503 Applied MacroeconomicsChapter 3. Endogenous Growth

Prof. M. El-SakkaProf. M. El-SakkaDept of Economics - Kuwait UniversityDept of Economics - Kuwait University

• The models we have seen so far do not provide satisfying The models we have seen so far do not provide satisfying answers to our central questions about economic growth. The answers to our central questions about economic growth. The models’ principal result is a negative one:models’ principal result is a negative one:

• - Capital accumulation does not account for a large part of either - Capital accumulation does not account for a large part of either long-run growth or cross-country income differenceslong-run growth or cross-country income differences. .

• - The only determinant of income is the - The only determinant of income is the “effectiveness of labor” “effectiveness of labor” ((AA), whose exact meaning is not specified and is taken as ), whose exact meaning is not specified and is taken as exogenous.exogenous.

• This chapter focuses on the accumulation of knowledge.This chapter focuses on the accumulation of knowledge.

• What do we need?

• we need to introduce a separate sector of the economy where we need to introduce a separate sector of the economy where new ideas are developednew ideas are developed. .

Framework and Assumptions

• We then need to model both We then need to model both how resources are divided how resources are divided between conventional output and between conventional output and new new research and research and developmentdevelopment (or (or R&DR&D) sector, and how inputs into R&D ) sector, and how inputs into R&D produce new ideas. We make two other major produce new ideas. We make two other major simplifications:simplifications:

• FirstFirst, both the R&D and goods production functions are , both the R&D and goods production functions are assumed to be assumed to be generalized Cobb–Douglas functions; generalized Cobb–Douglas functions; but the but the sum of their exponents on the inputs is not necessarily sum of their exponents on the inputs is not necessarily restricted to restricted to 11. .

• SecondSecond, the model takes the fraction of output saved and , the model takes the fraction of output saved and the fractions of the labor force and the the fractions of the labor force and the capital stock capital stock used in used in the R&D sector as the R&D sector as exogenousexogenous and constant. and constant.

• The model involves four variables: labor (The model involves four variables: labor (LL), capital (), capital (K K ), ), technology (technology (AA), and output (), and output (Y Y ). ).

• There are There are two sectorstwo sectors, a , a goods-producinggoods-producing sector where sector where output is produced and an output is produced and an R&DR&D sector where additions to the sector where additions to the stock of knowledge are made. Fraction stock of knowledge are made. Fraction aL aL of the labor force of the labor force is used in the R&D sector and fraction 1−is used in the R&D sector and fraction 1−aL aL in the goods-in the goods-producing sector. Similarly, fraction producing sector. Similarly, fraction aK aK of the capital stock is of the capital stock is used in R&D and the rest in goods production.used in R&D and the rest in goods production.

• Both Both aL aL and and aK aK are are exogenous and constantexogenous and constant. .

• Because the use of an idea or a piece of knowledge in one Because the use of an idea or a piece of knowledge in one place place does not prevent it from being used elsewheredoes not prevent it from being used elsewhere, both , both sectors use the full stock of knowledge, sectors use the full stock of knowledge, AA..

• The quantity of output produced at time The quantity of output produced at time t t is thusis thus

• Note that equation (3.1) implies Note that equation (3.1) implies constant returns constant returns to capital to capital and labor: with a given technology, doubling the inputs and labor: with a given technology, doubling the inputs doubles the amount that can be produced.doubles the amount that can be produced.

• The production of new ideas depends on the quantities of The production of new ideas depends on the quantities of capital capital and labor engaged in research and on the and labor engaged in research and on the level of technologylevel of technology..

• where where B B is a shift parameter is a shift parameter ))determines the position of a function). .

• Notice that the production function for knowledge is Notice that the production function for knowledge is not assumed not assumed to have constant returns to scaleto have constant returns to scale to capital and labor. Thus it is to capital and labor. Thus it is possible that there are possible that there are diminishing returns in R&D.diminishing returns in R&D. At the same At the same time, interactions among researchers, fixed setup costs, and so on time, interactions among researchers, fixed setup costs, and so on may be important enough in R&D that doubling capital and labor may be important enough in R&D that doubling capital and labor more than doubles outputmore than doubles output. We therefore also allow for the . We therefore also allow for the possibility of possibility of increasing returnsincreasing returns. .

• The parameter The parameter θθ reflects the effect of the existing stock of reflects the effect of the existing stock of knowledge on the knowledge on the success of R&Dsuccess of R&D. This effect can operate in . This effect can operate in either directioneither direction. On the one hand, past discoveries may . On the one hand, past discoveries may provide provide ideas ideas and tools that make future discoveries easier. In this case, and tools that make future discoveries easier. In this case, θ θ is positiveis positive. .

• On the other hand, the easiest discoveries may be made first. In On the other hand, the easiest discoveries may be made first. In this case, it is harder to make new discoveries when the stock of this case, it is harder to make new discoveries when the stock of knowledge is greater, and so knowledge is greater, and so θ θ is negative.is negative.

• Because of these conflicting effects, Because of these conflicting effects, no restriction no restriction is placed on is placed on θ θ in in (3.2).(3.2).

