Endogenous Growth Currently Updated Through Slide 43, Slides 32-35 Will Be Moved

1. Chapter 3 Endogenous Growth � Currently Updated Through Slide 43, Slides 32-35 Will Be Moved

2. 3.1 Framework and assumptions UNO, ECON 6204, Summer 2011, Dr. Tufte 2 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

3. Overview We�ll model technology as the output of a research and development industry So the economy produces two things: consumption goods and ideas Major simplification We�re going to go back to the idea that the saving rate is constant, and not optimally determined Continuous time UNO, ECON 6204, Summer 2011, Dr. Tufte 3 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

4. Specifics Same four endogenous variables: K, L, A and Y For simplicity, there is no depreciation Two sectors Goods Ideas Labor and capital are split between the two sectors You can only work in one sector Technology is not split Ideas can be used anywhere UNO, ECON 6204, Summer 2011, Dr. Tufte 4 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

5. Specifics Goods production function Y(t)=[(1-aK)K(t)]?[(1-aL)A(t)L(t)](1-?) The shares of labor and capital devoted to goods production, aK and aL, are exogenous. Ideas production function dA/dt=B[aKK(t)]�[aLL(t)]?A(t)? Note that this is production of new ideas (old ideas do not have to be replaced) This function allows for decreasing, constant or increasing returns to scale UNO, ECON 6204, Summer 2011, Dr. Tufte 5 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

6. Specifics Returns-to-scale restrictions are made to capture scalability of production processes Constant-returns-to-scale makes sense for most physical processes, so we impose it for the goods production function It isn�t clear what makes sense for the production of ideas, so �+?+? isn�t restricted to equal 1 Later, we spend a lot of time discussing what seems plausible UNO, ECON 6204, Summer 2011, Dr. Tufte 6 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

7. Specifics The key to understanding the model is the similarities and differences between output, capital and technology New output is produced every period, but it is all consumed in that period (its �depreciation� = 100%) New technology is produced every period, but it never depreciates, so it accumulates like capital Capital is output diverted from consumption Technology is produced instead of consumption UNO, ECON 6204, Summer 2011, Dr. Tufte 7 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

8. Digression: Similarity to the Solow Model In the Solow model: dK/dt = sY(t) = s[K(t)]a[A(t)L(t)](1-a) We didn�t divide the economy into 2 sectors, but we could have In this model dA/dt=B[aKK(t)]�[aLL(t)]?A(t)? We could equate some parameters and get back to the Solow model So, there isn�t much here that we couldn�t have done in Chapter 1. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 8

9. 3.2 The model without capital UNO, ECON 6204, Summer 2011, Dr. Tufte 9 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

10. The Dynamics of Knowledge Accumulation For simplicity, assume ?=�=0 The production functions are then Y(t)=(1-aL)A(t)L(t) dA/dt=B[aLL(t)]?A(t)? The growth rate of technology is then: gA(t)=(dA/dt)/A=B[aLL(t)]?A(t)?-1 UNO, ECON 6204, Summer 2011, Dr. Tufte 10 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

11. The Dynamics of Knowledge Accumulation What we want is the growth rate of the growth rate of technology, (dgA/dt)/gA You might call this the �acceleration� of technology Substitute in for L(t) and A(t) to get: gA(t)=B[aLL(0)ent]?[A(0)egt]?-1 Take logs to get: lngA(t)=lnB+?ln[aLL(0)]+nt?+(?-1)lnA(0)+gAt(?-1) Note the switch from g to gA in this chapter, and the notational switch just above because I can�t do a subscript of a superscript in Powerpoint Take the time derivative to get: (dgA/dt)/gA=n?+gA(?-1) UNO, ECON 6204, Summer 2011, Dr. Tufte 11 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

12. The Dynamics of Knowledge Accumulation Multiply through by gA to get: (dgA/dt) =n?gA+(gA)2(?-1) The behavior of this quadratic will depend on the size of ? The growth of technology is endogenous � if it�s growing then it will change its growth rate � so new growth models are sometimes called endogenous growth models UNO, ECON 6204, Summer 2011, Dr. Tufte 12 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

