Economic Growth and Demography

Yuri Yegorov, University of Vienna∗

This draft: 7 July 2011

Abstract

The goal of this article is to discuss the interaction between Rus-sian demographic problems and its specialization on the extractionof natural resources. We have several simultaneous processes since1990s: specialization in extraction of natural resources and demo-graphic problem caused by low fertility in 1990s. Russia has no la-bor scarcity at present, but will face it in 10 years. Also, its provenoil resources are only for 20 years, and this calls for a necessity ofeconomic diversification. While resource-extraction technology is lesslabor intensive, movement to technological development will requiremore labor, and this labor should be skilled. Thus, Russia faces aproblem of optimal transition from resource extraction to technologi-cal development with simultaneous labor training in the environmentof its growing scarcity. Capital from resource export can be used forboth investment in new technologies and demographic recovery. Fer-tility is also endogenous here, depending on consumption level. Thepolicy implications are very important. While there is little reasonfor boosting fertility initially (since extraction sector is not labor in-tensive), demography has high inertia, and it will be too late after,especially when oil resources will be close to depletion. The formalmodelling is done is the framework of two sector growth model. Asequence of models of dynamic growth, starting from more simpletowards more complex, is suggested.

∗E-mail:yury.egorov@univie.ac.at. The paper was prepared for presentation at DEGITconference, St.Petersburg, Russia, September 2011

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1 Introduction

The goal of this article is to discuss the interaction between Russian demo-graphic problems and its specialization on the extraction of natural resources.

At present, Russia still have enough workers and the share of retired peo-ple is relatively small (comparing to EU countries). However, Russia can facea serious demographic problem in the next 10 years. It comes from severefertility decline in 1990s, formation a gap in its demographic pyramid. Withthe ageing of older cohort (baby-boomers born in 1950-60s) there will be atremendous increase in the ratio of retired to working. According to the workof VID/IIASA 2008 (www.populationeurope.org), in 2006 Russia has fertil-ity rate of only 1.29 children per woman, and even the adjusted rate(thattakes into account postponement of birth) gives TFR of 1.52, well below thereproduction rate. Given this trend, Russian population will shrink from142.2 mln. in 2007 to 124.2 mln. in 2030. Moreover, old age dependencyrate (the ratio of sizes of the group 65+ to the group 15-64) will grow from19.7% to28.3%. The recent program to support mothers may improve thissituation only partially.

After transition of early 1990-s, Russian specialization in extraction ofnatural resources increased. This has both short and long term consequences.In the short term, this leads to Dutch disease [cite BOFIT] and higher ex-posure to world financial crisis of 2008. In the long run we should take intoaccount finiteness of oil reserves. In the case of Russia, extraction goes fasterthan for Arab countries. While Saudi Arabia has proven reserves for about100 years, Russia has only for 20, while world’s average in close to 40. Thismeans that Russia will be forced to change its profile of specialization on oilextraction and export faster than most of OPEC countries.

20 years ago, before dissolution of the USSR, Russia was a countrywith many specializations: industry, agriculture and extraction of naturalresources were giving employment (there was no unemployment) and con-tributed to GDP. With the transition to market economy most of industrybecame bankrupt (also due to suboptimality of transition path) and onlyextraction of natural resources (mostly oil, natural gas and metals) repre-sent almost all export and contribute to the core production of GDP today.The ratio of capital and workforce differs across sectors in the world and in

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Russia. Typically industry has the highest capital endowment per worker(K/L), and in services and agriculture it is much lower.1 The ratio of K/Lis relatively high for the sector of natural resource extraction, especially dueto necessity to build expensive infrastructure (like gas pipelines) to accessthe market and due to use of expensive technologies in regions with severeclimate. At the same time, the required labour force is relatively small andcannot give full employment. Since dissolution of the USSR, the size of bu-reaucracy and service sector grew a lot, but still could not not employ allworking force have previous job in industry and agriculture. It might betrue that centrally recovered industry (by protectionist policy) might pro-duce low added value, but it could reconstruct employment. Without thisstep, income redistribution policy is of crucial importance, and Russia has towatch OPEC countries. Privatization of resource industry made income po-larization very high, and this has not only social, but also demographic effect.

