Introduction to economic growth (5)

Transcription

1 Introduction to economic growth (5) EKN 325 Manoel Bittencourt University of Pretoria August 13, 2017 M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

2 Introduction By now we suspect that technological progress is the engine of economic growth and development, then what we need now is an explicit framework that models technological progress endogenously (technology is part of the model now) we have already seen, intuitively, that technological progress, or the creation of new ideas that are incorporated into the production function, and which will eventually result in higher output or higher utility, is the product of profit-maximising agents such a model was firstly proposed by Paul Romer, or Romer (1990) for short M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

3 The basic elements of the Romer model The Romer model introduces a sector which is exclusively dedicated to the creation of new ideas, or an R&D sector in this economy the main purpose of the model is to understand the sustained economic growth that has taken place in developed countries. (Is it relevant to developing countries? What about technological transfer amongst countries? What about the technological frontier of the whole world?) just like the Solow model, the Romer model consists of a production function and a set of equations describing how these inputs evolve over time M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

4 The basic elements of the Romer model The Romer production function is as follows; Y = K α (AL Y ) 1 α (1) capital and labour exhibit constant returns to scale, (just as before), however, this production function as a whole exhibits increasing returns to scale M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

5 The basic elements of the Romer model Because the production function incorporates the role of ideas A, and ideas are nonrivalrous, we have increasing returns to scale (think of Jobs and Wozniak and their idea of personal computers, one idea, multiple applications) the equations describing capital and labour are identical to the ones provided by the Solow model: K = s K Y dk L/L = n, capital is a function of savings, or how much people forego consumption, and also of the depreciation of capital. Labour, or population, grows exponentially at the rate n M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

6 The basic elements of the Romer model However, so far we are still in the realm of the neoclassical model, we now need an equation describing technological progress, remember that now A is supposed to be endogenous! So, Romer provides us with a production function for new ideas. Can we say at this stage that more researchers can produce more ideas? If so... Ȧ = θl A (2) Ȧ is the number of new ideas produced at any point in time, and it is a function of the number of people engaged in discovering new ideas L A and the rate at which they discover new ideas θ (or how productive researchers are) can we say that θ depends on the stock of ideas up to this point in time? If so, the stock of past ideas will raise the productivity of researchers now M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

7 The basic elements of the Romer model Hence, taking a step forward we can assume that the rate of new ideas is given by; θ = δa φ if φ > 0, it indicates that research productivity increases with the stock of existing ideas ( If I have seen farther than others, it is because I was standing on the shoulders of giants ), this is Newton referring to Kepler. What if φ < 0 or φ = 0? In those cases the fish gets harder to catch, and the productivity of ideas is independent of the stock of existing ideas can we also say that the productivity of research is a function of the number of people doing research at some point in time? If so, then we can have a more general production function for ideas; Ȧ = θl λ A Aφ (3) M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

8 The basic elements of the Romer model What if φ > 0? Then we have positive spillovers in research, the gains to society and future generations generated by Calculus are immense (social returns) ( standing on shoulders effect ), however the gains that Newton (private returns) got were not as large what about λ? If λ < 1 then there are duplications of ideas, for instance, one researcher at Oxon is developing an model of technological transfer, which is identical to the model created by another researcher at MIT ( stepping on toes effect ) lastly, labour in this economy is allocated as L Y + L A = L M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

9 Growth in the Romer model The question now is: what is the growth rate along a balanced growth path? The model assumes that if a constant fraction of the population is engaged in R&D then all per capita growth is due to technological progress. Alternatively, without technological progress there is no growth, g y = g k = g A so, if growth depends on technological progress, the next question is: what is the rate of technological progress along the balanced growth path in this economy? using equation (3) (the production function for ideas) above and dividing it by A, Ȧ/A = δ Lλ A A 1 φ = g A (4) M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

10 Growth in the Romer model Then by taking logs and derivatives of (4) we get, 0 = λ L A L A (1 φ)ȧ A the intuition of the above is that in the balanced growth path, the growth rate of the number of researchers is equal to the growth rate of the population. What if not? Then the number of researchers will eventually exceed the population, however according to the model L A /L A = n. Substituting L A /L A = n into (5) we get, g A = λn (6) 1 φ the intuition is that the long-run growth rate of this economy is given by the parameters in the production function for new ideas, λ and φ, and the rate of growth of researchers, or by the population growth rate M Bittencourt (University of Pretoria) EKN 325 August 13, / 34 (5)