13. Case 1: ?<1 This is not the same ? as in the CRRA function Here, it is the effect of existing technology on the creation of new ideas The steady state is where: (dgA/dt) =n?gA+(gA)2(?-1)=0, Or gA* = ?n/(1-?) So the rate of growth of technology converges to a positive value In the real world, this seems to be upward convergence UNO, ECON 6204, Summer 2011, Dr. Tufte 13 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

14. Case 1: ?<1 This is a model of endogenous growth Output growth occurs because of technological growth Technology grows because new ideas are created and old ideas don�t depreciate It�s kind of odd that the growth of technology depends on population growth This seems truer at larger scales, but not at smaller ones Technological growth does not depend on the proportion of people working in research and development UNO, ECON 6204, Summer 2011, Dr. Tufte 14 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

15. Case 1: ?<1 The steady state growth rate of technology depends on the production parameters for ideas (of course), and the population growth rate. But, it�s always the same sign as the population growth rate The actual growth rate of technology depends on the share of labor working in R&D, but the steady state growth rate does not. So putting more people into R&D can increase your growth rate in the short-run but not in the long-run. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 15

16. Case 1: ?<1 So, an increase in the share of the population working in R&D can produce a level effect on technology and output But, no growth effect For example, the R&D in a war effort might boost your output permanently, but would only produce a transitory effect on its growth rate This sounds a lot like the U.S. in World War II UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 16

17. Case 2: ?>1 In this case, there is no steady-state growth rate of technology The growth rate accelerates This implies that the overall economy never reaches a steady-state either This is implausible, but shouldn�t be completely dismissed. Growth rates of developed countries have been inching up over the decades. This is one type of fully endogenous model of growth UNO, ECON 6204, Summer 2011, Dr. Tufte 17 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

18. Case 3: ?=1 The model simplifies to: gA(t)=B[aLL(0)ent]? Growth of ideas depends on population Growth of ideas depends on the proportion of the population working in research and development (dgA/dt) =n?gA Growth of ideas is accelerating when population growth is positive, and has no steady state. Growth of ideas stops when population growth stops These models are simple and plausible Sometimes called AK models UNO, ECON 6204, Summer 2011, Dr. Tufte 18 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

19. The Importance of Returns to Scale to Produced Factors It�s reasonable for goods and/or capital capital to have constant returns to scale It�s not that clear that idea production should have constant returns to scale If it doesn�t, this will be transmitted to the rest of the economy UNO, ECON 6204, Summer 2011, Dr. Tufte 19 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

20. The Importance of Population Growth In all of these models, output growth rates Increase with the level of population Generally increase with the population growth rate Yet output growth rates will stabilize at a high level when population levels out But, to the extent that technology is easily transmitted across borders, what is critical is world population, and world population growth rates. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 20

21. Digression: Population Policies and Luddites Luddites were a 19th century social movement that believed in destroying technology and capital to preserve jobs Endogenous growth models suggest that severe enough population control policies are capable of producing technological regress without the need to destroy anything UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 21

22. 3.3 The general case UNO, ECON 6204, Summer 2011, Dr. Tufte 22 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

23. The Dynamics of Knowledge and Capital The instantaneous growth rate of capital is: dK/dt = sY(t) Substitute in the production function to get: dK/dt = s[(1-aK)K(t)]?[(1-aL)A(t)L(t)](1-?) Gather constants so that: cK=s(1-aK)?(1-aL)(1-?) dK/dt = cK[K(t)]?[A(t)L(t)](1-?) gK(t)=(dK/dt)/K = cK[K(t)]?-1[A(t)L(t)](1-?) UNO, ECON 6204, Summer 2011, Dr. Tufte 23 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

24. The Dynamics of Knowledge and Capital Take logs of both sides to get: lngK(t)=lncK+(1-?)ln[A(t)L(t)/K(t)] Now take the derivative with respect to time to get: dgK(t)/dgK(t)=(1-?)[gA(t)+n-gK(t)] UNO, ECON 6204, Summer 2011, Dr. Tufte 24 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

