Innovative Elites and Technology Diffusion Oliver Pardo Abstract This paper studies the effect of lags in technology diffusion on the growth process and the distribution of income. An overlapping generations model with heterogeneous agents is constructed with this purpose. Agents are distributed in a rectangular grid where they are able to observe and copy the technology of their neighbors. The potential GDP growth of the whole economy is driven by those agents who have the highest propensity to innovate. Therefore, the growth rate is a function of the characteristics of an innovative elite, instead of being a function of some average characteristics of the whole population. The closer you are to an agent who belongs to that elite, the more likely is that you have a relatively high level of income, given the possibility of free-ride on his ideas. Because of the possible dispersion of the innovative elite across the grid, this could lead to multimodal distributions of income. Therefore, the model sets an alternative hypothesis to explain the spatial distribution of income across regions and countries. Key words: innovative elites, technology diffusion, spatial distribution of income. JEL Classification: O31, O33, D90.
1
Introduction
At least a quarter of the per capita income differences around the world can be explained by the cross-country variation in the adoption of technologies (Comin and Hobijn (2008)). Nevertheless, traditional growth models tend to assume a common pool of technology, omitting a potential source for income inequality. The idea of a common pool of technology follows from an extreme interpretation of the non-rivalry of ideas: once applied by an agent, an innovation can be implemented immediately by everyone else. In reality, it takes time for innovations to arrive to other parts of the economy. This paper studies the dynamics of growth and income distribution when lags in technological diffusion are accounted for. With this purpose, a model with heterogeneous agents is constructed. With enough frictions in the diffusion of new 1
technologies, the model is able to replicate certain stylized facts of income distribution. Particularly, it allows for the endogenous appearance of geographical locations where income tends to concentrate. At the center of these clusters, there are those agents that lead the innovation process and whose characteristics eventually define the growth rate for the whole economy. Those agents are denominated the innovative elite of the economy. The rest of the paper is organized as follows. Section 2 presents the basic setup of the model and set the difference equations that define the equilibrium trajectory. Unfortunately, when particular frictions in the process of technological diffusion are accounted for, there is no analytical solution for the dynamics of the model. Therefore, section 3 turns to numerical simulations in order to describe the dynamics of the model. Section 4 realizes some comparative statics, focusing on the effects that population size and the dispersion of population characteristics have on economic growth. Then section 5 offers some final comments, point some limitations of the model and suggest some possible extensions. Finally, the appendix present a generalization of the model presented on section 2.
2 2.1
Model Basic setup
The following is an overlapping generations model where agents live for two periods. Each agent has a fixed position in a rectangular grid of dimension I × J. For each point in the grid there is a dynasty, defined a a sequence of agents where each one is the son of his predecessor. Each dynasty is indexed according to its position in the grid and each agent is indexed according to the dynasty they belong to and the date in which they were born. Therefore, agent (i, j, t) lives in the position (i, j) of the grid, was born in period t and is son of the agent (i, j, t − 1). Given that agents live for two periods, in each point of the grid there are simultaneously two agents (father and son). It is assumed that fathers and sons share the same preferences and the same technology. Each agent is able to observe the technologies used by other agents in their “neighborhood”. Two canonical cases are studied in this paper. In the first one, the neighborhood of any agent is formed by all the agents on the grid. In the second one, the neighborhood of an agent is composed by himself and by the agents who live just in front of him, just behind him or just at one of his sides (a von Neumann neighborhood of range 1). Let yi,j,t be the technology used by agent (i, j, t) and let zi,j,t be the most efficient of the technologies used in the neighborhood of agent (i, j, t). Thus in the first canonical case zi,j,t is given by zi,j,t = max{yk,l,t | k = 1, ..., I, l = 1, ..., J}
