7
Appendix: Not for publication
7.1
Proofs of Lemmas and of Proposition 2
That the capital-output ratio is higher in F firms follows
Proof of Lemma 1.
<
immediately from the fact that κE < κF (shown in the text), since kE /yE = κ1−α
E
= kF /yF . Similarly, that the capital-labor ratio is higher in F firms follows from
κ1−α
F
observing that
kF kE
κF κE
/
=
/
=
nF nE
A χA
µ ¶ 1−α
χ α
>1
χ
where the inequality again follows from Assumption 1.
Proof of Lemma 2. Due to constant-return-to-scale, aggregation holds, thus we can
replace individual-firm variables (lower case) by aggregate variables (upper case). Since
κE ≡ KEt / (χAt NEt ) is constant and NF t = Nt − NEt , then
NEt =
KEt
KEt
, NF t = Nt −
,
χAt κE
χAt κE
(23)
where κE is given by (9).
¡ l
¡ l
¢ E
¢ E
E
E
l
E
l
E
= sE
The next-period capital is given by kt+1
t +lt = R / R − ηρt st = R / R − ηρt ζ mt .
Using (4), and aggregating over all entrepreneurs yields:
¡
¢ E α
KEt+1 = Rl / Rl − ηρE
t ζ ψκE At NEt .
(24)
Dividing both sides of (24) by KEt , and substituting κE by its equilibrium expression,
we obtain (10). That NEt+1 /NEt = (KEt+1 /KEt ) / (1 + z) follows from (23).
Recall that the condition ν E > ν is equivalent to
Ã
µ
¶1−θ !−1
ψ ρE
(1 − η) ρE Rl
Rl
−θ
1+β
> (1 + ν) (1 + z) .
l
l
R − ηρE
R − ηρE
(1 − ψ) α
(25)
Using the fact that,
´
1
Rl − ηρE
1
η
1 ³
−α
− 1−α
α
(1
−
ψ)
=
−
=
χ
−
η
.
Rl ρE
ρE Rl
Rl
and rearranging terms allows us to rewrite (25) as
´
1
1
1 ³
ψ
−α
− 1−α
α
>
χ
−
η
(26)
(1
−
ψ)
(1 − ψ) α (1 + ν) (1 + z)
Rl
µ ³
´¶θ
1
1
1−θ
−α
−θ
− 1−α
α
+β (1 − η)
(1 − ψ) χ
−η
.
Rl
1
The right-hand side of equation (26) is monotonically decreasing in χ, while the lefthand side is constant. Moreover, since the right-hand side tends to ∞ (0) as χ → 0 (∞),
there exists a unique χ̂ such that
´
1
1
ψ
1 ³
−α
− 1−α
α
χ̂
−
η
=
(1
−
ψ)
(1 − ψ) α (1 + ν) (1 + z)
Rl
µ ³
´¶θ
1
1
1−θ
−α
−θ
− 1−α
α
+β (1 − η)
(1 − ψ) χ̂
−η
.
Rl
Therefore, the condition ν E > ν will be satisfied when χ > χ̂.
The following results are immediate.
1. The right-hand side of (26) is decresing in β, η and Rl so this inequality must hold
for sufficiently large β, η and Rl .
2. The left-hand side of equation (26) is decreasing in ν and z. Thus, the condition
ν E > ν is satisfied for sufficiently small ν and z.
Proof of Lemma 3. Using equation (23), and recalling that κE and κF are constant,
we can rewrite (12) as:
´
³
ηρ
Bt = ζ W wt−1 Nt−1 − KF t − El KEt
R
¶
µ
NF t ηρE χNEt
W wt−1 Nt−1
At Nt
− κF
− l κE
=
ζ
At Nt
Nt
R
Nt
µ
¶
¶
µ
NEt
ηρE χNEt
W
α At−1 Nt−1
− l κE
At Nt
=
ζ (1 − α) κF
− κF 1 −
At Nt
Nt
R
Nt
¶
µ
α−1
NEt
W (1 − α) κF
κF At Nt
=
ζ
− 1 + (1 − η)
(1 + z) (1 + ν)
Nt
which proves the Lemma.