11 Growth in the Romer model Moreover, for exponential and sustained growth to happen, the number of new ideas should be increasing over time, or the number of researchers should be increasing, which is given by the population growth. More researchers)more ideas. That is why we have n in (6) above a comparison with Solow is in place here: in the Solow model a higher population decreases K/L. In the Romer model, a higher population leads to more ideas and ideas are nonrivalrous, and therefore can generate increasing returns to scale furthermore, can we also assume that the productivity of researchers has been increasing over time? Given that the US economy has been growing on average at 1.8% per year, we can suggest that φ < 1. Why? If φ = 1 the US economy would be growing at much faster rates than in the last 100 years! Hence, φ is still positive, but 6= 1 M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

12 Comparative statics: a permanent increase in the R&D share The question now is: what happens if the government decides to increase the subsidy for R&D and that leads to an increase in the share of the population searching for new ideas? more researchers produce more ideas, so the growth rate of technology is higher. However, as Ȧ/A > n the ratio L A /A decreases over time and so does the rate of technological change until g A = n. There is only a temporary change, see Figures 5.1 and 5.2 what about the level of technology given the increase in R&D subsidy? It will permanently increase given the increased R&D subsidy, see Figure 5.3 M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

13 Comparative statics: a permanent increase in the R&D share M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

14 Comparative statics: a permanent increase in the R&D share M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

15 Comparative statics: a permanent increase in the R&D share M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

16 Comparative statics: a permanent increase in the R&D share Per capita output in the model is given by: y s k A = ( n + g A + δ ) 1 α α (1 sr ) s R = L A /L, s k Ȧ A = θ s RL A, A = θ s RL g A y (t) = ( n + g A + δ ) 1 α α (1 sr ) θs R L(t) (7) g A the intuition is: a larger economy comes from the nature of ideas, a larger economy means a larger market for ideas, and a more populous economy has more inventors. More researchers lead to more ideas, and that increases the productivity of an economy M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

17 The economics of the Romer model The Romer model benefits from developments in other branches of Economics, say, industrial organisation, for the analysis of imperfect competition (the microfoundations of macroeconomics), which were developed in the 1970s and 1980s a key contribution by Romer (1990) is that in his model there are profit-maximising agents plus endogenous technological progress the economy has 3 sectors: a final-goods sector, an intermediate sector, and a research sector the research sector creates new ideas which will be incorporated into new capital goods. Furthermore, the research sector sells the exclusive right of use of a particular idea to a firm in the intermediate sector, which in turn becomes during a period of time a monopolist. Lastly, the intermediate-goods sector sells the capital good to a firm in the final-goods sector, which produces the final output M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

18 The final-goods sector It consists of a large number of firms operating in a competitive environment, combining capital and labour, and producing a homogeneous good (just like Solow) the production function of this sector includes labour L and a number of capital goods x, from the intermediate sector ideas are incorporated into the capital goods x supplied by the intermediate sector and used by the final-goods sector to produce output Y M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

19 The final-goods sector The production function in this sector exhibits constant returns to scale, recall that this sector presents a large number of firms, so we have perfect competition. Hence, doubling the amount of labour L and capital x only doubles the output Y produced Y = L 1 Z A Y α 0 x α j dj as it is always the case with economics, firms have also to decide how much labour and capital are going to be used to produce output. Hence, we can write a standard profit-maximisation problem as follows; max L Y,xj L1 Y Z A α 0 x α j dj wl Y Z A 0 p j x j dj, (8) M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

20 The final-goods sector The first-order conditions are; w = (1 α)y/l Y p j = αl 1 Y α xα 1 j (9) the intuition is that firms pay a salary that is equal to the marginal product of labour w, and firms rent capital until the marginal product of capital equals the rental price p j, a perfect competitive outcome, by definition M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

21 The intermediate-goods sector This sector consists of monopolists, which produce capital goods and then sell those goods to the firms in the final sector they are monopolists because they purchase the patent, or the monopoly rights, from the research sector. Patent protection is the source of their monopoly rights, at least in the short and medium run the profit-maximisation problem for an intermediate-goods firm is as follows; max x j π j = p j (x j )x j rx j, M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

22 The intermediate-goods sector Where p j (x j ) is the demand function for the capital good by the final-goods sector from (9) above. The first-order condition is (we are making use of the chain rule here); p 0 (x)x + p(x) r = 0, which implies, after some rewriting, p = p 0 (x)x/p r, p 0 (x)x/p is the elasticity of demand, and it can be calculated from (9) above. It is α 1, hence the firm in this sector charges a price that is a markup over the marginal cost; recall that we have imperfect competition in this sector. After substituting the elasticity of demand α 1 above, p = 1 α r M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

23 The research sector In the research sector researchers are free to engage in the discovery of new ideas, and the incentive for that is that maybe one or two good ideas will be sold to the intermediate sector (patents or monopoly rights) recall that ideas are designs or instructions (blueprints) of how to (re)combine inputs into the production function so that more output is generated. These new instructions are discovered according to equation (3), Ȧ = δl λ A Aφ. once the new idea is invented and some firm from the intermediate sector realises that the idea is a good one, the inventor sells the patent to this firm in the intermediate sector M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

