1. Norton Media Library
Chapter 5
The Engine
of Growth
Charles I. Jones

2. Where Does Technological Progress Come From?
Technological progress plays a crucial role in the Solow-type growth models
However, so far technological progress and its growth were given exogenously
We are now asking this question: where does technological progress come from?
In other words, we are seeking to endogenize technological progress
Improvements in technology are understood in terms of the firms’ incentives to maximize their profits

3. Romer versus Schumpeterian Models
Romer models look at technological progress from the point of view of increasing the variety of products
Laptops in addition to desktops
Blue coke in addition to original red-colored coke
iPhones in addition to just handheld sets
Schumpeterian models focus on improving the quality of existing products
Faster computers with more memory
Safer cars
Creative destruction
It turns out the implications for growth are similar irrespectively of whether we focus on more variety or better quality

4. Romer Model: Production Function
Romer models explain why and how the advanced countries experience sustained growth.
The focus is on the technological frontier, i.e. the best practice.
Technology diffusion to the less developed countries is left aside.
Main equations are similar to the Solow-type models.
The production function is a little bit different:
A is now the endogenous stock of ideas
is labor input (index Y will be explained later)
There are three factors of production now since A is endogenously determinedincreasing returns to scale

5. Romer Model: Capital Accumulation
The capital accumulation equation looks very similar to that of the Solow models:
Labor grows exponentially, as in Solow:

6. Romer Model: Production of New Ideas
The key addition of Romer model to the Solow framework is the introduction of the production function of new ideas.
More researchers can produce more ideas, recall the discussion in chapter 4 about the benefits of population growth in terms of generating more ideas.
is the number of people attempting to discover new ideas.
is the rate at which new ideas are discovered.
Labor L is allocated between work and innovation.

7. Rate of Discovery of the New Ideas
The rate at which new ideas are discovered is modeled as follows:
Several cases are possible with respect to parameter representing the so-called externality effects:
Productivity of research increases with the stock of ideas A
Laser, calculus, electricity etc
The “fishing out” case: most obvious ideas discovered first, subsequent ideas are increasingly difficult to discover
Implies constant rate at which ideas are discovered:
Productivity of research is independent of the stock of A

8. Rate of Discovery of the New Ideas
Rate of discovery of the new ideas may also depend on , the number of innovators (inventors).
Consider
The term rather than just reflects the fact that with many researchers thinking about innovations, duplication of effort is possible.

9. Intuition Behind Innovation Equation
It follows that (why?)
An individual engaged in research creates new ideas.
Rate of discovering new ideas will depend on the aggregate research effort.
reflects the “stepping on toes”, or congestion, effect: too many cooks will spoil the broth (duplication effort)
is the positive externality effect: benefits of the discovery spill over to the other researchers (calculus, Newton’s laws). This is the “standing on shoulders” effect

10. Growth in the Romer Model
Initial endowments: we assume an economy starts with initial endowments of and
As in the Solow models, all per capita growth is due to technological progress.
Along the balanced growth path, all variables are growing at the same rate:

11. Rate of Technological Progress
What is the rate of technological progress along a balanced growth path?
Taking logs and differentiating the last equation implies
The growth rate of the number of researchers must equal the rate of growth of the population, otherwise researchers will outnumber the whole population:

12. Long-Run Growth Rate
Along the balanced growth path, the economy is growing at the rate of technological progress:
Suppose there is no duplication of research effort, and that research productivity is independent of the stock of ideas, i.e.
In this case ideas get produced according to
If the number of researchers stays the same, ideas do get generated since , but the rate of growth of ideas approaches zero with time because A is increasing:
However, if the growth rate in ideas is constant, too.

13. Population Growth and Long-Run Growth Rate
Higher population growth rates strains resources (Solow-type models), but it also increases the number of researchers , which increases the number of non-rivalrous ideas. These ideas benefit the whole economy, so the overall effect of the population growth is beneficial!
Long-run growth is impossible without the ongoing growth of the population since that would mean constant number of new ideas relative to the ever growing stock of ideas.
In this way, assuming results in a rather sour prediction.