• As in the Solow model, As in the Solow model, the saving rate is exogenous the saving rate is exogenous and constant. and constant. In addition, In addition, depreciation is set to zero depreciation is set to zero for simplicity. Thus,for simplicity. Thus,

• Likewise, we continue to treat Likewise, we continue to treat population growth population growth as exogenous as exogenous and constant. For simplicity, we do not consider the possibility that and constant. For simplicity, we do not consider the possibility that it is negative. This implies; it is negative. This implies;

• Finally, as in our earlier models, the initial levels of Finally, as in our earlier models, the initial levels of AA, , KK, and , and L L are are given and strictly given and strictly positivepositive. This completes the description of the . This completes the description of the model. model.

• Because the model has two state variables whose behavior is Because the model has two state variables whose behavior is endogenousendogenous, , K K and and AA, it is more complicated to analyze than the , it is more complicated to analyze than the Solow model. We therefore begin by considering the model Solow model. We therefore begin by considering the model without capitalwithout capital; that is, we set ; that is, we set α α and and β β to zeroto zero. This case shows . This case shows most of the model’s central messages. We then turn to the general most of the model’s central messages. We then turn to the general case.case.

The Model without Capital

• The Dynamics of Knowledge AccumulationThe Dynamics of Knowledge Accumulation

• When there When there is no capital is no capital in the model, the production function for in the model, the production function for output (equation [3.1]) becomesoutput (equation [3.1]) becomes

• Similarly, the production function for new knowledge (equation Similarly, the production function for new knowledge (equation [3.2]) is now[3.2]) is now

• Equation (3.5) implies that output per worker is Equation (3.5) implies that output per worker is proportional to proportional to AA, , and thus that the growth rate of output per worker and thus that the growth rate of output per worker equals the equals the growth rate of growth rate of AA. .

• We therefore focus on the dynamics of We therefore focus on the dynamics of AA, which are given by (3.6). , which are given by (3.6). This equation implies that the growth rate of This equation implies that the growth rate of AA, denoted , denoted gAgA, is:, is:

• Taking logs of both sides of (3.7) and differentiating the two sides Taking logs of both sides of (3.7) and differentiating the two sides with respect to time gives us an expression for the with respect to time gives us an expression for the growth rate growth rate of of gAgA::

• Multiplying both sides of this expression by Multiplying both sides of this expression by gAgA((t t ) yields) yields

• Equation (3.9) determines the subsequent behavior of Equation (3.9) determines the subsequent behavior of gAgA..

• To describe further how the To describe further how the growth rate of growth rate of A A behaves behaves (and thus to (and thus to characterize the behavior of output per worker), we must characterize the behavior of output per worker), we must distinguish among the distinguish among the cases cases θ <θ <1, 1, θ >θ >1, and 1, and θ θ = 1. = 1. We discuss We discuss each in turn.each in turn.

• Case 1: Case 1: θ < θ < 11

• Figure 3.1 shows the phase diagram for Figure 3.1 shows the phase diagram for gA gA when when θ θ is less than 1. is less than 1.

• In the diagram and equation (3.9) for the case of In the diagram and equation (3.9) for the case of θ < θ < 11, , ˙̇gA gA is is positive for small positive for small positive values of positive values of gA gA and and negative for large negative for large valuesvalues. .

• We will useWe will use gg∗∗A A to denote the unique positive value of to denote the unique positive value of gA gA that that implies that implies that ˙̇gA gA is zero. is zero.

• From (3.9), From (3.9), gg∗∗A A is defined by: is defined by: γn γn + (+ (θθ−1)−1)gg∗∗A A = 0. = 0.

• Solving this for Solving this for gg∗∗A A yieldsyields

• This analysis implies that regardless of the economy’s initial This analysis implies that regardless of the economy’s initial conditions, conditions, gA gA converges to converges to gg∗∗AA. If the parameter values and the . If the parameter values and the initial values of initial values of L L and and A A imply imply gAgA(0) (0) < g< g∗∗AA, for example, , for example, ˙̇gA gA is is positive; that is, positive; that is, gA gA is is risingrising. It continues to rise until it reaches . It continues to rise until it reaches gg∗∗AA..

• Similarly, if Similarly, if gAgA(0) (0) > g> g∗∗AA, then , then gA gA falls until it reaches falls until it reaches gg∗∗AA. Once . Once gA gA reaches reaches gg∗∗AA, both , both A A and and YY//L L grow steadily at rate grow steadily at rate gg∗∗AA. Thus . Thus the economy is on a the economy is on a balanced growth pathbalanced growth path..

• The model implies that the long-run growth rate of output per The model implies that the long-run growth rate of output per worker, worker, gg∗∗AA,, is an increasing function of the rate of is an increasing function of the rate of populationpopulation growthgrowth, , nn. .

• Equation (3.10) also implies that the fraction of the Equation (3.10) also implies that the fraction of the labor forcelabor force engaged engaged in R&D does not affect long-run growthin R&D does not affect long-run growth. This too may . This too may seem surprising: since growth is driven by technological progress seem surprising: since growth is driven by technological progress and technological progress is and technological progress is endogenousendogenous, it is natural to expect , it is natural to expect an increase in the fraction of the economy’s resources devoted to an increase in the fraction of the economy’s resources devoted to technological progress to increase long-run growth.technological progress to increase long-run growth.

• This change is analyzed in Figure 3.2. This change is analyzed in Figure 3.2. aL aL does not enter does not enter expression (3.9) for expression (3.9) for

• ˙̇gAgA: : ˙̇gAgA((tt) = ) = γngAγngA((tt) + () + (θ θ − 1)[− 1)[˙̇gAgA((tt)])]22. .