2
(1)
whereas in the second zi,j,t is given by zi,j,t = max{yi+1,j,t , yi−1,j,t , yi,j,t , yi,j+1,t , yi,j−1,t }
(2)
Both the own technology and the technology used by the neighbors can be improved withdrawing resources from the production of a perishable consumption good. In their first period of live agents observe the technologies used by their neighbors and allocate some of their time in improving the most efficient of them. Therefore, in his second period of live each agent could use the neighborhood’s leading technology of the previous period (or an improved version of it). Formally, let li,j,t ∈ [0, 1] be the fraction of resources allocated to innovation by agent (i, j, t) in his first period of life. The technology used by him in his second period of life will be then yi,j,t+1 = (1 − δ + φli,j,t )zi,j,t (3) The specification in equation (3) assume that knowledge depreciates at a rate δ. For simplicity, we assume for now onwards that δ = 0. The parameter φ > 0 express the sensibility of technological improvements to the time allocated to innovation. It is assumed that this parameter is the same for all agents, although the results of the model will hold if instead of assuming heterogeneity in the discount factor (see below), the parameter φ varies across agents. For agent (i, j, t) let ci,j,t and di,j,t+1 be the consumption in his first and in his second period of life, respectively. The endowment of time in both periods is normalized to one. Therefore, the time allocated to the production of the perishable consumption good will be 1 − li,j,t . It is assumed that all agents produce the same consumption good and that there is no credit market. This conditions imply that for each agent the consumption in each period of life can not exceed the amount produced by himself in that period. Therefore, for his first period of life, consumption of agent (i, j, t) will be given by 1 − li,j,t times his potential output: ci,j,t ≤ (1 − li,j,t )yi,j,t
(4)
It is assumed that agents do not have altruistic feelings toward their descendants. Therefore, in the second period of life there are no incentives to improve the technology. Thus all the available time in the second period of life will be allocated to the production of the consumption good. The restriction for the consumption in the second period of life of agent (i, j, t) will be then di,j,t+1 ≤ yi,j,t+1
(5)
For simplicity, assume that the instantaneous utility received by each agent is logarithmic1 . Let βi,j ∈ (0, 1) be the discount factor for the dynasty (i, j). The 1 The general results of the model will still hold if a more general CRRA function is assumed. The reason for using a logarithmic specification is that it allows to simplify the exposition of the model and its algebraic results.
3
optimization problem faced by each agent, omitting subindexes (i, j) for a while, will be: max log (ct ) + β log (dt+1 ) s.t. (6) lt ,ct ,dt+1 ,yt+1
ct ≤ (1 − lt )yt dt+1 ≤ yt+1 yt+1 = (1 + φlt )zt 0 ≤ lt ≤ 1
2.2
Solution
This subsection solves the agents maximization problem (6) in order to describe the equilibrium trajectory of the model with a set of difference equations. The Euler condition for the maximization problem in (6) will be 1 1 yt ≥ β φzt , with equality if lt > 0 (7) ct dt+1 Equation (7) states that if some time is devoted to innovation, then the marginal utility of time allocated to work must equal the marginal utility of time allocated to innovation. In equilibrium, it can not be the case that lt = 1, because in that case the marginal utility of consumption in the first period of life would be infinite. However, it could be optimal for an agent to allocate no time to innovation, in which case lt = 0. In equilibrium, equitations (4) and (5) hold with equality. Therefore, the Euler equation (7) can be restated as βφ 1 ≥ , (1 − lt ) (1 + φlt )
with equality if lt > 0
(8)
Let lt (yt , zt ) be the optimal fraction of time allocated to innovation. From inequality lt ≥ 0 and equation (8) it follows that φβ−1 } lt (yt , zt ) = max{0, φ(1+β)
(9)
From equation (9) it follows that some time is allocated to innovation if and only if φβ > 1. Given that β ∈ (0, 1), then the inequality φ > 1 is a necessary condition to observe innovation in the model. Introducing the solution described by (9) inside equation (3) and reintroducing subindexes (i, j), it is obtained that the dynamics for the technology used by dynasty (i, j) are given by ´ ³ φβi,j −1 yi,j,t+1 = 1 + max{0, 1+βi,j } zi,j,t ; (10) Equation (10), a set of initial conditions and either equations (1) or equation (2) define the dynamics of the model. The following subsection describes the dynamics when zi,j,t is given by equation (1), which is the simplest case. 4
2.3
Dynamics