Another problem is that oil and gas resources are finite. The knownreserves of oil in Russia are only for 20 years, although gas resources arefor much longer period. Hence, diversification of economy should become apriority in the next 10-20 years, and this needs not only investment in newcapital and technology, but also preparation of skilled labour force. In thecase of Russia, the last task has two components: a) to have enough labourin 20 years, b) to have skilled labour. While the second problem is relatedto investment in human capital and is linked to revival of highly depreciatededucation, the first problem is purely demographic.

Now Russia has no labour scarcity (and even some unemployment) be-cause of two reasons. First, resource-extraction technology is less labourintensive that more industrial technology of the former USSR. Second, thelargest cohorts, born in 1950-60s and in 1980s are still in labour force. Inthe next 10 years, Russia will face massive retirement of large cohort ofbaby-boomers and the entry of generation born in 1990s into labour force.However, due to tremendous fall in fertility in 1990s (partially as a conse-quence of reforms, high income polarization and inability of most of familiesto afford even two children) the replacement will be incomplete (cohort enter-ing labour force is about half size of one exiting it) and will cause substantial

1In fact, land as production factor is quite important in agriculture, but Russia so farhas no objective valuation of land there.

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rise of the ratio of retired to labour force. This might cause some labourshortage, but the effect will not be fatal for a country specialized on activi-ties like oil extraction, not labour intensive. However, if Russia would startits economic diversification, it will need more population than it has.

Kuznets has discovered the inverse-U relationship between income dis-parity and economic growth. Similar relationship might exist between con-sumption and population growth rate: while poor countries (and people)have higher mortality rate, the cohort of rich normally has lower fertilitysince females find it optimal to have fewer children. Applying these ideasto Russia, that currently has quite unequal distribution of income (with theratio of income in top-10 per cent to bottom-10 per cent at 15), it is possi-ble to suggests that high income disparity hampers both fertility and growth.

When we look at the dynamics of fertility rate [5] 2 in Russia and otherCIS countries, we can observe several phenomena. First, there is a substantialdifference between urban and rural TFR. In 1960-65 rural TFR was close to3 and urban close to 1.8, and in 1981-86 they stayed almost at the same level(2.9 and 1.8). The overall drop of fertility rate in Russia in that period wassmall (from 2.2 to 2.05) and mostly explained by continuing urbanization.However, after the transition, even rural fertility dropped to 1.6 in 1997, withurban TFR only at 1.1. Fig. 1 shows the dynamics of TFR between 1990and 2005 for 6 CIS countries. While EU also has demographic transition,it is not so sharp as in Russia (i.e. process has started earlier and drop peryear in recent 2 decades is not so pronounced).

In the first 10 years of transition the drop of GDP per capita was sub-stantial, but it recovered in the first decade of the 21st century. At the sametime, we observe growing income polarization. The index of average incomeof top-10 to bottom-10 per cent in income distribution in the USSR was rel-atively small (about 3.5, see Atkinson and Mickelwright [1]), well below thelevel in USA (6) and Latin America (10 or above). However, during the last 2decades it grew to about 15 and stays at this level without significant change.

There exists some literature about the link between fertility and devel-

2We consider total fertility rate (TFR) as an average number of children per womanduring her lifetime.

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opment. Easterlin (see [6]) in 1969 came with the hypothesis about positiverelationship between those variables. Now we observe negative relationshipbetween fertility and human development index. If we will try to put to-gether both effects, we can arrive to inverse-U shape. And if we take intoaccount the effect that high fertility in poor countries have the origin in ir-rational behaviour (from economic perspective) driven by religion in muslimcountries), then we can arrive to inverse-U relationship between fertility andincome at least for European group of CIS countries. Income polarizationcan only reinforce this effect. Suppose that for high income equality we stayat fertility above reproduction level. If we raise inequality, then both poorand rich group will produce less children, but for different reasons (problemof survival and different preferences, especially for rich women). Thus, wecan expect fertility drop in Russia to be potentially explained by high incomepolarization starting 1992.

