1
Appendix: Resources, Innovation and Growth in the Global Economy
2
Pietro F. Perettoa; , Simone Valenteb a
Duke University; b ETH Zürich. 12 July 2011
3
A
Agents’Behavior and World General Equilibrium Household problem: derivation of (11), (12), (13). In each instant t, each household
solves the static problem
f
max
J
XiHj ;XiF j
g
J
J
log u s.t. E =L =
Z
0
NH
PiH XiHj =LJ
di +
Z
NF
PiF XiF j =LJ di;
0
4
where J = H; F , j = h; f . Denoting by { H the Lagrange multiplier, the …rst-order conditions
5
in H are
6
7
8
9
10
11
12
XiHh = h
LH PiH
{H
R NH 0
XiHh =LH
1
LH PiF (1 i and XiF h = h R NF di {H 0 XiF h =LH
) 1
i : di
(A.1)
Multiplying both sides of the …rst (second) equation by PiH (PiF ), integrating both sides across varieties, and eliminating { H by means of the initial expressions in (A.1), we obtain R N H H Hh R NF F F h P X P X di di XiHh = R 0N H i 1i PiH and XiF h = R 0N F i 1i PiF : (A.2) H F (P ) di (P ) di i i 0 0 Taking the ratio between these two expressions and substituting XiHh and XiF h by means of
(A.1), we have R N H H Hh Pi Xi di 0 R NF F F h = 1 Pi Xi di 0
:
(A.3)
Appendix: Resources, Innovation and Growth in the Global Economy
1
2
3
Following the same steps for F , we have R N H H Hf R NF F F f Pi Xi di Pi Xi di Hf Ff H 0 0 Xi = R N H P and X = PiF R F i i N H 1 F 1 (Pi ) di (Pi ) di 0 0 R NH
0 R NF 0
PiH XiHf di PiF XiF f di
=
1
(A.5)
:
Market clearing yields the values of production in the two countries: Y
H
=
Z
NH
PiH XiHh di
+
0
Y
F
=
Z
NF
PiF XiF h di
+
0
6
7
8
9
(A.4)
and
4
5
;
2
Z
Z
NH
PiH XiHf di;
(A.6)
0 NF
PiF XiF f di:
(A.7)
0
R NF R NH ) E H ; from From (A.3) and (2), we have 0 PiH XiHh di = E H and 0 PiF XiF h di = (1 R NF R NH ) E F and 0 PiF XiF f di = E F . Combining (A.5) and (2), we have 0 PiH XiHf di = (1 these with the constraints (A.6)-(A.7), we obtain (12).
Substituting (A.3) and (A.5) in (1), we obtain the indirect utility u~J E J =LJ = log
10
(1
)1
+ log E J =LJ :
(A.8)
13
In each country the household chooses the time path of expenditure E J =LJ that maximizes R1 e t u~J E J (t) =LJ dt subject to the dynamic wealth constraint (4) re-written in terms 0
14
(11).
11
12
15
of assets per capita. The logarithmic form (A.8) implies the standard Keynes-Ramsey rules
From (A.2) and (A.4), we have: XiHh XiHf XiF h XiF f
R NH
= R0N H 0
R NF
= R0N F 0
PiH XiHh di PiH XiHf di PiF XiF h di PiF XiF f di
=
=
1 1
EH ; EF
(A.9)
EH ; EF
(A.10)
Appendix: Resources, Innovation and Growth in the Global Economy
1
3
where the last terms follow from the derivation of (12) above. Hence: XiHh
+
XiHf
=
XiHh
1+
XiF f + XiF h = XiF f 1 +
EF ; EH EH : EF
1 1
(A.11) (A.12)
2
Using (A.2) and (A.4) to eliminate, respectively, XiHh and XiF f from the right-hand sides of
3
(A.11) and (A.12), we obtain: XiHh + XiHf = XiF f + XiF h =
E H + (1 ) EF PiH R NH H 1 (Pi ) di 0 F
= PiH
"
H
E + (1 )E R NF F 1 (Pi ) di 0
PiF
"
= PiF
R NH
1
(PiH )
0
R NF 0
#
YH YF 1
(PiF )
di #
;
;
(A.13) (A.14)
di
4
where the left-hand sides represent the total demand for the i-th variety produced in country
5
H and F , respectively. Substituting XiH = XiHh + XiHf and XiF = XiF f + XiF h in (A.13)
6
and (A.14), respectively, we obtain the demand schedule (13).
