Dynamic Macroeconomics
Problem Set 1
Universität Siegen
Dynamic Macroeconomics
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Table of Contents
1
Cobb-Douglas Production Function
2
Simple Macroeconomic Model
3
Reading Exercise
Dynamic Macroeconomics
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Cobb-Douglas Production Function
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a) Properties i and ii
Consider the Cobb-Douglas production function
Y = F (K , L) = ĀK α L1−α
where K is the capital stock and L the labor force.
The Cobb-Douglas function is homogeneous of degree one, since
F (λK , λL) = Ā(λK )α (λL)1−α
= Āλα+1−α K α L1−α
= λF (K , L)
Both factors of production are necessary, since
F (0, L) = Ā × 0α L1−α = 0 = F (K , 0) = Ā × K α 01−α .
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a) Properties iii and iv
The marginal product of capital is
∂F (K , L)
= αĀK α−1 L1−α = |{z}
α Ā
∂K
>0 |
L 1−α
>0
K
{z
}
>0
For which
1−α
L 1−α
1
1−α
lim FK (K , L) = lim αĀ
= αĀL
lim
=∞
K →0
K →0
K →0 K
K
1−α
1−α
L
1
1−α
lim FK (K , L) = lim αĀ
= αĀL
lim
=0
K →∞
K →∞ K
K →∞
K
For L the derivation is similar.
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b) Elasticity of Y w.r.t. K
The elasticity of Y w.r.t. K is given by
Y ,K =
∂F (K , L) K
∂K
Y
= αĀK α−1 L1−α
= αĀK α L1−α
K
Y
1
Y
=α
If K increase by one percentage point, Y will increase by α
percentage points.
The elasticity of Y with respect to L is 1 − α.
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c) and d) Wages and rental rates
On competitive markets, factors of production are paid their marginal
product.
Therefore the wage is w = FL and the share of wage income in total
income is
FL L
(1 − α)ĀK α L−α L
(1 − α)F (K , L)
wL
=
=
=1−α
=
Y
Y
F (K , L)
ĀK α L1−α
Therefore the rental rate of capital is r = FK and the share of capital
income in total income is
FK K
αĀK α−1 L1−α K
rK
αF (K , L)
=
=
=α
=
Y
Y
F (K , L)
ĀK α L1−α
For Cobb-Douglas production function, the income share going to the
different factors of production is constant.
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Wage shares in some countries
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e) Per-capita production function
To derive the production function per capita, start with the
production function for total output and divide by the labor force:
Y = ĀK α L1−α
L1−α
Y
= ĀK α
L
L
y = ĀK α L−α
α
K
y = Ā
L
α
y = Āk
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|:L
|y ≡
Y
L
|k ≡
K
L
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f) Interpreting Ā
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Table of Contents
1
Cobb-Douglas Production Function
2
Simple Macroeconomic Model
3
Reading Exercise
Dynamic Macroeconomics
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a) Firms optimization problem
The firms optimization problem is
max ĀK α L1−α − wL − rK
K ,L
(1)
taking the factor prices w and r as given.
The first order conditions for K and L are
αĀK α−1 L1−α = r
(1 − α)ĀK α L−α = w
Interpretation?
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a) Firms optimization problem ii
How does demand for L change, if w changes?
Start with F.O.C. for general function F (K , L) and use the implicit
function theorem
FL (K , L) = w
FL (K , h(L)) = w
Take derivative w.r.t. wage (using the chain rule) gives
FL,L (K , h(L)) ∗ h0 (L) = 1
h0 (L) =
1
FL,L
Sign?
Dynamic Macroeconomics
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a) Firms optimization problem iii
What determines the trade-off between K and L?
From F.O.C get
α L
r
=
1−αK
w
First assume α = 0.5 and r = w . Then KL = 1. Interpretation?
Now assume α = 0.5 but r > w . Then wr > 1 and thus KL > 1.
Interpretation?
Now assume r = w but α < 0.5. Then
α L
=1
1−αK
L
1−α
=
1>1
K
α }
| {z
>1
Interpretation?
(General tip: Look at the extreme case to make sense of the
economic mechanisms at hand)
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b) Endogenous and exogenous variables
Table: Endogenous and Exogenous Variables
Symbol
Ls = L̄
K s = K̄
Ā
α
Ld
Kd
L∗
K∗
Y∗
Name
Supply of Labor by Households
Supply of Capital by Households
Productivity parameter
Production function parameter
Demand of Labor by firms
Demand of Capital by firms
Equilibrium Level of Labor
Equilibrium Level of Capital
Equilibrium Level of Production
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Type
Exogenous
Exogenous
Exogenous
Exogenous
Endogenous
Endogenous
Endogenous
Endogenous
Endogenous
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c) General Equilibrium I
Market clearing condition for Labor
Ls = Ld
Market clearing condition for Capital
Ks = Kd
Firm demand for capital and labor from first order conditions.
Solution for labor
L∗ = L̄
Solution for capital
K ∗ = K̄
Solution for production
Y ∗ = ĀK̄ α L̄1−α
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d) Observed and predicted y
Table: Model prediction for GDP per capita relative to USA
Country
United States
Japan
Italy
Brazil
China
India
South Africa
Observed k
1
1.0949
1.0385
0.2420
0.2299
0.0057
0.1410
Observed y
1
0.7276
0.6949
0.2129
0.1816
0.0812
0.1907
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Predicted y ∗
1
1.0307
1.0127
0.6232
0.6126
0.1786
0.5205
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e) Implied Ā
Table: Model prediction for Ā relative to USA
Country
United States
Japan
Italy
Brazil
China
India
South Africa
Observed k
1
1.0949
1.0385
0.2420
0.2299
0.0057
0.1410
Observed y
1
0.7276
0.6949
0.2129
0.1816
0.0812
0.1907
Dynamic Macroeconomics
Implied A
1
0.7059
0.6171
0.3402
0.2964
0.4546
0.3664
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f) Results from MatLab i)
Dynamic Macroeconomics
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f) Results from MatLab ii)
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Table of Contents
1
Cobb-Douglas Production Function
2
Simple Macroeconomic Model
3
Reading Exercise
Dynamic Macroeconomics
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Table 1 of Hall/Jones (1999)
Dynamic Macroeconomics
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