Finally, Russia has the largest territory in the world, which has rela-tively small population. The population density in eastern part of Russiais comparable one in rural Canada and Australia, and much lower than inEU countries. Yegorov (2009) has studied the optimal population densityfrom the perspectives of economic growth. If population density is too high(like in India), there are few land and natural resources per capita, and thishampers growth. But in another limit of underpopulated economy too highresource endowment per capita cannot generate growth, because of lack ofhuman resources to build infrastructure (roads, etc) to bring these resourcesto the market. If Russia would put more attention to exploitation of re-newable resources (like forestry and agriculture), it will face the problem ofroads, originated in this low density of population. There is no other way todeal with this problem except for increase in population. Thus, in all scenar-ios of transition of Russian specialization from extraction of non-renewableresources (that will eventually end) towards either extraction of renewableresources, or to industrialization, or towards services or hi-tech, it will needmore population.

This article presents formal models of dynamic growth of new sector thatwill bring economic diversification as well as associated demographic policiesto make this transition smooth.

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2 Models in the Basis

The present model is grounded on two streams of literature: dynamic modelsof economic growth and interplay between economics and demography. Heretwo basic models from both steams will be briefly described.

2.1 One Sector Model

We start from simple and move to more complex models. Here a well knownmodel based on one input and AK technology is presented.

Assume that oil resources are fully depleted. Then we have an environ-ment of one-sector growth model. The higher is initial capital stock, thebetter will be performance and obtained utility. Formally we can deal with amodel of A-K type. In a standard set up we have the following optimizationproblem:

maxI(t)

∫e−rtU(C)dt (1)

s.t. C = AK − f(I), K̇ = I(t)− δK.

As we know, for U(C) = C and linear cost of investment (f(I) = I),we indeed have unbounded growth. However, if we make investment costslinear-quadratic, f(I) = cI + dI2, we will get finite steady state. Formally,we consider the following optimization problem

maxI(t)

∫e−rt(AK − cI − d

2I2dt (2)

s.t. K̇ = I(t)− δK.

We have the Hamiltonian:

H = AK − cI − d

2I2 + λ(I − δK). (3)

The first order conditions give 2 differential and 1 algebraic equations:

I =λ− c

d, (4)

λ̇ = λ(r + δ)− A,

K̇ = I − δK. (5)

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There exists a unique steady state:

λ∗ =A

r + δ, K∗ =

1

δd(

A

r + δ− c). (6)

It is also easy to find comparative statics:

∂K/∂A > 0, ∂K/∂r < 0, ∂K/∂δ < 0, ∂K/∂c < 0, ∂K/∂d < 0.

2.2 Economic Influence of Population Density

Yegorov (2009) suggested a model capturing the influence of population den-sity on growth. The basic idea is interplay of two factors: natural endowmentof resources per capita and cost of building infrastructure. The technology isCobb-Douglas in capital and land, and consists of many small spatially dis-tributed firms, managed by one farmer, or worker. There also exists state firmthat collects all output and ten exports it. The major choice is the selectionof spatial grid covering the country. It has a form of vertical and horizontalroads that access all spatial points (production takes place everywhere). Themore dense is road network, the higher is fixed cost of its construction. Onthe other hand, it decreased the average distance of transportation withoutroad. If population density is low, there is high land endowment per capita(which is a proxy for natural resource endowment). At the same time, thecost per capita of building road network of fixed density is also higher.3

Let T denotes unit distance transport cost without road, B is spatialdensity of production, δ is depreciation rate of infrastructure, and f is thefixed cost of constructing unit distance of road. Then it was shows that theoptimal density of road network is given by ε = (δf/BT )1/2, where ε is theside of elementary square, formed by road network.