7
Manufacturing …rms: the monopolist problem, derivation of (14) and (15). The producer
8
of the i-th variety in country J solves the following problem. Given technology (5), the cost-
9
minimizing conditions over rival inputs, LJXi and MiJ , yield
10
11
12
13
14
15
16
17
WJ PM
=
1
MiJ LJ X
, which in turn
i
yields total cost J W J LJXi + PM MiJ = W J + CX W J ; PM
ZiJ
XiJ ;
(A.15)
where J CX W J ; PM
(PM )
W
J 1
"
+
1
1
1
#
(A.16)
is the standard unit-cost function homogeneous of degree one. From (A.15) instantaneous h i J XiJ W J . Since the monopolist knows the pro…ts JXi read PiJ CX W J ; PM ZiJ demand schedule (13), instantaneous pro…ts can be written as J Xi
=
h
PiJ
J CX
J
W ; PM
ZiJ
i
aj PiJ
WJ :
(A.17)
Appendix: Resources, Innovation and Growth in the Global Economy
R NJ
PiJ
1
1
where aJ
2
as given by the single monopolist. The problem is to maximize ViJ (t) de…ned in (6), with
3
instantaneous pro…ts given by (A.17). The problem reduces to a static one, where the …rst
4
order condition determines the price-setting rule of each monopolist,
5
[Y J =
4
PiJ =
1
0
di] only contains aggregate variables and is therefore taken
J CX W J ; PM
ZiJ
(A.18)
;
6
which implies a positive mark-up of over the marginal cost. Given (A.16), the conditional
7
factor demands for LJXi and MiJ of each …rm are LJXi MiJ
J J W J ; PM CX @CX W J ; PM J J Zi Xi = + (1 ) = + @W J WJ J J J J @CX W ; PM CX W ; PM = ZiJ ZiJ XiJ = XiJ : @PM PM
8
J Substituting CX W J ; PM
9
W J LJXi = W J + (1
10
11
)
1
= 1
XiJ(A.19) ; (A.20)
PiJ from (A.18), equations (A.19)-(A.20) imply 1
PiJ XiJ and PM MiJ =
PiJ XiJ . Integrating across varieties in
both these expressions yields (14). J . Time-di¤erentiating ViJ in (6) Denote the rate of return to horizontal innovations as rN V_ iJ ViJ
12
and imposing symmetry, we have
13
expressions and solving for rJ gives
J rN
14
where we can substitute
15
from (6), to obtain (15).
16
ZiJ
ZiJ
J Xi
(t) =
J Xi ViJ
J + = rN
Y_ J = J Y 1 YJ NJ
and
V_ iJ ViJ
J N_ J Xi (t) + NJ ViJ (t)
=
Y_ J YJ
N_ J . NJ
Combining these
;
W J from (A.17)-(A.18), as well as ViJ = Y J =N J
Resource-processing in Home: derivation of (16)-(17). In the resource-processing sector, 1
17
the cost-minimizing conditions over LM and R yield
18
function is
19
H
h
CM W ; p = (&)
W
H 1
+ (1
&) p
1
WH p
i11
;
=
& 1 &
R LM
. The associated cost
(A.21)
Appendix: Resources, Innovation and Growth in the Global Economy
1
5
and the conditional factor demands for raw resource and labor read @CM W H ; p p R pR = PM M = SM W H; p H @p CM (W ; p) H @CM W ; p WH L W H LM = PM M = SM W H; p @W H CM (W H ; p)
PM M; PM M;
2
R where we have de…ned the elasticities of CM (:; :) to resource price and wage as SM W H; p
3
L L and SM p; W H , respectively. Recalling that SM W H; p = 1
4
expressions yield (16). Log-di¤erentiating (A.21) we have (17).