• Thus the rise in Thus the rise in aL aL does not affect the curve does not affect the curve showing showing ˙̇gA gA as a as a function of function of gAgA. But . But aL aL does enter expression (3.7) for does enter expression (3.7) for

• gAgA: : gAgA((tt) = ) = BaBaLLγγ L L((tt))γ γ AA((t t ))θθ−1−1. .

• The increase The increase in in aL aL therefore causes an immediate increase in therefore causes an immediate increase in gA gA but but no change in no change in ˙̇gA gA as a function of as a function of gAgA. This is shown by the . This is shown by the dotted arrow in Figure 3.2.dotted arrow in Figure 3.2.

• As the phase diagram shows, the increase in the growth rate of As the phase diagram shows, the increase in the growth rate of knowledge knowledge is not sustainedis not sustained. When . When gA gA is above is above gg∗∗AA, , ˙̇gA gA is is negative. negative. gA gA therefore therefore returns gradually returns gradually to to gg∗∗A A and then remains and then remains there. Intuitively, the fact that there. Intuitively, the fact that θ θ is less than 1 is less than 1 means that the means that the contribution of additional knowledge to the production of new contribution of additional knowledge to the production of new knowledge is not strong enough to be self-sustaining.knowledge is not strong enough to be self-sustaining.

• This analysis implies that, the increase in This analysis implies that, the increase in aL aL results in a rise in results in a rise in gA gA followed by a gradual return to its initial level. That is, it has followed by a gradual return to its initial level. That is, it has a level a level effect effect but not a growth effect on the path of but not a growth effect on the path of AA. This information is . This information is summarized in Figure 3.3.summarized in Figure 3.3.

• Case 2: Case 2: θ > θ > 11

• This corresponds to the case where the production of new This corresponds to the case where the production of new knowledge knowledge rises more than proportionally with the existing stockrises more than proportionally with the existing stock. . Recall from equation (3.9) that Recall from equation (3.9) that

• ˙̇gA gA = = γngA γngA + (+ (θ θ − 1)− 1)gAgA22..

• When When θ θ exceeds 1exceeds 1, this equation implies that , this equation implies that ˙̇gA gA is positive is positive for all for all possible values of possible values of gAgA. Further, it implies that . Further, it implies that ˙̇gA gA is increasing in is increasing in gA gA (since (since gA gA must be positive). The phase diagram is shown in must be positive). The phase diagram is shown in Figure 3.4.Figure 3.4.

• The The impact of an increase in the fraction of the labor force impact of an increase in the fraction of the labor force engaged in R&D is now dramaticengaged in R&D is now dramatic. From Equation (3.7), an . From Equation (3.7), an increase in increase in aL aL causes an immediate increase in causes an immediate increase in gAgA, as before., as before.

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• But But ˙̇gA gA is an increasing function of is an increasing function of gAgA; thus ; thus ˙̇gA gA rises as well. And rises as well. And the more rapidly the more rapidly gA gA rises, the more rapidly its growth rate rises. rises, the more rapidly its growth rate rises. Thus the Thus the increase in increase in aL aL causes the growth rate of causes the growth rate of A A to exceed to exceed what it would have been otherwise by an ever-increasing amount.what it would have been otherwise by an ever-increasing amount.

• Case 3: Case 3: θ θ = 1= 1• When When θ θ is exactly equal to 1, is exactly equal to 1, the production of new knowledge is the production of new knowledge is

proportional to the stockproportional to the stock. In this case, expressions (3.7) and (3.9) . In this case, expressions (3.7) and (3.9) for for gA gA and and ˙̇gA gA simplify to: simplify to:

• If population growth is positive, If population growth is positive, gA gA is growing over timeis growing over time; in this ; in this case the dynamics of the model are similar to those when case the dynamics of the model are similar to those when θ >θ >11. If . If population population growth is zerogrowth is zero, on the other hand, , on the other hand, gA gA is constant is constant regardless of the initial situation.regardless of the initial situation.

• The Importance of Population GrowthThe Importance of Population Growth

• consider equation (3.7) for knowledge accumulation:consider equation (3.7) for knowledge accumulation:

• gAgA((tt) = ) = BBaLaLγγ L L((tt))γ γ AA((tt))θθ−1−1. .

• Built into this expression: when there are Built into this expression: when there are more people to make more people to make discoveriesdiscoveries, more discoveries are made. And when more , more discoveries are made. And when more discoveries are made, the stock of knowledge grows faster, and so discoveries are made, the stock of knowledge grows faster, and so (all else equal) output per person grows faster. In the particular (all else equal) output per person grows faster. In the particular case of case of θ θ = 1 and = 1 and n n = 0, this effect operates in a special way: = 0, this effect operates in a special way: long-long-run growth is increasing in the level of population.run growth is increasing in the level of population.

• When When θ θ is greater than 1, the effect is even more powerful, as is greater than 1, the effect is even more powerful, as increases in the level or growth rate of population lead to ever-increases in the level or growth rate of population lead to ever-rising increases in growth.rising increases in growth.

• When When θ θ is less than 1, there are is less than 1, there are decreasing returns decreasing returns to scale to to scale to produced factors. In this case, although knowledge may be helpful produced factors. In this case, although knowledge may be helpful in generating new knowledge, the generation of new knowledge in generating new knowledge, the generation of new knowledge rises less than proportionally rises less than proportionally with the existing stock. with the existing stock.