14. Population Growth and Long-Run Growth Rate
Let us assume now that there is no duplication problem (as before), but that research productivity depends on the stock of existing ideas, that is, assume that
In this case, the productivity of researchers grows over time even if the number of researchers is constant.
Caveat: If , that would imply ever increasing growth rates of the economy since has been growing rapidly. However, the U.S. growth rates have been rather modest at 1.8% for the last century.
Hence, we assume that

15. Romer and Solow Models
In Solow (neoclassical) models, government policies produce level, but not growth effects.
Romer model produces the same result even if technological progress is endogenous, rather than exogenous, as it is in the Solow models.
Even increasing the share of researchers will not change the growth in Romer model since

16. Comparative Statics: R&D Share Increase
Suppose the government extends subsidy to the R&D institutions so that the share of researchers goes up.
Assume for simplicity that
Along the balanced growth path,
If the share of researchers increases relative to L, an increased number of new ideas is produced, which increases the growth rate of technology.
Rate of technological progress will exceed population growth n, so starts declining, and so does .

17. Comparative Statics: R&D Share Increase
Initially, the rate of technological progress exceeds population growth
However, as time goes by
starts declining back to the initial level

18. Comparative Statics: R&D Share Increase
A permanent increase in the share of researchers increases the rate of technological progress only temporarily

19. Comparative Statics: R&D Share Increase
However, the level of technology increases permanently as a result of a permanent increase in
Note: the level and growth effects in the Romer model produced by an increase in are qualitatively similar to the effects produced by an increase in the investment share in the Solow models.

20. Solving Romer Model The only difference with Solow is the term
Technology level increases with labor force as:
Substituting into the first equation results in:
A larger economy is also a richer economy: scale effect.

21. Economics of Romer Model
We saw in Chapter 4 how R&D can be the result of profit-maximizing behavior.
What is the microeconomics behind this link?
Consider a three-sector economy: final goods, intermediate goods, and the research sector.
Research sector produces ideas that take the form of the new variety of capital goods: new computer chips, industrial robots, or printing presses.
Researchers sell the exclusive right to produce a specific capital good to an intermediate goods firm, a monopolist, that sells intermediate good to the final-goods sector that produces consumer output.

22. Final-Goods Sector
A large number of perfectly competitive firms combine labor and capital to produce a homogeneous final output good, Y.
The production function in the final goods sector is:
Output Y is produced using labor and a number of intermediate goods
Notice that the level of technological progress A in this specification is exactly the number of intermediate goods!
For any given level of A, the production function exhibits constant returns to scale.

23. Profit Maximization Problem
Final Goods Sector:
Profit Maximization Problem
Replace for convenience the summation sign with an integral:
Normalizing the final good price to unity, we come up with the following profit-maximization problem:
where w and are wages and capital rental prices, respectively.

24. First-Order Conditions
Final Goods Sector:
First-Order Conditions
Firms hire labor until the marginal product of labor equals the wage
Firms rent capital goods (intermediate goods) until the marginal product of each kind of capital equals the rental price

25. Intermediate Goods Sector
Produced by monopolists
Are a variation of the same capital good that faces a downward-sloping demand curve because of the specific design
Producers of intermediate goods
Sell their output to the producers of final goods
Are monopolists because they purchase a specific design from the research sector
Patent protection ensures monopolistic power
Production technology in the intermediate goods sector
One unit of raw capital is transformed into exactly one unit of the intermediate capital good with a specific design
It costs r (dollars) to rent one unit of raw capital
Raw capital technically comes from the primary sector which we do not consider

26. Intermediate Goods Sector: Profit Maximization Problem
is the demand function for the capital good. Remember it is equal to from the final-goods sector profit maximization problem.
First-order conditions:
is the inverse price elasticity of demand which is equal to

27. Intermediate Goods Sector: Producers’ Markups and Demand
The intermediate-goods firm simply charges a markup over the raw materials’ price r.
Since the demand function for capital good j is given by
, it follows that
The demand for capital good j will be identical for all firms in the intermediate goods sector:
The demand for capital good j will be a function of the raw materials price r, technology parameter , and the number of manual workers

28. Intermediate goods sector producers earn positive profits which enables them to buy patents from the research sector.