Let’s assume as an initial condition that yi,j,0 = 1 for all (i, j). If for every dynasty it happens that φβi,j ≤ 1, then no dynasty has incentives to innovate and therefore yi,j,t = 1 for every (i, j) and every t. In order to obtain a process of innovation and growth, it is necessary and sufficient that φβi,j > 1 for at least one dynasty. Let’s assume that this is the case. The improvements of each agent on his neighborhood’s best technology can be measured in relative terms through γi,j ≡ yi,j,t+1 /zi,j,t − 1. From equation (10) it follows that in equilibrium γi,j = max{0,
φβi,j −1 } 1+βi,j
(11)
The right hand side of equation (14) will be denominated the propensity to innovate of dynasty (i, j). This propensity is a non-decreasing function of the marginal effect of time allocated to innovation φ and the discount factor β. Any agent such that φβi,j > 1 make improvements on his neighborhood’s best technology. Among those agents, there is a subset of them such that their neighborhood’s best technology is their own technology. Those agents that make improvements based on their own technology in period t will be denominated the innovative elite of that period. Formally, an agent (i, j, t) belongs to the innovative elite of period t if and only if φβi,j > 1 and zi,j,t = yi,j,t . Consider the case where technology diffuses all around the grid with one period lag, i.e., zi,j,t is given by equation (1). Given the initial condition yi,j,0 = 1 for all (i, j) and the assumption that φβi,j > 1 for at least one pair (i, j), the dynasty(ies) with the highest propensity to innovate will take the lead on technological improvements. Thus in this case the innovative elite is time-invariant and formed in every period by the dynasty(ies) with the highest propensity to innovate. Particularly, given the assumption that φ is the same across all dynasties, the innovative elite is formed by the dynasty(ies) with the highest discount factor β. Let (m, n) be the coordinates on the grid of (one of) the dynasty(ies) with the highest propensity to innovate. From the previous analysis it follows that when equation (1) holds, zi,j,t = ym,n,t for all (i, j) an all t ≥ 1. This result and equation (10) imply ³ ´ φβi,j −1 yi,j,t+1 = 1 + max{0, 1+βi,j } ym,n,t for all (i, j, t) (12)
Equation (12) imply that for all (i, j), yi,j,t grows at the same rate that ym,n,t . Now the variable ym,n,t grows at a constant rate equal to γm,n . Then the technology of every dynasty -and therefore the potential output of the economy- grows at a rate given by the propensity to innovate of the innovative elite. Thus this model can be seen as a first step to formalize the claim of Mokyr (2005) that “the best model to explain technological progress ... is not the mean level of human capital ... but just the density in the upper tail of the distribution, that is, the level of education and sophistication of a small and pivotal elite ... ”[page 51]. 5
Discount factor
1 0.8 0.6 0.4 0.2 0 0 5
20
10
15
15
10
20 j
5
25
i
Figure 1: Distribution of the β’s across the grid Summing up, when technological diffusion is specified as it is in equation (1), the economy is always in a steady state where it grows at a rate equal to the highest propensity to innovate in the population. However, when technological diffusion is given by equation (2), numerical simulations are required to illustrate the dynamics of the model.
3
Simulations
This section illustrates the dynamics of the model through numerical simulations. Section 3.1 presents a numerical example of the case described in section 2.3. It is just presented to serve as a benchmark for the other simulations. Section 3.2 leaves aside the technology diffusion implied by equation (1) and assumes equation (2) instead. Section 3.3 still assumes that equation (2) holds, but adds a new friction: just a fraction of the neighbor’s technology can be copied without any cost. The size of the grid in all three simulation is 20 × 20, so there is a total of 400 dynasties. The marginal effect of time allocated to innovation φ is always equal to 1.2. The discount factor βi,j is a continuous random variable identically and independently distributed in [0, 1]. Particularly, for every dynasty (i, j) it is assumed that 1 βi,j = (13) 1 + exi,j where xi,j comes from a standard normal distribution. 6
For all the three simulations, it is taken the same realization for the β’s. The distribution of this realization over the grid is presented in figure 1. The average β in this realization is 0.5024, its standard deviation is 0.2192, the minimum 0.0482 and the maximum is 0.9303. The dynasty with the highest β is the one in the position (16, 16) of the grid. It should be noted that given that the parameter φ is assumed equal to all dynasties, the distribution of the propensity to innovate mimics the one presented in figure 1.
