1. Engine of Growth
ECON 401: Growth Theory

2. Engine of Growth
Objective: develop an explicit theory of technological progress and answer the questions:
Where does the technological progress come from?
Can we expect the growth to continue?
Is there a limit to economic growth?
The theories that tries to answer these questions are known as endogenous growth theory or new growth theory.

3. Engine of Growth
Basic Elements: Model is designed to explain why and how the advanced countries sustained growth
We will introduce search for new ideas to endogenize technological progress
Technological progress is driven by R&D
Production function is given by
Y=Ka(ALY)1-a (5.1)
Where K is capital stock, LY is labor, and A is the stock of ideas.

4. Engine of Growth
For a given level of technology, this production function exhibits constant returns to scale in K and L.
 It must exhibit increasing returns with respect to all three inputs.
Capital accumulation is given by
Labor and population grows at a constant rate of n.

5. Engine of Growth
A(t) is the stock of knowledge or the number of ideas that have been invented until time t. Change in A, then , will be determined by the number of people working to discover new ideas, LA, and the rate at which they discover new ideas.
Rate of discovering new ideas might
- be a constant, or
- might depend on the stock of ideas
- it might be an increasing function of A if the level of A positively impacts productivity of researcher
- it might be a decreasing function of A if the obvious ideas are discovered first, leaving the most difficult ones

6. Engine of Growth Hence, we can model What does it tell us if f>0?
indicates that productivity of research increases with the stock of ideas.
f<0? f=0?
It is also possible that average productivity of research depends on the number of people searching for new ideas at any point in time (duplication).
 LAg , where 0<g<1, should enter the production function and not LA.

7. Engine of Growth General production function for ideas
We will assume f<1. Individual researchers take the discovery rate of new ideas as given and see constant returns to research. However, for the economy as a whole, production function for ideas may not be characterized by constant returns to scale.
Allocation of resources:
Constant fraction of output is invested in capital
Allocation of labor between output production and idea production is determined by utility maximization and markets.
LA+LY=L
However, we will assume a constant fraction: sR=LA/L
We assume economy starts out with K0 and L0, and A0.

8. Engine of Growth Growth In Romer Model:
Per capita growth is due to technological progress
Balanced growth path has:
gy= gk= gA
What is the rate of technological progress along a balanced growth path?
Rewrite the production function for ideas as
Growth rate of A is gA along a balanced growth path.
happens if both numerator and denominator grows at the same rate
 Along a balanced growth path, growth rate of number of researchers must be equal to the growth rate of population.

9. Engine of Growth
If the growth rate of researchers is higher, the number of researchers would eventually exceed the population which is impossible.
Solving for gA : gA=(ln/1-f) – long run growth rate of this economy is determined by the parameters of the production function for ideas and the rate of growth of researchers (or population)
What if l=1 and f=0?
Productivity of researcher is constant
No duplication problem
Production function looks like:
Economy generates a constant number of new ideas each period. Over time, this constant number becomes a smaller fraction of A. Hence, growth rate of new ideas falls over time, approaching to zero. However, technological progress never ceases.

10. Engine of Growth
To generate exponential growth number of ideas must be expanding over time  number of researchers is increasing. So growth of ideas is clearly related to growth in population
How is this different from Neoclassical model?
In neo-classical model, higher n reduces level of income along a balanced growth path
Here, an additional effect – input to creative process  ideas are non-rivalrous and hence everybody benefits
Do we expect long-run growth to cease if the population stops growing?

11. Engine of Growth
Special case: If l=1 and f=1 we can have sustained growth in the presence of a constant research effort. In this case, productivity of researcher is proportional to the existing stock of ideas.
Drawback: World research effort has increased enormously over the last 40 years, but the growth rate of advanced economies did not rise as predicted by the Romer model.
- So we can reject f=1 (knowledge spillover parameter) is rejected by this evidence. It can be positive and large but less than 1.
As in neoclassical model, changes in government policy and changes in investment rate do not have long-run effect on growth rate.

12. Engine of Growth
The idea-based models in which changes in policy can permanently increase the growth rate all rely on the assumption that f=1. But thıs generates a counterfactual prediction. Setting f<1, however, eliminates the long run growth effect of policy as well.
What happens if the share of population searching for ideas increase permanently?
(1) Consider the effect on technological progress and on stock of ideas
(2) Analyze the model
To simplify, assume l=1 and f=0. Figure 5.1 shows what happens to technological progress when sR increases.

13. Fig. 5.1

14. Engine of Growth
A permanent increase in the share of the population devoted to research raises the rate of technological progress temporarily, but not in the long run.
Figure 5.2.
What happend to the level of technology? Figure 5.3.
- The level of technology is permamnently higher as a result of the permanent increase in R&D.

