Homework 4 Solution (pdf)

Solution to Homework 4. AGEC 840
Hecksher-Olhin model with fixed factor requirements: general equilibrium
a. By definition
c ( w,
r)
min wL rK subject to f ( L , K ) ≥ 1.
i
=
Li , Ki
i
+
i
i
, K
i
) = min[ Li
/ aiL
, K
i
/ aiK
Note that f ( L
] ≥ 1 implies that L ≥ a and K ≥ a . And
i
so, wL
i
+ rK i
is minimized when L
i
= aiL
and K
i
= aiK
. Hence, the unit-cost functions
are
c
i
( w,
r)
= waiL
+ raiK
, i = 1, 2 .
The aggregate demands for goods i = 1, 2 are
D
1
= θ ( wL + rK)
/ p1
and D
2
= ( 1−θ )( wL + rK)
/ p2
,
where θ is the share of income spent on good 1.
b. Equilibrium conditions are
(1) p1 = wa1L
+ ra1K
(zero-profits)
(2) p2 = wa2L
+ ra2K
(3) L = a1L y1
+ a2L
y2
(the resources are fully employed)
(4) K = a1K y1
+ a2K
y2
(5) y
1
= θ ( wL + rK)
/ p1
(markets clear)
(6) y
2
= ( 1−θ
)( wL + rK)
/ p2
c. Dividing (5) by (6) yields
y1 / y2
= θ /(1 −θ)[
p2
/ p1]
, or
(7) p1 / p2
= θ /(1 −θ
)[ y2
/ y1]
.
Dividing (3) by (4) yields
L / K = ( a1L y1
+ a2L
y2
) /( a1K
y1
+ a2K
y2
) , or
L / K = ( a1L
y1
/ y2
+ a2L
) /( a1K
y1
/ y2
+ a2K
) , or
(8) y1 / y2
= ( a2K
/ a1K
)( L / K − a2L
/ a2K
) /( a1L
/ a1K
− L / K)
.
Substituting (8) in (7) yields
(9) p
1
/ p2
= θ /(1 −θ)
( a1K
/ a2K
)( a1L
/ a1K
− L / K) /( L / K − a2L
/ a2K
) .
Differentiating yields
(10) d ( p1 / p2
) / dθ > 0 and
(11) d p / p ) / d(
L / K)
0 .
(
1 2
<
As the share of income spent on good 1 increases, its relative price increases. By the
Rybczynski theorem, as the endowment of labor becomes more plentiful relative to
capital, the output of labor-intensive good 1 increases by more than the output of good 2
decreases. And so, good 1 becomes less scarce relative to good 2, and the relative price
of good 1 falls.
i
i
i
i
iL
i
iK

d. From (1)
1 = [ w/
p1 ] a1L
+ [ r / p1]
a1K
.
From (2)
p2 / p1
= [ w/
p1
] a2L
+ [ r / p1]
a2K
,
where p / p 2 1
is given by (9). Solving these system of equations yields
⎛ w/
p1
⎞ 1 ⎛a2K
− a1K
[ p2
/ p1]
⎞
(12) ⎜ ⎟ =
⎜
⎟
⎝ r / p1
⎠ a1La2
K
− a2La1K
⎝ a1L[
p2
/ p1
] − a2L
⎠
1
⎛a2K
− a1K
[ p2
/ p1]
⎞
=
⎜
⎟ ,
a1 K
a2K
( a1L
/ a1K
− a2L
/ a2K
) ⎝ a1L[
p2
/ p1]
− a2L
⎠
where we assume that a2 L
/ a1L
< [ p2
/ p1]
< a2K
/ a1K
. Differentiating and using (10) and
(11) yields
−2
a1K
( p1
/ p2
) d(
p1
/ p2
)
(13) d ( w/
p1)
/ dθ =
> 0 ,
a a ( a / a − a / a ) dθ
1K
2K
1L
1K
2L
2K
−2
1K
( p1
/ p2
) d(
p1
/ p2
a
)
(14) d( w/
p1) / d(
L / K)
=
< 0 ,
a a ( a / a − a / a ) d(
L / K)
1K
2K
−2
a1L
( p1
/ p2
) d(
p1
/ p2
)
(15) d ( r / p1)
/ dθ = −
< 0 ,
a a ( a / a − a / a ) dθ
1K
2K
1L
1L
1K
2L
2K
1K
2L
2K
−2
1L
( p1
/ p2
) d(
p1
/ p2
a
)
(16) d( r / p1) / d(
L / K)
= −
> 0 .
a a ( a / a − a / a ) d(
L / K)
1K
2K
1L
By the Stolper-Samuelson theorem, the real income of a factor increases (decreases) as
the relative price of the good that uses that factor intensively increases (decreases).
Having pointed this out, we can apply the reasoning from (c).
e. Dividing equations in (12) yields
w a2K
− a1K
[ p2
/ p1
]
= .
r a1L[
p2
/ p1
] − a2L
Differentiating and using (10) and (11) yields
d ( w / r) / dθ > 0 and d ( w / r) / d(
L / K ) < 0 .
Note that the derivatives can also be determined from (13)-(16) because
d w/
r)
= d([
w/
p ]/[ r / ]) .
(
1
p1
f. By (12), w > 0 and r > 0 if and only if a2 L
/ a1L
< [ p2
/ p1]
< a2K
/ a1K
. Susbtituting
(9) yields
a2 L
/ a1L
< [( 1− θ ) / θ ] ( a2K
/ a1K
)( L / K − a2L
/ a2K
) /( a1L
/ a1K
− L / K)
< a2 K
/ a1K
,
or
a2L
/ a2K
L / K − a2L
/ a2K
< [ 1/ θ −1]
< 1, or
a / a
a / a − L / K
1L
1K
1L
1K
1K
2L
2K