24 The research sector The price P A paid by the firm in the intermediate sector for this monopoly right is a function of the present discounted value of the profits to be made by this firm. Making use of the standard arbitrage equation; rp A = π + Ṗ A, r = π + Ṗ A P A the LHS is the interest earned from putting P A in the bank, the RHS is the profits π plus the capital gain coming from any change in the price of the patent Ṗ A. The above implies; P A = π r n, which is the price of a patent along a balanced growth path. The intuition is that P A is determined by the expected profits to be made π, the interest earned from putting the money in the bank (r is constant along the BGP, by definition) and the population growth n M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

25 Solving the model We have seen the microeconomics behind the model. There are constant returns to scale to capital and labour, and we know that ideas (also an input to production) display increasing returns to scale (they are nonrivalrous), therefore in the aggregate this model exhibits increasing returns to scale moreover, if we have increasing returns to scale, we also have imperfect competition. It is the intermediate-goods sector and the source of imperfections, or monopoly rights, are the patents acquired from the research sector. Furthermore, the price charged by the intermediate sector, given the existence of imperfect competition, will be higher than the marginal cost. Finally, the capital goods from the intermediate sector will be used by the final-goods sector to produce output however, the very existence of profits in the model is to compensate the research sector for its efforts in creating new ideas, which will be incorporated in the production function M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

26 Solving the model So, the question that remains at this stage is: how is labour allocated between the sectors, say, research and final-goods sector. For that we have to talk about the remuneration of labour in both sectors, (arbitrage again) the remuneration obtained by labour in the final-goods sector is given by the first-order conditions from equation (8) above, w Y = (1 α) Y L Y, which is the marginal product of labour in that sector M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

27 Solving the model The remuneration earned by researchers depends on the value of their inventions, and the productivity of inventions is given by θ. Moreover, researchers ignore the possibility of duplication of ideas, remember the parameter λ! hence, the remuneration earned by researchers is equal to the marginal product θ of the research sector times the value of the new idea P A, w R = θp A also take into consideration that in the research and final-goods sectors there is free entry, so making use of arbitrage conditions we have w Y = w R M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

28 Solving the model Finally, the share of population working in the research sector s R = L A /L is given by, s R = r n, αg A and the economic intuition is that the faster the economy grows, given by a higher g A (recall that g y = g k = g A ), the higher is the fraction of the population that works in R&D the Romer model breaks with Solow not only because ideas are now an input into production and given the nonrivalrous nature of ideas increasing returns to scale will happen, but because there is imperfect competition and profits, and those profits exist as an incentive device so that more ideas can be created and incorporated in the economy M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

29 Optimal R&D The question now is: is the share s R of the population working in R&D optimal? In general, and according to the Romer model, the answer is no. That means that the market alone is not giving the right amount of incentives so that more ideas can be invented and eventually incorporated into the production function the first distortion takes place because the market does not reward researchers for the prospective φ > 0 spillovers which will be generated from their ideas for future researchers; remember Newton and his Calculus! Hence, with φ > 0 ( standing on shoulders ) the market will not create enough incentives for research the second distortion comes from the duplication of ideas, λ < 1 ( stepping on toes ). Specifically, with duplication the productivity of research will be diminished, or alternatively, there will be too much (useless) research M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

30 Optimal R&D The third distortion is related to the consumer-surplus effect. The inventor of a new idea captures the monopoly profit. However, the society as a whole captures the consumer surplus. When we compare both areas, it becomes clear that the incentive to innovate is too little, see Figure 5.4 M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

31 Optimal R&D M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

32 Optimal R&D Hence, distortions 1 and 3 above justify the fact that governments end up financing research, otherwise too little research or not so many new ideas would be created in the first place therefore, the patent system, although important, is a necessary but not sufficient condition for the creation of new ideas. The same goes with profits a word of caution is in place here: some would argue that monopolies are bad and naturally inefficient. However, Romer is telling us that without a temporary monopoly ideas will not be rewarded and therefore underproduced, which in turn has a detrimental effect to growth M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

33 Summary Technological progress is the engine of growth, and it arises because researchers seek to gain something from their inventions, and this something is a reward for their efforts ideas are nonrivalrous, therefore we end up with increasing returns to scale a higher number of researchers can create more ideas, and ideas are behind per capita economic growth M Bittencourt (University of Pretoria) EKN 325 August 13, / 34

34 Summary An increase in the R&D subsidy will create a temporary growth effect and a permanent level effect intellectual property rights matter, as do patents, monopoly rights (all ideas themselves) and population lastly, there are differences between social and private returns to innovation M Bittencourt (University of Pretoria) EKN 325 August 13, / 34