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2010 26 1.255 Innovation, diffusion and the distribution of income

2010 26 1.255 Innovation, diffusion and the distribution of income
2010 26 1.255 Innovation, diffusion and the distribution of income

J Evol Econ(2010)20:689–714

DOI10.1007/s00191-009-0170-8

REGULAR ARTICLE

Innovation,diffusion and the distribution of income

in a Malthusian economy

Mark Staley

Published online:20November2009

?Springer-Verlag2009

Abstract Between5000bce and1800,the population of the world grew120-fold despite constraints on the total amount of land available for production. This paper develops a model linking population growth to increasing pro-ductivity driven by random innovation and diffusion.People are endowed with a set of skills obtained from their parents or neighbours,but those skills are imperfectly applied during their lifetimes.The resulting variation in productivity leads to a distribution of income and to a process of diffusion whereby high-income activities spread at the expense of low-income activities. An analytic formula is derived for the steady-state distribution of income.The model predicts that the rate of growth of population approaches an asymptotic limit,whereupon there are no scale effects.The model also predicts that if the rate of diffusion of knowledge is increased,the growth rate will increase. Keywords Malthusian·Innovation·Diffusion·Selection

JEL Classi?cation C02·O15·O31·O33·O40·N00

1Introduction

For thousands of years prior to1800,average per-capita income was very stable and very low.According to Thomas Malthus,incomes were stagnant for so long because“the constant effort towards population,which is found even in M.Staley(B)

University of Ontario Institute of Technology,2000Simcoe Street North,

Oshawa,ON,Canada,L1H7K4

e-mail:staleymd@https://www.wendangku.net/doc/8310153555.html,

690M.Staley the most vicious societies,increases the number of people before the means of subsistence are increased”(Malthus1826).The two main assumptions of Malthus’model were that the rate of population growth was increasing in per-capita income,and that there were diminishing returns to labour because land was in?xed supply.He showed that under these two assumptions income would be mean reverting.Population would also be mean reverting unless there were improvements to skills or technology that allowed more people to subsist off the same amount of land.Hence according to Malthus,innovation led to an increase in population,but did not lead to any increase in income per capita.

Historical evidence supports a link between population growth,innovation, and a stagnant level of income per capita.Phillip Hoffman has constructed an index of total factor productivity(TFP)for agricultural land in the Paris Basin between1500and1800,showing a steady increase in TFP accompanied by a similar increase in the labour force over that period of time(Hoffman 1996,Table4.10).Galor(2005)has summarized evidence showing that tech-nologically superior regions had higher population but not higher income per capita in the pre-industrial epoch.And Ashraf and Galor(2008)have tested the Malthusian assumptions for the years1,1000and1500and have found that societies characterized by higher land productivity and an earlier onset of agriculture had higher population densities.

During the nineteenth century the western world escaped from the Malthusian trap.How this occurred is a question of fundamental importance not only for the history of the developed countries,but also for an under-standing of the barriers faced by less-developed countries today.Recently, a new approach to modeling this transition,called Uni?ed Growth Theory, has appeared in the economic growth literature(Galor and Weil2000;Galor and Moav2002).According to Uni?ed Growth Theory,the escape from the Malthusian tap was an inevitable by-product of the slow evolution of relation-ships between technology,population,and human capital.These relationships, which are only partially understood,are the focus of this paper.

We present a model of the pre-industrial economy in which random inno-vation,diffusion and selection give rise to a steady increase in knowledge, a steady increase in population,but nevertheless a stationary distribution of income.The model is designed to capture the following three“stylized facts”of the pre-industrial economy:

1.The Malthusian trap.Innovation leads to higher population but does not

lead to higher income per capita.

2.A very slow rate of population growth.Figure1shows world population at

around?ve million in the year5000bce(when agriculture was taking hold), increasing to600million on the eve of the industrial revolution(Kremer 1993).In the context of a Malthusian economy this120-fold increase in population represents an enormous amount of innovation,but in any given year the rate of improvement would have been glacial.A120-fold increase in population between5000bce and1800implies a growth rate of only

Innovation,diffusion and the distribution of income in a Malthusian economy691

-5,000-4,000-3,000-2,000-1,00001,0002,000

Year

Fig.1World population(millions),5000bce to2000,log scale.Source is Kremer(1993).The regression line is based on data between5000bce and1800.The slope of the regression line is 0.075%,which is slightly higher than the?gure of0.07%quoted in the text,which was based on the ratio of the endpoints

0.07%per annum,which in turn implies a very low rate of technological

advancement.

3.A lack of observed scale effects.An important characteristic of Fig.1is

that the rate of growth of population appears to be independent of the level of population between5000bce and1800.

The assumptions of the model are as follows:

?There are a large number of production units,each of which is small(e.g.

a family,or a tribe).The economy is competitive.

?Production functions are?rst-order homogeneous in land and labour, and productivity is a function of human capital(knowledge).There is no physical capital in the model.

?The supply of land is?xed,and is allocated to the units of productions in such a way that the marginal product of land is same for all units.The assumption of equal marginal product can be justi?ed in the context of

a society in which there is clear title to land and a competitive rental

market.In the context of a more“primitive”society the assumption can be justi?ed by considering the optimal level of military spending for each unit.

Equilibrium is obtained when there is a military stalemate,which occurs when the marginal products are equalized across all units.?Knowledge is transmitted through two mechanisms:vertical transmission,

i.e.from parent to child;and horizontal transmission,i.e.from neighbour

to neighbour.

?Learning is error-prone,which leads to variation in productivity across units.

?High-income units produce more children than low-income units.