The General Case

• The Dynamics of Knowledge and CapitalThe Dynamics of Knowledge and Capital• As mentioned above, when the model includes capital, there are As mentioned above, when the model includes capital, there are

two endogenous state variables, two endogenous state variables, A A and and KK.. Paralleling our analysis Paralleling our analysis of the simple model, of the simple model, we focus on the dynamics of the growth rates we focus on the dynamics of the growth rates of of A A and and KK. Substituting the production function, (3.1), into the . Substituting the production function, (3.1), into the expression for capital accumulation, (3.3), yieldsexpression for capital accumulation, (3.3), yields

• Dividing both sides by Dividing both sides by KK((tt) and defining ) and defining

cK cK = = s s (1 − (1 − aK aK ))αα(1− (1− aLaL))1−1−α α

• gives us:gives us:

• Taking logs of both sides and differentiating with respect to time Taking logs of both sides and differentiating with respect to time yieldsyields

• gK gK is rising if is rising if gA gA + + n n − − gK gK is positive, falling if this expression is is positive, falling if this expression is negative, and constant if it is zero. negative, and constant if it is zero.

• This information is summarized in Figure 3.5. In (This information is summarized in Figure 3.5. In (gAgA,,gK gK ) space, ) space, the locus of points where the locus of points where gK gK is constant has an intercept of is constant has an intercept of n n and and a slope of 1. Above the locus, a slope of 1. Above the locus, gK gK is falling; below the locus, it is is falling; below the locus, it is rising.rising.

• Similarly, dividing both sides of equation (3.2), Similarly, dividing both sides of equation (3.2),

˙ ˙A A = = BB((aKaK K K))ββ((aLaL L L))γγAAθθ, by , by A A yields an expression for the yields an expression for the

growth rate of growth rate of AA::

• where where cA cA ≡ ≡ BaBaKK β β a aLL

γγ. Aside from the presence of the . Aside from the presence of the KKββ term, term,

this is essentially the same as equation (3.7) in the simple this is essentially the same as equation (3.7) in the simple version of the model. Taking logs and differentiating with version of the model. Taking logs and differentiating with respect to time givesrespect to time gives

• Thus Thus gA gA is rising if is rising if βgK βgK + + γ n γ n + (+ (θθ−1)−1)gAgA is positive, falling if it is positive, falling if it is negative, and constant if it is zero. This is shown in Figure is negative, and constant if it is zero. This is shown in Figure 3.6. 3.6. The set of points where The set of points where gA gA is constant has an intercept is constant has an intercept of −of −γ nγ n//β β and a slope of (1− and a slope of (1− θθ)/)/ββ. Above this locus, . Above this locus, gA gA is is rising; and below the locus, it is falling.rising; and below the locus, it is falling.

• Case 1: Case 1: ββ++θ < θ < 11• If If β β + + θ θ is less than 1,is less than 1, (1 − (1 − θθ)/)/β β is greater than 1is greater than 1. Thus the locus of . Thus the locus of

points where points where ˙̇gA gA = 0 is steeper than the locus where = 0 is steeper than the locus where ˙̇gK gK = 0. This = 0. This case is shown in Figure 3.7. The initial values of case is shown in Figure 3.7. The initial values of gA gA and and gK gK are are determined by the parameters of the model and by the initial determined by the parameters of the model and by the initial values of values of AA, , KK, and , and LL. Their dynamics are then as shown in the . Their dynamics are then as shown in the figure.figure.

• Figure 3.7 shows that regardless of where Figure 3.7 shows that regardless of where gA gA and and gK gK begin, they begin, they converge to Point E converge to Point E in the diagram. Both in the diagram. Both ˙̇gA gA and and ˙̇gK gK are zero at are zero at this point. Thus the values of this point. Thus the values of gA gA and and gK gK at Point E, which we at Point E, which we denote denote gg∗∗A A and and gg∗∗K K , must satisfy, must satisfy

• Rewriting (3.18) as Rewriting (3.18) as gg∗∗K K = = gg∗∗A A + + n n and substituting into (3.19) and substituting into (3.19) yieldsyields

• From above, From above, gg∗∗K K is simply is simply gg∗∗A A + + nn. Equation (3.1) then implies . Equation (3.1) then implies that when that when A A and and K K are growing at these rates, output is growing at are growing at these rates, output is growing at rate rate gg∗∗K K . Output per worker is therefore growing at rate . Output per worker is therefore growing at rate gg∗∗AA..

• This case is similar to the case when This case is similar to the case when θ θ is less than 1 in the version is less than 1 in the version of the model without capital. Here, as in that case, of the model without capital. Here, as in that case, – the long-run growth rate of the economy is endogenous, the long-run growth rate of the economy is endogenous, – long-run growth is an increasing function of population growth long-run growth is an increasing function of population growth – is zero if population growth is zero. is zero if population growth is zero.

• The fractions of the labor force and the capital stock engaged in The fractions of the labor force and the capital stock engaged in R&D, R&D, aL aL and and aK aK , do not affect long-run growth; nor does the , do not affect long-run growth; nor does the saving rate, saving rate, ss. .

• Case 2: Case 2: ββ++θ θ = 1 and = 1 and n n = 0= 0• When When ββ++θ θ is 1 and is 1 and n n is 0, both expressions simplify to is 0, both expressions simplify to gK gK = = gAgA. .