29. Intermediate Goods Sector:
Producers’ Profits
The intermediate goods producers’ profit is:
Since , and ,
it can be shown that the profit of capital-goods producers is given by:
How do we derive this expression?

30. Intermediate Goods Sector: Demand for Capital and
Aggregate Production Function
Total demand for capital from the intermediate sector:
where K is the total capital stock in the economy
Amount of capital used by each intermediate sector firm:
We now have the link with the Solow model:

31. Research Sector Researchers are mining for new ideas
Faster computer chips
Genetically altered seeds
Novel organizational practices
Etc
New designs are discovered according to
On discovery of a new design, the inventor receives a patent from the government
Patent gives the exclusive right to produce the design
Patents are sold to the intermediate goods producers

32. Research Sector: Price of a Patent
What is the price of a patent?
The price of a patent must be equal to the present discounted value of the future profits by the intermediate-goods firm. (why?)
Present discounted value: a recap
Present discounted value of a stream of incomes given the discount factor r is the sum of these future incomes discounted by r taking into account the number of the periods.
In case of the research sector, let us assume there is just one period.

33. Patent Price and the Arbitrage
Arbitrage is the method of profiting by exploiting price differences across different places or investment opportunities.
The practice of arbitrage leads to the equalization of prices among different places or investment opportunities. (why?)
Suppose the patent costs If you invest it in the bank for one period (year), your gain will be equal to If you use the same amount of money to buy a patent, your gain will be
The arbitrage condition requires that
Rewriting the above equation results in
Along a balanced growth path, r is constant, and

34. Patent Price and the Arbitrage
The arbitrage condition for the price of the patent now says that this price is equal to the present discounted value of the future profits plus the possible future change in the patent’s price
We have to discount the sum of these future values by the discount factor r.

35. Patent Price and the Arbitrage
Along the balanced growth path:
The interest rate r is constant
Rewrite the arbitrage equation in the following fashion:
Since r is constant, must be constant as well
Constant implies both profits and the patent price are growing at the same rate, which is equal to n (why?)
It follows that

36. Features of Romer’s Model
The aggregate production function exhibits increasing returns:
Increasing returns require imperfect competition
Firms in the intermediate goods sector are monopolists
Capital goods sell at a price exceeding marginal cost
Profits earned by the intermediate producers are extracted by the inventors who get compensated for the time they spend looking for new ideas
There are no economic profits in the model since all rents are compensating some factor input
Markets no longer result in the most desirable outcome due to the presence of imperfect competition

37. Solving for the Share of Researchers
What is the fraction of total population L that decides to engage in research to develop new designs of capital goods? Remember that
and
Labor working in the final goods sector receives a wage w that is equal to its marginal product in this sector:
Labor in the research sector earns the wage equal to the value of its marginal product in that sector as well:
By the principle of arbitrage, wages in the two sectors must be the same: It follows that

38. Growth and Share of Researchers
The faster the economy grows, the greater the share of researchers in the economy
The higher the interest rate obtainable from the bank, the lower the fraction of researchers
Interest rate can be shown to be equal to , which is lower than the marginal product of capital,
which is equal to
In the Solow model, perfect competition ensures that all factors are paid their marginal products. In the Romer model, imperfect competition requires to pay factors less in order to free funds for research.

39. Creative Destruction Romer model
Technological level A is equal to the number of intermediate goods
The increase in A is brought about by the profit-maximizing behavior of innovators and firms
Steam engines and electric motors
Since the lifetime of a patent is forever, older inventions never get obsolete
Typewriters must then coexist with computers
Schumpeterian view
Creative destruction: 1940, Joseph Schumpeter
New technologies are replacing the older ones
Productivity grows due to replacement of old designs by the new ones

40. Schumpeterian Model: Discrete Innovations
Aggregate production function:
Innovation occurs in steps since
Think of as “walking technology”, then will be a horse cart, a steam engine, and a modern car.
Each time we innovate, we get more productive: if i>j.
New innovations occur according to
where is the size of the innovation.
The growth rate of A is then equal to The length of the
time period between any two innovations is uncertain.

