155 APPENDIX E
DERIVATION OF THE PRIMARY FACTOR DEMAND
Producer in each production sector combines labor and capital in order to
produce the value added product or the effective input combination of primary factors
such that total cost of production is the least which can be shown by the problem:
Min
Wi .Li + Ri .K i
st.
VAi =
Li , K i
where
(
L−i ρi + K i− ρi
)
−
1
; for i ∈ IND
ρi
Li and Wi are the quantity and wage rate of labor
Ki and Ri are the quantity and rental rate of capital
VAi is the value added product or effective input combination of primary
factors used in producing the output of production sector i
⎛
1 ⎞
⎟ is an elasticity of substitution between labor and capital in
⎝ 1 + ρi ⎠
production sector i
σ iLK = ⎜
From this minimization problem, Lagrangian function can be written as
⎡
ξ = Wi .Li + Ri .K i − Λ. ⎢ L−i ρi + K i− ρi
⎣
(
)
−
1
ρi
⎤
− VAi ⎥
⎦
(E.1)
Taking derivative of the Lagrangian function in (E.1) with respect to
Li , K i and Λ yields the following necessary conditions:
ξL = 0 ;
⎛ 1
Wi = Λ. ⎜
⎝ ρi
⎞ − ρi
− ρi
⎟ . Li + K i
⎠
ξK = 0 ;
⎛1
Ri = Λ. ⎜
⎝ ρi
⎞ − ρi
− ρi
⎟ . Li + K i
⎠
i
i
ξΛ = 0 ;
VAi =
(L
− ρi
i
(
(
+ K i− ρi
)
−
155
)
⎛ 1+ ρi ⎞
−⎜
⎟
⎝ ρi ⎠
)
⎛ 1+ ρi ⎞
−⎜
⎟
⎝ ρi ⎠
. ( − ρi ) .L−i ρi −1
(E.2)
. ( − ρi ) .K i− ρi −1
(E.3)
1
ρi
(E.4)
156 Divide (E.2) by (E.3), then
Wi
Ri
=
⎛ Li ⎞
⎜
⎟
⎝ Ki ⎠
− (1+ ρi )
(E.5)
Rearrange the term in (E.5), then
Li =
⎛ Wi ⎞
⎜ ⎟
⎝ Ri ⎠
⎛ 1 ⎞
−⎜
⎟
⎝ 1+ ρi ⎠
(E.6)
. Ki
Substitute (E.6) into (E.4), then
ρi
⎡
⎤
1+ ρi
⎛
⎞
W
− ρi
− ρi ⎥
⎢
i
VAi = ⎜ ⎟ .K i + K i
⎢⎝ Ri ⎠
⎥
⎣⎢
⎦⎥
VAi =
ρi
⎡
⎤
1+ ρi
⎛
⎞
W
⎢ i
+ 1⎥
⎢⎜⎝ Ri ⎟⎠
⎥
⎢⎣
⎥⎦
−
−
1
ρi
1
ρi
. Ki
1
Ki
ρi
⎡
⎤ ρi
1+ ρi
⎛
⎞
W
= VAi . ⎢⎜ i ⎟
+ 1⎥
⎢⎝ Ri ⎠
⎥
⎢⎣
⎦⎥
Ki
⎛ W + Ri ⎞1+ ρi
= VAi . ⎜ i
⎟
⎝ Ri ⎠
1
Ki =
⎛ Ri ⎞
VAi . ⎜
⎟
⎝ Wi + Ri ⎠
⎛ 1 ⎞
−⎜
⎟
⎝ 1+ ρi ⎠
(E.7)
⎛ 1 ⎞
Substitute σ iLK = ⎜
⎟ into (E.7), then
⎝ 1 + ρi ⎠
Ki
⎛ Ri ⎞
= VAi . ⎜
⎟
⎝ Wi + Ri ⎠
−σ iLK
(E.8)
157 Substitute K i in (E.7) into (E.6), then
⎛ 1 ⎞
−⎜
⎟
⎝ 1+ ρi ⎠
Li =
⎛W ⎞
VAi . ⎜ i ⎟
⎝ Ri ⎠
Li =
⎛ Wi ⎞
VAi . ⎜
⎟
⎝ Wi + Ri ⎠
⎛ Ri ⎞
.⎜
⎟
⎝ Wi + Ri ⎠
⎛ 1 ⎞
−⎜
⎟
⎝ 1+ ρi ⎠
⎛ 1 ⎞
−⎜
⎟
⎝ 1+ ρi ⎠
(E.9)
⎛ 1 ⎞
Substitute σ iLK = ⎜
⎟ into (E.9), then
⎝ 1 + ρi ⎠
Li
⎛ Wi ⎞
= VAi . ⎜
⎟
⎝ Wi + Ri ⎠
−σ iLK
(E.10)
Thus, labor and capital demand of the production sector i is given by
−σ iLK
Li
⎛ Wi ⎞
= VAi . ⎜
⎟
⎝ Wi + Ri ⎠
Ki
⎛ Ri ⎞
= VAi . ⎜
⎟
⎝ Wi + Ri ⎠
; for i ∈ IND
−σ iLK
; for i ∈ IND