1
Proof of Lemma 4. Part (i). We start by proving that ρlE = (1 − ψ) α χ
1−α
α
R. To this
l
,
aim, observe that, since (assuming that the incentive constraint is binding) mt = ψPtl yEt
then
¡ l ¢
©
¡ l ¢α
ª
(AEt nEt )1−α − wt nEt
= max (1 − ψ) Ptl kEt
Ξlt kEt
nEt
The first order condition yields:
nEt
kl
= Et
AlEt
µ
(1 − ψ) (1 − α) Ptl AlEt
wt
2
¶ α1
Then, plugging (19) and (20) into the first order condition yields
nEt = ((1 − ψ) χ)
1
α
µ
Ptl α
R
1
¶− 1−α
l
kEt
AlEt
(27)
Finally, plugging the optimal nEt into the profit function, and simplifying term, yields
the value of a E firm in the labor-intensive sector:
³
´1−α ³ α ´−1
³ α ´−1
¡ l ¢
1
1
l
l
((1 − ψ) χ) α
− (1 − α) χ−1 ((1 − ψ) χ) α
kEt
= (1 − ψ) kEt
Ξt kEt
R
R
1
1−α
l
l
= (1 − ψ) α χ α RkEt
≡ ρlE kEt
,
(28)
where ρlE is identical to ρE in the one-sector model of section 3 (see equation (6)). This
is the rate of return for E firms when F firms are active in the labor-intensive industry.
Next, we show that, when F firms are active in both industries, the return to investment in the capital-intensive sector for E firms, ρkE , is lower than ρlE . When F firms
are active in the capital-intensive industry, the value of a E firm in the labor-intensive
sector is
¡ k ¢
¡
¢1−α k
kEt
= (1 − ψ) Ptk AkEt
Ξkt kEt
k
k
= (1 − ψ) χ1−α RkEt
≡ ρkE kEt
where we have used equation (21) to eliminate Ptk . Finally, Assumption 1 ensures that
1
1
ρlE > ρkE (since (1 − ψ) 1−α χ > 1 ⇔ (1 − ψ) α χ
1−α
α
> (1 − ψ) χ1−α ). Thus, E firms will
not invest in the capital-intensive sector. This completes the proof of part (i) of the
Lemma.
Part (ii). We prove the argument by constructing a contradiction. Suppose that,
l
k
> 0 and KEt
> 0, KFl t > 0. Then, (19) and (20) hold true, and ρlE =
when KEt
1
(1 − ψ) α χ
1−α
α
R as shown in the first part of the proof, see (28). Moreover, ρlE = ρkE =
(1 − ψ) χ1−α R, since otherwise E firms would not invest in both industries. Solving for
Ptk yields
1
R
R
χα ¡
>¡
¢
¢
k 1−α
k 1−α
AF t
AF t
¡
¢1−α
where the inequality follows from Assumption 1, and Ptk = R/ AkF t
is the condition
Ptk = (1 − ψ)
1−α
α
that guarantees that F firms make zero profits in the capital-intensive industries. Thus,
the inequality establishes that F firms would be making positive profits in the capitalintensive sector, which is impossible in a competitive equilibrium. Thus, KFl t = 0 when
E firms are active in both sectors. This concludes the proof of part (ii) of the Lemma.
3
Proof of Proposition 2. The problem of the monopolist is:
¢
¡
max P k − R K k ,
Kk
subject to (16), and the equilibrium conditions, (17), (18) and (20). Replacing K k with
¡
¢σ
Y k = ϕP l /P k Y l (by equation 17), we can rewrite the problem as
³¡ ¢1−σ
¡ ¢−σ ´ ¡ l ¢σ l
P Y .