5
R SM W H ; p , the above
Proof of Proposition 1. Recalling that the global demand for the intermediate is M = 1
Y H + Y F . Substituting this
6
M H + M F , the …rst expression in (14) implies PM M =
7
equation in the …rst expression in (16), and imposing the market-clearing condition R = ,
8
we obtain
9
1
R p = SM W H; p
YH +YF :
10
Since (1
11
trade condition (10) can be re-written as PM M F = (1
12
PM M F =
) is the share of expenditures on imported goods in both countries, the balanced
1
1 13
(A.22)
) EH
(1
Y F from (14), we have
Y F = (1
) EH
(1
) EF :
(A.23)
14
From (12), substituting Y F = E F + (1
15
(18). Substituting (E F =E H ) back in (A.23) to eliminate E F we have
16
) E F . Substituting
E H =Y F = 1 +
1 1
) E H in (A.23) we obtain the …rst expression in
(A.24)
: 1
17
Combining (E F =E H ) in (18) with (A.24), we obtain E F =Y F
18
second expression in (19). From (12), we also have Y H = E H + (1
19
can be substituted by means of (E F =E H ) in (18) to obtain E H =Y H , that is, the …rst
20
expression in (19). Combining this result with (A.24) yields the second expression in (18).
21
=1
, which is the
) E F , where E F
Appendix: Resources, Innovation and Growth in the Global Economy
1
B
6
Resource Booms in World Equilibrium Production, Wages and Resource Price: derivation of (20), (21), (22), (23). Equation
(20) follows directly from the second expression in (18). Equation (22) is derived as follows. Using
M
= 0 and the free-entry condition N H ViH = W H Y H , we rewrite the wealth
constraint (4) as
N_ H NH
+
V_ iH ViH
= rH +
LH YH
p N H ViH
+
EH . N H ViH
Using
V_ iH ViH
=
Y_ H YH
N_ H NH
from the
free entry condition, we obtain W H LH p Y_ H H = r + + H H Y Y YH
EH : YH
2
Substituting rH = (E_ H =E H ) + from (11), and recalling that E_ H =E H = Y_ H =Y H from (18),
3
we have
4
EH = YH
+
W H LH p + : YH YH
(B.1)
5
Solving this expression for Y H yields (22). Equation (21) is obtained following the same
6
steps for country F : since Foreign has no resource endowment, the analogous expression
7
of (B.1) for J = F is E F =Y F
8
R re-writing (A.22) as p = SM (1; p)
9
we obtain (23).
+ W F LF =Y F , from which we have (21). Finally,
=
1
Y H + Y F =LH , and using (20) to eliminate Y F ,
Proof of Proposition 2. Equations (20)-(23) form a static system of four equations in four unknowns determining constant equilibrium values E H , Y H , E F , Y F . Di¤erentiating (24) we obtain d ( p ) =d in expression (25). Equation (17) then implies that the price elasticity of demand for the raw resource,
R M
(1; p)
1
R dSM (1; p ) p ; R dp SM (1; p )
10
is less than (greater than, equal to) unity if
11
expressions for dY H =d , dY F =d , dW F =d in (27) follow directly from (20), (21) and (22).
12
is less than (greater than, equal to) unity. The
Appendix: Resources, Innovation and Growth in the Global Economy
J Horizontal innovation and resource booms: derivation of (28). Plugging rN =r=
7
and
setting Y_ J = 0 in (15), we have N_ J (t) = N J (t) 1
2
3
( + )+
1 1
N J (t) W J (t) ; Y J (t)
which can be re-arranged to yield (28). Proof of Proposition 3. The proof follows immediately from Proposition 2: see the text above Proposition 3.
4
Reallocation of labor in home: derivation of (31), (32), (33). Denoting total employment
5
in start-up operations by LJN = (N_ J + N J ) LJNi , the growth rate of the number of …rms
6
implied by the free entry condition (7) is
7
8
9
N_ J (t) W J (t) LJN (t) = N J (t) Y J (t)
(B.2)
:
where we can substitute N_ J (t) =N J (t) by (28) to obtain LJN
(t) =
1
Y J (t) W J (t)
N J (t) :
(B.3)
10
Setting J = H and W H = 1, we obtain (31). Setting W H = 1 in the second equation in (14),
11
we obtain (32). Substituting (31)-(32) in the market clearing condition LM = LH
12
we obtain (33).