41. Schumpeter versus Romer
The lifetime of a patent in the Romer model is forever
As a result, the Romerian innovators have no fear of being sometime replaced by an entrant
Schumpeter emphasizes the fact and the fear of being eventually replaced by another innovator
As a result, the fear of replacement decreases incentives to do research in the Schumpeterian models of creative destruction
The long-run results are similar between Romer and Schumpeter

42. Schumpeter: Production and Technology
Aggregate production function:
Technological progress is discrete:
The technological level is indexed by i
An example of discrete jumps in technology:
is a quill pen
: a fountain pen
: a typewriter
: a printer

43. Schumpeter: Technological Progress
Technology grows in steps
The size of innovation is captured by parameter
Growth is largely unpredictable
Since innovations at any given moment in time occur with a specific probability, they may not arrive for a long period of time, then appear suddenly one by one
When an innovation occurs, it increases the stock of knowledge in a discrete fashion:

44. Technological Growth and Growth over Time
Technological level grows discretely
It is wrong to interpret the above equation as the one representing growth over time
Innovations happen randomly, so there are some periods when innovations do not occur at all

45. Schumpeterian Model: Probability of Innovation
The process of innovation
Innovations are uncertain and made with probability at any point in time
Shoulders and Toes
Same effects as before, but now they work through the probability of invention by one researcher:
For the economy as a whole, the probability of innovation equals:

46. Schumpeterian Model: Probability of Innovation
The effects of on innovation probability are twofold:
Increasing stock of knowledge increases the chance of finding a new innovation, assuming : standing on shoulders
Making new innovations becomes difficult as knowledge advances: alternatively, it takes more time to discover something really new (which kind of effect is this?)

47. Schumpeterian Model: Remaining Parts
The remaining parts of the Schumpeterian model are identical to the Romer model:
Capital accumulation:
Labor force growth:
Labor division:
Initial endowments:

48. Schumpeterian Growth Growth is irregular
Innovations happen with a certain probability
As a result, economic growth is stepwise
Due to the random nature of innovations in the Schumpeterian model, we can only make statements about growth in terms of averages, or alternatively, in terms of mathematical expectations.
Expected growth rate of A over time:
The expected growth rate of A over time is equal to the probability of innovation times the size of the innovation :

49. Schumpeterian Balanced Growth Path
Along the balanced growth path, the actual growth rate of the economy will converge to the expected growth rate:
Similarity with Solow and Romer:
Growth rate of technology plays a key role in Schumpeterian growth, just like it does in the Solow and Romer models
The growth rate in a schumpeterian economy is equal to the expected growth rate of technology

50. Schumpeterian Expected Growth Rate of Technology
Taking logs and differentiating, we obtain the following:
Since the number of researchers cannot exceed total population,
, which implies that
The average long-run growth rate in the Schumpeterian model is identical to the growth rate in Romerian model!

51. Growth Path in the Schumpeterian Model
The average growth is governed by the growth of population n, the duplication of effort , and the spillovers parameter .
Flat sections in the graph indicate those time periods when no innovation occurs
Log income per capita jumps by the value of each time an innovation occurs

52. Innovation Size and Growth Rate
The innovation size does not affect the growth rate
While the larger innovation size increases the absolute size of technological level A, increasing productivity, it also reduces the probability of finding next innovation since
Such reduction of probability occurs because we are assuming
The productivity effect of a larger innovation size gets offset by the longer period of time that has to pass before the next innovation arrives.

53. Economics of Schumpeterian Model
Imperfect competition is again necessary to justify monopolistic profits needed to compensate researchers for their work.
Differences are in how intermediate goods are used, and in the nature of innovation.
Three sectors: final goods, intermediate goods, and research.
Only one single intermediate good produced by a monopolist who owns a patent.