− R Pk
max P k
Pk
l
Here, Y is given by
α
¶−1
µ l ¶ 1−α
l
1−α
α
α
P
P
ψ (χ (1 − ψ)) α KEl +
AF N.
Yl =
R
R
as proven in Section 7.4. The first-order condition yields:
³
¡ ¢−1 ´
0 = (1 − σ) + σR P k
¶
µ
³
¡ k ¢−1 ´
dP l P k dY l P k
.
σ k l +
+ 1−R P
dP P
dP k Y l
µ
dP l P k
dP k P l
(29)
dY l P k
.
dP k Y l
Differentiating (18) w.r.t. P k yields:
¡ ¢1−σ
−ϕσ P k
dP l P k
=
.
dP k P l
1 − ϕσ (P k )1−σ
Now we compute the elasticities
and
Differentiating (29) w.r.t. P k yields:
¶
µ
1 YFl dP l P k
dY l P k
=− 1−
.
dP k Y l
1 − α Y l dP k P l
Therefore, the first-order condition can be rewritten as
³
¡ k ¢−1 ´
(1 − σ) + σR P
¡ ¢1−σ
¶¶
µ
µ
³
¡ k ¢−1 ´
ϕσ P k
1 YFl
= 1−R P
,
σ− 1−
1−αYl
1 − ϕσ (P k )1−σ
which is expression (22) in the text.
7.2
Post-Transition Equilibrium (Section 3.5)
In this section, we provide the details of the analysis in Section 3.5 Under log utility, the
equilibrium wage, rate of return on capital, output and foreign balance are given by:
wt = AEt (1 − α) (1 − ψ) (κEt )α
ρt = ρE,t = α (1 − ψ) (κEt )α−1
Yt = AEt Nt (κEt )α
Bt
Nt
β
β
=
=
wt
χ (1 − α) (1 − ψ) (κEt )α
At Nt
1 + β At Nt
1+β
4
If
α (1 − η) (1 − ψ) >
ψR
β
,
1 + β (1 + z) (1 + ν)
(30)
then capital in E firms evolves according to (14) and eventually converges to a steady
state, where
κ∗E
=
µ
ψ
ηα (1 − ψ)
β
+
1 + β (1 + z) (1 + ν)
R
1
¶ 1−α
.
Here we let Rl = R in the steady state. The steady state rate of return to capital is
thus equal to
ρ∗E =
α (1 − ψ)
ψ
β
1+β (1+z)(1+ν)
+
ηα(1−ψ)
R
.
Condition (30) ensures that ρ∗E > R; i.e., entrepreneurs never invest in bonds. Otherwise,
entrepreneurs will eventually place part of their savings in bank deposits..
5
7.3
Analysis of Footnote 33 in Section 4.3
In this section, we provide a complete formal argument of the discussion in footnote 33.
Assume, for simplicity, log preferences and η = 0. Let χi denote firm i0 s productivity
and Ki be the corresponding capital stock. Then, the rate of return to capital for firm
i is
1−α
1
1
ρiE = (1 − ψ) α χi α Rl = ω ρ · χiα ,
where ωρ is a unimportant constant. The law of motion of capital for firm i can be
written as
1
KiEt+1
= ω K χiα ,
KiEt
where ω K is also a unimportant constant. Denote ρEt =
(31)
P
ρiE KiEt /KEt the average
rate of return of E firms. We now show that ρEt grows over time, since the growth
rate of Kit is increasing in χi as shown by (31). Specifically, using (31), the next-period
average rate of return of E firms is equal to:
ρEt+1
Standard algebra establishes that
P
2
P
2
ω ρ χ α KiEt
= P 1i
.
α
χi KiEt
2
ω ρ χiα KiEt X ω ρ χiα KiEt α1 X ω ρ KiEt α1
P
=
χi >
χ =
P α1
P α1
KiEt i
χi KiEt
χi KiEt
P
1
ωρχ α K
P i iEt ,
KiEt
implying that ρEt+1 > ρEt . Thus, the average rate of return of E firms increases over
time in this case.