692M.Staley Together,these assumptions lead to a model that is similar to Darwin’s theory of Natural Selection,but where the unit of selection is knowledge instead of genes.1In the case of vertical transmission,selection occurs through a combination of the Malthusian mechanism and competition for land.Higher skilled individuals enjoy a higher income and thus have more children than lower-skilled individuals.The higher-income individuals are motivated to increase their holdings of land(e.g.they bid up the rent)while the lower-income individuals are forced to reduce their holdings of land,which lowers their income and leads to a reduction in their numbers.Since children learn most of their skills from their parents,this mechanism of selection leads to the preferential spread of useful knowledge.Horizontal transmission also gives rise to a selection effect because old skills are dropped in favour of newer more pro?table skills.

There are two offsetting forces working to maintain a static distribution of income.On the one hand there is a tendency for highly pro?table skills to spread at the expense of less pro?table skills as described above.Given slow random innovation and selection one would expect the average income of the population to slowly rise.On the other hand,the Malthusian mechanism works to limit income per capita through population pressure and a?nite supply of land.Hence there is a tendency for income to revert to the mean. Continuous random innovation and population pressure eventually balance, leading to a stable distribution of income.The mean of the income distribution is just slightly higher than the subsistence wage,which translates into a slow but steady increase in population.

The key prediction of the model is that the rate of population growth is constant,i.e.there are no scale effects.This prediction is consistent with the pattern seen in Fig.1,but is not what one might at?rst expect from intuition.Kremer(1993)presents a proto-typical model of population growth showing how scale effects arise naturally when innovation is assumed local.In Kremer’s model,each person’s chance of inventing something is independent of others,so the aggregate rate of invention is proportional to population. When combined with the Malthusian mechanism linking population and in-come,Kremer’s model predicts that the rate of growth of population should be proportional to the level of population.In the present model,although individual(random)innovation is also assumed to be local and independent of others,the?nite rate of diffusion acts as a“speed limit”on the aggregate rate of innovation and hence eliminates scale effects.As the population grows, more people discover new skills that have already been discovered elsewhere 1From Darwin(1883):“...I saw,on reading Malthus on Population,that natural selection was the inevitable result of the rapid increase of all organic beings...”.The type of selection considered here has been variously termed cultural selection or behavioural selection,to distinguish it from genetic selection(Jablonka and Lamb2005).

Innovation,diffusion and the distribution of income in a Malthusian economy693 but have not yet diffused across society,i.e.they end up“re-inventing the wheel”.Hence a?nite rate of diffusion eliminates scale effects.

The prediction that the rate of growth of population is constant is not consis-tent with the Boserupian point of view(Boserup1965).The main assumption of Boserupian theory is that increases in population spur increases in the rate of innovation.In a Malthusian economy all technological progress is absorbed by population growth,so the rate of growth of population serves as a measure of the aggregate rate of invention.Our model predicts that the rate of growth of population is independent of the level of population,which implies that the aggregate rate of invention is independent of the level of population,in contradiction to Boserupian theory.2

Several recent papers have presented models of pre-industrial growth based on Malthusian assumptions(Kremer1993;Jones1999;Galor and Moav2002; Hansen and Prescott2002;Lucas2002).These papers are mainly concerned with the transition from Malthusian income stagnation to modern growth, while the present paper is concerned only with the Malthusian era.The role of selection in the diffusion of innovation was previously discussed by Galor and Moav(2002),by Clark and Hamilton(2006),and by Clark(2007).Whereas those authors explored the possibility that genetic selection may have driven increases in productivity,the present paper considers only what may be termed cultural selection,i.e.changes in knowledge and skills.Many of the results in this paper have been obtained using the tools of continuous-time stochastic calculus,originally applied to the study of economic growth by Bourguignon (1974)and Merton(1975).

The paper is organized as follows.Section2presents a model of population growth with random innovation and diffusion of knowledge(both horizontal and vertical).Section3presents an analytic formula for the distribution of income and shows how one may compute the population growth rate as a function of demographic and economic factors.Finally,Section4summarizes the predictions of the model,proposes some tests,and suggests an extension to the model that may have relevance for modern growth.

2The model

To set the stage we?rst present a simple macroeconomic model that captures the phenomenon of a steadily growing population and a constant level of income per capita.The detailed model to be presented in the following sections will be shown to be reducible to this simple macroeconomic model in the aggregate.

2Note that it is not Malthusian theory that is in contradiction with Boserupian theory.Kremer was able to successfully marry Boserupian theory with Malthusian theory by postulating that the rate of growth of technology is proportional to the size of population.

694M.Staley Let Y,X,L stand for output,?xed land,and labour respectively.Let A stand for labour ef?ciency(or human capital per person).The simple model consists of three equations:

Cobb–Douglas Production:

Y=Xα(AL)1?α,0<α<1,

Exogenous Innovation:

A=e gt,

Malthusian Dynamics:

·

L=B Y?δL

The symbol B in the last equation represents the number of new labourers that survive to adulthood per unit of economic output,andδrepresents the natural death rate of labourers.This last equation is analogous to the savings equation used by Solow in this1956model of industrial growth,but with labour substituted for capital(Solow1956).Following the technique used by Solow,a steady-state solution to these equations can be obtained:

Population Growth:

g L≡

·

L

L

=1?α

αg,

Income Per Capita:

y≡Y

L

=

δ+g L

B

Given the success of the above model in capturing the essence of the pre-industrial economy one might simply stop at this point.But the model as it stands is incomplete because it treats innovation as if it were some macroeconomic effect by which improvements in ef?ciency descend upon the entire population in a coordinated fashion.The goal of the following sub-sections is to present a detailed model incorporating individual innovation and diffusion of knowledge,but which reduces to the above model at the aggregate level.