25. The Dynamics of Knowledge and Capital In gA-gK space, the locus where this equals zero is: dgK(t)/dgK(t)=0=(1-?)[gA(t)+n-gK(t)] This must slope upward, and Intercept the gK axis at n. The implied dynamics are that when we are above the line gK(t) is relatively large and must be declining UNO, ECON 6204, Summer 2011, Dr. Tufte 25 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

26. The Dynamics of Knowledge and Capital Recall that the growth rate of technology is: gA(t)=B[aKK(t)]�[aLL(t)]?A(t)?-1 Factor out constants to get: gA(t)=B[aK]�[aL]?[K(t)]�[L(t)]?A(t)?-1 Take logs to get: lngA(t)=ln{B[aK]�[aL]?}+�lnK(t)+?lnL(t)+(?-1)lnA(t) Now take the derivative with respect to time: [dgA(t)/dt]/gA(t)= �gK(t) + ?n + (?-1)gA(t) UNO, ECON 6204, Summer 2011, Dr. Tufte 26 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

27. The Dynamics of Knowledge and Capital In gA-gK space, the locus where this equals zero is: [dgA(t)/dt]/gA(t) = 0 = �gK(t) + ?n + (?-1)gA(t) The intercept on the gK axis is at -?n/� <0 The slope, (1-?)/�, is indeterminate There are several cases, most of them implausible The direction of movement is to the right, dgA> 0, when above the line Since this is where gK is relatively large UNO, ECON 6204, Summer 2011, Dr. Tufte 27 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

28. Case 1: � + ? < 1 In this case the slope of the dgA/dt locus is steeper than the dgK/dt locus The direction of the arrows indicates a sink at the steady-state This means that sunspots can lead to one country being on one (short-run) path and another being on a different (short-run) path without it really making much difference to the long-run outcome UNO, ECON 6204, Summer 2011, Dr. Tufte 28 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

29. Case 1: � + ? < 1 At the steady state: dgK(t)/dgK(t)=0=(1-?)[gA(t)+n-gK(t)] gA(t)+n = gK(t) [dgA(t)/dt]/gA(t) = 0 = �gK(t) + ?n - (?-1)gA(t) �[gA(t)+n] + ?n - (?-1)gA(t) = 0 gA(t)[� + (?-1)] = -?n � �n gA(t) = (? + �)n/[1 � � - ?] So, technological and capital growth depend positively on ?, �, n, and ? They do not depend on the labor and capital shares devoted to research and development, or the saving rate UNO, ECON 6204, Summer 2011, Dr. Tufte 29 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

30. Case 1: � + ? < 1 Models like this are sometimes called semi-endogenous growth models, because Growth is endogenous, but There isn�t much interesting that we can say about pro-growth policies because the growth rates at the sink only depend on the production parameters and the population growth rate. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 30

31. Case 2: � + ? = 1, and n=0 In this case the two loci coincide All points on the intersection are steady states All of them are sinks Now sunspots make a big difference, because they can take you to a steady-state where growth is higher or lower than in other countries This is another case of a fully endogenous growth model This is similar to Romer �90 UNO, ECON 6204, Summer 2011, Dr. Tufte 31 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

32. Digression: Scale Effects and Growth There is a distinction that needs to be made between two mathematical possibilities. � + ? < 1 This leads to a sink in which rates of growth converge upward to the steady-state, so that growth rates accelerate � � ? + ? > 1 Increasing-returns-to-scale in idea production Note that in this model you do not need to have increasing-returns-to-scale in order to have high and accelerating growth rates UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 32

33. 3.4 the nature of knowledge and the determinants of the allocation of resources to r&d UNO, ECON 6204, Summer 2011, Dr. Tufte 33 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

