Dynamics of Cumulative Culture with
Microfoundation
Kensuke Ito*

1．Introductioni)

Theories on cumulative culture have been

methods . However, if we focus on thedeveloped mainly in the field of cultural

cumulative aspects, there are actually several

evolution which is an application of darwinian

similarities including the following two points.

evolutinary process to cultural phenomena. To

First, both deal with the macro-scale dynamics

find the reason why only human-beings appear

resulting from the accumulation of some micro-to achieve high cultural complexity, they havescale activities. Economics has also provided

not simply quoted the concepts of preceding

various microfounded dynamic models forstudies represented by Boyd and Richerson

capital accumulation, whereas static aspects(1985) but focused on several aspects which

are often emphasized by cultural evolutionists.

might be related to the accumulation of cultural

Second, both have a strong tendency oftraits. For example, Henrich(2004) presented

prediction or purpose-orientation. Although thethe contribution of population size totendency has been treated carefully in cultural

cumulative culture; Mesoudi(2011a) clearfied

evolution as“guided variation”, cumulative

the restriction caused by the acquistion cost ofculture is especially the field where it contains

accumulated knowledge; Lehman et al.(2013)

since cultural accumulation is almost peculiar

calculated the optimal strategy of time

to human-beings who seem to be more rational

allocation for learning schedules.

than other species. Therefore, we can say that

The aim of this study is to add a diﬀerent

it is worth applying the methodology ofdynamics by using the methodology of

economic growth theory to cumulative cultural

economic growth theory which values

evolution.

rationality and expectation of individuals. InThis paper is composed of five chapters

general, economics in the mainstream has been

including this introduction. Chapter 2 provides

considered to be diﬀerent from cultural

an explanation of the structure of our model.

evolution in terms of both concepts andChapter 3 deals with the derivation of its* A doctoral student of Graduate School of Interdisciplinary Information Studies in The University of Tokyo
Keywords：cultural evolution, cumulative culture, economic growth theory47steady-state with several interpretations. Then,

Chapter 5 summarizes implication andChapter 4 covers the confirmation of itsconclusion.

stability by means of phase diagrams. Finally,

2．ModelTo emphasize the role of rationality andthat academic papers are often evaluated byexpectation, our model is constructed of two

the number of citations. Therefore, the function

main components: cultural stock and individuals

intuitively means the process by whichseeking to maxmize their utility.

individuals make new culture by mixing

Cultural stock is the state variable which

existing culture and their eﬀort just likedenotes the amount of accumulated knowledge

researchers write new papers by referring toii)

in a society . Note that it is assumed to beprevious studies.

homogeneous for simplification; in other words,

It is important to note that we do not impose

the eﬀect of cultural diversity is eliminatedany constraints on α. In addition to ordinary

here. Individuals, on the other hand, have theincreasing function, for the model deals withrole of amplifying cultural stock in each period

culture, Y t can be a decreasing function withby using existing stock and their eﬀort as therespect to Kt if we assume the easiest culture

control variable. In addition, there is no-humanis likely to be made first and cultural

capital and no-uncertainty; that is, individuals

accumulation gradually lessens the room forcan precisely predict the amount of cumulative

future reproduction

culture even though they cannot memorize

α to be both positive and negative. This

what they have learned.

assumption allows for richer transitional

Specifically, their reproduction is accordingiii)

to the following Cobb-Dougras function ,iv)v). Therefore, we consider

dynamics which we shall confirm later.
Then, only the fraction of Y t is assumed to(1)where Y t is the reproduced culture at time t,

be evaluated and inherited to the future asvi)

ﬂow ,K t is cultural stock, and h t is the amount of(2)eﬀort allocated for reproduction. While it iswhere the amount of cultural flow iscalled capital share in economics, we here

represented by M t , and p is the exogenous

define α as the parameter of cultural quality:

variable for its probability, 0 < p < 1. Although

how existing culture can contribute toexogenous probability is a strong simplification,

reproduction. This is an analogy of the caseit does not aﬀect the main implications of the48

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model.