2.1Production

The production function for each unit follows the Cobb–Douglas form:

Y i=Xαi(A i L i)1?α.(2.1) Here i labels a unit of production in which people have attained a certain level of knowledge A i,and Y i,X i,and L i stand for the levels of output,land and labour associated with that unit.We assume that the quality of land is homogeneous across all units.Note that there is no capital in this model(or

Innovation,diffusion and the distribution of income in a Malthusian economy 695equivalently,capital is tied to land or to labour in some ?xed proportion 3).A set of people may be a tribe,a manor in pre-industrial Europe,or just a family.We assume that the size of a unit is small in comparison with the whole population,so the economy is competitive.One may think of the quantity (A i L i )as representing the amount of effective labour,or human capital.Total output for the economy is simply Y = i

Y i .

Since income drives population in a Malthusian economy,our immediate goal is to derive an expression for per-capita income applicable to each unit.In order to do so we must ?rst determine how land is distributed across the various units of production.Two assumptions are suf?cient.First,we assume that the marginal product of land is the same across all units:

?Y i ?X i

=constant.Second,we assume that the total amount of land is ?xed (normalized to 1for convenience): i X i =1.

With these two assumptions one may show that land is distributed in propor-tion to human capital:

X i =

A i L i AL

,(2.2)where L is the total quantity of labour and A is the labour-weighted average productivity across all units.Per-capita income is then proportional to produc-tivity:4y i ≡Y i L i =A i (AL )α,(2.3)

and aggregate production follows Y =(AL )1?α,consistent with the macroeco-nomic model presented at the beginning of Section 2(for X =1).

The assumption of a constant marginal product of land can be justi?ed in the context of a society where there is clear title to land and a competitive rental market.In that case,the marginal product of land is equal to the rent,and rent is the same for everyone because the quality of land is assumed homogeneous across all units of production.

In a society without formal land ownership the distribution of land is more likely determined by military strength.But even then one can argue that 3The assumption that capital is tied to other factors seems reasonable for a pre-industrial economy.For example,draught animals were an important form of capital but required pasture for grazing so the potential for accumulation was limited.

4An interpretation of Eq.2.3is that each unit earns its average product of labour,i.e.y i = Y AL A i .

696M.Staley as long as people are rational,land will be distributed as described above. Consider the situation where there are two neighbouring units of production, one of which enjoys a high marginal product of land(labeled“H”),and the other of which has a low marginal product of land(labeled“L”).All units wish to expand their holdings of land because according to(2.1)that will allow them to expand output.It is pro?table for a given unit to invade its neighbour only if the military budget required to defend the acquired piece of land is less than its marginal product.It turns out that unit“H”can economically expand its territory at the expense of unit“L”if it spends an amount on defense that is intermediate between the marginal products of the two units.In that case unit “L”will?nd it uneconomical to match the military spending of unit“H”and will be forced to retreat.A stalemate will be obtained when land is divided between the two communities such that their marginal products are the same.

There remains the question of how much of income goes to fuel population growth.An extreme Ricardian view might be that only the labour share of income fuels population growth because the remainder of income(rent) is squandered by a small land-owning elite on luxury goods and military adventures.But luxury goods makers and soldiers presumably have children, so some of that rental income will support the“effort towards population”.It is not the aim of this paper to develop a full theory of land ownership,so instead we will simply assume that all income generated by a unit of production goes to support the raising of children in that unit.Equation2.3can then be used as the basis for a Malthusian model of population dynamics.

2.2Fertility

Following Hansen and Prescott(2002)we assume that the rate of growth of population in a production unit is a linear function of income:

·

L i

=B y i?δ.(2.4)

L i

Here y i is given by Eq.2.3and B andδare constants.Since the size of the labour force is proportional to total population,Eq.2.4also describes the rate of growth of labour.The?rst term in Eq.2.4then represents the rate of entry into the labour force,which is roughly equal to the number of children that survive to adulthood.Here adulthood means the ability to both work and reproduce.A natural interpretation of the?rst term in Eq.2.4is that wealthy parents produce more children than poor parents.5The second term in Eq.2.4 represents the natural death rate of labourers.

5In a paper entitled“Survival of the Richest”,Clark and Hamilton provide evidence based on parish records from pre-industrial England showing a positive correlation between the number of heirs listed in wills and the total assets of testators,the later presumably a good proxy for income (Clark and Hamilton2006).

Innovation,diffusion and the distribution of income in a Malthusian economy 697Equation 2.4can be derived by assuming a constant elasticity of parent’s marginal utility with respect to both net consumption c (after child-rearing expenses)and the number of children n .E.g.

U (c ,n )= cn φ 1?θ?11?θ,c =y ?kn .

Here k is the expenditure required to raise a single child.For a given level of family income y ,utility is maximized when n =B y ,B =φk (1+φ).If we assume that knowledge is passed down through the generations,then Eqs.2.3and 2.4together de?ne a system that exhibits the characteristics of Darwinian selection.Consider a hypothetical situation where there are two types of labourers,one representing the majority (labelled “L”),and the other representing a small minority having above-average skills (labelled “H”).Figure 2shows schematically what happens to these two populations over time.Since the population of type “H”individuals is small at ?rst,the equilibrium of the economy is initially dictated by the properties of type “L”individuals.The net income of type “L”individuals is just high enough to allow the population to remain stable (each couple produces on average two children that survive to reproduce).But the type “H”individuals enjoy a higher income and so are able to grow in number.The marginal product of land in type “H”communities is temporarily higher than that of the land controlled by the type “L”communities.Hence by the logic of the previous section the territory controlled by type “H”individuals grows at the expense of the territory controlled by type “L”communities until the marginal products of land are equalized.In the new equilibrium the net income of type “L”people is lower than before,so their numbers start to shrink (e.g.the rate of infant mortality goes up).In the meantime,the population labelled “H”continues to rise,which according to Eq.2.2triggers further expansion of territory.The process continues until type “H”individuals have taken over the economy.As expected,the higher level of population absorbs the higher income of the more

Fig.2Selection in a Malthusian economy.The population of individuals having superior skills eventually displaces the population of individuals having inferior skills,leading to a net increase in population.However income per capital always reverts to the subsistence level

I n c o m e P o p u l a t i o n

698M.Staley productive people,to the extent that disposable per-capita income once again reverts to its original subsistence level.

