Economic growth and economic development 74

Introduction to Modern Economic Growth
∗
The next proposition shows that sgold and kgold
are uniquely defined and the

latter satisfies (2.22).
Proposition 2.4. In the basic Solow growth model, the highest level of con∗
sumption is reached for sgold , with the corresponding steady state capital level kgold

such that
(2.22)

¡ ∗ ¢
f 0 kgold
= δ.

Proof. By definition ∂c∗ (sgold ) /∂s = 0. From Proposition 2.3, ∂k∗ /∂s > 0,
thus (2.21) can be equal to zero only when f 0 (k∗ (sgold )) = δ. Moreover, when
f 0 (k∗ (sgold )) = δ, it can be verified that ∂ 2 c∗ (sgold ) /∂s2 < 0, so f 0 (k∗ (sgold )) = δ
is indeed a local maximum. That f 0 (k∗ (sgold )) = δ is also the global maximum is
a consequence of the following observations: ∀ s ∈ [0, 1] we have ∂k∗ /∂s > 0 and

moreover, when s < sgold , f 0 (k∗ (s)) −δ > 0 by the concavity of f , so ∂c∗ (s) /∂s > 0

for all s < sgold , and by the converse argument, ∂c∗ (s) /∂s < 0 for all s > sgold .
Therefore, only sgold satisfies f 0 (k∗ (s)) = δ and gives the unique global maximum
Ô

of consumption per capita.

In other words, there exists a unique saving rate, sgold , and also unique corre∗

, which maximize the level of steady-state consponding capital-labor ratio, kgold
∗
, the higher saving rate will increase
sumption. When the economy is below kgold
∗
, steady-state consumption
consumption, whereas when the economy is above kgold

can be increased by saving less. In the latter case, lower savings translate into higher
consumption because the capital-labor ratio of the economy is too high so that individuals are investing too much and not consuming enough. This is the essence of
what is referred to as dynamic inefficiency, which we will encounter in greater detail
in models of overlapping generations in Chapter 9. However, recall that there is no
explicit utility function here, so statements about “inefficiency” have to be considered with caution. In fact, the reason why such dynamic inefficiency will not arise
once we endogenize consumption-saving decisions of individuals will be apparent to
many of you already.
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