EC9A2 Advanced Macro Analysis Class #1
Jorge F. Chávez
University of Warwick
October 29, 2012
Outline
1. Some math
2. Shocking the Solow model
3. The Golden Rule
4. CES production function (more math)
5. A few remarks about PS1
Properties of concave and convex functions
Proposition Let f : R2 → R be a C 2 (twice-differentiable) function
defined over the open convex set S. Then:
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′′
′′
′′ ′′
′′
f is convex ⇐⇒ f11
≥ 0, f22
≥ 0 and f11
f22 − (f12
) ≥0
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′′
′′
′′ ′′
′′
f is concave ⇐⇒ f11
≤ 0, f22
≤ 0 and f11
f22 − (f12
) ≥0
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′′
′′ ′′
′′
f11
> 0 and f11
f22 − (f12
) > 0 ⇐= f is strictly convex
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′′
′′ ′′
′′
f11
< 0 and f11
f22 − (f12
) > 0 ⇐= f is strictly concave
2
2
2
2
A corollary of the above proposition is that for a C 2 single-variable
real-valued functions (f : R → R) to be strictly concave we need
f ′′ (x) < 0 (sufficient condition). Note that f strictly concave is not a
′′
necessary condition for f11
< 0.
Proposition If f is concave, then concavity implies that f is continuous
at the interior of its domain. This does not necessarily holds for the
boundary points.
“Shocking” the Solow model
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Consider the basic discrete-time version of the Solow model as seen
in the lecture notes.
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We start from a steady-state at time 0.
Suppose there is a one-time increase in the depreciation rate at the
end of time 0 (δ1 < δ2 )
How does the economy converges to the new steady-state?
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How is {yt }∞
t=0 affected?
How is the path of log Kt affected?
“Shocking” the Solow model
A “shock” to δ
Figure: Solow 1
kt+1
(n + δ2 kt )
f (kt )
(n + δ1 kt )
f (k)
sf (k)
sf (kt )
A
B
k2
k1
kt
“Shocking” the Solow model
Some transitions
Figure: Some transitions in relevant variables
log Kt
Transitory growth
y1ss
y2ss
Kt = k 1 L t
t∗
t=0
(a) Path of yt
t
t=0
Kt = k 2 L t
Transition
t∗
(b) Path of log Kt
t
“Shocking” the Solow model
A “shock” to s
Figure: Solow 1
kt+1
(n + δkt )
f (kt )
s2 f (kt )
B
s1 f (k)
s2 f (k)
s1 f (kt )
A
k1
k2
kt
The Golden Rule
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Consider the basic model in discrete time with population growth.
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The law of motion of the stock of capital per worker is:
kt+1 =
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sf (kt ) + (1 − δ) kt
≡ ϕ (kt )
1+n
And the steady state (fixed-point) condition is:
sf (k) = (n + δ) k
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Thus, at a steady-state
c = (1 − s)f (k) = f (k) − (n + δ)k
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Therefore we can maximize consumption. The condition is:
f ′ (k) = (n + δ)
The Golden Rule
Figure: The Golden Rule
f (k ss )
(n + δ)k ss
c1
s1 f (k ss )
k1
k ss
The Golden Rule
Figure: The Golden Rule
f (k ss )
(n + δ)k ss
c2
s2 f (k ss )
k2
k ss
The Golden Rule
Figure: The Golden Rule
f (k ss )
c1
cGR
(n + δ)k ss
s1 f (k ss )
c2
sGR f (k GR )
s2 f (k ss )
k2
k GR
k1
k ss
CES technology
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Consider the production function:
[
]σ/(σ−1)
F (K, L) = γK (σ−1)/σ + (1 − γ) L(σ−1)/σ
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We can check that the elasticity of substitution between K and L is
constant and equal to σ
log(K/L)
Recall that the elasticity of substitution is εKL = − d dlog(F
(the
K /FL )
percentage change in relative inputs K/L in response to a
percentage change in relative factor prices)
For the function above:
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FK
= γK −1/σ F 1/σ
FL
= (1 − γ) L−1/σ F 1/σ
Using these results:
FK
γ
=
FL
1−γ
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(
K
L
)−1/σ
⇒
K
=
L
(
FK 1 − γ
FL γ
)−σ
In the last expression, think of k = K/L as a function of
p = FK /FL . This relationship has the form k = Cp−σ for some
constant C.
CES as a general case
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CES approximates a linear production function as σ → ∞
σ−1
= 1 ⇒ lim F (K, L) = γK + (1 − γ) L
σ→+∞
σ→+∞
σ
lim
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CES approximates a fixed-proportions technology (Leontief) as
σ → 0. To see this we need to analyze two cases:
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Suppose K > L. Then:
F (K, L)
lim
σ→0
L
=
=
)σ/(σ−1)
( ( )
(σ−1)/σ
K
+ (1 − γ)
lim γ
σ→0
L
lim (1 − γ)σ/(σ−1) = 1
σ→0
where the last equality follows because K/L > 1 and
lim (σ − 1)/σ → −∞. Thus lim F (K, L) = L
σ→0
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σ→0
Suppose K < L. Then, following a similar argument we can
establish that lim F (K, L) = K.
σ→0
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Combining both results:
lim F (K, L) = min (K, L)
σ→0
CES as a general case
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CES converges to the Cobb-Douglas function when σ → 1.
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To see this take the logarithm of F and evaluate the limit using
L’Hospital rule:
(
)
log γK (σ−1)/σ + (1 − γ) L(σ−1)/σ
lim log F (K, L) = lim
σ→1
σ→1
(σ − 1) /σ
=
−γK (σ−1)/σ log K/σ 2 −(1−γ)L(σ−1)/σ log L/σ 2
γK (σ−1)/σ +(1−γ)L(σ−1)/σ
lim
σ→1
1/σ 2
= γ log K + (1 − γ) log L
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This implies:
lim F (K, L) =
σ→1
exp (γ log K + (1 − γ) log L)
= K γ L1−γ
CES and Inada conditions
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CES for σ ∈ (0, +∞) satisfies assumption 2 in the lecture notes: it
is strictly increasing and strictly concave in each input. For example,
we can check concavity:
FK (K, L) = γK −1/σ F 1/σ
(
1/σ
( )(σ−1)/σ )(σ−1)/σ
L

=  γ + (1 − γ)
K
It is easy to show that FK (K, L) is decreasing in K which implies
that FKK < 0. Similarly one can show that FLL < 0.
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CES for σ = 1 (Cobb-Douglas case) satisfies Inada conditions.
For σ ̸= 1 CES does not satisfy some parts of the Inada conditions:
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For σ > 1:
lim Fk = +∞, but , lim Fk = γ 1/(σ−1) > 0
K→0
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K→+∞
For σ < 1:
lim Fk = 0, but , lim Fk = γ 1/(σ−1) > 0
K→0
K→+∞
Some variations of the Solow model
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Consider the Solow model with CES production function given by:
]σ/(σ−1)
[
(σ−1)/σ
(σ−1)/σ
+ (1 − γ) Lt
F (Kt , Lt ) = γKt
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For the above case, characterize the equilibrium using both
competitive markets and a social planner.
Add a government: (i) Passive government, (ii) Active government
I’ll cover some of these in a handout to be posted soon!