The mechanism of selection just described is similar to the evolutionary mechanisms presented by Nelson and Winter (1982)but with the roles of capital and labour reversed.In Nelson and Winter’s models,?rms grow by reinvesting capital while competing for ?nite supplies of other resources such as labour.To ?nd something even closer to the present model one must look to the theoretical population ecology literature.6Ecological models typically capture competitive dynamics by assuming the existence of a common limiting resource,and a population growth equation similar to Eq.2.4,but with a “crowding term”C (L ),e.g.

·L i L i

=βi C (L )?δ.C (L )is a decreasing function of the total population L ,and βi is a population speci?c birth parameter.We can relate Eq.2.4to the above ecological model by mapping A i →βi and B (AL )α→C (L ).In the model of population growth given by Eqs.2.3and 2.4,people are competing for a ?xed quantity of land and the most productive people can survive crowded conditions that are too onerous for other types of people.The main difference between the ecology models and our model of Malthusian dynamics is that in the former case fertility rates are determined by genetics,whereas in our case fertility rates are determined by skills,which are nevertheless passed to descendents as if they were something like genes.

2.3Horizontal diffusion

One of the key assumptions of the present model is that knowledge diffuses through a Malthusian economy at a ?nite rate.In the case of vertical diffusion,knowledge transfer is slow because it takes many generations for a new skill to spread through the population.In this section we propose a model for horizontal diffusion (the transfer of knowledge from neighbour to neighbour)that is also not instantaneous.

Consider two populations,labeled i and j ,and assume that the productivity of any person in population j is greater than that of a person in population i .That is,A j >A i .According to Eq.2.3this productivity difference manifests itself as a difference in income,i.e.y j >y i .One would expect that if a person in population i came into contact with a person in population j and was able to observe the superior techniques used by the person in population j ,then there would be a transfer of knowledge.This type of knowledge transfer can be captured using epidemic models,which generally assume that the rate of 6See Vandermeer and Goldberg (2003)Ch.1,for example.

Innovation,diffusion and the distribution of income in a Malthusian economy699 transfer between two populations is proportional to the product of the two populations(see Giroski2000for a review).The resulting dynamics gives rise to an S-shaped pattern of diffusion as seen by Griliches in his pioneering study of hybrid corn(Griliches1957).The Bass model of diffusion(Bass1969), widely used by marketers to forecast the spread of technology,is also based on this type of rule.

Another aspect of diffusion observed by Griliches,and also emphasized by Rogers(1995),is that the speed of diffusion appears to be proportional to the economic bene?t that is obtained by switching to the superior technique or technology.This aspect of diffusion can be captured by assuming that the speed of diffusion is an increasing function of the difference in income.

We now postulate the following dynamics for the horizontal diffusion of knowledge between two populations,labeled i and j:

·

L j=νL i L j

L

y j?y i

,(2.5)

·

L i=?

·

L j=νL j

L i

L

y i?y j

,(2.6)

where L is the total population.The?ow of knowledge is always from lower-

income activities to higher-income activities.The constantνcaptures the speed

of diffusion.Note the symmetry between Eqs.2.5and2.6:only one of these

equations is needed to specify the model.The intuition behind Eq.2.5is that

the rate of increase in the population with superior knowledge A j is propor-

tional to the number of potential learners(L i),and is also proportional to the percentage of labourers that have already attained that level of knowledge

(L j/L).This second factor represents the likelihood that a potential learner will be neighbours with a potential“teacher”,which captures the epidemic nature of diffusion.

In an economy with many different levels of productivity,the rate of dif-

fusion away or towards a given level of knowledge can be obtained by summing

the effects of diffusion over all relevant pairs of types.Hence to obtain the total

rate of change of L i,we can sum Eq.2.6over j to obtain

·

L i=νL i(y i?y),(2.7) where y is the labour-weighted average wage across the economy.

It is useful at this point to place the above model of knowledge diffusion

in the context of technology diffusion models.Giroski(2000)classi?es dif-

fusion models into four categories:epidemic models,probit models,density-

dependent population models,and information cascade models.Epidemic

models have already been discussed.Probit models postulate that units of

production(?rms in modern parlance)are heterogeneous in their ability to

adopt new technologies.For example,a?rm may adopt a new technology only

if the pro?t in doing so exceeds some threshold,sayπ*.Let’s say the dis-

tribution ofπ*across?rms is f(π*).The proportion of?rms adopting the

technology is then equal to the area under f(π*)whereπ*is less than the

700M.Staley increase in pro?t obtained by switching to the new model.Density dependent population models include the model of selection described in the previous https://www.wendangku.net/doc/8310153555.html,rmation cascade models describe the phenomena whereby “herd mentality”may cause ?rms to adopt a certain technology even when there are other more pro?table alternatives.

The model of diffusion described by Eqs.2.5,2.6and 2.7is a hybrid of epi-demic and probit models.The factor νL i L j /L in Eq.2.5captures the epidemic dynamics,while the dependence on y j ?y i captures the heterogeneity of capabilities across units of production.One can derive the y j ?y i factor from a probit model by assuming that each unit of production adopts a new technique only if the accompanying increase in income exceeds some threshold.If that threshold is uniformly distributed across units,then the proportion of units adopting the given technique will be linear in y j ?y i .

