Economic growth and economic development 390

Introduction to Modern Economic Growth
of the representative household at time t by A (t). Then we have the following law
of motion for the total assets of the household

A˙ (t) = r (t) A (t) + w (t) L (t) − c (t) L (t)
where c (t) is consumption per capita of the household, r (t) is the risk-free market
flow rate of return on assets, and w (t) L (t) is the flow of labor income earnings of
the household. Defining per capita assets as
a (t) ≡

A (t)
,
L (t)

we obtain:
(8.7)

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

In practice, household assets can consist of capital stock, K (t), which they
rent to firms and government bonds, B (t). In models with uncertainty, households
would have a portfolio choice between the capital stock of the corporate sector and
riskless bonds. Government bonds play an important role in models with incomplete
markets, allowing households to smooth idiosyncratic shocks. But in representative
household models without government, their only use is in pricing assets (for example
riskless bonds versus equity), since they have to be in zero net supply, i.e., total
supply of bonds has to be B (t) = 0. Consequently, assets per capita will be equal
to the capital stock per capita (or the capital-labor ratio in the economy), that is,
a (t) = k (t) .
Moreover, since there is no uncertainty here and a depreciation rate of δ, the market
rate of return on assets will be given by

(8.8)

r (t) = R (t) − δ.

The equation (8.7) is only a flow constraint. As already noted above, it is not
sufficient as a proper budget constraint on the individual (unless we impose a lower
bound on assets, such as a (t) ≥ 0 for all t). To see this, let us write the single
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