That is, in this case That is, in this case the two loci lie directly on top of each otherthe two loci lie directly on top of each other: : both are given by the 45-degree line. Figure 3.8 shows the both are given by the 45-degree line. Figure 3.8 shows the dynamics of the economy in this case.dynamics of the economy in this case.

• As the figure shows, regardless of where the economy begins, the As the figure shows, regardless of where the economy begins, the dynamics of dynamics of gA gA and and gK gK carry them to the 45-degree line. Once carry them to the 45-degree line. Once that happens, that happens, gA gA and and gK gK are constant, and the economy is on a are constant, and the economy is on a balanced growth path. balanced growth path.

• As inAs in the case of the case of θ θ =1 and =1 and n n = 0 in the model without capital, the = 0 in the model without capital, the phase diagram does not tell us what balanced growth path the phase diagram does not tell us what balanced growth path the economy converges to.economy converges to.

The Nature of Knowledge and the Determinants of the Allocation of Resources to R&D

• So far we have simply described the “So far we have simply described the “AA” variable produced by ” variable produced by R&D as knowledge. But R&D as knowledge. But knowledge comes in many formsknowledge comes in many forms..

• Many of these different types of knowledge play important roles in Many of these different types of knowledge play important roles in economic growth. Imagine, for example, that 100 years ago there economic growth. Imagine, for example, that 100 years ago there had been a halt to basic scientific progress, or to the invention of had been a halt to basic scientific progress, or to the invention of applied technologies. it seems likely that all of them would have applied technologies. it seems likely that all of them would have led to substantial reductions in growth.led to substantial reductions in growth.

• There is no reason to expect the determinants of the accumulation There is no reason to expect the determinants of the accumulation of these different types of knowledge of these different types of knowledge to be the sameto be the same. There is thus . There is thus no reason to expect a no reason to expect a unified theory of the growth unified theory of the growth of knowledge. of knowledge. Rather, we should expect to find Rather, we should expect to find various factors various factors underlying the underlying the accumulation of knowledge.accumulation of knowledge.

The Nature of Knowledge and the Determinants of the Allocation of Resources to R&D

• At the same time, all types of knowledge share one essential At the same time, all types of knowledge share one essential feature: they are feature: they are nonrivalnonrival. . That is, the use of an item of knowledge That is, the use of an item of knowledge in one application makes its use by someone else no more in one application makes its use by someone else no more difficult. difficult.

• Conventional private economic goods, in contrast, are Conventional private economic goods, in contrast, are rivalrival: : the the use of, say, an item of clothing by one individual precludes its use of, say, an item of clothing by one individual precludes its simultaneous use by someone else.simultaneous use by someone else.

• In the case of knowledge, In the case of knowledge, excludabilityexcludability depends both on the depends both on the nature of the knowledge nature of the knowledge itself and on itself and on economic institutions economic institutions governing property rights. Patent laws, for example, give inventors governing property rights. Patent laws, for example, give inventors rights over the use of their designs and discoveries. Under a rights over the use of their designs and discoveries. Under a different set of laws, inventors’ ability to prevent the use of their different set of laws, inventors’ ability to prevent the use of their discoveries by others might be smaller.discoveries by others might be smaller.

The Nature of Knowledge and the Determinants of the Allocation of Resources to R&D

• In some cases, excludability is more dependent on the In some cases, excludability is more dependent on the nature of nature of the knowledgethe knowledge and less dependent on the legal system. The and less dependent on the legal system. The recipe for Coca-Cola is sufficiently complex that it can be kept recipe for Coca-Cola is sufficiently complex that it can be kept secret without copyright or patent protection. secret without copyright or patent protection.

• The degree of excludability is likely to have a strong influence on The degree of excludability is likely to have a strong influence on how the development and allocation of knowledge depart from how the development and allocation of knowledge depart from perfect competitionperfect competition. If a type of knowledge is entirely non. If a type of knowledge is entirely non--excludable, there can be excludable, there can be no private gain in its developmentno private gain in its development..

• When knowledge is excludable, the producers of new knowledge When knowledge is excludable, the producers of new knowledge can can license the right to use the knowledge license the right to use the knowledge at positive prices, and at positive prices, and hence hope to earn positive returns on their R&D efforts.hence hope to earn positive returns on their R&D efforts.

•

The Nature of Knowledge and the Determinants of the Allocation of Resources to R&D

• The major forces governing the allocation of resources to the The major forces governing the allocation of resources to the development of knowledge are four forces: development of knowledge are four forces: 1.support for basic 1.support for basic scientific research, 2.private incentives for R&D and innovation, scientific research, 2.private incentives for R&D and innovation, 3.alternative opportunities for talented individuals, and 4.learning-3.alternative opportunities for talented individuals, and 4.learning-by-doing. by-doing.

• 1.Support for Basic Scientific Research1.Support for Basic Scientific Research

• Basic scientific knowledge are available relatively Basic scientific knowledge are available relatively freelyfreely; this ; this research research is not motivated is not motivated by the desire to earn private returns in by the desire to earn private returns in the market. The economics of this type of knowledge are relatively the market. The economics of this type of knowledge are relatively straightforward. Since it is useful in production and is given away straightforward. Since it is useful in production and is given away at at zero costzero cost, it has a positive externality. Thus its production , it has a positive externality. Thus its production should be should be subsidizedsubsidized..