6
7.4
Equilibrium in Section 5.1
In this section, we provide a formal characterization of the equilibrium in the two-sector
economy of Section 5.1. The equilibrium entails are four stages, described in the text.
For notational convenience, we let AkJt = AlJt = AJ .
Proposition 3 Stage 1 is defined as
1
KEt
< ((1 − ψ) χ)− α
AE N
µ
Ptl α
R
1
¶ 1−α
(32)
,
where
Ptl
1
³
¡ k ¢1−σ ´ 1−σ
σ
= 1 − ϕ Pt
,
Ptk =
(33)
R
.
A1−α
F
(34)
In the first stage, both of the E and F firms are active in the labor-intensive industry
while only the F firms produce capital-intensive goods. Specifically, prices of labor- and
capital-intensive goods are determined by (33) and (34). Labor, capital and output in
the labor- and capital-intensive industries are such that
l
l
, NEt
= ((1 − ψ) χ)
NFl t = N − NEt
KFl t
=
µ
Ptl α
R
1
¶ 1−α
AF NFl t ,
l
KEt
= KEt ,
Ytl
=
µ
Ptl α
R
1
α
µ
¶−1
Ptl α
R
1
¶− 1−α
l
KEt
,
AE
ψ (χ (1 − ψ))
¢σ
¡
Ytk
k
, KEt
= 0,
Ytk = ϕPtl /Ptk Ytl , KFk t = 1−α
AF
respectively. Moreover, capital of E firms evolves according to
µ
¶α
βψ l KEt
KEt+1
=
P
,
AE N
1 + β t AE N
1−α
α
(35)
µ
l
KEt
+
¶ α
Ptl α 1−α
AF N,
R
(36)
(37)
(38)
and the aggregate output is equal to
¡ ¢σ
Yt = Ptl Ytl ,
(39)
k
Proof. When KFl t > 0 and KFk t > 0, it is straightforward from Lemma 4 that KEt
= 0.
(33) follows immediately from (18), whereas (34) follows from the zero-profit condition
(20) for F firms in the capital-intensive industry. The first part of (36) comes from (20).
7
The first part of (37) follows from (17). Using the condition that final-good firms make
zero profits, together with, (17) and (18) leads to
Yt = Ptl Ytl + Ptk Ytk
Ã
µ k ¶1−σ !
¡ ¢σ
Pt
Ptl Ytl = Ptl Ytl ,
=
1 + ϕσ
l
Pt
which establishes (39). (35) follows immediately from (27). To derive (36), observe that
µ
¶
¡ l ¢α
ψ NEt
l
+ 1 AF N
Yt = κF
1−ψ N
!
α Ã
1
µ l ¶− 1−α
µ l ¶ 1−α
l
1−α
1
KEt
Pt α
Pt α
ψχ α (1 − ψ) α
+ N AF
=
R
R
AE
Ã
1 !
µ l ¶ 1−α
l
1−α K
1
α
R
P
t
Et
ψχ α (1 − ψ) α
AF
=
+N
AE
R
Ptl α
α
µ l ¶−1
µ l ¶ 1−α
1−α
α
Pt α
P
t
l
=
ψ (χ (1 − ψ)) α KEt +
AF N.
R
R
Finally, (32) ensures that KFl t > 0, according to (27). The rest is immediate.
Proposition 4 Stage 2 is defined as
1
µ l ¶ 1−α
µ
¶ 1
1
1 Ptl α 1−α
KEt
Pt α
−α
((1 − ψ) χ)
<
≤
.
R
AE N
χ
R
(40)
In the second stage, F firms disappear in the labor-intensive industry. Specifically, prices
l
= N,
of labor- and capital-intensive goods are determined by (33) and (34). NFl t = 0, NEt
capital and output in the labor-intensive industries are such that
¡ l ¢α
l
KFl t = 0, KEt
= KEt , Ytl = KEt
(AE N)1−α ,
capital and output in the capital-intensive industry is identical to (37) in Stage 1. Moreover, capital in E firms also evolves according to (38) as in Stage 1.