σ is the elasticity of intertemporal substitution

Finally, we can set the following diﬀerentialvii)

equation for the cultural stock ,and σ > 0, η is the acquisition cost per a unitof culture and η > 0. Since the curvature of u1(3)

increases as σ approaches zero, we can also

where δKt denotes depreciation and 0 < δ <interpret σ as the parameter of creativity: how

1. In the cumulative culture, obsolescence ofmuch incentive do individuals have for culturalprevious knowledge or physical depreciation ofreproduction

storage media would be practical examples.

viii)

.

That is all of the assumptions. Individuals inIn addition to the dynamics of state variable,

the model reproduce new culture by using

objective function needs to be defined forexisting cultural stock and their eﬀort for themicrofoundation. We assume it is composed ofutility stemming from its evaluation. However,two functions; u 1 : positive utility throughon the other hand, they have to decide theevaluation and u 2 : negative utility throughoptimal amount of eﬀort due to the acquisitionacquisition cost of cultural stock. If individuals

cost which increases proportionately to cultural

wish to maximize their utility over an infinite

stock. Particularly in a model with no-humanhorizon, therefore, objective function can be set

capital, dynamic optimization problem clearly

as follows:

appears since current evaluation leads to(4)increasing future acquisition cost. Thus, we cansay that their learning schedule is based on awhere ρ is time preference and ρ > 0.

preference for“evaluated smoothing”underSpecifically, let u1 and u2 be CRRA and linear,

the constraint of acquisition cost.

respectively.
(5)

3．Steady State
According to the above settings, the dynamic
optimization problem is given by

The Hamiltonian expression can be written as,ix),(7)where v t denotes costate variable associated
4with K. By solving the problem with(6)x)

substituting for (3) , we can obtain the Eulerxi)

equation .

Dynamics of Cumulative Culture with Microfoundation

49

(8)

contrast, we have to take account of thepossibility for the eﬀect of time preference toThen, the following equations are also derived

be less in reality. Impatience in the model

which determine the transitional dynamics of

certainly increases K* since it makes

44

the model by assuming K = 0 in (3) and M /M

individuals care relatively little about future= 0 in (8) and using (2),acquisition cost; however, this should be(9)

(10)
4

4

where (9) and (10) denote K = 0 and h = 0 loci,weakened if individuals can memorize andreuse what they have learned without incurring
additional costs. In other words, the eﬀect ishighly attributed to the simplication of themodel: no-human capital.

respectively. Finally, we get the steady-state

Third, δ works both positively andvalues of control and state variables from thenegatively on K*. Intuitively, even the highabove simultaneous equations.

depreciation rate directly decreases cultural(11)

Three explicit properties can be found in K*:xii)

the steady-state amount of cultural stock .

stock, it can also contribute to the increaseindirectly by reducing future acquisition costwhich results in stimulating reproduction. Thisis the reason why only the positive eﬀect is

First and most importantly, α and p are notinfluenced by σ. We can then confirm theincluded here. This means, in the steady-state,

following specific condition by calculating under

the amount of cultural stock would not change

which the positive eﬀect exceeds the negativeeven if we controlled its quality and theeﬀect.
(12)

probability of evaluation.
Second, for the other included parameters,

Therefore, contrary to our intuition, high* > 0, Kρ
* > 0, and Kη
* < 0 hold, respectively.
Kσdepreciation rate could increase K* if

It would be natural that high creativity and low

individuals were enough creative to satisfy theacquisition cost provide more cultural stock. In

conditionxiii).

4．Dynamics
Although we have dealt thus far with K* andconfirmed. As shown in (9) and (10), the form of

its properties, they are not meaningful until theloci varies depending on the exogenous

stability of each transitional dynamics isparameter. Specifically, it diﬀers according to50

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情報学研究

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the following range: α < －1, －1 < α < 0, 0 <surpasses positive ﬂow.
4. Optimal K* increases again by the decrease

α < 1, and 1 < α.

of acquisition cost, and back to 1.

Case1: 0 < α < 1
Therefore, on the assumption of diminishing
If we assume 0 < α < 1, that is, diminishingreturns, cultural stock eventually converges toreturns to existing cultural stock, transitional

the steady state while repeating boom anddynamics for the model takes the oscillation

recession.

path as depicted in

Case2: 1 < α
On the other hand, in the case of increasing
returns to existing cultural stock, our modeltakes the dynamics with saddle-path stability
shown in figure 2xiv). Hence, under this case,there is a slight possibility for the steady-statexv)

to be unstable .