The Nature of Knowledge and the Determinants of the Allocation of Resources to R&D

• 2.Private Incentives for R&D and Innovation2.Private Incentives for R&D and Innovation

• Many innovations receive little or no external support and are Many innovations receive little or no external support and are motivated almost entirely by the motivated almost entirely by the desire for private gaindesire for private gain..

• As described above, for R&D to result from economic incentives, As described above, for R&D to result from economic incentives, the knowledge that is created must be at least somewhat the knowledge that is created must be at least somewhat excludableexcludable. Thus the developer of a new idea has some degree of . Thus the developer of a new idea has some degree of market power.market power.

• Typically, the developer is modeled as Typically, the developer is modeled as having exclusive control having exclusive control over the use of the idea and as licensing its use to the producers over the use of the idea and as licensing its use to the producers of final goods. of final goods.

The Nature of Knowledge and the Determinants of the Allocation of Resources to R&D

• There are in fact three distinct externalities from R&D: the There are in fact three distinct externalities from R&D: the consumer-surplus consumer-surplus effect, the effect, the business-stealing business-stealing effect, and the effect, and the R&D R&D effect. effect.

– The consumer-surplus The consumer-surplus effect is that the individuals or firms effect is that the individuals or firms licensing ideas from innovators licensing ideas from innovators obtain some surplusobtain some surplus, since , since innovators cannot engage in perfect price discrimination. Thus innovators cannot engage in perfect price discrimination. Thus this is a this is a positive externality positive externality from R&D.from R&D.

– The business-stealing The business-stealing effect is that the introduction of a effect is that the introduction of a superiorsuperior technology typically makes existing technologies less technology typically makes existing technologies less attractive, and therefore harms the owners of those attractive, and therefore harms the owners of those technologies. This externality is negative.technologies. This externality is negative.

The Nature of Knowledge and the Determinants of the Allocation of Resources to R&D

– The R&D effect The R&D effect is that innovators are generally assumed not is that innovators are generally assumed not to control the use of their knowledge in the to control the use of their knowledge in the production of production of additional knowledgeadditional knowledge. Innovators are assumed to earn returns . Innovators are assumed to earn returns on the use of their knowledge in goods production (equation on the use of their knowledge in goods production (equation [3.1]) but not in knowledge production (equation [3.2]). Thus [3.1]) but not in knowledge production (equation [3.2]). Thus the development of new knowledge has a positive externality the development of new knowledge has a positive externality on others engaged in R&D.on others engaged in R&D.

• The net effect of these three externalities is ambiguous. It is The net effect of these three externalities is ambiguous. It is generally believed, however, that the normal situation is for the generally believed, however, that the normal situation is for the overall overall externality from R&D to be positiveexternality from R&D to be positive. In this case the . In this case the equilibrium level of R&D is inefficiently low, and R&D subsidies equilibrium level of R&D is inefficiently low, and R&D subsidies can increase welfare.can increase welfare.

The Nature of Knowledge and the Determinants of the Allocation of Resources to R&D

• 3.Alternative Opportunities for Talented Individuals3.Alternative Opportunities for Talented Individuals

• Major innovations and advances in knowledge are often the result Major innovations and advances in knowledge are often the result of the work of of the work of extremely talented individualsextremely talented individuals. Such individuals . Such individuals typically have choices other than just pursuing innovations and typically have choices other than just pursuing innovations and producing goods. producing goods.

• Hence economic Hence economic incentives and social forcesincentives and social forces influencing the influencing the activities of highly talented individuals may be important to the activities of highly talented individuals may be important to the accumulation of knowledge.accumulation of knowledge.

• three factors three factors that influence talented individuals’ decisions whether that influence talented individuals’ decisions whether to pursue activities. to pursue activities.

• 1. the 1. the size of the relevant marketsize of the relevant market: the larger is the market from : the larger is the market from which a talented individual can reap returns, the greater are the which a talented individual can reap returns, the greater are the incentives to enter a given activity. incentives to enter a given activity.

The Nature of Knowledge and the Determinants of the Allocation of Resources to R&D

• 2. the degree of 2. the degree of diminishing returnsdiminishing returns. .

• 3.the 3.the ability to keep the returns ability to keep the returns from one’s activitiesfrom one’s activities

• 4. Learning-by-Doing 4. Learning-by-Doing

• The simplest case of learning-by-doing is when learning occurs as The simplest case of learning-by-doing is when learning occurs as a a side effect side effect of the production of new capital, the stock of of the production of new capital, the stock of knowledge is a function of the stock of capital. Thus there is only knowledge is a function of the stock of capital. Thus there is only one state variable. one state variable.

• To analyze this economy, begin by substituting (3.23) into (3.22). To analyze this economy, begin by substituting (3.23) into (3.22). This yields: This yields:

• Remember before that the key determinant of the economy’s Remember before that the key determinant of the economy’s dynamics is how dynamics is how θ θ compares with 1, compares with 1, equivalent is how equivalent is how φ φ compares with 1.compares with 1.

• If If φ φ is less than 1, the long-run growth rate of the economy is a is less than 1, the long-run growth rate of the economy is a function of function of nn. .

• If If φ φ is greater than 1, there is explosive growth. is greater than 1, there is explosive growth.

• if if φ φ equals 1, there is explosive growth if equals 1, there is explosive growth if n n is positive and steady is positive and steady growth if growth if n n equals 0.equals 0.

