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 ,

Appendix: Resources, Innovation and Growth in the Global Economy

Jul 12, 2011 - Appendix: Resources, Innovation and Growth in the Global Economy .... Market clearing yields the values of production in the two countries: 5 ..... Since (1 ξ) is the share of expenditures on imported goods in both countries, the ...

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