34. Overview Goods are rival If you are using it, other people can�t use it Knowledge is non-rival If you are using it, other people still can use it Non-rivalry implies marginal costs that are zero in most situations. Thus either: Knowledge production can�t be profitable, or The price of knowledge is marked up in imperfect markets UNO, ECON 6204, Summer 2011, Dr. Tufte 34 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

35. Overview Goods tend to be excludable You can prevent someone else from using your (private) good Non-excludability is why we learn about public goods Knowledge can be excludable or non-excludable Excludability can be due to Complexity Legal and social restrictions Excludability helps make knowledge production profitable UNO, ECON 6204, Summer 2011, Dr. Tufte 35 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

36. Support for Basic Scientific Research If you support basic research with a subsidy, you can get around the non-rivalry and non-excludability problems It makes sense to do this because there is a positive externality to research UNO, ECON 6204, Summer 2011, Dr. Tufte 36 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

37. Private Incentives for R&D and Innovation Models exist in which innovators can earn monopoly profits (or at least positive but sub-optimal economic profits) from their new ideas Outcomes are not Pareto-optimal due to imperfect markets Positive externalities lead to too little R&D Negative externalities lead to too much R&D UNO, ECON 6204, Summer 2011, Dr. Tufte 37 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

38. Private Incentives for R&D and Innovation Consumer Surplus Effects The positive externality associated with knowledge spillovers Business-Stealing Effects The negative externality from new technology displacing old technology R&D Effects The positive externality from innovators profiting from making new goods, but not from others expanding on their ideas UNO, ECON 6204, Summer 2011, Dr. Tufte 38 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

39. Alternative Opportunities for Talented Individuals Rent seeking can include lots of activities that are not beneficial for growth We should see more rent seeking in: Small (and/or artificially closed) market Innovators profit more in bigger markets Markets in which severely diminishing returns are protected Why innovate if you can still profit in a field that isn�t scalable? Regions in which property rights are poorly defined Why innovate when you might not get to keep the money? UNO, ECON 6204, Summer 2011, Dr. Tufte 39 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

40. Learning-by-Doing (AK Models) The Solow production function is used But, the aggregate stock of technology is now driven by the aggregate stock of capital: A(t)=BK(t)f, f > 0 The more capital in existence, the more ideas there are about how to use capital So, dK/dt=sK(t)aB(1- a)K(t)f(1- a)L(t)(1- a) The behavior of this equation is similar to the model with no capital; f here and ? there work similarly When f<1, output growth depends on population growth UNO, ECON 6204, Summer 2011, Dr. Tufte 40 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

41. Digresson: 3 Romers David: macroeconomist at U.C.-Berkeley, author of this text. Married to Christina Christina: macroeconomist at U.C.-Berkeley, first head of Obama�s council of economic advisors Married to David Paul: macroeconomist at NYU, has an absolute lock on a Nobel prize for his seminal work in endogenous growth Founder of Aplia (an economics homework site) Advocate for charter cities. Not related to David Romer UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 41

42. 3.5 The Romer Model This section is based on his 1990 paper (similar to slide 18), not his 1986 paper (similar to slide 43).

43. Overview Ideas have: Fixed costs to develop, and Are non-rival, while excludability is difficult This means the developers of ideas won�t develop any at all without some monopoly power. But, you also can�t have monopoly power for �ideas�. You need monopoly power for each �distinct idea� so that substitution is limited. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 43

44. Overview The model of this section is most similar to the simplified example on Slide 18. This eliminates having to worry about transition dynamics. We use transition dynamics (phase diagrams) all the time, but the Romer text is trying to fill space with new ideas, not rehash ones you�ve already learned. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 44

45. The Ethier Production Function and the Returns to Knowledge Creation Ethier production: Infinite number of (potential) inputs You can use one or more Critical feature: Suppose you use m units of n different inputs, divided evenly You will get more output if you divide the m units over n+1 different inputs (divided evenly) Romer assumes that each input is an idea, and that a certain amount of labor uses each idea UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 45