Note that Eq.2.7is similar to the equation for population dynamics pre-sented in the previous section (Eq.2.4).The correspondence can be seen if we map v →B and v y →δ(νy is a constant since y is constant in a Malthusian economy).Hence our model can also be viewed as a selection model.In epidemic models,selection acts on different variants of pathogens that are competing for hosts.If skills are something like pathogens,then these skills are “competing for people”and only the most communicable will survive,communicability in this case being related to differences in income.7

Finally,Eq.2.7can be combined with Eq.2.4to obtain the total rate of change of population of a given type:

·L i L i =By i ?δ+ν(y i ?y )(2.8)

This last equation combines the effects of vertical diffusion and horizontal diffusion.If one sums Eq.2.8over i one obtains the equation of Malthusian Dynamics ·

L =B Y ?δL ,Y =yL ,consistent with the macroeconomic model presented at the beginning of Section 2.

2.4Innovation

Our model of innovation is very simple:

dA i A i =σdz i .(2.9)Here σis a constant and dz i represents a draw from a standardized iid normal process:dz i ~N (0,dt).There is no direction to innovation,and productivity is as likely to decrease as it is to increase.To simplify matters we assume that 7Dawkins concept of a meme comes closest to capturing the idea of skills competing for people (Dawkins 1976).

Innovation,diffusion and the distribution of income in a Malthusian economy701 individuals innovate independently of one another,so the dz i are uncorrelated across i.One complication that we will need to address later is that the de?nition of a group may change over time.For example,a group may start out as a single“tribe”,but after the population expands,the tribe may split into two independent tribes each pursuing their own innovation according to Eq.2.9.

Equation2.9admits frequent small changes in productivity as well as rare large changes in productivity.An example of the later might be the Neolithic Revolution.It is conceivable that major improvements in cultivation took place in only a few locations,and that those improvements slowly diffused, vertically or horizontally,to the rest of the world.Equation2.9is consistent with such large local jumps in productivity.

2.5Summary of the model

Income:y i≡Y i

L i

=A iα,A=

j

L j

L

A j,L=

i

L i,(2.3)

Diffusion/Selection:

·

L i

L i

=B y i?δ+ν(y i?y),y≡

i

L i

L

y i,(2.8)

Innovation:dA i

A i

=σdz i.(2.9)

3Aggregate growth

3.1Simulation of the model

Before developing an analytic model of aggregate growth we present the results of a simulation exercise,designed to replicate the pattern seen in Fig.1. The purpose of the simulation is to highlight some key properties that will need to be captured in the analytical solution.Some of the parameters of the simulation have been chosen to be consistent with historical data;others have been chosen based on plausibility.Together,they are designed to produce a rate of growth of0.075%per year,consistent with the slope of the regression line shown in Fig.1.

First,we assume that there are a large number of“tribes”,each of which is restricted in size.Initially there are1,000tribes,each containing5,000people; hence there are5million people to start.Whenever a tribe grows beyond 10,000people,it splits into two.This step is necessary to prevent any single tribe from taking over the entire economy,contradicting the assumption that there are a large number of units of production.We are assuming that as

702M.Staley

387415443472500529557586

Income (1990 U.S. Dollars)

Fig.3The distribution of income.The bars show the empirical distribution based on Monte Carlo simulation,and the solid line is the best-?t normal (Gaussian)distribution

population expands in the pre-industrial world,the number of communities expands with it,instead of each community becoming larger.

Parameters Using data from Hansen and Prescott (2002),along with our estimate of growth g L =0.075%,we can infer that δmust be approximately 1/18in units of years ?1,which implies that in the pre-industrial era the average person could expect to live an additional 18years upon reaching adulthood.8Summing (2.8)over i we have

g L ≡·L L =B y ?δ,(3.1)and hence B =(g L +δ) y =0.05625 y .According to ?gures contained in

Maddison (2007),world GDP per capita prior to 1800was roughly US $500in 1990terms.Therefore B 0.05625 500=0.0001125.For simplicity let us assume initially that there is no horizontal diffusion,so v =0and all diffusion occurs vertically through a process of selection.The share of land in production,α,is set to 0.3.The one parameter left to be determined is σ.By a process of trial and error it was determined that σ=0.0022results in a rate of growth that is close to 0.075%.This value of σimplies that productivity ?uctuates with a standard deviation of 0.22%per annum,or just less than 1%over the average working life of an individual.

The main ?nding of the simulation exercise is that the distribution of income quickly adopts the form shown in Fig.3,even when starting from an arbitrary shape.The mean of the distribution ($500)is quite stable,which translates into a constant population growth rate and hence no scale effects.Note that the mean income of $500is close to the subsistence level of δ/B =$4938Annualising Eq.15in Hansen and Prescott,δ=2 1 35?g where g =0.075%,hence δ=0.0555.

Innovation,diffusion and the distribution of income in a Malthusian economy703 (substitute g L=0into Eq.3.1).The standard deviation of the distribution is approximately$38.Over99%of the population has an income that lies somewhere between$400and$600.The solid line in Fig.3represents a normal distribution with the same mean and standard deviation as the simulated distribution.It turns out that the normal distribution?ts the simulated results very well,although as we shall see in the next section the exact distribution is actually related to a Bessel function.

Although the average level of income is almost constant after a few hundred years of simulation,there is still a small residual dependence on population. Figure4shows that the dependence appears to be approximately of the form a?b/N,where a and b are constants and N is the number of tribes.As N→∞the average income(and hence the growth rate)becomes independent of population.

The observed dependence of the growth rate(and mean income)on the parameters B,δ,v andσis as follows:

1.When v=0,the population growth rate is independent of B.However,

average income is negatively correlated with B as expected in a Malthusian economy.When v>0,the population growth rate is negatively correlated with B.

2.If one increases the death rateδ,the level of population drops and the

average level of income increases,as expected in a Malthusian economy.

But interestingly,the population growth rate goes up.

3.The population growth rate and the average income are both positively

correlated with v.

4.The population growth rate and the average income are both positively

correlated withσ.

