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 kgold = δ Proof By definition ∂c∗ (sgold ) /∂s = From Proposition 2.3, ∂k∗ /∂s > 0, thus (2.21) can be equal to zero only when f (k∗ (sgold )) = δ Moreover, when f (k∗ (sgold )) = δ, it can be verified that ∂ c∗ (sgold ) /∂s2 < 0, so f (k∗ (sgold )) = δ is indeed a local maximum That f (k∗ (sgold )) = δ is also the global maximum is a consequence of the following observations: ∀ s ∈ [0, 1] we have ∂k∗ /∂s > and moreover, when s < sgold , f (k∗ (s)) −δ > by the concavity of f , so ∂c∗ (s) /∂s > for all s < sgold , and by the converse argument, ∂c∗ (s) /∂s < for all s > sgold Therefore, only sgold satisfies f (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 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 60