LH X
LH N
13
Total factor productivity, growth and welfare: derivation of (34)-(38). Imposing sym-
14
metry in (13) and substituting pricing rule (A.18), we obtain (34). Substituting (34) in
15
(1) yields (35). Using (20) to eliminate from
16
17
18
country K 6= J, we obtain (36). De…ning
J
J
the value of manufacturing production in N J =N0J
1, the innovation rate (29) can
be re-written as N J (t) = N0J
1+ J 1 + Je
t
in each t:
(B.4)
Appendix: Resources, Innovation and Growth in the Global Economy 1
N J (t)
By de…nition of TFP, we have T J (t) = ZiJ (t)
8
1
, where we can substitute (B.4)
together with ZiJ =ZiJ = z, to get log T J (t) = log T J (0) +
1
t+
1
1+ J 1 + Je
log J
:
t
1
Without loss of generality, we can approximate log 1+1+J e
2
log T J (t) as in (37). Substituting (36) and (37) in (35), we obtain (38).
5
J
(1
t
e
) and thus write
Proof of Proposition 4. Substituting the equilibrium level of instantaneous utility (38) in
3
4
'
t
the welfare function (3) we have Z 1 J e t log J + log T J (t) + (1 U =
) log T K (t) dt:
(B.5)
0
6
7
Integration of (37) yields Z 1 1 1 log T J (t) e t dt = log T J (0) + 2
+
1
0
J
( + )(
1)
(B.6)
:
8
Substituting (B.6) in (B.5), and setting log T J (0) = 0 without loss of generality, we obtain
9
(39).
10
11
C
The Role of Trade: Introducing Tari¤s Suppose that both countries impose tari¤s on imported goods. We denote by
h
(
f)
12
the ad-valorem tari¤ imposed by the Home (Foreign) government on the units of imported
13
consumption goods produced by Foreign (Home), and by
14
the Foreign government on imported resource-based intermediates produced in Home. The
15
tari¤s on manufacturing goods
16
and imply, after utility maximization, the following rules for expenditure allocation
17
YH =
EH +
1 1+
h
and
f
the ad-valorem tari¤ imposed by
modify the consumers’ expenditure constraints
E F and Y F = f
m
EF +
1 1+
EH : h
(C.1)
Appendix: Resources, Innovation and Growth in the Global Economy
1
The tari¤ on resource-based intermediates
2
Foreign so that
3
PM M H =
4
1
6
1+ |
m
{z
Y H and PM M F (1 +
0
1+
1+
1
h
1+
m
13
2
positive, "
2
+
(C.3)
f
)2 ) 1+ h (
(1
2 (1+
m
1+
=
1
=
(1
+
)
f
+
1 1+
f
)
+
+
1 1+
h
1 1+
f
h
f
m
+
1 1+
f
+
1 1+
h
f
1 2
(1
f)
(C.7)
;
)2 = 2 #
> 0 implies that the term in square brackets is strictly
(C.8)
> 0:
As a consequence of (C.8), the e¤ects of the tari¤ on intermediates, >0
(C.6)
;
1 1+
1 1+
)2 (1 + h ) (1 +
@ (E F =E H ) @ m
(C.5)
;
1 1+
2
) (1+ h )(1+
m
f
1 1+
m
(1
2
1
(C.4)
;
1 1+
1
EF YF where
m
+
1+
12
(C.2)
PiH XiHf di
0
=
EH YH
11
Y F:
1
1
1 1+
=
Y YH
10
R NH
PiF XiF h di
1
9
1
=
Following the same steps as in the proof of Proposition 1, we obtain the modi…ed ratios
F
8
m)
1 EF : 1+ f | {z }
1 YF = EH 1+ h } | {z } R NF
PM M F
EF EH
7
instead, modi…es the conditional demand of
The balanced trade condition thus reads 1
5
m,
9
@ (Y F =Y H ) @ m
Rewriting (C.6) as " (1 )2 EH = + YH 1+ h
1+ f m
m
1
+ (1
@ (E F =Y F ) @ m
<0
1
1 1+ 1+
@ (E H =Y H ) @ m
<0
)
#
>0
m,
are (C.9)
1
;
(C.10)
Appendix: Resources, Innovation and Growth in the Global Economy
1
we also obtain
(1 + YH = YF
2
3
4
we have h
@ (Y F =Y H ) @ h
and
f,
@(
E F =E H @
7
8
9
@ (E H =Y H ) @ f
> 0 and 1
h)
1+
+
m
(1
1 1+
> 0. Rewriting (C.5) as +
f
)
+
(1 )2 1+ f
1 1+
1
1
1+
m
(C.11)
;
f
> 0. As a consequence, the e¤ects of the tari¤s on manufacturing goods,
read
@ (E F =E H ) @ h
5
6
@ (E H =Y H ) @ h
10
f
)
<0 >0
@ (Y F =Y H ) @ h @ (Y
F =Y H
@
>0
)
@(
E H =Y H
=?
