14.451 Lecture Notes Economic Growth

George-Marios Angeletos
Let κ ≡ k/l denote the capital-labor ratio at the stead state. By CRS,
F (k, l) = lf(κ)
FK(k, l) = f 0 (κ)
FL(k, l) = f(κ) − f 0 (κ)κ
F (k, l)
k
= φ(κ)
where f(κ) ≡ F (κ, 1) and φ(κ) ≡ f(κ)/κ. The Euler condition then reduces to
1=β[1 − δ + f 0 (κ ∗ )]
That is, the capital-labor ratio is pinned down uniquely by the equation of the MPK, net of
depreciation, with the discount rate
f 0 (κ ∗ ) − δ = ρ
where ρ ≡ 1/β − 1 or, equivalently, β ≡ 1/(1 + ρ). The gross rental rate of capital and the
netinterestratearethus
while the wage rate is
r ∗ = ρ + δ and R ∗ = ρ,
w ∗ = FL:(κ ∗ , 1)
The average product of labor and the average product of capital are given by
y ∗
l ∗ = f(κ∗ ) and y∗
k ∗ = φ(κ∗ ),
while, by the resource constraint, the consumption-capital ratio is given by
c ∗
k∗ = φ(κ∗ ) − δ = y∗
k
The comparative statics are then trivial. QED
68
∗ − δ.