3.1
Perfect copying all around the grid
The following simulation is a numerical example of the dynamics described in section 2.3. Innovations spread all around the grid with just one period lag, so zi,j,t is defined by equation (1). Figure 2 shows the potential output of each dynasty distributed across the grid, for periods 1, 20, 100 and 200. Because the potential output of the young and the effective income of their parents is the same, figure 2 represents the spatial distribution of income for several periods. Because all the dynasties start with the same level of technological knowledge and the dynasty (16,16) has the highest propensity to innovate, the potential output of each dynasty grows at a rate equal to φβ16,16 − 1 1.2 × 0.9303 − 1 γ16,16 = = = 6.03% (14) 1 + β16,16 1 + 0.9303 The whole economy grows at this rate and reaches its steady state immediately, as shown in figure 3. There is almost no inequality in the distribution of potential income, because any gap caused by some innovation is closed up with just one period lag. The variation coefficient for yi,j,t equals 0.0073 for every t, as figure 4 illustrates.
3.2
Perfect copying in the immediate neighborhood
Now assume that zi,j,t is given by equation (2). Therefore, the agents can observe the technology used by just their immediate neighbors. The distribution of income across the grid is illustrated in figure 5, for the same periods as in figure 2. As time passes on, the spatial distribution of income looses uniformity and multiple modes appear. The variation coefficient of income increases, as figure 4 shows. This is a consequence of the asymmetric propensity to innovate and the inability -for most of the agents- to copy the most efficient technology on the grid. It should be noted that the spatial distribution of income is barely linked to the spatial distribution of the propensity to innovate implied by figure 1. The income of each dynasty is more linked to their proximity to the dynasties that form the innovative elite than to other of their idiosyncratic characteristics, like their propensity to innovate. Therefore, the distance to the innovation centers would delay the adoption of new technologies. This is supported by empirical studies like
7
Figure 2: Spatial distribution of income in simulation 3.1.
8
0.07 0.06
Growth Rate
0.05 0.04 Simulation in 3.1 Simulation in 3.2 Simulation in 3.3
0.03 0.02 0.01 0
0
50
100 Time
150
200
Figure 3: GDP growth rates in the three simulations. 0.7
Variation coefficient for y
0.6 0.5
Simulation in 3.1 Simulation in 3.2 Simulation in 3.3
0.4 0.3 0.2 0.1 0
0
50
100 Time
150
200
Figure 4: Variation coefficients of old agents’ income in the three simulations. 9
Figure 5: Spatial distribution of income in simulation 3.2. Keller (2002) and Bottazzi and Peri (2003), who find a statistically significant effect of geographical distance on technology diffusion. Each mode of the distribution in figure 5 represents a dynasty that belongs to the innovative elite of the corresponding period. Agents around them try to copy or improve the technology used by those dynasties instead of copying or improving the technology inherited from their parents. Nevertheless, as times passes on, other agents can bring to the neighborhood a more efficient technology. In that moment the agents placed around a dynasty from which they used to copy its technology could cease to do that. At the same time, those dynasties that used to be part of the innovative elite but start to loose followers also start to find better to copy the technology from somewhere else. Thus the number of modes in the spatial distribution of income and the number of dynasties that belong to the innovative elite eventually decline. In the particular example of figure 5, the number of dynasties that survive as part of the innovative elite is just two when t = 100. Those dynasties are the ones located at (16, 16) and (1, 11). Not surprisingly, the dynasty (1, 11) is the second with the highest β and therefore the second with the highest propensity 10
Figure 6: Spatial distribution of income in simulation 3.3. to innovate. In contrast with the simulation in section 3.1, this economy does not reach an steady state immediately. Instead, the growth rate of the economy increases and approaches asymptotically to 6.03%, as shown in figure 3. Still, the highest propensity to innovate is the determinant of the long-run growth rate for the economy.
3.3
Imperfect copying in the immediate neighborhood
In the model described in section 2 and the numerical simulations of sections 3.1 and 3.2, any technology observed in the previous period can be replicated in the current period without any cost. Instead, assume that just a fraction α ∈ [0, 1] of the neighbor’s technology can be used without any time assigned to the adaptation of a foreign technology. This general version of the model is presented in the appendix. Now this section presents a numerical simulation where copying is restricted to the immediate neighborhood -so zi,j,t is given by equation (2)- and α = 0.9. All other parameters and the realization of the β’s are the same as in sections 3.1 and 3.2. 11
Figure 6 shows the evolution of the spatial distribution of income in this new case. As in section 3.2, the distribution looses uniformity as time passes on. Nevertheless, inequality increases faster, as showed by figure 4. The asymmetries in income distribution are more highlighted, as can be seen from the skewness around the modes of the distribution. Again, the dynasties with the highest propensity to innovate push development around them, but to less extent than in the case of section 3.2. This is a consequence of the additional friction introduced in the process of technological diffusion. The number of dynasties that belong to the innovative elite still decreases, but a slower pace than in section 3.2. The dynamics of aggregate output reveal a more interesting phenomenon. As in section 3.2, growth starts increasing below the previous steady state level. But when t is around 140 and the growth rate is near enough to 6.03%, it starts to fluctuate in an endogenous cycle. The reason for this is that for some dynasties is optimal to wait for a wider gap between their technology and the best technology on their neighborhood. Only when the gap is wide enough, it is optimal to them to invest resources in the adaptation of other technologies. Still, the upper bound for the growth rate is given by the highest propensity to innovate in the population.