The last observation conforms to the expectation that in a“Darwinian”economy the rate of increase in mean productivity should be a positive function of the variance of productivity.The more variance there is,the more that

1/(Number of Tribes)

Fig.4Dependence of mean income on population.The x-axis is the reciprocal of the number of tribes(or units of production).The graph shows that the mean income slowly converges to a limiting value,in this case approximately$500,as the number of tribes approaches in?nity

704M.Staley selection has to operate upon.Some of the other results are surprising.For

example,the second bullet point says that if the death rate increases,the rate of

growth of population increases,and it is not at all obvious why this relationship

should hold.

By studying the workings of the simulation one can obtain a picture of how

diffusion and selection work in this economy and so gain some intuition around

the relationships reported above.The picture that emerges is as follows.

There are a large number of units,some of which are operating close to the

frontier of knowledge,while others are further behind.The more units there

are operating near the frontier,the more likely it is that one of them will

accidentally discover something that increases the overall productivity of the

economy.Furthermore,the faster the speed of diffusion,the more units will

be located near the frontier of knowledge.Therefore an increase in the speed

of diffusion should increase the growth rate.Regarding the second point,an

increase in the death rate leads to a higher rate of selection(i.e.the slope of the

curves in Fig.2are steeper),and therefore a higher rate of(vertical)diffusion.

Finally,we can gain some intuition around why there are no scale effects

in this model.One might expect that as the population increases,the rate of

discovery should go up because there are more units drawing independent

samples from the productivity distribution(dz i).However,not everyone is operating near the frontier of knowledge.Those that are lagging the frontier

are making discoveries just as fast as those that are ahead,but the laggards are

effectively“re-inventing the wheel”.As the economy expands this phenom-

enon becomes more and more common,counteracting the increased rate of

discovery.

3.2Analytic solution

We now wish to?nd an expression for the growth rate of population g L as a

function of the parametersσ,B,δ,v andσ.According to Eq.3.1,the overall

growth rate of population is a simple linear function of the average income y,

so we can direct our efforts towards?nding y.

In the summary of the model shown in Section2.5,we listed some differ-

ential equations for L i and A i;and we expressed y i as a function of A i.So it

should be possible to?nd a differential equation for y and?nd its?xed-point

solution.To carry out this procedure we make use of Ito’s lemma(Ito1951):9 Ito’s Lemma:

Given the process dx i=a(x i)dt+b(x i)dz i,and function f({x i},t),then

df=?f

?t dt+

i

?f

?x i

{a(x i)dt+b(x i)dz i}+1

2

ij

?2f

?x i?x j b(x i

)b

x j

ρij dt,

whereρij is the correlation between dz i and dz j.

9A non-rigorous derivation of Ito’s lemma can be found in Hull(2003).

Innovation,diffusion and the distribution of income in a Malthusian economy705 Since innovations are assumed to be independent,ρij is a matrix with ones down the diagonal and zeros everywhere else,i.e.ρij=δij.Hence

df=?f

?t dt+

i

?f

?x i

{a(x i)dt+b(x i)dz i}+1

2

i

?2f

?x2i

b(x i)2dt

From Eq.2.3we have

y=A1?α

L

.(3.2)

Applying Ito’s lemma to(3.2)using(2.3,2.8)and(2.9)(see Section2.5)we can obtain dy in several steps.First:

dA=A

(B+ν)

σ2y

y

dt+σ

H

1

N

dZ

(3.3)

where

σ2y=1

N

i

(y i?y)2is the variance of income,

H

1

N

=

i

L2

i

y2

i

i

L i y i

2is a Her?ndahl index(a measure of concentration),

dZ~N(0,1)is a standardized normal,

and N is the number of production units(e.g.tribes,manors).One can

readily see that the Her?ndahl index is of order1/N by scaling the number

of production units by a positive factor.Note that according to Eq.3.3,if σy and y are static then A grows exponentially,which is consistent with the macroeconomic model presented at the beginning of Section2.One of the

goals of the following analysis is to show that the distribution of y is indeed

asymptotically static.

From Ito’s lemma we then have:

dA1?α=(1?α)

dA?

1

2

α(1?α)σ2A1?αH

1

N

dt.(3.4)

Combining Eqs.3.3and3.4with dL=L(B y?δ)dt we?nally obtain:

dy=

(1?α)(B+ν)σ2y+αy(δ?B y)?α(1?α)

σ2

2

yH

1

N

dt

+(1?α)σy

H

1

N

dZ,(3.5)

The equation for y is mean reverting,with the point of attraction being a neg-ative linear function of H(1/N),which explains the pattern shown in Fig.4.In

706M.Staley the limit that N →∞,H vanishes,as does the stochastic term.The vanishing of the stochastic term is analogous to the elimination of unsystematic risk in a large diversi?ed portfolio of assets.We are then left with a deterministic portion only,which simpli?es to:dy dt =(1?α)(B +ν)σ2y +αy (δ?B y ).(3.6)

In deriving Eq.3.6we have not considered the issue of changing group structure,such as occurred in the simulation exercise where we continuously split old tribes into new tribes.However,it turns out that this complication is irrelevant for Eq.3.6because at any given time,quantities such as y and σy are invariant under splitting.

A stable ?xed-point solution can be obtained by setting both sides of Eq.3.6

equal to zero.10At the ?xed point,σ2y is related to y as follows:

σ2y =α1?αy (B y ?δ)(B +v).(3.7)

Using the parameter values listed in Section 3.1,we obtain σy =$37.80,which is close to the value of $38obtained in the simulation exercise.

In order to proceed further in deriving an expression for y in terms of the

parameters of our model,we need to ?nd another equation for σ2y .We can apply Ito’s lemma to derive an equation for d σ2y dt and try to ?nd its ?xed

point,but it turns out that the ?xed-point equation for σ2y then has a term

containing the third moment.Going further we could derive a whole set of recurrences relations for the higher moments,but that would only lead to an in?nite regress.Clearly we need to determine the entire density function for income.