15. Fig. 5.2

16. Fig. 5.3

17. Engine of Growth Along a balanced growth path,
- Per capita output is proportional to the population of the world economy along a balanced growth path. That is, model has a scale effect: larger world economy will be a richer worl economy  nonrivalrous nature of ideas  larger market for an idea and larger potential creators of ideas
Economies that invest more will be richer
Share of labor devoted to research has two effects:
More researchers mean fewer workers producing output
More researchers mean more ideas

18. Engine of Growth Economis of the Model: Romer economy has
a final goods sector
produce output
an intermediate goods sector
Reason: presence of increasing returns to scale
a research sector
produce ideas
Research sector creates new ideas, which take the form of new varieties of capital goods. Research sector sells the exclusive right to produce a specific capital good to an intermediate goods firm. The intermediate goods firm, as a monopolist, manufactures the capital good and sells it to the final goods sector.

19. Engine of Growth Final Goods Sector:
Large number of perfectly competitive firms
Uses capital and labor to produce final output Y
Final output is produced using labor and a number of different capital goods (intermediate goods)
A measures the number of capital goods that are available to be used in the final goods sector. For a given A, production function exhibits constant returns to scale.
Firms decide on how much labor and how much of each capital good to use in producing final good by solving the profit maximization problem.

20. Engine of Growth
Firms hire labor until marginal product of labor equals wage.
Firms rent capital goods until the marginal product of each kind of capital equals the rental price.
Intermediate-goods Sector:
Consists of monopolists. They gain their monopoly power by purchasing the design for a specific capital good. Because of patent protection, only one firm manufactures each capital good.
Simple production function: one unit of raw capital can be automatically translated into one unit of capital good.
Firm charges a price that is simply a markup over marginal cost. All capital goods sell for the same price and hence each capital good is employed by the final goods sector by the same amount. Each capital goods firm earn the same profit.

21. Engine of Growth Research Sector:
Ideas are new designs for capital goods (e.g., a faster computer chip)
Inventor receives a patent for a new design (patent lasts forever). Inventor sells the patent to an intermediate goods firm and uses the proceeds to consume and save.
How much will a potential bidder pay for the patent?
Present discounted value of the profits to be earned by an intermediate goods firm.

22. Engine of Growth Solving the Model: Remember:
Aggregate production function exhibits increasing returns to scale
Increasing returns require imperfect competition.
Capital goods sell at a price that is greater than marginal cost
Profits earned by these firms are extracted by the inventors
This framework is called monopolistic competition
(no economic profits in the model).
Because of arbitrage, individuals are indifferent between working in the final goods sector or in the research sector.

23. Engine of Growth
Labor earns a wage that is equal to marginal product in that sector.
Researchers earn a wage based on the value of the designs they discover = marginal product*value of the new ideas created.
 Because of arbitrage, these wages has to be equal.
The share of the population that works in the research sector is (after some algebra):
Note that faster the economy grows, the higher the fraction of the population that works in research. The higher the discount rates that applies to current profits to get the present discounted value (r-n), the lower theis fraction.

24. Engine of Growth
Interest rate in this economy is less than the marginal product of capital.
How is this different from Solow model?
In the Romer model, production in the economy is characterized by increasing returns and all factors cannot be paid their marginal products.
In Solow, rK+wL=Y – all factors are paid their marginal product – no output remains to compensate individuals for their effort in creating new A. In this model, capital is paid less than its marginal product, and the remainder is used to compensate researchers for creation of new ideas.

25. Engine of Growth Optimal R&D:
Is the share of the population that works in research optimal?
Three distortions:
(1) Market values research according to the stream of profits that are earned from the new design
- ignores the impact of new invention on the productivity of future research. Hence, researchers are not compensated for this effect. This distortion is often called “knowledge spillover”.
(2) “Stepping on toes” effect. Researchers do not take into account the lower research productivity through duplication.

26. Engine of Growth
(3) “Consumer surplus effect”. Figure 5.4. Potential gain to society is larger than the monopoly profit and hence too little innovation is generated.
In general, these distortions can be very large. Consider the cure for malaria or cholera – consumer surplus and thge knowledge spillovers has to be large - governments fund research in general.
Empirically, positive externality of research outweigh the negative externalities, social rate of return far exceeds the private rate of return – market tend to produce too little innovation.
Classical economic theory argues that monopolies (or P>MC) are bad for welfare and efficiency (deadweight losses). However, economics of ideas suggest that it is critical for firms to set their prices above MC for innovation.

27. Fig. 5.4