46. The Ethier Production Function and the Returns to Knowledge Creation So: Y=[?L(i)fdi](1/f) , where 0<f<1 Y=[A(LY/A)f](1/f)=[A1](1/f) [(LY)f](1/f) [(A)-f](1/f) Y=A(1- f)/f LY Constant returns to scale in labor Increasing, constant, or diminishing marginal productivity from ideas Examples (where A is the total number of ideas) LY=12, A=2, f=1/3 Y = [(2)(12/2)(1/3)][1/(1/3)] = 48 LY=12, A=3, f=1/3 Y = [(3)(12/3)(1/3)][1/(1/3)] = 108 Bottom line: the producer of final output wants to incorporate as many ideas as possible. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 46

47. The Ethier Production Function and the Returns to Knowledge Creation An output producing firm takes the price set by the R&D sector, so their constrained cost minimization Lagrangian is: ?p(i)L(i)di-?{[?L(i)fdi](1/f)-1} There�s a �trick� here. Since there are no scale effects, the production quota can be set to any value without changing the optimal solution. So, we can choose to set it to 1 for convenience, and [?L(i)fdi](1/f) = 1 ?L(i)fdi = 1 UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 47

48. The Ethier Production Function and the Returns to Knowledge Creation The FOC of the problem are then of the form: p(i) = ?{[?L(i)fdi](1/f-1)L(i) (f-1) But, from the previous slide, the term in the middle goes to 1, so: p(i) = ?L(i) (f-1) Without making a full solution, since f<1, it should be clear the derived demand for the labor using the �inventors� idea, L(i), will have downward sloping demand UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 48

49. The Ethier Production Function and the Returns to Knowledge Creation Later, we�ll need the elasticity of demand This is the same for each of the i inputs, so I�ll drop the i index From p = ?L(f-1) Then: dp = ? (f-1) L (f-2)dL + L(f-1)d ? Holding ? constant yields: dL/dp=1/{? (f-1) L (f-2)} Multiply this by p/L to obtain the elasticity: (dL/dp)(p/L)=1/{? (f-1) L (f-2)}{?L(f-1)/L}=1/ (f-1) On pg. 128, Romer states this as an absolute value UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 49

50. The Ethier Production Function and the Returns to Knowledge Creation It is difficult to show, but as f approaches 1 the marginal product of an idea gets flatter This implies that the ideas are closer substitutes, and therefore The elasticity of demand for each input is greater UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 50

51. The Rest of the Model Total population is fixed (in this simplified model) at L-overbar Some labor works in the R&D sector, LA, and some works in the goods producing sector LY These amounts are not fixed, so L-overbar = LA(t)+LY(t) UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 51

52. The Rest of the Model New ideas are created in proportion to the number of existing ideas, and the number of people working in R&D dA/dt = BLA(t)A(t), where B>0 So, ideas will grow at a constant rate unless we add labor to the R&D sector. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 52

53. The Rest of the Model Individuals optimize using a simplified version of the Ramsay-Cass-Koopmans models Max U=?e-?tlnC(t)dt, for ?>0 The text assumes log-utility (a special case of CRRA) for simplicity. s.t. ?e-rtC(t)dt = X(0) + ?e-rtw(t)dt UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 53

54. The Rest of the Model There is free entry in idea creation To create 1 new idea Hire 1/[BA(t)] workers So new ideas actually require less labor as technology expands Pay them w(t) You do not need to pay the holders of patents on previous ideas to create your new idea, but You can�t sell your new idea either It needs to be �embodied� in labor that knows how to use it. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 54

55. The Rest of the Model An idea can�t be sold. Instead it is embodied in workers who know it. The patent holder decides how many workers to embody, and what price to charge for them This monopoly profit maximization problem requires that the monopolist know the wage of workers, the prices of other embodied ideas, and the amount of labor used to produce goods to yield its derived demand UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 55

56. The Rest of the Model Free entry and exit will drive the firms in the R&D sector to a zero profit condition They are monopolistically competitive Dynamically, this condition is that the present value of future discounted profits equals the cost of creating the idea: ?e-r(t -t)p(i, t)dt = w(t)/[BA(t)] UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 56