The most direct approach to ?nding the density function is to ?rst derive it for each y i ,and then sum over i (weighting by L i /L ).We can sum the individual distributions because we are assuming no correlation between the various stochastic processes driving the changes in productivity.First,we apply Ito’s lemma to Eq.2.3using Eqs.2.8and 2.9to obtain,in the limit of an in?nite number of tribes:dy i =y i ?α(B +ν)σ2y ˉy

+α(δ?B y ) dt +y i σdz i .(3.8)This describes a simple process of geometric Brownian motion.

Next,to compute the density function for y i we use the Fokker–Planck Equation,also known as the Kolmogorov Forward Equation (Cox and Miller 1995):

10The ?xed point is stable by inspection of the last term in Eq.3.6.

Innovation,diffusion and the distribution of income in a Malthusian economy 707Fokker–Planck–Kolmogorov Equation:

Given the process dy i =a (y i )dt +b (y i )dz i ,the density function ρi (y i ,t )satis?es

?ρi (y i ,t )?t =???y i a (y i )ρi (y i ,t ) +12?2?y 2

i b (y i )2ρi (y i ,t ) ,Application of this equation to Eq.3.8leads to the following partial differential equation for ρi (y i ,t ):

?ρi = σ2+α(B +ν)σ2y y

?α(δ?B y ) ρi + 2σ2+α(B +ν)σ2y y ?α(δ?B y ) y ?ρi ?y +σ22y 2?2ρi ?y 2.(3.9)

Here we have dropped the index i on y i because the income scale is common across all units.Now de?ne f (y ,t )as the distribution of income across all units of production:

f (y ,t )≡ i

L i (t )L (t )ρi (y ,t ).We may now derive a partial differential equation for f using (3.9),along with (2.9)and (3.1):

?f ?t = (B +ν)(y ?y )+σ2+α(B +ν)σ2y y

?α(δ?B y ) f + 2σ2+α(B +ν)σ2y ˉy ?α(δ?B y ) y ?f ?y +σ22y 2?2f ?y 2

The term (B +ν)(y ?y )in the above expression captures the effect of knowl-edge diffusion,while the rest of the expression is identical to Eq.3.9.

A steady-state distribution of income is obtained when ?f /?t =0.Substitut-

ing for σ2y from Eq.3.7we obtain the following ordinary differential equation for f (y ):

0=y 2f +ayf +(b y +c )f (3.10)

708M.Staley where a =2σ2 2σ2?α1?α

(δ?B y ) b =22(B +ν)c =2σ2 ?(B +ν)y +σ2?α1?α

(δ?B y ) Following the suggestion of Polyanin and Zaitsev (2003,p.228),we make the substitutions z =2 b y and f (y )=z 1?a u (z ).

Equation 3.10then becomes

0=z 2d 2u dz 2+z du dz

+ z 2?γ2 u ,γ= (1?a )2?4c (3.11)

which is Bessel’s equation.It has the solution u (z )=C 1J γ(z )+C 2Y γ(z ),

where J γand Y γare γ-order Bessel functions of the ?rst and second kind respectively,and C 1,C 2are arbitrary constants.Hence

f (y )= 2 b y

1?a C 1J γ 2 b y +C 2Y γ 2 b y .In order to prevent f (0)from blowing up,C 2must be zero.To see why,expand J γ(z )near z =0(Abramowitz and Stegun 1972,p.360):J γ(z )= 12z γ∞ k =0

1k ! (γ+k +1) ?14z 2 k ,and Y γ(z )=J γ

(z )cos (γπ)?J ?γ(z )sin (γπ).When γ>0,lim z →0J γ(z )=0and lim z →0J ?γ(z )=∞,hence lim z →0

Y γ(z )=?∞.So in order for the function f (y )to be bounded at the origin,we must set

C 2=0.Therefore our solution is f (y )=C 2 b y 1?a J γ 2 b y ,(3.12)where C is a normalization constant.

Now that we have the functional form for f (y ),the ?nal step of our analysis to compute y as a function of the parameters of our model:α,B,δ,v and σ.Since the coef?cients of the function f (y )themselves contain y (see Eq.3.10),we must solve for y using the consistency relation: D yf (y )dy

D f (y )dy =y ,(3.13)

2020年高考政治哲学原理和方法论知识点整理(表格版)