4
Scale Effects
The previous analysis shows that the existence of an innovative elite is a prerequisite for growth: In order to observe growth, it should be the case that φβ should be greater than one for at least one dynasty. If φ > 1 and β is a random variable whose support is [0, 1], then the probability that φβ > 1 for at least one agent is increasing in the population size. For example, if βi,j is distributed uniformly over [0, 1] and φ > 1, then the probability that φβi,j > 1 for at least one (i, j) is 1 − (1/φ)I×J , which is increasing in I × J. Therefore, a society starting with a low but increasing population could be initially stagnated and then suddenly starts to growth2 . Similarly, the highest β value -and therefore the highest propensity to innovateis non-decreasing in the population size. In other words, better innovators can be found on larger populations3 . This implies the existence of a scale effect: the larger the population, the highest the growth rate. Figure 7 presents a numerical example of this situation for the case where zi,j,t is given by equation (1). The horizontal axis represent the size of the grid -i.e., the population size-. The vertical axis represent the average growth rate in 100 experiments per each number in the horizontal axis. The distribution of the β’s is given by equation (13) and the value of φ is 1.2. As 2
For a model where stagnation endogenously disappear an the role of population size on it, see Galor and Weil (2000). Their model endogenize fertility decisions, but is less explicit on the link between population size and innovation. 3 This idea can be traced back to Phelps (1968):“If I could re-do the history of the world, halving population size each year from the beginning of time on some random basis, I would not do it for fear of losing Mozart in the process” [pages 511-512].
12
0.09 0.08 0.07
Growth Rate
0.06 0.05 0.04 0.03 0.02 0.01 0
0
2000
4000 6000 Population size
8000
10000
Figure 7: Growth rate vs. population size the figure shows, average growth increases with population. However, the growth rate is bounded by (φ − 1)/2, for the reason that β is always in [0, 1]. A similar result would be obtained if, instead of increasing the population size, we increase the variance of the β’s. Figure 8 show the average growth rate in 100 experiments against variance of xi,j , as defined in equation (13). The number of dynasties is constant and equal to 400. Although the average propensity to innovate remains constant, an increase on its variance is enough to increase the growth rate. The reason for this is that as the variance increases, the highest propensity to innovate increases as well. Moreover, if the variance of the propensity to innovate is low enough, there is no growth at all. It should be noted that a representative-agent perspective would not be able to reveal this relation, for the reason that it would only focus on the average propensity to innovate. Thus the model highlights the significance, for growth analysis, of considering higher moments in the distribution of population characteristics.