Figure 1

Phase diagram in 0 < α < 1

Intuitively, this oscillation implements thefollowing iteration;

1. K* starts to accumulate by reproduction.
2. Optimal K* gradually decreases by theincrease of acquisition cost.
3. K* finally declines since negative flow

Figure 2

Phase diagram in 1 < α

Table 1 Growth paths and their stability
PathStability

α<1stable
stableDynamics of Cumulative Culture with Microfoundation－1 < α < 0stablestable0<α<1stablestable1<αsaddle
almost stable

51

Case3: −1 < α < 0

Case4: α < −1

Then, if we let cultural reproduction be aFinally, even if α is less than negative one,decreasing function of existing cultural stockstability is the same with Case 3 thoughand α be －1 < α < 0, transitional dynamics

equation (10) switches to convex as represented

follows the stable paths depicted in figure 3.

in figure 4.

Thus, the steady-state is stable regardless ofinitial values.

Figure 4

Phase diagram in α < −1

Table 1 summarizes all results derived
Figure 3

Phase diagram in −1 < α < 0above. Although each transitional dynamic
takes a diﬀerent path, they are all stable except
a certain situation in Case 2. Therefore, weconclude that the steady-state is almost stable.

5．Conclusion
As a result of the dynamic analysis, theHence, quality and evaluation are actuallyfollowing two main conclusions were obtained:

neutral to the amount of cultural stock which
finally accumulates. This implies, practically,• The steady-state value of cultural stock isthat indirect policies would be more eﬀective to

not aﬀected by its quality and evaluation.

cultural stock by supporting an environment• The steady-state is stable except a certain

where individuals can easily utilize well-case with increasing returns.

archived culture and thereby exert theircreativity, rather than direct policies which52

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情報学研究№91

interfere in the quality and evaluation of

more realistic by loosening the aformentioned

contents themselves. We could consider digitalstrong simplifications: homogeneous culture, no-archiving as an example of the former, andhuman capital and no-uncertainty, and empiricalawarding or certification system as that of thedata would also make our model morelatter.

persuasive by supporting the existence ofThis reseach predicts some elemental

concrete cultural traits suitable for proposed

dynamics of cumulative culture resulting fromdynamics. Despite those limitations, however,rationality and expectation and suggests someour model could work as a benchmark inessentially eﬀective factors to the amount oftackling more complex issues on cumulative

cultural stock in the long run. Further

culture since its structure and implications are

improvements can be considered in both

suﬃciently generalized and robust.

theoretical and empirical fields. Needless to say,theoretical extentions would make our model

Appendix
Derivation of the Euler Equation(A-6)

For optimization, the Hamiltonian mustsatisfy the following first-order conditions,

In terms of (A-2), the same rate can also be(A-1)

derived by dividing both sides by v andsubstituting (A-3), (A-4) and (A-5).
(A-7)

(A-2)
Derivatives associated with objective function

Finally, we obtain the Euler equation as (8)from (A-6) and (A-7).

can be both obtained from (3).
(A-3)

Optimal intellectual property rights
Our model has an additional implication for(A-4)
By using (A-3), (A-1) is simplied as follows.

intellectual property rights if we assume their
excludability increases both creativity and(A-5)

acquisition cost per a unit of culturexvi)－1. WeThen, taking logarithms and time derivatives ofshall find it convenient here to set σ(A-5) leads to the negative growth rate ofand θ decreases as the excludability getscostate variable.

strong (That is, curvature of u 1 approaches

Dynamics of Cumulative Culture with Microfoundationas θ53linear). Thus, if we focus on K*, it shifts to(A-10) is the condition for intellectual property(A-8)

rights to be eﬀective in the long run, whereθ ip/θ is the increasing rate of creativity andwhere Kip* is the steady-state amount of culturalηip/η is that of acquisition cost; each denotes

stock with intellectual property rights.

positive and negative contribution

Accordingly, K* < K ip* requires the following

Therefore, we can say that the rights arecondition.