f
where the sign of @ Y F =Y H =@
@ (E H =Y H ) @ h @
@ (E F =Y F ) @ h
>0
)
@(
E F =Y F
>0
f
@
)
f
=0
(C.12)
>0
is ambiguous.
f
Notice that, by (C.2), the total value of intermediate production now reads 1
PM M = PM M H + PM M F =
YH +
YF 1+ m
(C.13)
;
R W H; p On the basis of (C.4)-(C.7) and (C.13), and recalling that pR = SM
PM M , we can
10
recalculate the four-equations static system (20)-(23) augmented by the presence of tari¤s –
11
and thus determine manufacturing production values, resource price and Foreign wage: 1 F
Y YH
1+
(1+
=
+ F
p
E =Y LF H L +p = (E H =Y H ) R = SM (1; p)
where
13
with (C.16), we obtain p =
f
+
)
1 1+
+
1 1+
h
1 1+
f
(C.14) h
YF
(C.15) (C.16)
1
1+
1+
YH
(C.17)
m
Y F =Y H is the production ratio de…ned in (C.14). In particular, combining (C.17)
12
14
1 1+
f
F
WF = YH
)2 ) 1+ h (
(1
2
m
R LH SM (1; p) ; R SM (1; p)
E H =Y H 1
1+
; 1+
m
(C.18)
Appendix: Resources, Innovation and Growth in the Global Economy
11
1
which is essentially expression (24) augmented for the presence of tari¤s. We can now analyze
2
the comparative static e¤ects of variations in the tari¤ levels.
3
Variation in
h.
Suppose that the Home government increases
h.
From (C.9), there
4
is a decrease in
Y F =Y H . From (C.10), there is also an increase in E H =Y H . Hence,
5
the net e¤ect on
is positive. This implies a reduction in p in (C.18). Considering (C.16),
6
the reduction in p combined with the increase in E H =Y H
7
of Home’s manufacturing production, Y H . This implies that the asymptotic mass of …rms
8
in Home, N H , decreases, that is, a negative e¤ect on TFP growth in Home. Considering
9
Foreign: by (C.9), the increase in
h
implies a decrease in the value
does not modify E F =Y F . Hence, considering (C.15),
10
the wage-to-production ratio W F =Y F is una¤ected and there is no e¤ect on Foreign TFP
11
growth.
12
Variation in
f.
Suppose that the Foreign government increases
f.
From (C.10), there
13
is an increase in E H =Y H . From (C.11), however, the e¤ect on
14
the net e¤ect on
15
TFP growth. Considering Foreign: by (C.9), the increase in
16
considering (C.15), the wage-to-production ratio W F =Y F increases. This implies a reduction
17
in the asymptotic mass of …rms in Foreign, N F , and therefore a negative e¤ect on Foreign
18
TFP growth.
19
Variation in
is ambiguous. Hence,
is ambiguous. This implies ambiguous e¤ects on Home’s production and
m.
f
increases E F =Y F . Hence,
Suppose that the Foreign government increases
m.
and a decrease in E H =Y H . Hence, the net e¤ect on
From (C.9), there is
20
a decrease in
is ambiguous. This
21
implies ambiguous e¤ects on Home’s production and TFP growth. Now consider (C.15):
22
since E F =Y F
23
increases. This implies a reduction in the asymptotic mass of …rms in Foreign, N F , and
24
therefore a negative e¤ect on TFP growth in Foreign.
increases (by (C.9)), the wage-to-production ratio in Foreign, W F =Y F ,