If A is the spatial size of a country (assumed to be a square with side A),ρ is population density (spread uniformly; model is rural rather than urban),p is world price of output, c is local production cost before transportation,t is unit distance transport cost along road and k is capital endowment perworker, then the profit of export-transport firm is given by the following

3In fact, roads were always a problem of Russia, and here mathematical justification ispresented.

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expression:

Π = kaρ−b(p− c− At/2)− ka/2ρ−(1+b)/2(δfT )1/2, (7)

where a, b are parameters of Cobb-Douglas production function. It is shownthat there exists a neighbourhood of zero population density, where profitsare negative. They reach maximal value for some optimal population density:

ρ2 = (C2(1 + b)/bC1)2/(1−b), C1 ≡ (p− c− At/2)ka, C2 = (δft)1/2ka/2. (8)

3 Adding Demography

Demography can be added either using explicit cohort structure (and corre-sponding delays) or by simple accounting of consumption (or even maternalpremium) on fertility. If we use Leontieff production function instead of A-K,then imbalance between capital and labour has an adverse impact on growth.There exists optimal fertility policy associated with optimal growth of capi-tal stock, that brings no unemployment and at the same time does not havelabour scarcity.

Formally we can assume production function of Leontieff type:

Y = min[K, L]. (9)

Then any deviation from disproportional growth of capital and population(which is a proxy for labour force) will result in slower growth.

It is also possible to consider some power of this function. If the powerexceeds one, then we have scale economies. However, such function with sumof powers exceeding one, is not very realistic. It may be better to suggestthe following production function:

Y = AK1−b min[Kb, L], b ∈ (0, 1), (10)

Here we get several effects. First of all, for the evolution of L(t) = Kb(t)we get a simple A-K production function. Second, simultaneous growth ofcapital and population leads to higher capital endowment per worker, andthus to higher production and consumption. Third, we get scale economies,and one person endowed with the same capital becomes more productive, if

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total population grows.

While this can sound counter-intuitive to the world in general, this maybe true for Russia and other countries with low population density. Yegorov(2009) have shown that for production technologies using resources dispersedin space (agriculture, forestry, mining) there exists population density opti-mal for economic growth. The basic idea is the balance between land en-dowment per capita ( proxy for natural resources) the cost of building infras-tructure. If initial population density is too small, then its growth is a scalefactor to overall growth.

3.1 Factors Influencing Population Growth

It is well known that population growth is determined by fertility and mortal-ity rates. Over short time horizons each of these factors matter. However, fora sustainable reproduction of population (given low infant mortality, achievedfor more developed economies) we need total fertility rate of about 2.1 chil-dren per average woman. How this factor has evolved in Russia and otherCIS countries over the last two decades? Fig. 1 shows that in 1990 TFRwas 1.9-2 (only slightly below sustainable level) for Russia and 5 other Euro-pean CIS republics4, while in 2000 it has dropped to 1.1-1.4 and was in therange 1.2-1.5 in 2005. Many EU countries also have TFR below 2, but inmost of EU countries adjusted TFR is in the range 1.5-2. Moreover, in somecountries, like France, Norway, Denmark and Ireland, TFR slightly exceeds 2in 2004-05.5 Also, it decline was slow and not so dramatic as in CIS countries.

There were many changes in transition economies. The GDP per capitahas declined in CIS countries after the reforms of 1992, but after 2000 it grewand has reached quite high levels, especially for Russia. Another importantfactor is income disparity, that also grew, but did not change much after-wards. It can be measured by Gini coefficient, that measures the integralbelow income distribution curve. Gini varies between 0 (full equality) and 1(maximal inequality). In 1992 Gini in Russia was 0.26, but grew to 0.48 in1998. Its change was very fast: 0.26 between 1980 and 1992, 0.35 in 1993,0.45 in 1994 (Yudaeva, 2002, Fig.5). Its change in Ukraine was less (from

4We want to eliminate Asian republics that have completely different fertility pattern.It is often based on muslim traditions, with less rationality and lower influence of income.