57. The Rest of the Model Additional general equilibrium assumptions The wage is the same in both sectors Initial wealth is the present value of the endowment of ideas at time zero. Output is only used for consumption, and In a (simple) model with identical agents, they all consume the same amount. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 57

58. Solving the Model The framework of the D. Romer text has assumed away a lot of the detail in the P. Romer paper. Specifically, D. Romer has constructed the textbook model so that the share of income in each sector is constant UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 58

59. Solving the Model The patent-holder is a monopolist. The optimal gross mark-up (the ratio of price to marginal cost) is a well-known result from managerial economics: ?/(?-1), where ? is the elasticity of demand Given the earlier result that the elasticity of demand is 1/(f-1), the price the embodied labor can be sold for is then w(t)/f UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 59

60. Solving the Model Since each patent-holder has 1 idea to embody, the units of labor they can sell is (L-overbar-LA)/A(t) And the profit on each of those laborers is their rental price minus their wage: {[w(t)/f]-w(t)} So total profit to each patent-holder is [(L-overbar-LA)/A(t)]{[w(t)/f]-w(t)}, or [(1- f)/f][(L-overbar-LA)/A(t)]w(t) UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 60

61. Solving the Model We now have instantaneous profits for the patent-holder But, we need to find the present value of this in perpetuity So we need to discount it, and We need to account for the fact that its future value will grow with the economy Recall from basic finance that there is a formula for the value of a growing perpetuity If we can figure out the growth rate of profit, and the appropriate discount rate, we can get the present value. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 61

62. Solving the Model The growth rate of the economy will be related to the growth rate of ideas By definition: dA/dt = BLA(t)A(t) So: (dA/dt)/A(t) = BLA(t) This is constant in this model And Y=A[(1- f)/f]LY So lnY = [(1- f)/f]lnA + lnLY And dY/dt = [(1- f)/f](dA/dt) = [(1- f)/f]BLA(t) UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 62

63. Solving the Model Wages will also grow at the same rate as output Since all output is used to pay factors of production, and embodied ideas are the only one Pay for embodied ideas is divided between labor and monopoly profits by a constant mark-up The amount of overall labor in the R&D market is constant, so the wage is never further divided up UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 63

64. Solving the Model The growth rate of profit only depends on the growth rate of output/wages, and the growth rate of technology Rearranging profit a bit yields: p = [(1- f)/f][L-overbar-LA]w(t) /A(t) So: lnp = ln[ ] + ln[ ] +ln[w(t)] � ln[A(t)] I didn�t type what�s inside the brackets to save space Then dp/dt = dw/dt � dA/dt dp/dt = [(1-f)/f]BLA(t)- BLA(t) = [(1-2f)/f]BLA(t) UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 64

65. Solving the Model The growth rate of consumption must also be the same as the growth rate of output Since all output is consumed But, the Keynes-Ramsay rule tells us how consumption growth is related to rates of return dC/dt = r � ? (for log utility) r = dC/dt + ? = ? + [(1- f)/f]BLA(t) This is constant since the share of labor employed in R&D is constant UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 65

66. Solving the Model Now we can apply the (continuous time) formula for a growing perpetuity to get the present value of the infinite profit stream from an idea PV=cash flow/(r-g) PV = {[(1- f)/f][(L-overbar-LA)/A(t)]w(t)}/{? + [(1- f)/f]BLA(t) � [(1-2f)/f]BLA(t)} This simplifies to: PV = [(1- f)/f]{(L-overbar-LA)/[? + BLA(t)]}[w(t)/A(t)] UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 66

67. Solving the Model Recall that the present value of the profit stream must equal the cost of an idea PV = w(t)/bA(t) This can be solved for LA LA = (1-f)L-overbar-??/B Note that this might be negative, so the realistic answer is LA = max{(1-f)L-overbar-??/B,0} And dY/dt = max{[(1-f)2/f]BL-overbar � (1-?)?,0} UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 67

68. Implications This formula can tell us which exogenous factors are important for long-run growth The model can also tell us if decentralized decision-making will lead to optimal growth (or just growth) UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 68