Proof. The first inequality of (40) implies that KFl t = 0. Now the wage rate is determined by the marginal product of labor in E firms.
wt =
Ptl
µ
l
KEt
AE N
¶α
l
KEt
− ψ)
AE N
¶α−1
.
(1 − α) (1 − ψ) AE
It is then easy to show that
ρlEt
=
Ptl α (1
µ
8
.
Suppose that E firms are active in the capital-intensive industry. We have
ρkEt = (1 − ψ) χ1−α R.
k
However, the second inequality of (40) implies that ρlEt > ρkEt . Therefore, KEt
= 0 in
the second stage. Finally, (40) is non-empty by Assumption 1.
Corollary 1 If
α (1 + β)
,
(41)
βψR
then there are only two stages in the economy (E firms never produce capital-intensive
χ1−α <
goods).
µ
³
σ
Proof. Define P̃ ≡ 1 − ϕ (A
R
1−α
F)
l
1
´1−σ ¶ 1−σ
as the constant price of labor-intensive
goods in Stage 2. The law of motion (38) implies a upperbound of capital stock during
the second stage of transition:
KEt
≤
AE N
Ã
βψP̃ l
1+β
1
! 1−α
.
This gives the lowerbound of the rate of return:
Ã
!−1
l
βψ
P̃
α (1 − ψ) (1 + β)
ρlEt > P̃ l α (1 − ψ)
=
.
1+β
βψ
Recall that ρkEt = (1 − ψ) χ1−α R. Therefore, ρlEt > ρkEt always holds under the assump-
tion of (41).
Proposition 5 Stage 3 is defined as
1
χ
µ
Ptl α
R
1
¶ 1−α
KEt
1
≤
<
AE N
χ
µ
Ptl α
R
1
¶ 1−α
+
1
A1−α
E
1 !α
µ
¶σ Ã µ l ¶ 1−α
1 Pt α
Ptl
ϕ k
.
χ
R
Pt
(42)
In the third stage, the E firms start to produce capital-intensive goods. Specifically,
prices of labor- and capital-intensive goods are determined by (33) and (34). NFl t = 0,
l
= N, capital and output in the labor- and capital-intensive industries are such that
NEt
KFl t
Ytk
= 0,
l
KEt
1
=
χ
µ
Ptl α
R
1
¶ 1−α
¡ l ¢α
AE N, Ytl = KEt
(AE N)1−α ,
k
¡ l k ¢σ l
Ytk − A1−α
k
k
l
E KEt
= ϕPt /Pt Yt , KF t =
, KEt
= KEt − KEt
,
1−α
AF
9
(43)
(44)
respectively. Moreover, the total capital of E firms evolves according to the law of motion
µ
µ µ l ¶α
¶¶
l
KEt+1
KEt
KEt
βψ
KEt
l
k 1−α
=
Pt
−
.
(45)
+ Pt AE
AE N
1+β
AE N
AE N
AE N
k
Proof. Lemma 4 implies that KFl t = 0.37 KEt
> 0 implies equalized rates of return
across two industries.
ρkEt
=
ρlEt
1−α
⇒ (1 − ψ) χ
l
KEt
1
=
χ
Ptl
R=
µ
Ptl α
R
µ
l
KEt
(1 − ψ) α
AE N
1
¶ 1−α
¶α−1
⇒
AE N.
l
will be allocated to the capital-intensive inGiven total capital of E firms KEt , KEt −KEt
¢
¡ ¡ l ¢α
k
(AE N)1−α + Ptk A1−α
dustry. Enterpreneurs’ total income is equal to ψ Ptl KEt
E KEt ,
k
which gives the law of motion of capital (45). Finally, we need Ytk > A1−α
E KEt to ensure
KFk t > 0. This is given by the second inequality of (42).