5Source: www.populationeurope.org, VID/IIASA.

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0.25 to 0.31), and even Belarus it changed from 0.24 to only 0.28 (Yudaeva,2002, Table 4). However, Ukraine and Belarus has lower GDP per capitathan Russia, and that is why real income per capita in the population groupthat excludes rich does not differ much among those countries.

If we look only at European CIS countries, we can see the correlationbetween decline in fertility rate and growth of income inequality. However,we cannot observe whether a decline in Gini will raise TFR in those countries.At the global scale, effect is not so obvious, since we have different cultures.For example, Brazil has Gini between 0.5 and 0.6 for the last 50 years, andstill has positive population growth. However, most of EU countries havemore income equality today, than Russia. In our model below we will employthe assumption about negative influence of Gini coefficient on fertility ratefor CIS countries.

3.2 Endogenous Population Growth

Further we will not make distinction between population and labour force,assuming them proportional to each other. Consider the following equation

L̇ = εα(C, G)L, (11)

where α = α(C, G) has inverse-U shape in C (positive population growth fora subset of middle consumption level). It also depends on Gini coefficient G(G ∈ [0, 1]: dα/dG < 0. This can be justified by considering separate fertilitydecision in different income groups. Deviation from the mean gives lower fer-tility, since both rich and poor have such behaviour, but for different reasons.6

In order to capture these stylized facts, we introduce the following func-tional form for α:

α(C,G) = C(2− C)−G. (12)

It has the following mathematical properties:

α(0) = −G, maxC

(α(C; G)) = 1−G for C = 1,

α(C) = 0, for C1,2 = 1±√

1−G. (13)

The coefficient ε is a positive small number and is introduced for calibrationpurposes.

6This may be not applied for muslim countries, but is definitely applicable for Russia.

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4 Modelling Interaction between Economics

and Demography

Classical economic growth models (see, for example, Barro and Sala-i-Martin,1995) consider constant population. In some cases population growth is givenby exogenous function. There are two methodological problems for incorpo-rating endogenous population change: a) no clear relationship between eco-nomics and demography, b) the problem of objective function. The first issuewas discussed above. As for the objective function, it is possible to have twoformalizations that will be exposed below.

4.1 General Problem Formulation

Consider first the following dynamic optimization problem:

maxI(t)

∫e−rtC(t)dt (14)

s.t. K̇ = I(t)− δK,

L̇ = α(C; G) (15)

C =Y (K,L)− I

L,

with Y (K, L) denoting production function. Here C denotes averageconsumption level per capita. Here we have two independent states, K andL, and one control I. There is a function of states and control, C, thatdetermines the evolution of L. Control K determines the evolution of K.For simplicity, labour force coincides with the population.Here the objectivefunction is consumption per capita. Thus, the central planner cares aboutthe average level of life in a country but not about its population.

In the second formalization central planner maximizes the product of con-sumption level and population, and thus cares not only about consumptionlevel but also about the size of a nation. While it is possible to consider alsoany positive functions of C and L, we consider just their product:

maxI(t)

∫e−rtC(t)L(t)dt (16)

s.t. K̇ = I(t)− δK,

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L̇ = α(C; G) (17)

C =Y (K,L)− I

L,

There are several differences from classical growth models here. Firstof all, there exist externality in a form of influence of economic activity onpopulation growth. There are two factors that influence the dynamics ofpopulation: a) consumption per capita, b) income distribution. We alsohave to calculate consumption per capita, and thus the output is divided bypopulation L. The decision of central planner is only about investment path,but it triggers two processes: a) dynamics of capital, b) dynamics of popula-tion. There is also an external parameter G in this model, measuring incomeinequality, but it cannot be influenced by the central planner. Nevertheless,policy issues related to it, will be also discussed.