5
Conclusion
Technology diffusion is a force toward convergence of income across countries. Nevertheless, the model presented in this paper predict that, once enough frictions in 13
0.09 0.08 0.07 Growth Rate
0.06 0.05 0.04 0.03 0.02 0.01 0 −0.01
0
0.5
1
1.5 2 2.5 Variance of x(i,j)
3
3.5
4
Figure 8: Growth rate vs. Dispersion of the discount factor technology diffusion are accounted for, there is no need of convergence. Instead, the spatial distribution of income is characterized by clusters of development, agglomerations of individuals where income tends to be concentrated. At the core of these clusters, there are those individuals who have the highest propensity to innovate in the economy. These agents are the ones who lead technological progress and propel economic growth. Their propensity to innovate, and not the average propensity to innovate in the population, is the determinant of the long-run growth rate of the whole economy. However, with the exception of this innovative elite, the income of each agent is more linked to his proximity to the elite than to his own propensity to innovate. This last point highlights that innovators do not internalize the marginal benefit of their activity. Therefore, the equilibrium is not Pareto optimal. However, the enforcement of intellectual property rights would not be necessarily more efficient, because here exists the classical trade-off between incentives to innovation and diffusion possibilities. Those who adopt technologies generate externalities as well, because their adaption allow others to do the same. Thus it is necessarily to evaluate the growth enhancing possibilities of policies that subsidize not just innovation, but technology imitation as well. There are a series of possible extensions for the model. One of these could be relaxing the assumption of a fixed position on the grid. Actually agents would want 14
to reallocate themselves in the geographical space in order to take advantage of a better position relative to the innovative elite. This raises the possibility of trading positions along the grid. The price of each position would take into account the flow of rents generated by being relatively close to the elites. Would this market internalize all the innovation externalities and deliver efficient outcomes? Finally, the long-run growth path of the whole economy is defined by just one individual. The reason for this is that the model assumes that once an agent has the better technology, he has nothing to learn from other agents. In reality, innovations are frequently a result of mixing ideas from different sources. This suggests that a more reasonable assumption would be to consider convex combinations of ideas as a source of innovations, as Olsson (2000) tries to model. That could break the hyperdeterminacy on growth of just one individual but still allow for a significant role for a subset of elite individuals, as Mokyr (2005) suggests it should be.
Appendix: Imperfect Copying Model The following extended version of the model presented in section 2 introduces the possibility that just a fraction of the neighbors’ technologies could be copied without any cost. As a consequence, it could be the case that an agent find optimal to innovate based on his own technology rather than spending time adapting a more efficient technology. In other words, it is not longer necessarily true that every agent implements the most efficient technology available in his neighborhood. For agent (i, j, t), let zi,j,t be the most efficient technology used by his neighbors. If he can observe the technologies used by all the agents on the grid, then zi,j,t is given by zi,j,t = max{yk,l,t |k = 1, ..., i − 1, i + 1, ..., I , l = 1, ..., j − 1, j + 1, ..., J}
(15)
If instead he can observe just the technologies used by his immediate neighbors, then zi,j,t is given by zi,j,t = max{yi+1,j,t , yi−1,j,t , yi,j+1,t , yi,j−1,t }
(16)
It is necessary to differentiate between the time allocated to improving the own technology and the time allocated to improving/adapting a foreign technology (The difference between improving or adapting a foreign technology vanishes in this model). For agent (i, j, t), denote h • li,j,t as the time allocated to improving his own technology f as the time allocated to adapting/improving his neighbors’ best technology • li,j,t
15
It is again assumed that every agent has one unit of time for each period of life. Therefore, the budget constraints for the first and second periods of life are, respectively f h ci,j,t ≤ (1 − li,j,t − li,j,t )yi,j,t (17) di,j,t+1 ≤ yi,j,t+1
(18)
Let α ∈ [0, 1] be the fraction of the neighbors’ technology that can be imitated without any allocation of time to technology adoption. The analogue of equation 3 for the technology in the second period of life is f h yi,j,t+1 = max{(1 + φli,j,t )yi,j,t , (α + φli,j,t )zi,j,t }
(19)
If α = 1, then the model coincides with the one presented in section 2. As previously mentioned, the difference between adaptation and innovation based on a foreign f is greater or less than 1. technology vanishes away. It depends on whether α + φli,j,t The problem of each agent, omitting the (i, j) subindexes, is now: max
lth ,ltf ,ct ,dt+1 ,yt+1
log (ct ) + β log (dt+1 )
s.t.
(20)
ct ≤ (1 − lth − ltf )yt dt+1 ≤ yt+1 yt+1 = max{(1 + φlth )yt , (α + φltf )zt } 0 ≤ lth + ltf ≤ 1 0 ≤ lth 0 ≤ ltf Problem (20) can be solved in two steps. First, for a given yt+1 , we minimize the amount of time allocated to the innovation or adaptation of technologies: min lth ,ltf
lth + ltf
s.t.
(21)
max{(1 + φlth )yt , (α + φltf )zt } = yt+1 lth ≥ 0 ltf ≥ 0 Problem (21) is a typical cost minimization problem. Solving it we obtain the optimal levels for lth and ltf conditioned on yt+1 . Let lth (yt+1 , yt , zt ) and ltf (yt+1 , yt , zt ) be those solutions.
