1 Economics of Innovation GPTs II: The Helpman-Trajtenberg Model Manuel Trajtenberg 2005

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 i : The productivity level of GPT i D i : aggregate of components for GPT i If there are m GPTs, The final good Q produced with GPT i:

6 Building blocks – cont. The elasticity of substitution between any two components: x i (j) : component j used with GPT i

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. Producers of components equate MR = mc = w => Given symmetry:

10 Production of the final good X i : 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, n i }, which means the smallest b i : Since it is a perfectly competitive sector, p = mc :

12 Switching GPTs Suppose first GPT, with “enough” components n 1 ; second GPT appears, when will switch to it? When b 2 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 GPT i GPT i+1 time Phase 2: produce with new GPT, keep developing components for it GPT i-1 switch

14 The value of a component firm The quantity of each component produced is x i =b i Q i /n i, 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 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 v i. Thus, free entry into R&D implies,

16 Labor constraint, consumption Consumption: Consumers maximize:

17 Consumption – cont. 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 GPT i GPT i+1 time Phase 2: produce with new GPT, keep developing components for it GPT i-1 switch

19 The time line of a GPT – 2d case Phase 1: production with old GPT, develop components for new GPT time Phase 2: produce with new GPT, keep developing components for it Phase 3: keep producing with this GPT, but stop developing components for it GPT i GPT i+1

20 Dynamics during phase 1 The variables to keep track of: w and n Phase 1: Keep producing with (old) GPT i-1; number of components for it constant at n i-1 (T i ) Develop components for (new) GPT i; profits for “startups” = 0.

21 Phase 1 dynamics – cont. Two differential equations for phase 1:

22 Evolution of w and n over phase 1 Both w and n rise over phase 1.

23 Phase 2 dynamics Now components are produced with new GPT i, hence they make positive profits, and the following holds:

24 Phase 2 dynamics – cont. The evolution of n as in phase 1, hence characterized by the differential equations, (ignore phase 3…)

25 Connecting equations 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 Profits: Nominal GDP: Real GDP (dividing by price of final output): continuous Depends on w Average growth rate over the whole cycle:

28 Reminder…

29 The evolution of GDP GPT 2 GPT 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

34 Stock market value – cont. 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