Economic Growth and Stability: Theory and Evidence

ECONOMIC GROWTH AND STABILITY1
Economic Growth and Stability: Theory and Evidence
Course
Instructor’s Name
Institutional Affiliation
The City and State
The Date

ECONOMIC GROWTH AND STABILITY2
Part A: Analytical Work
a) It is divided between labor and capital income according to their marginal
productivity. That is to say, from; yt = F (Kt, AtLt) = Kαt (AtLt) 1-α
So, from St = It = sYt . By making everything into the per effective worker terms, we
divide by AtLt:
We have, Kt+1 /At+1Lt+1 = s F (Kt, Lt)/AtLt AtLt/At+1Lt+1 + (1- α) Kt/AtL AtLt/Bt+1Lt+1
Yt+1 = αK αt + (1- α) Kt/ {(1 + g) (1 + n) = φ (Kt)
Therefore, 𝑦 ; t= (1- α) Kt/AtLt = Yt/AtLt = Kt / AtLt
b)
From the law of motion it can be asserted that; Kt+1 – Kt = sYt – 𝛿Kt
From the transformation; Kt+1 – Kt≡ change in Kt ≡ it – (n + g + 𝛿)Kt = s Kαt – (n +g+𝛿)Kt
It< (n + g + 𝛿) Kt ⤇⧍Kt< 0
It = (n + g + 𝛿) Kt ⤇⧍Kt = 0 ⤇ Kss
It > (n + g + 𝛿) Kt ⤇⧍Kt> 0
C)
At steady state, dkt+1 = 0,
K⧍ t = 0 s K⤇ α – (n + g + gn + 𝛿) Kt = 0
Now solving for the value of K gives:
K = [s/ (n + g + gn + 𝛿)] 1/ (1-α).
To effectively check for the overall stability of the steady sate, we need to check its
limit if it is less than unity. That is to say; lim K as Kt tends to infinity should be less than one
(Acemoglu 2009)

ECONOMIC GROWTH AND STABILITY3
For output per worker
From the growth rate of output per worker, yt – Yt/ Lt in steady state is gives the
following expression (Acemoglu 2009)
Yt/Lt = [Kαt (AtLt) 1-α]/Lt = (Kt/Lt) α At1-α = (Kt/AtLt) α At = KtαAt
Therefore, as the economy reaches the steady state then;
Ytss = KαBt
Now, from above we get;
yt+1ss/ytss – 1 = At+1/At – 1 =g. Similarly, by taking natural logs on both terms gives the
following;
Logyt+1ss – log ytss – log At+1 – log At ≡ g
The consumption per effective worker
Under the steady state, the consumption per effective worker is obtained from the general
equation as (Acemoglu 2009); yt = ct + it, where it = syt and also ct = (1-s) yt. Therefore, in the
steady state;
C = (1-s) y = (1 – s) A1/1-α (s/ 𝛿 + n) α/1- α
Capital per worker at steady state is obtained from; Kt+1ss = Ktss = K. it implies that at
steady state, yk = 0. Now getting the K from the expression of ykt gives (Noel and Mark 2017)
D)
K = (sA/ 𝛿 + n) 1/ (1-α)
Therefore, the growth rate of output per worker is given as; Kt+1 = 1-α (sA/ 𝛿 + n)

ECONOMIC GROWTH AND STABILITY4
Also from the capital per effective worker, K = [s/ (n + g + gn + 𝛿)] 1/ (1-α). Taking natural
logs where Kt/Lt = AtKt , where the Kt = Kt/ (AtLt).Now defining the per capita stock of capital as
kt =Kt / Lt:
It then gives;
Kt/Lt = kt = ktAt. By taking logarithms to get the growth rate of capital gives;
Log kt+1 – log kt = log At+1 – log At ≡ g(Noel and Mark 2017
E) The Golden –Rule Steady State
The value of 𝑘 ; at the golden rule in the steady state is the capital stock per worker which
maximizes the consumption at the steady state (Haine etal 2006). Therefore, from;
Ct = (1-s) yt = f (kt) –sf(kt).
It is believed that there is now way one can maximize consumption in all the states
(Haine etal 2006). This is because consumption is a function of f (kt) that is not bounded (Haine
et.al, 2006). Therefore, with such a reason, a corner solution is only obtained as kt = 0 that is true
when stationary points are obtained first. But the steady state condition is given as;
Sf (k) = (n + 𝛿) k. Hence it is true for all steady states because ct = f(kt) – sf(Kt) in steady
states (Halsmayer etal 2016).
In simple terms; c = f (k) – (n + 𝛿) k. The maximization problem now becomes;
∂c/ ∂k = f’ (k) – (n + 𝛿) = 0 f’ (k’) = n +⤇ 𝛿
Where y = Akα a closed form of solution for K’ can be obtained as’
f’ (k’) = αA(k’)α-1 = n + 𝛿