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Technological Progress
Some Definitions
A capital saving technological progress (or invention) allows producers
to produce the same amount with relatively less capital input.
A labor saving technological progress (or invention) allows producers
to produce the same amount with relatively less labor input.
A neutral technological progress allows producers to produce more with
same capital labor ratio ( do not save relatively more of either input)
i) "Hicks neutral" : Ratio of marginal products remain the same for a
given capital labor ratio. Hicks neutrality implies the production function
can be written as:
Y = T (t)F (K,L)
ii) "Harrod neutral": relative input shares K.Fk/LFLremain the same
for a given capital output ratio
Harrod neutrality implies the production function can be written as:
Y = F [K,LT (t)] (labor-augmenting form)
where T (t) is the index of the technology and·T (t) > 0
labor-augmenting: it raises output in the same way as an increase in the
stock of labor.
iii) "Solow neutral": relative input shares LFL/K.Fkremain the same for
a given labor output ratio
Solow neutrality implies the production function can be written as:
Y = F [KT (t), L] (capital-augmenting form)
Solow Model with labor augmenting technological progress
Suppose the technology T (t) grows at rate x
·K(t) = I(t)− δK(t) = sF (K(t), T (t)L(t))− δK(t) (29)
Dividing by L(t)·k = sF (k, T (t))− (n + δ)k and
·kk = s
F (k,T (t))k − (n + δ)
The average product of per capita capital F (k,T (t))k now increases over
time because the T (t) grows at a rate x.
Steady state growth rate: By definition the steady state growth rateµ ·kk
¶∗is constant
sF (k∗,T (t))
k∗ − (n + δ) =constant
Since F is CRS, F (k,T (t))k = F (1,
T (t)k )This implies that T (t) and k grow
at the same rate x, because s,n and δ are constantsµ ·kk
¶∗= x
Moreover since y = F (k, T (t)) = kF (1,T (t)k )
à ·y
y
!∗= x (30)
and c = (1− s)y,³ ·cc
´=
µ(1−s) ·y(1−s)y
¶= x
Transitional Dynamics
Define: effective amount of labor=physical quantity of labor× efficiency
of labor = L× T (t) ≡∧L
∧k = K
LT (t)= k
T (t)= capital per unit of effective labor.
∧y = Y
LT (t)= F (
∧k, 1) = f(
∧k) = output per unit effective labor
We can rewrite·K = sF (K,T (t)L)− δK divide both sides by T (t)L
·K
T (t)L= sf(
∧k)− δ
∧k (31)
·∧k =
·³K
T (t)L
´=
LT (t)·K−K(
·LT (t)+L
·T (t))
T (t)2L2=
·K
T (t)L−
∧kn
T (t)L−
∧kx
T (t)L
Therefore·K
T (t)L=
·∧k +
∧kn
T (t)L+
∧kx
T (t)L
substituting in (31)
·∧k = sf(
∧k)− (x + n + δ)
∧k (32)
and
·∧k∧k
= sf(∧k)∧k
− (x+ n + δ) (32)
where x + n + δ is the effective depreciation rate
The effective per capita capital depreciates at the rate x + n + δ
Macro Lecture Notes– Ozan Hatipoglu
)( tkγ
tk
)( n+− δ
*k
Behavior of the growth rate in the Solow Model with labor augmenting technological progress (1)
Macro Lecture Notes– Ozan Hatipoglu
)( tk∧
γ
tk∧
)( xn ++δ
*∧
k
Behavior of the growth rate in the Solow Model with labor augmenting technological progress (2)
∧
∧
t
t
k
kfs )(
Speed of Convergence:
The speed of convergence is given by
β = −∂(
·∧k∧k)
∂ log∧k
(33)
For the CD production fucntion·∧k∧k= sA(
∧k)−(1−α) − (x + n + δ)
or·∧k∧k= sAe−(1−α) log(
∧k) − (x+ n + δ)
β = (1− α)sA(∧k)−(1−α) (declines monotonically)
Near the steady state
sA(∧k)−(1−α) = (x+ n + δ)
β∗ = (1− α)(x+ n + δ) (34)
What does the data say about convergence?