第二单元探究世界与追求真理(唯物论) 标题原理方法论 世界的物质同一性原理世界的本质是物质,世界上先有物质后有意识,物质决定意识, 意识是客观存在在人脑中的反映 想问题、办事情时,要一切从实际出发,实事求是。理论联系实际,解放思 想,与时俱进,在实践中检验和发展真理.(有时候会考察课本“一切从实际 出发的内容”) 物质运动的辩证关系原理物质和运动不可分割。物质是运动的物质,运动是物质的根本属性和存在方 式, 世界上不存在脱离运动的物质;运动是物质的运动,物质是运动的承担者。 ①善于在运动中把握事物,不能静止地看问题,一切以时间、地点、条件为转 移。 ②既要反对离开物质谈运动的唯心主义观点(仁者心动);③又要反对离开运动 谈物质的形而上学观点(刻舟求剑); 运动和静止的关系原理①运动是绝对的、无条件的、永恒的②静止是相对的、有条件的、暂时的, ③物质世界是绝对运动和相对静止的统一。 ①既要承认事物的绝对运动,又要看到事物在运动中存在着相对静止,坚持绝对 运动和相对静止的统一②既要反对只承认静止而否认运动的形而上学的不变论, 又要反对只承认绝对运动而否认相对静止的相对主义和诡辩轮; 规律的客观性和普遍性原理①所谓规律,就是事物运动过程中固有的、本质的、必然的、稳定的联系; ②规律是客观的,是不以人的意志为转移的,它既不能被创造,也不能被消 灭; ③规律是普遍的,任何事物在其运动变化和发展中,都遵循其固有的规律。 ①必须遵循规律,按客观规律办事,而不能违背规律。否则就会受到规律的惩 罚。②在客观规律面前,人并不是无能为力的,人可以在认识和把握规律的基础 上根据规律发生作用的条件和形式利用规律,改造客观世界,造福于人类。 尊重客观规律和发挥主 观 能动性的辩证关系原理尊重客观规律是正确发挥主观能动性的前提和基础,发挥主观能动性是认识和 利用规律的必要条件。 想问题、办事情,既要尊重客观规律,按规律办事,又要充分发挥主观能动 性,把尊重客观规律和发挥主观能动性有机地结合起来。 意识的能动作用原理人能够能动地认识世界:①意识活动具有目的性和计划性② 主动创造性和自 觉选择性 ①发挥主观能动性, 自觉树立正确的思想意识,克服错误的思想意识②反对否认 意识能动作用的形而上学观点和片面夸大意识能动作用的唯心主义观点 人能够能动地改造世界:① 意识对改造客观世界具有指导作用(注意两重 性); ②意识对于人体生理活动具有调节和控制作用(高昂的精神、萎靡的精神) 物质和意识的辩证关系原理①物质决定意识 ②意识对物质具有能动作用:正确意识对事物发展起促进作用错误意识对事物 发展起着阻碍作用。 ①要坚持一切从实际出发,实事求是 ②要发挥主观能动性,能动地认识和改造世界,树立正确的思想意识,克服错误 思想意识 一切从实际出发:做是什么;为什么怎么做:①坚持客观规律②发挥主观能动性③把两者结合起来

马克思主义哲学的基本原理和方法论总结

马克思主义哲学的基本原理和方法论总结 一、唯物论(辩证唯物论) 1.世界的本原是物质(世界的物质性原理) (1)原理:世界的本质是物质,物质决定意识。 (2)方法论:这要求我们看问题办事情要一切从实际出发,实事求是。 反对从主观出发,反对“上帝创世说”。 2.自然界的物质性原理 (1)原理:自然界是物质的,它的产生、存在和发展是客观的。 (2)方法论: ①人们在利用自然、改造自然的同时,应当尊重、保护自然,学会与自然和谐相处。 ②承认自然界的客观性是我们正确处理人与自然关系的基本前提。 3.规律的普遍性和客观性 (1)原理: ①规律具有客观性和普遍性,它不以人的意志为转移,既不能被创造也不 能被消灭。一旦违背客观规律就会受到规律的惩罚。 ②规律是客观的、普遍的,但是人在客观规律面前并不是无能无力的,人 可以发挥主观能动性,认识和利用规律,改造客观世界为人类造福。(2)方法论: ①人必须遵循规律,而不能违背规律,按照客观规律办事; ②人还要发挥主观能动性,认识和利用规律,改造客观世界为人类造福; ③要把尊重客观规律性和发挥人的主观能动性结合起来。 4.物质和意识的辩证关系原理 (1)原理: ①世界是物质的,物质决定意识; ②意识具有能动性,对物质有反作用:正确的意识促进客观事物的发展, 错误的意识则阻碍客观事物的发展。 (2)方法论: ①我们必须坚持一切从实际出发,实事求是,使主观符合客观; ②充分发挥意识的能动作用,树立正确的意识,克服错误的意识。 5.人的主观能动性和客观规律性的辩证关系原理 (1)原理:

①规律是事物运动过程中固有的、本质的、必然的、稳定的联系。规律具 有客观性和普遍性,规律的存在与发生作用不以人的意志为转移,既不 能被创造也不能被消灭,是不可违抗的; ②人在客观规律面前不是无能为力的,人可以通过发挥主观能动性,认识 和利用客观规律,改造客观世界为人类造福。 ③尊重规律、按规律办事,离不开发挥人的主观能动性;发挥主观能动性, 也必须以尊重客观规律为基础。 (2)方法论: ①这就要求我们既要尊重规律,按客观规律办事,即一切从实际出发,实事求是; ②又要发挥主观能动性,认识和利用规律; ③把尊重客观规律和发挥主观能动性结合起来,坚持解放思想、实事求是, 做到主观和客观具体的历史的统一; ④我们既要反对片面夸大主观能动性的唯意志主义和唯心主义;又要反对 片面强调客观规律的机械唯物主义和片面强调客观条件,安于现状、因 循守旧、无所作为的宿命论思想。 二、马克思主义认识论 1.实践和认识的辩证关系原理 (1)原理: ①实践决定认识,实践是认识的基础。实践是认识的来源;实践是认识发 展的动力;实践是检验认识的真理性的唯一标准;实践是认识的目的和 归宿。 ②认识对实践具有反作用。正确的认识、真理和科学理论能指导人们有效 地开展实践活动,推动实践的发展;错误的认识会阻碍实践的发展。(2)方法论: ①坚持实践第一的观点,自觉参与实践活动; ②重视认识对实践的反作用,重视科学理论对实践的指导作用。 2.真理观 (1)真理是标志主观同客观相符合的哲学范畴,是人们对客观事物及其规律的正确反映。真理最基本的属性是客观性,真理还有条件性和具体性,真理和谬误往往相伴而行。这就要求我们热爱真理,坚持真理一元论,真理面前人人平等;同时,我们还要正确对待谬误。 (2)人类的认识具有反复性。因此,人类追求真理的过程并不是一帆风顺的。 人们对一个事物的正确认识往往要经过从实践到认识,再从认识到实践的多次反复才能完成。人类的认识需要不断深化。 (3)人类的认识具有无限性和上升性。从实践到认识、从认识到实践的循环是一种波浪式的前进或螺旋式的上升。因此,人类追求真理是一个永无止境的过程。

TRIZ创新方法尔雅满分答案

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