54. Creative Destruction
Researchers work to produce a new version of the capital good that is more productive for the final good.
Intermediate good producers are eventually replaced when the new patent is sold by the researchers to another capital goods producer.
The intermediate-goods producer is always in danger of being replaced (destruction) by a new, more productive, supplier (creation).
The understanding of possibility of such replacement will be reflected in the value that the intermediate goods producer will be willing to pay for a patent.

55. Final Goods
Unlike Romer, in the Schumpeterian model there is only one single intermediate good, and is the level of technological progress that changes discretely over time, i=1,2,3,…..
The production function for final goods is:
Output Y is produced using labor , and a single capital good , the intermediate good at step i.
A large number of firms are competing in the final-goods sector.
The production function is constant returns to scale since doubling
and labor will double the output.
Example: i=1 is mainframe computers, i=2 is modern servers. One server is more productive than one old mainframe, i.e.

56. One Version at a Time
Later versions of capital goods will ensure a higher productivity
Similarly to Romer, we’ll see that all intermediate firms will charge the same price for their capital inputs
The final goods firms will, of course, choose inputs with higher productivity at the same price
Therefore, at any given period of time between the two innovations only one single version (the latest one) of the capital input will be used by the final goods sector

57. Final Goods: Profit Maximization
Final goods producers are maximizing their profits as follows:
where w is wages, and is the rental price per unit of capital good
First-order conditions:
Compare to Romer FOC:
In both models, the economic reasoning is similar: hire factors until what you pay per unit is equal to the marginal product.

58. Intermediate Goods
A single intermediate good is produced by a single intermediate-goods firm that bought a design from the research sector. Monopolistic position is ensured by the patent protection.
Similarly to Romer, one unit of raw capital will produce one unit of the capital good, so that the profit maximization and the resulting price of the intermediate good is the same in the Schumpeterian model:
The first-order conditions imply that
Similarly to Romer model, the intermediate firm charges
Again we have a markup over the production cost r.

59. Intermediate Goods: Why Only One Version?
Why is it that the final goods firms only purchase one version of the capital good, and why that version is always the latest version?
Since, as we have shown, is the same irrespectively of the particular version, i.e. since doesn’t really depend on i, the version number, the final goods producers will naturally buy only the latest, most productive, version of the capital good.
For that reason, at any moment in time, only one single firm producing intermediate goods will be operating.

60. Intermediate Goods: Profits and Aggregate Output
The intermediate goods producer’s profits are given by
which is very similar to Romer’s Technically, it’s Romer’s profits if A=1.
However, in the Schumpeterian model profits are not divided over many intermediate goods producers, rather all going to just one firm.
Capital stock in the economy must equal the stock of the single intermediate good:
Aggregate output will then be

61. Research Sector
New versions of the capital good arrive with constant probability
A successful inventor receives a patent from the government. The patent lasts forever. The patent is sold to the intermediate good’s producer.
A patent number i+1 will not be sold to an intermediate good producer that uses patent number i, so the existing producer will have to go (destruction) to give way to the new firm (creation).
Why? Basically, patents cost money, and a new entrant doesn’t have operating costs, so it can offer a better price compared to the incumbent.

62. Arbitrage and Patent Value
The amount of money equal to , the value of the patent, can be deposited at the bank. In this case the increase in value will be equal to
The same amount of money can be paid to obtain the patent. In this case, the increase in value will be profits from producing an intermediate good, plus the change in the value of the patent, minus the expected value of the loss of due to somebody producing an innovation with probability
The arbitrage equation:
Rearranging,
Denote to be the aggregate probability of a new innovation occurring.

63. Patent Value and Balanced Growth Path
Along a balanced growth path, rental rate of capital r is constant.
The aggregate probability of innovation occurring is constant as well along the balanced growth path.
For the ratio to be constant, profits and patent value have to grow at the same rate.
Since , profits are growing at rate
From prior analysis, we know that along the balanced growth path
The arbitrage equation implies then:

64. Patent Values in Romer and Schumpeterian Models
The patent value in Schumpeterian model
is lower compared to the patent value in Romer model,
In the Schumpeterian model, since sooner or later the existing producer of intermediate goods must exit the market, the patent’s value is declining with the increase in probability of a new innovation
As the size of the innovations increases, the value of a patent increases, too.

