1

Economics of Innovation

GPTs II:The Helpman-Trajtenberg Model

Manuel Trajtenberg2005

2

based on:

Helpman, E. and M. Trajtenberg, "A Time to Sow and a Time to Reap: Growth Based on General Purpose Technologies". In Helpman, E. (ed.), General Purpose Technologies and Economic Growth. Cambridge: MIT Press, 1998.

3

GPTs in a Growth Model

Analyze the role of GPTs in the context of models of endogenous growth a la Romer (1990) and Grossman and Helpman (1991).

Very complex, hence scaled-back version: assume exogenous advances in the GPT itself (ignore feedback from user sectors to GPT).

Each (new) GPT prompts the development of “compatible components” to work with it: any kind of complementary investments, e.g. new generation computers that require compatible software packages.

4

A sketch of the model• New, more productive GPTs arrive at fixed time intervals.

• Each new GPT prompts the development of (new) compatible components, which require a fixed investment (R&D).

• Once enough such components become available, there is a switch in production of the final goods with the new GPT and its components.

• This process generates cycles of growth, including a downturn at first…

5

Building Blocks

1 , ii

i DQ

m

i iim

i i DQQ11

i : The productivity level of GPT i

Di: aggregate of components for GPT i

If there are m GPTs,

The final good Q produced with GPT i:

6

Building blocks – cont.

10 ,)(/1

0

in

ii

ii

i

djjxD

DQ

The elasticity of substitution between any two components:

xi(j) : component j used with GPT i

1)1/(1

7

Comparing the elements of the BT and HT models

In the BT model: applications sectors that integrate the GPT in their production.

In the HT model: final good sector that integrates both components and the GPT in its production.

Thus “components” “applications sectors”

8

The production of components• Each component supplied by a firm that owns the property right to the patent; got it by investing in R&D.

• Suppliers of components engage in monopolistic competition (can’t have p = mc, need margins to cover fixed cost).

• Each component produced with one unit of labor (no capital in the model), regardless of which GPT…Hence mc = w.

• Symmetry, hence same quantities, same price.

9

Components – cont.

wpjpi 1

)(

Producers of components equate MR = mc = w =>

,

)(

/)1(

/1/1

0

iiiii

iin

ii

xnXXn

xndjjxD i

Given symmetry:

10

Production of the final good

/)1(

/)1(

1

ii

i

ii

iii

iii

nQ

Xb

XnDQ

Xi : aggregate use of components by GPT i; also

labor employment (since one to one). The production of the final good is thus:

Where b is the amount of components per unit of final output, which equals also labor input per unit of final output (decreases with n, and with i):

Define:

11

Final good – cont.Competitive suppliers of the final good minimize (unit) costs, hence choose which GPT to adopt to do that, i.e. the most efficient combination {i, ni },

which means the smallest bi :

/)1(

1 ,

iiii

i nbbminb

Since it is a perfectly competitive sector, p = mc :

w

jpbw

p iQ )( since ,

12

Switching GPTs

1 ,

1

1

)1/(1

ii

i

i

nn

n

n

Suppose first GPT, with “enough” components n1 ;

second GPT appears, when will switch to it? When b2

< b1 =>

)1/(1

2/)1(1

/)1(2

2 1

n

nnn

More generally, switch from GPT i to GPT i+1 whenever,

13

The time line of a GPT

Phase 1: production with old GPT, develop components for new GPT

GPTi GPTi+1

time1Δ 2Δ

21 ΔΔΔ

Phase 2: produce with new GPT, keep developing components for it

1 ii TTΔ

1 ii nn GPTi-1

switch

14

The value of a component firm

i

iiiii n

Qbwx

wxwp

)1()1(

)(

detv it

tRi )()( ),(

rv

v

v i

i

i

i

The quantity of each component produced is xi=biQi/ni ,

hence the profits are:

Assuming the firm holds a patent forever, then the (stock market) value of the firm is:

At each point in time the non-arbitrage condition is satisfied (differentiate w.r.t. t):

15

Development of new components

0

ii

i

nwav

wav

Assume it takes a units of labor to develop a new component (i.e. a R&D workers).

Thus, the (fixed) cost of developing a new component is wa.

The value of a firm doing that is vi . Thus, free entry

into R&D implies,

16

Labor constraint, consumption

LQbna

Production

m

iii

DR

m

ii

1

&

1

deC

QC

t

m

ii

)(log max

1

Consumption:

Consumers maximize:

17

Consumption – cont.