69. Implications A higher discount rate leads to lower growth [d(dY/dt)]/d? = -(1-f)= f-1 < 0 Because the present value of profits from ideas is lower UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 69

70. Implications If ideas are closer substitutes (f is closer to 1) then growth is lower [d(dY/dt)]/df = (BL-overbar){-2(1-f)f-1-(1-f)2f-2} + ? Strictly speaking this is ambiguous, but it is unlikely that ? would be large enough to make it positive Because the Ethier production function is like a CES, higher f denotes greater substitutability, so there is less opportunity for mark-up and monopoly profits, and thus less incentive to create new ideas that drive growth UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 70

71. Implications Increased productivity in the R&D sector will increase growth rates [d(dY/dt)]/dB = (1-f)2L-overbar/f > 0 The text says there are two reasons for this That growth of ideas is higher Labor in the R&D sector goes up I think these are just 2 reflections of the same thing It is cheaper to produce a new idea, so people will think up more of them UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 71

72. Implications An increase in the population causes higher growth rates [d(dY/dt)]/dL-overbar = (1-f)2B/f > 0 The reason is that there are more people to sell output to, so more reason to buy embodied ideas as inputs. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 72

73. Implications All 4 exogenous factors work by making the creation of new ideas more profitable, thereby increasing the fraction of labor working in the R&D industry in equilibrium. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 73

74. Implications The model has imperfect markets This means that it violates the First Welfare Theorem, or alternatively It is consistent with the Greenwald-Stiglitz theorem Either way, this means that the model�s equilibrium will not be Pareto optimal Therefore we can improve everyone�s outcome without hurting anyone Would we want to? Why not? UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 74

75. Implications Why is the equilibrium not Pareto-optimal? Because R&D is too small in a decentralized market This is because R&D is driven by monopoly profits, and monopolists systematically restrict output to maximize their profits So � if a central planner could direct R&D, we might get to a higher level of growth I�m not too concerned about the proof on pp. 131-2, but it turns out that employment in the R&D sector is too low by a factor of 1-f under decentralized exchange This is interesting because it says that central planning will be most beneficial when ideas are the most substitutable UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 75

76. Implications If an outcome is not Pareto-optimal, it is because there are some identifiable externalities. In this model Final goods producers earn consumer surplus from buying the innovations of the R&D sector Patent-holders are hurt by innovation if the innovation is a close substitute The borderline is f > � Existing patents make new innovation easier, but require no compensation from the R&D sector. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 76

77. Extensions P. Romer�s paper includes capital Capital has only level effects if it is not involved in R&D, but growth effects if it is involved in R&D The exponent of 1 on A in dA/dt = BLA(t)A(t) seems to be too large Smaller values lead to results more like semi-endogenous growth models There is an alternative model in which technological improvement leads to better not different inputs � but it doesn�t make too much difference to the results other than to uncover some other microeconomic causes of growth UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 77

78. 3.6 Empirical Application: Time-Series Tests of Endogenous Growth Models Fully endogenous growth models give the most radically new results, so initial tests looked at these. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 78

79. Are Growth Rates Stationary? P. Romer asserted in the seminar I saw at the University at Buffalo in 1986-7 that growth rates appeared to be accelerating Also, the assertion of Robin Hanson about how output will behave when our �avatars� can control capital and become effective labor suggests jumps in growth rates rather than smooth acceleration. UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 79

80. Are Growth Rates Stationary? Jones asserts that because exogenous variables can have growth effects (in addition to level effects) in fully endogenous growth models, that growth rates should display this by not being stationary around a central value. This runs into the two problems I mentioned earlier in the semester Tests of non-stationarity have low power Rejecting stationarity wouldn�t imply that fully endogenous growth are correct UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 80

81. Are Growth Rates Stationary? It turns out that Jones found a small upward trend in growth rates (as indicated by P. Romer), but with a very wide confidence interval. So, is there no trend, or A huge trend we don�t have a powerful enough test to find? UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 81