65. Arrow Replacement Effect
Why is it that a new innovation always results in a replacement (destruction) of the existing producer of an intermediate good?
Arrow replacement effect due to Kenneth Arrow (1962): the new firm will always bid more for a patent since acquiring a new patent will not involve loss in value due to the obsolescence of a previous patent.
For the existing firm, buying a new patent will obliterate the value of the existing patent.

66. Allocation of Labor to Research
Individuals working in the final goods sector are earning
Individuals can also work as researchers and earn
The arbitrage condition (i.e. the requirement that the two types of wage be the same in both sectors) implies
It can be shown that as a result,

67. Allocation of Labor to Research
The Schumpeterian fraction of researchers
is similar to the Romer fraction:
The term r-n appears in both formulae, saying that the higher the discount rate that applies to profits, the lower the fraction of researchers.
Two effects of : (1) as the chance of innovation increases, the value of patent declines, in (2) innovations are more lucrative if they are more probable:

68. Allocation of Labor to Research
Since the aggregate probability of innovation exerts two opposite effects on , the combined effect can be seen by taking a derivative of with respect to
The combined effect is positive since the gains from innovation are large relative to the losses from eventual replacement (show it).

69. Comparing Romer and Schumpeter
In both models, the endogenous growth rate is given by
The innovation-increasing effect of population growth works in both models to the same extent irrespectively of the exact type of the innovation-producing process.
Schumpeterian approach reflects the dynamics of firm behavior by exploiting the concept of creative destruction.
The Schumpeterian model will have a higher if , or if the discount rate applied to profits is relatively large: people do more research if the future destruction carries little weight.
In both Schumpeterian and Romer models, since growth only depends on the population growth rate n, all policy changes produce only level effects, and no growth effects.

70. Optimal R&D: Knowledge Spillover
In Romer and Schumpeterian models, there are three distortions to research: lack of accounting for the spillovers to the future research, the “stepping on toes” effect that allows duplication of effort, and the consumer surplus effect since monopolists value innovation only according to their profits.
The knowledge spillover effect
Market values research according to profits from new design.
However, means productivity of research increases with the stock of ideas: the standing on shoulders effect.
Since researchers are not compensated for increasing the productivity of the future researchers, the market produces too little innovation: Newton, Maxwell, Einstein…
This is a classic positive externality problem

71. Optimal R&D: Stepping on Toes
Stepping on toes effect
means that researchers lower their research productivity because they duplicate effort.
Duplication of effort happens when two or more researchers are working to solve the same problem, or to produce the same design.
In this case, the externality is negative, and the market is producing too much research.

72. Optimal R&D: Consumer Surplus
Consumer surplus effect
An inventor only captures the monopolistic profit, which is smaller than the consumer surplus representing social gains from inventing the good. Too little innovation is generated.

73. Importance of Basic Research
The discrepancy between private gains due to innovation, and the social gains can be very large in case of fundamental, or basic, research: calculus, vaccination, the Internet etc
In this case the government should fund basic research since otherwise researchers will not find it profitable to engage in fundamental science
Non-rivalrous ideas
No direct short-term profits
Griliches (1991) finds rates of return in the fundamental inventions area to be around 40% and 60%, which is much higher than the private returns.

74. Too Little Research
Among the three effects associated with research, only one (stepping on toes) results in too much research because of the duplication of effort. The other two (knowledge spillovers and consumer surplus) work to result in too little research.
The combined result is, too little research is produced.

75. The Importance of Monopolies
Classical economics views monopolies as undesirable since monopolies charge above marginal cost, and since they result in deadweight losses to the society.
However, in the economics of ideas monopolies are crucial since it is monopolistic profits that motivate researchers to produce innovations.
If regulators destroy imperfect competition and monopolies in the market for innovations, there will be no gains to productivity, and as a result, no growth.