$©$ September 21, 2020, Christopher D. Carroll GrowthAndGrossSaving

Gross Saving and Growth in the RCK Model

In the neoclassical growth model with labor-augmenting technological progress at rate $γ$ , utility function $u(c) = c1− ρ∕(1 − ρ)$ , time preference rate $𝜗$ and depreciation rate $δ$ the steady-state will be at the point where the growth rate of consumption is equal to the growth rate of labor-augmenting technological progress, $γ$ ,

˙ C--= ρ− 1(f ′(k ) − δ − 𝜗) C = γ

(1)

which implies that

f ′(k) = ργ + 𝜗 + δ αk α − 1 = ργ + 𝜗 + δ [ ] -1- ρ γ + 𝜗 + δ α−1 k = ------------- . α

(2)

The aggregate gross saving rate is deﬁned as

s = y-−--c y α = k--−--c- k α α = 1 − c ∕k .

(3)

In steady-state by deﬁnition

k˙ = 0

(4)

but from the capital accumulation equation we know that

α ˙k = k − (δ + γ)k − c

(5)

so in steady-state

α c = k − (δ + γ)k.

(6)

This can be substituted into (3) to obtain

k α − (δ + γ )k s = 1 − ---------------- k α = (δ + γ )k1− α

(7)

and the expression for the steady-state level of capital per capita can be substituted in to yield

( ) − 1 s = (δ + γ) ργ--+-𝜗-+--δ- α ( ) --α(γ-+--δ)-- = δ + 𝜗 + ργ

(8)

The derivative of this expression with respect to $γ$ is

( ) ds- α(δ-+--𝜗-+--ργ-)-−-α-(γ-+--δ)ρ- dγ = (δ + 𝜗 + ργ )2 ( ) α(δ (1 − ρ ) + 𝜗 ) = ----------------2-- . (δ + 𝜗 + ργ)

(9)

This will be positive if its numerator is positive, i.e. if

ρδ < 𝜗 + δ ρ < 1 + 𝜗∕ δ.

(10)

A typical assumption is $𝜗 = .04$ and $δ = .08$ , implying that the steady-state relationship between saving and growth in the neoclassical model is positive only if the coeﬃcient of relative risk aversion $ρ$ is less than 1.5. Typically we assume values of $ρ$ in the range from 2 to 5, so the model leads us to expect a negative relationship between saving and growth.