Consider the benchmark case with
x = 0.02, n = 0.01 and δ = 0.05 (for US)
where x is the long term growth rate of GDP/ per capita
β∗ = (1− α)(x+ n + δ) = (1− α)(0.08) depends on α
Suppose α = 1/3(, based on data) then β∗ = 5.6% (half life of 12.5 years)
But the data says that β∗ ' 2 − 3% which implies α = 3/4 (too high for
physical capital)
-A broader definition of capital is needed to reconcile theory with the
Extended Solow Model with human capital
Y = AKαHη [T (t)L]1−α−η and∧y = A
∧kα∧hη
(35)
·∧k +
·∧h = sA
∧kα∧hη
− (x + n + δ)
µ∧k +
∧h
¶(36)
It must be the case that returns to each type of capital are equal.
α∧y∧k− δ = η
∧y∧h− δ and
∧h = η
α
∧k
Using in (36)·∧k = s
∼A∧kα+η
− (δ + n + x) where∼A =constant
β∗ = (1− α− η)(x+ n + δ)
What’s Wrong with Neoclassical Theory??
-Does not explain long-term consistent per capita growth rates .
-Can not maintain pefect competition assumption when technological
progress is not exogenous.
The AK Model
Y = AK·kk = sA− (n + δ) > 0 for all k if sA > (n + δ)
does not exhibit conditional convergence.. How?
How about
Y = AK +BKαL1−α whereA > 0, B > 0 and 0 < α < 1
Constant Elasticity of Substitution (CES) Production Functions
y = F (K,L) = Ana(bK)ψ + (1− a) [(1− b)L]ψ
o 1ψ
0 < a < 1
0 < b < 1
and
ψ < 1
The elasticity of substitution is a measure of the curvature of the iso-
quants where the slope of an isoquant is given bydLdK = −∂F/∂K
∂F/∂Lh∂(slope)∂(L/K)
L/KSlope
i−1= 11−ψ
Properties of CES production function
1) The elasticity of substitution between capital and labor, 11−ψ ,is con-
stant
2)CRS for all values of ψ
3) As ψ →−∞, the production function approaches Y = min [bK, (1− b)L] ,
As ψ → 0, Y = (constant)KaL1−a (CD)
For ψ = 1, Y = abK +(1− a)(1− b)L (linear) so that K and L are perfect
substitutes (infinite elasticity of substitution)
Proof: Take log of Y and apply L’Hospital’s rule. i.e.find limψ→0 [log Y ]
Transitional Dynamics with CES production function·kk = s
f(k)k − (n + δ)
Note that for CES
f 0(k) = Aabψhabψ + (1− a)(1− b)ψk−ψ
i1−ψψ
andf(k)k = A
habψ + (1− a) (1− b)ψ k−ψ
i 1ψ
i) 0 < ψ < 1
limk→∞£f 0(k)
¤= limk→∞
hf(k)k
i= Aba
1
ψ > 0
limk→0£f 0(k)
¤= limk→0
hf(k)k
i=∞
Therefore, CES can exhibit can generate endogenous growth for 0 <
ψ < 1, if savings rates are high enough.
Macro Lecture Notes– Ozan Hatipoglu
tk
)( n+δ
CES Model with and
ψ1
sAbakkfs )(
)10( <<ψ nsAba +>δψ1
0>kγ
ii) ψ < 0
limk→∞£f 0(k)
¤= limk→∞
hf(k)k
i= 0
limk→0£f 0(k)
¤= limk→0
hf(k)k
i= Aba
1
ψ <∞
No endogenous growth, negative growth rates are possible for low val-
ues of sMacro Lecture Notes– Ozan Hatipoglu
tk
)( n+δ
CES Model with and
ψ1
sAba
kkfs )(
)0( <ψ nsAba +< δψ1
0<kγ
Poverty TrapsMacro Lecture Notes– Ozan Hatipoglu
tk
1+tk
*k
Poverty Traps in the Solow Model
)( tkG
tt kk =+1
*trapk
for some range of k, the average product of capital is increasing in k
- non-constant savings rates
- increasing returns with learning by doing and spillovers
Asimple model
Suppose the country has access to two technologies
yA = Akα
yB = Bkα − b
where B>A. To employ yB gov’t has to incur a setup cost of b per worker.
Under what conditions will the government incur b?
yA ≤ yB
or Akα ≤ Bkα − b
and k ≥∼k where
∼k =
hb
B−Ai 1α
·kk = sAk
α
k − (n + δ)·kk = sBk
α−bk − (n + δ)
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