• Empirical Application: Time-Series Tests of Endogenous Empirical Application: Time-Series Tests of Endogenous Growth ModelsGrowth Models

• Jones (1995b) raises a critical issue about these models: Jones (1995b) raises a critical issue about these models: Does Does growth in fact vary with the factors identified by the models in the growth in fact vary with the factors identified by the models in the way the models predict? way the models predict?

• Jones considers two approaches to testing the predictions of fully Jones considers two approaches to testing the predictions of fully endogenous growth models about changes in growth. endogenous growth models about changes in growth.

• The first starts with the observation The first starts with the observation that the models predict that that the models predict that changes in the models’ changes in the models’ parameters permanently affect growthparameters permanently affect growth. .

• For example, in the model of Section 3.3 with For example, in the model of Section 3.3 with β β + + θ θ =1 and =1 and n n = 0, = 0, changes in changes in ss, , aLaL, and , and aK aK change the economy’s long run growth change the economy’s long run growth rate. rate.

• He therefore asks whether the actual growth rate of income per He therefore asks whether the actual growth rate of income per person person is is stationary stationary or or non-stationarynon-stationary and considers several tests and considers several tests of stationarity versus non-stationarity. of stationarity versus non-stationarity.

• A simple one is to regress the growth rate of income per person, A simple one is to regress the growth rate of income per person, on a on a constant and a trend, constant and a trend, and then test the null hypothesis that and then test the null hypothesis that b b = 0. = 0.

• a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time. Consequently, parameters such as the mean and variance, if they are present, also do not change over time and do not follow any trends.

• A second test is an A second test is an augmented Dickey-Fuller testaugmented Dickey-Fuller test. Consider a . Consider a regression of the formregression of the form

• If growth has some normal level that it reverts to when it is pushed If growth has some normal level that it reverts to when it is pushed away, away, ρ ρ is negative. If it does not, is negative. If it does not, ρ ρ is 0is 0..

• Jones examines data on U.S. income per person over the period Jones examines data on U.S. income per person over the period 1880–1987. His statistical results seem to provide powerful 1880–1987. His statistical results seem to provide powerful evidence that evidence that growth is stationarygrowth is stationary..

• The augmented Dickey-Fuller test overwhelmingly rejects the null The augmented Dickey-Fuller test overwhelmingly rejects the null hypothesis that hypothesis that ρ ρ = 0, thus appearing to indicate stationarity.= 0, thus appearing to indicate stationarity.

•

• The Magnitudes and Correlates of Changes in Long-Run The Magnitudes and Correlates of Changes in Long-Run GrowthGrowth

• Jones’s second approach is to examineJones’s second approach is to examine the relationships between the relationships between the determinants of growth the determinants of growth identified by endogenous growth identified by endogenous growth models and actual growth rates. models and actual growth rates.

• He begins by considering learning-by-doing models with He begins by considering learning-by-doing models with φ φ = 1. = 1. Recall that that model yields a relationship of the formRecall that that model yields a relationship of the form

• This implies that the growth rate of income per person isThis implies that the growth rate of income per person is

• where where gx gx denotes the growth rate of denotes the growth rate of xx. . gK gK is given byis given by

• where where s s is the fraction of output that is invested and is the fraction of output that is invested and δ δ is the is the depreciation rate.depreciation rate.

• Jones observes that Jones observes that YY//KK, , δδ, and , and gL gL all both appear to be fairly all both appear to be fairly steadysteady, , while investment rates have been trending upwhile investment rates have been trending up. Thus the . Thus the model model predicts an upward trend in growthpredicts an upward trend in growth. .

• Jones reports that in most major industrialized countries, Jones reports that in most major industrialized countries, YY//K K is is aboutabout 0.4 0.4 and the ratio of investment to GDP has been rising by and the ratio of investment to GDP has been rising by about about 1% per decade1% per decade. .

• Empirical Application: Population Growth and Technological Empirical Application: Population Growth and Technological Change since 1 Million B.C.Change since 1 Million B.C.

• Kremer (1993), argues that endogenous growth provide insights Kremer (1993), argues that endogenous growth provide insights into the dynamics of population, technology, and income over the into the dynamics of population, technology, and income over the broad sweep of human history.broad sweep of human history.

• Kremer begins his analysis by noting that essentially all models of Kremer begins his analysis by noting that essentially all models of the endogenous growth of knowledge predict that the endogenous growth of knowledge predict that technological technological progressprogress is an is an increasing function of population sizeincreasing function of population size. .

• He then argues that over almost all of human history, technological He then argues that over almost all of human history, technological progress has led mainly to progress has led mainly to increases in population increases in population rather than rather than increases in output per personincreases in output per person..

• A Simple ModelA Simple Model

• Kremer’s formal model is a straightforward. First, output depends Kremer’s formal model is a straightforward. First, output depends on technology, labor, and land:on technology, labor, and land:

• where where T T denotes the fixed stock of landdenotes the fixed stock of land. .

• Second, additions to knowledge are proportional Second, additions to knowledge are proportional to populationto population, , and also depend and also depend on the stock of knowledgeon the stock of knowledge::

• And third, population adjusts so that output per person And third, population adjusts so that output per person equals the subsistence levelequals the subsistence level, denoted , denoted yy::

• Aside from this Malthusian assumption about the determination of Aside from this Malthusian assumption about the determination of population, population, γ γ = 1.= 1.