4.2 Dynamical System

Consider the second problem, with maximization of CL. Its Hamiltonianhas the following form:

H = Y (K, L)− I + λ(I − δK) + µα((Y − I)/L; G). (18)

The first order conditions include one static equation λ = 1 + ∂α/∂Cµ/L,coming from the equation HI = 0 and four differential equations, includingtwo dynamic constraints and the conditions λ̇ = λr − ∂H/∂K, µ̇ = µr −∂H/∂L. Finally we get:

λ̇ = λ(r + δ)− Y ′K(1 + µ

α′CL

), (19)

µ̇ = µr − Y ′L − µα′C

LY ′L − Y + I

L2,

K̇ = I − δK,

L̇ = α(C; G),

λ = 1 +µ

Lα′C , C =

Y − I

L.

In principle, it is possible to exclude the variables I and C from two staticequations, and consider the system of 4 dynamic equations for variables K, L

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and two Lagrange multipliers λ, µ. Steady states of this system are given bythe algebraic system:

λ(r + δ)− Y ′K(1 + µ

α′CL

), (20)

µr − Y ′L − µα′C

LY ′L − Y + I

L2,

I = δK,

α(C; G) = 0.

(21)

It is difficult to make further analysis without particular specifications.The complexity requires using partly analytical and partly numerical tech-niques. In the case of α(C, G) specified above, we have two solutions forthis equation, C1,2 = 1 ± √

1−G. Thus, such a system is likely to havemultiplicity of equilibria.

4.3 Analytical Analysis for Simple Function

Consider the case α(C) = C, that corresponds to the early ideas of Easter-lin [6]. With Cobb-Douglas production function, Y = KbL1−b, we get thefollowing optimization problem:

maxI(t)

∫e−rtC(t)L(t)dt (22)

s.t. K̇ = I(t)− δK,

L̇ = C,

C = (K/L)b − I/L.

The Hamiltonian is:

H = KbL1−b − I + λ(I − δK) + µεL[(K/L)b − I/L]. (23)

From HI = 0 we get λ = 1− µε. The dynamical system is as follows:

λ̇ = λ(r + δ)− b(1 + µε)(L/K)1−b, (24)

µ̇ = µr − (1− b)(1 + µε)(K/L)b − I/L2,

K̇ = I − δK,

L̇ = εKbL1−b − I.

(25)

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While the system itself does not look trivial, it is possible to find its steadystates analytically:

λ∗ =2bδ

ε(r + δ) + bδ, µ∗ =

ε(r + δ)− bδ

ε[ε(r + δ) + bδ],(26)

L∗ =δ(ε/δ)1/(1−b)

µ∗r − (1− b)(1 + µ∗ε)(δ/ε)−b/(1−b), K∗ = L∗(

δ

ε)1/(1−b), I∗ = δK∗.

This shows that the simple system has a unique steady state, which oftencan be attracting saddle point.

4.4 Particular Specification

Let us consider the production function given by (10) and α given by (12).Then the optimization problem takes the following form:

maxI(t)

∫e−rtC(t)L(t)dt (27)

s.t. K̇ = I(t)− δK,

L̇ = C(2− C)−G,

C = AK1−bL−1(min[Kb, L])− cI.

This specification captures the following stylized facts: a) inverse-U shapefor the relationship between consumption level and fertility, b) existence ofscale economies in underpopulated country. While Leontieff production func-tion brings non-differentiability, it captures the optimal relationship betweencapital stock and population, that influences growth (see Yegorov, 2009).

It is possible to consider several regimes:a) underpopulated economy (when L < Kb),b) overpopulated economy (when L > Kb),c) economy with balanced population growth (when L = Lb).