16
In the second and last step, we replace lth and ltf in (20) for lth (yt+1 , yt , zt ) and and solve for the optimal yt+1 , denoted by yt+1 (yt , zt ) subsequently:
ltf (yt+1 , yt , zt )
max
ct ,dt+1 ,yt+1
log (ct ) + β log (dt+1 )
s.t.
(22)
ct ≤ (1 − lth (yt+1 , yt , zt ) − ltf (yt+1 , yt , zt ))yt dt+1 ≤ yt+1 max{yt , αzt } ≤ yt+1 ≤ max{(1 + φ)yt , (α + φ)zt } There are three possible parameter configurations: zt < yt , yt < αzt and αzt ≤ yt ≤ zt . It is intuitively clear that if no neighbor has a better technology, it is not optimal to allocate some time adapting/improving a foreign technology. In other words, if zt < yt then ltf should be 0. Similarly, if the fraction of technology that can be implemented without any cost is better than the own technology, then it is not optimal to allocate some time improving the own technology. In other words, if yt < αzt then lth should be 0. However, whenever αzt ≤ yt ≤ zt , there is no straightforward answer. Figures 9, 10 and 11 present the graphic analysis of the cost minimization problem (21), for each of the three parameter configurations aforementioned. Next we solve problem 20 for each parameter configuration, using the two-step procedure and the figures aforementioned. Case 1: zt < yt Figure 9 highlights the solution to problem (21) with a circle, for the case where zt < yt . In this case it is optimal to allocate no time to adapt or improve a foreign technology. Then the solution for the time allocated to each activity, conditional on yt+1 , is: ´ ´ ³ ³ f h 1 yt+1 (23) (lt (yt+1 , yt , zt ), lt (yt+1 , yt , zt )) = φ yt − 1 , 0 Now replacing (23) on (22), we obtain the utility maximization problem when zt < yt . The problem of time-allocation conditional on yt+1 has been solved and just is left to know the optimal level for yt+1 : max
ct ,dt+1 ,yt+1
ct ≤
log (ct ) + β log (dt+1 )
µ
1 1− φ
µ
yt+1 −1 yt
¶¶
dt+1 ≤ yt+1 yt ≤ yt+1 ≤ (1 + φ)yt 17
s.t. yt
(24)
Figure 9: Cost minimization when zt < yt The solution to (24) is given by yt+1 (yt , zt ) =
½
β(1+φ) yt 1+β
yt
if φβ ≥ 1; if φβ < 1.
Replacing (25) in (23) yields the optimal allocation of time when zt < yt : ³ ´ f φβ−1 h (lt (yt , zt ), lt (yt , zt )) = max{0, φ(1+β) }, 0 .
(25)
(26)
Case 2: yt < αzt Figure 10 highlights the solution to problem (21) with a circle, for the case where yt < αzt . In this case it is optimal to allocate no time to improve the domestic technology. Then the solution for the time allocated to each activity, conditional on yt+1 , is: ³ ´´ ³ (lth (yt+1 , yt , zt ), ltf (yt+1 , yt , zt )) = 0, φ1 yt+1 − α (27) zt Now replacing (27) on (22), we obtain the utility maximization problem when yt < αzt . The problem of time-allocation conditional on yt+1 has been solved and just is left to know the optimal level for yt+1 : max
ct ,dt+1 ,yt+1
log (ct ) + β log (dt+1 ) 18
s.t.
(28)
Figure 10: Cost minimization when yt < αzt ct ≤
µ
1 1− φ
µ
yt+1 −α zt
¶¶
yt
dt+1 ≤ yt+1 yt ≤ yt+1 ≤ (α + φ)zt The solution to (28) is given by yt+1 (yt , zt ) =
½
β(α+φ) zt 1+β
zt
if φβ ≥ α; if φβ < α.
Replacing (29) in (27) yields the optimal allocation of time when yt < αzt : ³ ´ φβ−α (lth (yt , zt ), ltf (yt , zt )) = 0, max{0, φ(1+β) } .
