APPENDIX D MARKET CLEARING FOR INTERMEDIAT GOODS ...

APPENDIX D

MARKET CLEARING FOR INTERMEDIAT GOODS (SUPPLY SIDE)

At world market if assuming 2-economy, demand for intermediate goods is equal to supply of intermediate goods. That is, Good 1:

Z1Nt + Z1St = X 1Nt + X 1St

Good 2:

Z 2Nt + Z 2St = X 2Nt + X 2St

We can solve relative price from only one of the above equation. For good 2; ⎛ α 3 p2t ⎞ ⎜ ⎟ ⎝ 1 − α 3 p1t ⎠

−α 3

⎛ α p2t ⎞ Yt N + ⎜ 3 ⎟ ⎝ 1 − α 3 p1t ⎠

⎛ α3 ⎞ ⎜ ⎟ ⎝ 1 − α3 ⎠

−α 3

−α 3

⎛ ⎞ (Yt N + Yt S ) ⎜ pp2t ⎟ ⎝ 1t ⎠

α2

α2

⎛ p ⎞ α1 −α 2 ⎛ p ⎞ α1 −α 2 + LS2t Bα 2 ⎜ 2t ⎟ Yt S = LN2t Bα 2 ⎜ 2t ⎟ ⎝ p1t ⎠ ⎝ p1t ⎠ −α 3

α2

⎛ p ⎞ α1 −α 2 = ( LN2t + LS2t ) Bα 2 ⎜ 2t ⎟ ⎝ p1t ⎠

(D.1)

Since YtW = Yt N + Yt S , and LWt = LtN + LSt ⎛ α3 ⎞ ⎜ ⎟ ⎝ 1 − α3 ⎠

Then, Let ytw =

−α 3

YtW ⎛ p2t ⎞ ⎜ ⎟ LWt ⎝ p1t ⎠

−α 3

α2

⎛ p ⎞ α1 −α 2 LW = W2t Bα 2 ⎜ 2t ⎟ Lt ⎝ p1t ⎠

(D.2)

YtW LW2t w and l = . 2t LWt LWt

From factor market clearing condition, ktw =

K tW K1Wt LW1t K 2Wt LW2t = + = κ1t l1wt + κ 2t l2wt , N tW LW1t N tW LW2t N tW

(D.3)

K itW and l = l + l = 1 , where κ it = W . Lit w t

w 1t

w 2t

The price ratio as a function of the ratio of intermediate good at world production can be expressed as

p1t α 3 X 2t = p2t 1 − α 3 X 1t 74

.

(D.4)

75

If

X 2t can be expressed in terms of state variable kt , then the equilibrium price can X 1t

be stated as the function of kt . l1t =

From equation (B.6), and from (B.7), l2t = 1 − l1t =

α 3 (1 − α1 ) , 1− γ

1 − α1α 3 − α 2 (1 − α 3 ) − α 3 − α1α 3 (1 − α 2 )(1 − α 3 ) = . 1− γ 1− γ

That is, labor used in each sector can be expressed only in parameters. ⎛ α ⎞ (1 − α 2 )(1 − α 3 ) into ⎜ 3 ⎟ Substituting l2wt = 1− γ ⎝ 1 − α3 ⎠ then,

⎛ p2t ⎞ ⎜ ⎟ ⎝ p1t ⎠

α 2 +α1α 3 −α 2α 3 α1 −α 2

=

−α 3

⎛p ⎞ ytw ⎜ 2t ⎟ ⎝ p1t ⎠

−α 3

α2

⎛ p ⎞ α1 −α 2 = l2wt Bα 2 ⎜ 2t ⎟ , ⎝ p1t ⎠

(1 − γ ) ytw Bα 2 (1 − α 2 )α 3α3 (1 − α 3 )(1−α3 )

⎤ p2t ⎡ (1 − γ ) ytw = ⎢ α2 α3 (1−α 3 ) ⎥ p1t ⎣ B (1 − α 2 )α 3 (1 − α 3 ) ⎦

α1 −α 2 γ

(D.5)

Since the integrated economy is considered as closed economy, then the production of final good can be solved as all intermediate goods produced in the integrated world are used in the production of world final good. Substituting ytw = f (ktw ) from above result into price ratio, then α1 −α 2 ⎤ p2t ⎡ α1α1 (1 − α1 )1−α1 ⎛ 1 − γ ⎞ = ⎢ α2 ⎥ ktα1 −α 2 , ⎟ 1−α 2 ⎜ p1t ⎢⎣ α 2 (1 − α 2 ) ⎝ γ ⎠ ⎥⎦

which is equal to equation (3.3.1).

(D.6)