)(

,1)()(1

tr

tQtpm

iiQ

Normalization:

Results from optimal allocation of consumption over time:

18

The time line of a GPT

Phase 1: production with old GPT, develop components for new GPT

GPTi GPTi+1

time1Δ 2Δ

21 ΔΔΔ

Phase 2: produce with new GPT, keep developing components for it

1 ii TTΔ

1 ii nn GPTi-1

switch

19

The time line of a GPT – 2d case

Phase 1: production with old GPT, develop components for new GPT

time1Δ 2Δ

321 ΔΔΔΔ

Phase 2: produce with new GPT, keep developing components for it

Phase 3: keep producing with this GPT, but stop developing components for it

3Δ

GPTi GPTi+1

20

Dynamics during phase 1The variables to keep track of: w and n

Phase 1: • Keep producing with (old) GPT i-1; number

of components for it constant at ni-1(Ti )

• Develop components for (new) GPT i; profits for “startups” = 0.

w

w

v

vawvwav

v

vr

v

v

v

i

iii

i

ii

i

i

i

i

,

0 ;

21

Phase 1 dynamics – cont.

0)( ,for )(1

1)()( , ,

iii

i

m

iQi

m

iiQ

TnL

ww

La

n

tQtpLQbnabw

p

)(1

)2(

)1(

wL

an

w

w

i

Two differential equations for phase 1:

22

Evolution of w and n over phase 1

)(

)(

1)(

)()(

)()(

)(1

(2) ,)1(

i

i

Tt

iii

Tti

i

eTaw

Tta

Ltn

eTwtw

wL

an

w

w

Both w and n rise over phase 1.

23

Phase 2 dynamics

ii

i

i

i

i

ii

QQi

i

anwwwav

v

v

vnbQ

w

Qpbw

pn

wbQ

1 ,

, ;)1(

1

1 , ,)1(

Now components are produced with new GPT i, hence they make positive profits, and the following holds:

24

Phase 2 dynamics – cont.

)(1

1

wL

an

anww

i

The evolution of n as in phase 1, hence characterized by the differential equations,

(ignore phase 3…)

25

Connecting equations

)1()(

)(

)()(

)()(

)1()(

1)(

1

1

1

11

11

1

1

Δ

iii

iiii

Δii

Δ

iii

eTaw

Δa

LΔTn

TnΔTn

eTwΔTw

eTan

Tw

Continuity of w at end of phase 1

Switching condition

26

Dynamic system

Whole system: 2 differential equations for phase 1, 2 for phase 2, 4 connecting equations => long-run stationary equilibrium with cycles of constant length.

When a new GPT appears the wage jumps up, makes it unprofitable to further invest in components for the old GPT => start investing in components for new GPT.

27

Computing GDP

/)1()()(

1)(

tn

twLtY k

k

/logYg

Profits:

Nominal GDP:

Real GDP (dividing by price of final output):

continuousDepends on w

Average growth rate over the whole cycle:

Ltw

QPQ

)()1(

)1()1(

28

Reminder…

w

jpbw

p iQ )( since ,

/)1(

1 , min

iiii

i nbbb

QPn

QP

n

Qbwx

wxwp

Qi

iQ

i

iiiii

)-(1 )1(

)1()1()(

29

The evolution of GDP

GPT 2GPT 1

GPT 3

time

Y

30

Comments on growth

Real growth starts only in the second phase, although the new GPT “has been around” for a while (recall Solow’s saying in the 1970s: “we see computers everywhere except for the economic statistics”). See Paul David on electricity.

It is only when there are enough complementary investments that economy-wide growth occurs (notice that the components sectors develop before…).

Decline during the first phase: diversion of resources from production (with the old GPT) to R&D for the new one.

31

Further comments - extensions

If allow for coexistence of successive GPTs, that would smooth transition, and may do away with initial downward jump in output and decline during phase 1. But in any case: faster growth during second phase.

Extensions:

• stock market

• Diffusion

• Relative wages of skilled and unskilled workers

32

From Paul David: “Computer and Dynamo”…

33

Stock market value

)()( :2 1

0)( :2

11

)(

)1(

)()( :1

)()()()(

1

)(

1

1)(

1

11

1

1

tawtvandPhase

tvPhase

eTn

detvPhase

tvtntvntS

i

i

tT

ii

T

t

it

i

iiii

i

i

34

Stock market value – cont.

2 )()(

1 )()(1)1(

)()( 1

phasetawtn

phasetawtnetS

i

itTi

Ltw

tS

tGDP

tS

)()1(

)(

)(

)(

Simulate S/GDP

35

Stock Market Value to GDP Ratio

S falls faster than GDP in phase 1, but starts recovering before phase 2

36

Price earnings ratio