Ec 117 – GROWTH THEORY
LECTURE NOTES
Foster, UCSD
1-Jul-09
TOPIC 3 – HARROD-DOMAR MODEL
A.
Production Functions and Returns to Scale
1.
Introduction:
a)
The Harrod-Domar and Solow-Swan growth
models incorporate production and well as
consumption, so we will need a macroeconomic production function.
b)
Aggregate production function:
Q = F(K, L).
1)
K and L are capital and labor employed.
2)
Q = total output (GDP).
3)
Technology reflected in algebraic form of F(●).
c)
Average and marginal factor products.
2.
Homogeneity and Returns to Scale:
a)
For some λ ≥ 0, let Q
0
= F(K
0
, L
0
) and Q
1
=
F(K
1
, L
1
) = F(λK
0
, λL
0
).
b)
A production function is said to be “homogeneous of degree h
” if:
•
Q
1
≡ λ
h
Q
0
•
That is, F(λK
0
, λL
0
) ≡ λ
h
F(K
0
, L
0
)
•
Note the identity (≡).
c)
If h = 1, the production function is “homogeneous of degree 1
” and is said to exhibit
constant returns to scale
(CRS).
1)
If it is CRS, then F(λK
0
, λL
0
) ≡ λ F(K
0
, L
0
).
2)
For λ = 2, this means that doubling all inputs (K and L) will result in exact doubling
of output Q.
3.
Properties of CRS Production Functions:
a)
If Q = F(K, L) is CRS, then average and marginal products are “scale invariant” (
i.e.
,
“homogeneous of degree zero”).
1)
Average product proof (divide thru identity by λK and by λL):
Notation (Production Functions)
Q
Output, GDP
L
Labor force employed
K
Capital stock employed
)
,
(
;
)
,
(
;
L
K
F
K
Q
MPK
K
Q
APL
L
K
F
L
Q
MPL
L
Q
APL
K
L
=
∂
∂
=
=
=
∂
∂
=
=
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
)
,
(
)
,
(
)
,
(
APL
L
Q
L
L
K
F
L
L
K
F
L
L
K
F
L
Q
APL
=
=
=
≡
=
=
λ
λ
λ
λ
λ