82. The Magnitudes and Correlates of Changes In Long-Run Growth Jones shows that the investment share has been rising without growth rates following. AK models indicates that the two should rise together Jones also shows that employment in the R&D sector has gone up quite a lot in developed economies, without their per capita growth rates rising UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 82

83. Discussion Jones argues that it is unlikely that there are increasing returns to scale in research and development We�ve spend so much more on these over the last 50 years that growth rates would�ve gone way up if this case were true Jones suggests that returns to research and development are decreasing That ? + � + ? < 1 in Section 3.3 UNO, ECON 6204, Summer 2011, Dr. Tufte 83 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

84. Discussion Others argue that the number of sectors in which R&D is performed increases with the size of the economy Each sector still has increasing-returns-to-scale, but doesn�t get the resources for that to make much difference These models require implausible parameter restrictions UNO, ECON 6204, Summer 2011, Dr. Tufte 84 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

85. Discussion With decreasing returns to scale, increases in R&D share or the saving rate do improve growth rates, but the effect is transitory Remember that this doesn�t necessarily mean it will be short This is a transitory effect on growth rates. There will still be a permanent effect on the levels of output. UNO, ECON 6204, Summer 2011, Dr. Tufte 85 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

86. Discussion These results point away from fully endogenous growth and towards semi-endogenous growth models. These are models in which there are decreasing returns to investment in innovation It�s still good, but there are limits to how much of it we can sensibly do UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 86

87. 3.7 empirical application: population growth and technological change since 1 million b.C. UNO, ECON 6204, Summer 2011, Dr. Tufte 87 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

88. Population Growth and Technological Change Since 1 Million B.C. Really only to the onset of per capita income growth In a subsistence society, endogenous growth (pushed by population growth) makes two predictions Population growth rates rise with population Population density, in isolated regions, is proportional to the land available to support the population Alternatively, more advances are made in more populous regions, but measuring this would require evaluating which advances were actually important UNO, ECON 6204, Summer 2011, Dr. Tufte 88 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

89. A Simple Model (with ?=1) Cobb-Douglas production with fixed land, T: Y(t)=Ta[A(t)L(t)](1-a) Technological growth is population driven: dA/dt=BL(t)A(t) Per capita incomes are constant at a subsistence level, y Then: y=Ta[A(t)L(t)](1-a)/L(t) Or: L(t) is proportional to A(t)(1-a)/aT So: (dL/dt)/L=[(1-a)/a]{(dA/dt)/A}=[(1-a)/a]BL(t) UNO, ECON 6204, Summer 2011, Dr. Tufte 89 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

90. Results Regression results show a good linear fit between growth rates and levels of population prior to 1700 Regression results also indicate population density is highest in larger regions UNO, ECON 6204, Summer 2011, Dr. Tufte 90 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

91. Discussion This offers a new �solution� to a set of outstanding anthropological problems: Why were dense areas important? Why were �the hills� filled with backward people all over the globe How could China make so many advances without becoming rich? Why did the Islamic world decline? There were always barbarians, but why did barbarian hordes always seem to arrive equipped with better technology? UNO, ECON 6204, Summer 2011, Dr. Tufte 91 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

92. Population Growth vs. Growth In Income Per Person over the Very Long-Run Why did incomes explode after 1700 then? Because population growth rates can�t change as quickly as technological growth rates, and eventually the latter swamped the former Why won�t this persist? People prefer to be less fertile when richer UNO, ECON 6204, Summer 2011, Dr. Tufte 92 Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed.

93. 3.8 Models of Knowledge Accumulation and the Central Questions of Growth Theory Growth Good: the Solow residual is very large, so endogenizing technology growth is a good thing Cross-Country Income Differences Bad: the huge differences in income would need to be explained by huge differences in technology Bad: technology is non-rival, so why can�t capitalists just take it overseas for higher returns? UNO, ECON 6204, Summer 2011, Dr. Tufte Notes drawn from Chapter 3 of Romer's "Advanced Macroeconomics", 4th Ed. 93