets of the representative household at time t, r (t) is the
rate of return on assets and w (t) is the equilibrium wage rate (and thus the wage
earnings of the representative household). The constraint, (5.20) is the flow budget

constraint, meaning that it links tomorrow’s assets to today’s assets. Here we need
an additional condition so that this flow budget constraint eventually converges (i.e.,
so that a (t) should not go to negative infinity). This can be ensured by imposing
a lifetime budget constraint. Since a flow budget constraint in the form of (5.20) is
both more intuitive and often more convenient to work with, we will not work with
the lifetime budget constraint, but augment the flow budget constraint with another
condition to rule out the level of wealth going to negative infinity. This condition
will be introduced below.
5.9. Optimal Growth in Continuous Time
The formulation of the optimal growth problem in continuous time is very similar. In particular, we have
(5.21)

max

[c(t),k(t)]∞
t=0

Z

∞

exp (−ρt) u (c (t)) dt

0

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