Proposition 6 Stage 4 is defined as
KEt
1
≥
AE N
χ
µ
Ptl α
R
1
¶ 1−α
+
1
A1−α
E
1 !α
µ
¶σ Ã µ l ¶ 1−α
1 Pt α
Ptl
ϕ k
.
χ
R
Pt
In the fourth stage, economic transition is complete in the sense that F firms vanish
even in the capital-intensive industry. Specifically, prices of labor- and capital-intensive
goods are determined by (33) and (46).
Ptk =
R
.
(1 − ψ) A1−α
E
(46)
l
NFl t = 0, NEt
= N, capital and output in the labor- and capital-intensive industries are
identical to (43) and (44), except that KFk t = 0. The law of motion of capital in E firms
also follows (45) in the third stage.
The proof is immediate and is omitted.
Finally, we revisit the foreign balance. The balance sheets of the banks must take
into account the investments of F firms in both industries:
KFk t+1 + KFl t+1 + Bt =
37
β
wt Nt .
1+β
Alternatively, KFl t = 0 can be ensured by the first inequality of (42).
10
Proposition 7 In the first stage, the country’s asset position in the international bond
market increases if
¡
¢σ−1
α (1 − ψ)
> ϕσ P l /P k
,
ψ
(47)
where P l and P k follow (33) and (34), respectively.
Proof. Using (19) and (20), standard algebra shows that:
Bt+1
AF N
Kk
Kl
β
wt − F t+1 − F t+1
1+β
AF N
AF N
α
µ l ¶ 1−α
α
β
P
=
P l (1 − α) AF
1+β
R
⎛
⎞
1
³ l ´− 1−α
1
1
P α
µ l ¶ 1−α
l
α
KEt+1 ⎟
((1 − ψ) χ)
R
Pα
⎜
−
⎝1 −
⎠
R
AE N
=
¡ l k ¢σ õ
α !
¶−1
µ l ¶ 1−α
l
1−α K
ϕP /P
P lα
P
α
Et+1
.
+
ψ (χ (1 − ψ)) α
−
R
A
R
A1−α
FN
F
l
Since KEt+1
is increasing, Bt+1 is an increasing sequence if (47) holds.
The main results of Proposition 1 therefore carry over to this extended model economy.
11
Figure A1 (Panel 2) Capital-Output Ratios by Ownership and Sector in Manufacturing in 2006
Manufacture of Artwork and Other Manufacturing
Manufacture of Communication Equipment, Computers and Other Electronic Equipment
Manufacture of T ransport Equipment
Manufacture of General Purpose Machinery
Smelting and Pressing of Non-ferrous Metals
Manufacture of Non-metallic Mineral Products
Manufacture of Rubber
FIE
Manufacture of Medicines
DPE
Processing of Petroleum, Coking, Processing of Nuclear Fuel
SOE
Printing,Reproduction of Recording Media
Manufacture of Furniture
Manufacture of Leather, Fur, Feather and Related Products
Manufacture of T extile
Manufacture of Beverages
Processing of Food from Agricultural Products
0
0.5
1
Note: We use net value of fixed assets as a proxy for capital. Data source: CSY 2007
1.5
2
2.5
3
3.5
Figure A1 (Panel 1) Capital-Labor Ratios (thousand yuan per worker) by Ownership and Sector in Manufacturing in 2006
Manufacture of Artwork and Other Manufacturing
Manufacture of Communication Equipment, Computers and Other Electronic Equipment
Manufacture of T ransport Equipment
Manufacture of General Purpose Machinery
Smelting and Pressing of Non-ferrous Metals
Manufacture of Non-metallic Mineral Products
Manufacture of Rubber
FIE
Manufacture of Medicines
DPE
Processing of Petroleum, Coking, Processing of Nuclear Fuel
SOE
Printing,Reproduction of Recording Media
Manufacture of Furniture
Manufacture of Leather, Fur, Feather and Related Products
Manufacture of T extile
Manufacture of Beverages
Processing of Food from Agricultural Products
0
100
200
300
400
500
600
700
800
900