useful only if positive per negative contribution
(A-9)

xvii)

.

is less than the composition of exogenous
parameters: ln

By rearranging and taking logarithms,
approximately we get,
(A-10)Notesi)

See Mesoudi(2011b), p.21. and pp.177-188., for example.

ii)

Specifically, this means the sum of all information stored in goods or individuals which corresponds to the term “genotype” inbiology. Because of the diﬃculty of its quantification, however, most empirical studies analyze cultural “phenotype” which is
the observable characteristics as a result of background information such as shape, color and motion.

iii)

Linearity in ht is just for the simplification. We can derive the same main conclusions by using more generalized forms including
Cobb-Douglas and even CES production function.

iv)

Romer(2011), pp.103-104. assumes the same condition in the field of R&D.

v)

Hence, ￨α￨ would be more accurate for the parameter of quality rather thanα.

vi)Csikszentmihalyi(2014) proposes similar framework form a viewpoint of creativity named systems model.

vii)

Equation (3) has non-constant solutions for any α ≠ 0.

viii)

Concretely, u1 becomes logarithmic when σ equals one and approaches linear as σ increases.

ix)

The transversality condition is not required here since individuals get utility from the evaluation which corresponds not toconsumption but to savings in macroeconomic models.

x)

See in Appendix.

xi)

Note that this equation does not explicitly indicate individuals’ dynamic decision-making because, unlike general macroeconomic models, they obtain utility not directly from its control variable but indirectly from evaluated culture.

xii)

Notice that, even while the steady-state, contents still continue to change because K* just denotes the equivalence between Mtand δKt.

xiii)

Since the right-hand side of the condition must be greater than one, Kδ* < 0 holds if we assume u1 to be logarithmic.

xiv)

We can verify the slope of each loci in the neighbor of the steady-state by log-linearization. If we set ht = (ht － h* )/h* and Kt =^^^^^^(Kt － K* )/K*, equation (9) and (10) can be linearized as ht = (1 － α)Kt and ht = －αKt , respectively. Thus, (10) has the steeper
slope in the case 1 < α.
xv)

In addition, we shall find that unstable regions are getting smaller as α increases by the ratio between the slopes of bothlinearized loci: limα→∞ (1 － α)/－α = 154

東京大学大学院情報学環紀要

情報学研究№91xvi)

Note that this analysis is just from the viewpoint of social planner. The one who gets more incentive and the one who owes
additional acquisition cost would be diﬀrent in reality.

xvii)

If we adhere to using σ, (A-9) becomes< ln.

References
Boyd, R., and P. J. Richerson. (1985). "Culture and the evolutionary process," Chicago: University of Chicago Press. pp.147-151,
Manchester, U.K., Aug. 1988.
Csikszentmihalyi, M. (2014). "The systems model of creativity and its applications. In D. K. Simonton (Ed.)," The Wiley Handbook ofGenius (no.17525-17862). Wiley-Blackwell. (Kindle edition).
Henrich, J. (2004). "Demography and cultural evolution: How adaptive cultural processes can produce maladaptive losses―The Tasmanian case." American Antiquity, 69(2), 197214.
Lehmann, L., Wakano, J. Y., Aoki, K. (2013). "On optimal learning schedules and the marginal value of cumulative cultural evolution."
Evolution. 2013 May; 67(5):1435-45. doi: 10.1111/evo.12040. Epub 2013 Jan 23.
Mesoudi, A. (2011a). "Variable cultural acquisition costs constrain cumulative cultural evolution." PLos One, 6, e18239. doi:10.1371/
journal.-pone.0018239.
Mesoudi, A. (2011b). "Cultural evolution: how darwinian theory can explain human culture and synthe-size the social sciences," Chicagoand London: University of Chicago Press.
Romer, D. (2011). "Advanced Macroeconomics," 4th ed. New York: McGraw-Hill Education.

伊東謙介（いとう・けんすけ）［生年月］1991/12/05［出身大学または最終学歴］早稲田大学 ( 学士 ), 東京大学 ( 修士 )［専攻領域］文化進化理論 , 経済成長理論［所属］東京大学大学院学際情報学府社会情報学コース

Dynamics of Cumulative Culture with Microfoundation

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