1−
L / K /( a1L
/ a1K
)
a1L
/ a1K
− L / K
< 1/
θ −1
< .
L / K /( a / a ) −1
L / K − a / a
2L
2K
g. Equilibrium conditions under free trade are
T T T
(1T) = w a r a (zero-profits)
(2T)
p1 1L
+
1K
2L
2K
T T
T
p2 = w a2L
+ r a2K
L = a1L y1
+ a2L
y2
K = a1K y1
+ a2K
y
* T * T
* T
y1 + y1
= θ ( w ( L + L ) + r ( K + K )) / p1
*
T * T
* T
y2 + y2
= ( 1−θ
)( w ( L + L ) + r ( K + K )) / p2
(3T)
(4T)
2
(5T)
(6T)
(the resources are fully employed)
(markets clear)
Note that FPE obtains if both factors are fully employed in both countries (we need
* *
a2 L
/ a2K
< L / K < a1L
/ a1K
). In what follows I drop the superscript “T” (free trade) on
the endogenous variables, but remember that equilibrium output prices, wages, and rents
are different in autarky and under free trade (we know that outputs are the same in
autarky and free trade because we assumed that both factors are fully employed, and
there is only one point on the PPF where this is possible). Using (1T) and (5T) we can
determine the real wage and rental rate under free trade:
1 = [ w/
p1 ] a1L
+ [ r / p1]
a1K
*
*
*
( y + y ) / θ = [ w / p ]( L + L ) + [ r / p ]( K + ), or
1 1
1
1
K
⎛ a1L
a1K
⎞⎛
w/
p1
⎞ ⎛ 1 ⎞
⎜
⎟⎜
⎟ = ⎜ ⎟
*
*
*
⎝ L + L K + K ⎠⎝
r / p1
⎠ ⎝(
y1
+ y1
) / θ ⎠
Solving these system of equations yields
w
*
⎛ w/
p1
⎞ 1 ⎛ K − a + ⎞
1K
( y1
y1
) / θ
⎜ ⎟ =
⎜
⎟
w
w
−
*
w
⎝ r / p1
⎠ a1L
K a1K
L ⎝ a1L
( y1
+ y1
) / θ − L ⎠
And so, demand for good 1 in the home country is
D
1
= θ ( wL + rK)
/ p1
θ
w
*
*
= [( K − a1K
( y1
+ y1
) / θ ) L + ( a1L
( y1
+ y1
) / θ − L
w
w
a1L
K − a1K
L
The excess supply is
w
*
* w
[( θK
− a1K
( y1
+ y1
)) L + ( a1L
( y1
+ y1
) −θL
) K]
y1
−
w
w
a K − a L
=
=
y
y
1L
1K
w
w
w
*
* w
1
( a1L
K − a1K
L ) −[(
θK
− a1K
( y1
+ y1
)) L + ( a1L
( y1
+ y1
) −θL
) K]
w
w
a1L
K − a1K
L
w
w
*
w w
1
( a1L
K − a1K
L ) + ( a1K
L − a1L
K)(
y1
+ y1
) −θ
( K L − L K)
w
w
a1L
K − a1K
L
w
) K]
.

( a
=
2K
L − a
2L
K)(
a
1L
K
w
− a
1K
w
L ) + ( a
1K
Δ(
a
1L
L − a
K
w
1L
− a
K)(
a
1K
2K
w
L )
L
w
− a
2L
K
w
) −θ
( K
w
w
L − L K)
Δ
( l − a
=
2L
/ a
2K
)( a
1L
/ a
1K
− l
w
) + ( l − a
1L
(1/( a
2K
/ a
1K
)( l
w
K))
Δ(
a
− a
1L
2L
/ a
1K
/ a
2K
− l
w
) −θ
( l − l
)
w
)( a
1L
/ a
1K
− a
2L
/ a
2K
)
( l − l
)( a
/ a1K
− a2L
/ a2K
)(1 − θ )
w
> ( < )0 depending on whether l > ( l (because / a < L / K < a a ).
1 L 2K
2L
1K
>
1 L
/
1K
a2 L 2K
1L
/
1K
The Hecksher-Olhin theorem predicts that the labor-abundant country will export the
labor-intensive good, and this is exactly what we find.