(29)
(30)
Case 3: αzt ≤ yt ≤ zt Figure 11 highlights the two possible solution to problem (21) with a circle, for the case where αzt < yt < zt . In this case it is not necessarily suboptimal to allocate some time to improve the domestic technology, despite the fact that it is not the most efficient in the neighborhood. The solution depends on the desired level of yt+1 . If it is high enough, the best the agent can do is to allocate his time to improve the foreign technology, because yt < zt . However, if yt+1 is not quite high, it could 19
Figure 11: Cost minimization when αzt ≤ yt ≤ zt be better to allocate his time improving the domestic technology, because αzt < yt . Then the solution for the time allocated to each activity, conditional on yt+1 , is: ³ ³ ´ ´ 1 yt+1 − 1 , 0 if yt+1 ≤ ´´ ³ φ ³yt (lth (yt+1 , yt , zt ), ltf (yt+1 , yt , zt )) = 0, 1 yt+1 − α if yt+1 > φ zt
(1−α)zt yt ; zt −yt (1−α)zt yt . zt −yt
(31)
Now replacing (31) on (22), we obtain the utility maximization problem when αzt < yt < zt . The problem of time-allocation conditional on yt+1 has been solved and just is left to know the optimal level for yt+1 : max
ct ,dt+1 ,yt+1
³ 1− ³ ct = 1−
log (ct ) + β log (dt+1 )
³
yt+1 ³ yt 1 yt+1 φ zt 1 φ
−1
´´
−α
´´
yt
if yt+1 ≤
yt if yt+1 >
s.t.
(32)
(1−α)zt yt ; zt −yt (1−α)zt yt . zt −yt
dt+1 = yt+1
yt ≤ yt+1 ≤ max{(1 + φ)yt , (α + φ)zt } There are two possible solutions to yt+1 in problem (32): the one given by equation (25) and the one given by equation (29). Any possible solution must satisfy 20
the restriction yt ≤ yt+1 ≤ max{(1 + φ)yt , (α + φ)zt }. Whenever both (25) and (29) satisfy this restriction, the solution is given by the one that yields the highest utility. Summing up, the solution for (lth , ltf ) in problem (20) is either (26) or (30). We can compute the utility of these two possible solutions and then look for the one that yields the highest utility. Let U h and U f be the utility levels of (26) and (30), respectively. Given the agent has a instantaneous logarithmic utility, U h and U f are given by: ³ ³ ³³ ´ ´ ´ ´ φβ−1 φβ−1 Uth ≡ log 1 − max{0, φ(1+β) } yt + β log max{ 1 + φ max{0, φ(1+β) } yt , αzt } Utf ≡ log
³³ ´ ´ ´ ´ ³ ³ φβ−α φβ−α 1 − max{0, φ(1+β) } yt + β log max{yt , α + φ max{0, φ(1+β) } zt }
The solution for (20) is then given by: ³ ´ max{0, φβ−1 }, 0 if Uth = max{Uth , Utf }; φ(1+β) ³ ´ (lth (yt , zt ), ltf (yt , zt )) = 0, max{0, φβ−α } if Utf = max{Uth , Utf }. φ(1+β)
(33)
Introducing (33) in equation (19) and reintroducing the (i, j) subindexes, we obtain the dynamics for the technology of the (i, j) dynasty: ³ ´ f h h 1 + φ max{0, φβi,j −1 } yi,j,t if Ui,j,t = max{Ui,j,t , Ui,j,t }; φ(1+βi,j ) ´ ³ (34) yi,j,t+1 = f h α + φ max{0, φβi,j −α } zi,j,t if U f = max{Ui,j,t , U }. i,j,t i,j,t φ(1+βi,j )
Equation (34), the initial conditions and either equation (15) or equation (16) define the dynamics of the extended model.
References Bottazzi, L. and Peri, G. (2003). Innovation, demand and knowledge spillovers: Evidence from european patent data. European Economic Review, 47:687–710. Comin, D. and Hobijn, B. (2008). An exploration of technology diffusion. Harvard Business School Working Paper No. 08-093. Galor, O. and Weil, D. (2000). Population, technology and growth: From malthusian stagnation to demographic transition and beyond. American Economic Review, 90:806–829. Keller, W. (2002). Geographical localization of technology diffusion. American Economic Review, 92:120–142. Mokyr, J. (2005). Long-term economic growth and the history of technology. In Aghion, P. and Durlauf, S., editors, Handbook of Economic Growth. Elsevier. 21
Olsson, O. (2000). Knowledge as a set in idea space: An epistemoligical view. Journal of Economic Growth, 5:253–275. Phelps, E. (1968). Population increase. Canadian Journal of Economics, 1:497–518.
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