• We solve the model in two steps. The first is to find the size of the We solve the model in two steps. The first is to find the size of the population that population that can be supported on the fixed stock of land can be supported on the fixed stock of land at a at a given time. Substituting expression (3.57) for output into the given time. Substituting expression (3.57) for output into the Malthusian population condition, (3.59), yieldsMalthusian population condition, (3.59), yields

• Solving this condition for Solving this condition for LL((t t ) gives us) gives us

• This equation states that the population that can be This equation states that the population that can be supported is decreasing in the subsistence level of output, supported is decreasing in the subsistence level of output, increasing in technology, and proportional to the amount of increasing in technology, and proportional to the amount of land.land.

• The second step is to find the dynamics of technology and The second step is to find the dynamics of technology and population. Since both population. Since both y y and and T T are constant, (3.61) implies are constant, (3.61) implies that the growth rate of that the growth rate of L L is is (1 − (1 − αα)/)/α α times the growth rate of times the growth rate of AA::

• Thus population growth is increasing in the size of the Thus population growth is increasing in the size of the population unless population unless α α is large or is large or θ θ is much less than 1 (or a is much less than 1 (or a combination of the two). Intuitively, Kremer’s model implies combination of the two). Intuitively, Kremer’s model implies increasing growth even with diminishing returns to increasing growth even with diminishing returns to knowledge in the production of new knowledge (that is, even knowledge in the production of new knowledge (that is, even with with θ <θ <1)1)

• because labor is now a produced factorbecause labor is now a produced factor: improvements in : improvements in technology lead to higher populationtechnology lead to higher population, which in turn leads to , which in turn leads to further improvements in technology.further improvements in technology.

• Kremer tests the model’s predictions using population Kremer tests the model’s predictions using population estimates extending back to 1 million B.C. that have been estimates extending back to 1 million B.C. that have been constructed by archaeologists and anthropologists.constructed by archaeologists and anthropologists.

• • Figure 3.9 shows the resulting scatter plot of population Figure 3.9 shows the resulting scatter plot of population

growth against population. Each observation shows the level growth against population. Each observation shows the level of population at the beginning of some period and the of population at the beginning of some period and the average annual growth rate of population over that period.average annual growth rate of population over that period.

• • The figure shows a strongly positive, and approximately The figure shows a strongly positive, and approximately

linear, relationship between population growth and the level linear, relationship between population growth and the level of population. A regression of growth on a constant and of population. A regression of growth on a constant and population (in billions) yieldspopulation (in billions) yields

• A regression of growth on a constant and population (in A regression of growth on a constant and population (in billions) yieldsbillions) yields

• where where n n is population growth and is population growth and L L is population, and where is population, and where the numbers in parentheses are standard errors. the numbers in parentheses are standard errors.

• Thus there is an overwhelmingly statistically significant Thus there is an overwhelmingly statistically significant association between the level of population and its growth association between the level of population and its growth rate. rate.

• Population Growth versus Growth in Income per Person Population Growth versus Growth in Income per Person over the Very Long Runover the Very Long Run

• As described above, over nearly all of history As described above, over nearly all of history technological technological progress has led almost entirely to higher population rather progress has led almost entirely to higher population rather than to higher average income.than to higher average income.

• But this has not been true over the past few centuriesBut this has not been true over the past few centuries: the : the enormous technological progress of the modern era has led enormous technological progress of the modern era has led not only to vast population growth, not only to vast population growth, but also to vast increases but also to vast increases in average income.in average income.

• The evidence suggests that preferences are such that once The evidence suggests that preferences are such that once average income is average income is sufficiently highsufficiently high, population growth is , population growth is decreasing in incomedecreasing in income. the model predicts that . the model predicts that population population growthgrowth peaks at some point and then declines. peaks at some point and then declines.

•

• Models of Knowledge Accumulation and the Central Models of Knowledge Accumulation and the Central Questions of Growth TheoryQuestions of Growth Theory

• Explaining cross-country income differences Explaining cross-country income differences on the basis of on the basis of differences in knowledge accumulation faces a fundamental differences in knowledge accumulation faces a fundamental problem, however: problem, however: the non-rivalry of knowledgethe non-rivalry of knowledge. .

• producers in poor countries can use the same knowledge as producers in poor countries can use the same knowledge as producers in rich countries. If the relevant knowledge is producers in rich countries. If the relevant knowledge is publicly available, publicly available, poor countries can become rich poor countries can become rich by having by having their workers or managers read the appropriate literature. their workers or managers read the appropriate literature.

• If the relevant knowledge is If the relevant knowledge is produced by private R&Dproduced by private R&D, poor , poor countries can become rich by instituting a credible program countries can become rich by instituting a credible program for respecting foreign firms’ property rights.for respecting foreign firms’ property rights.

• With such a program, With such a program, the firms in developed countries the firms in developed countries with with proprietary knowledge proprietary knowledge would open factories would open factories in poor in poor countries, hire their inexpensive labor, and produce output countries, hire their inexpensive labor, and produce output using the proprietary technology. The result would be that using the proprietary technology. The result would be that the the marginal product of labor in poor countries, and hence marginal product of labor in poor countries, and hence wages, would rapidly rise wages, would rapidly rise to the level of developed countries.to the level of developed countries.

• poor countries, have FDI Yet we poor countries, have FDI Yet we do not see incomes do not see incomes in those in those areas jumping to the levels of industrialized countries.areas jumping to the levels of industrialized countries.