In real case we can have a shift between different regimes, and this makesfull analysis quite complicated. That is why it may be useful to start fromsome numerical experiments that show co-dynamics of capital stock and pop-ulation size under standard policies. One of such policies is related to con-stant saving rate and was suggested by Solow (see Barro and Sala-i-Martin,1995).

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4.5 Dynamics with Population below Optimal

The policy of constant saving rate is typical for macroeconomic analysis (al-though not always is optimal). In numerical simulations it was applied forthe model (27) under the assumption od underpopulated economy(L < Kb).Since the existence of demographic externalities of economic growth is notyet a common knowledge and economic theorists, the policies of constantsaving rates are often applied by governments. The goal was to see the in-fluence of dynamics of consumption and inequality on population dynamics.

The following set of equations was used for these simulations:

Y = ALK1−b, I = sALK1−b, C = (1− sc)AK1−b, (28)

K(t + 1) = K(t)(1− δ) + sALK1−b,

L(t + 1) = L(t) + εL(t)(2C(t)− C(t)2 −G).

In this simulations the condition of underpopulation was maintained dur-ing simulation interval, and thus the model was correct. Fig.2 shows thedynamics of population in the case of moderate inequality7, G = 0.4. Herethe population grows from the beginning, since for moderate G the intervalof consumption levels for positive population growth is relatively large. Fig.3 shows the co-dynamics of population, capital and consumption level. Capi-tal shows the highest growth rate, but consumption also grows substantially.The reason for population decline in the end of simulations is due to over-consumption (C is close to 2).

Fig. 4 shows what happens under the same set of parameters, except forG, that is now 0.6. Here we have even some decline of population (negativereproduction) in the beginning due to higher inequality. However, economicgrowth makes C larger, and later population begins to grow. At the end ofsimulation period C stays close to one, that gives the highest reproductionlevel.

However, further increase in inequality demonstrate catastrophic effect.Fig. 5 shows the simulations for G = 0.75. Here we observe economic

7One should not think that this value is real Gini, but rather a positive function of it,since the model was not calibrated. The author thinks that G = 0.4 is rather high Gini,while 0.3 is more normal.

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growth in the beginning, and even consumption grows. However, due to highG this is not sufficient to maintain reproduction, and population starts todecline. At the initial stages this process is rather slow, but its persistencehampers further economic growth. When even capital stock starts to de-cline, consumption level falls down catastrophically, triggering fast decline ofpopulation size.

4.6 Dynamics with Population above Optimal

Here the set of equations describing discrete model was as follows:

Y = AK, I = sAK, C = (AK − I)/L, (29)

K(t + 1) = K(t)(1− δ) + sALK1−b,

L(t + 1) = L(t) + εL(t)(2C(t)− C(t)2 −G).

Fig. 6 illustrates one of simulations for initially overpopulated economy(L(0) > Kb(0)). Here we have rapid accumulation of capital in the beginning,leading to overconsumption. This results in population decline after certainstage, due to model specification. Another problem was a dynamic breakof the condition L > Kb at a certain stage of evolution. This brings thenecessity to shift to model with underpopulated economy.

4.7 Comments for Further Research

One of the problems revealed by this numerical research is related to parabolicshape of function α. Since it vanishes to negative infinity for large C, weobserve too pronounced negative externality of overconsumption. In reality,the effect should be finite. That it why it makes sense to modify functionα for further research, so that it keeps desired properties without negativeeffects. Consider the function

α = C exp(1− C). (30)

Its derivative is α′ = (1 − C) exp(1 − C). It is always positive for 0 <C < +∞ and has a unique maximum at C = 1, that equals to 1. Here itsproperties coincide with the function 2C−C2, used before. If we subtract G,varying between 0 and 1, there will be always two positive roots C exp(1 −C) = G, giving the interval of positive population growth. However, thenegative effect on population dynamics for large C now becomes finite.

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5 Conclusions and Policy Implications

1. The recent negative demographic process in Russia and other CIS coun-tries call for urgent policy measures. The goal of this article was to attractmore attention to this problem as well as to propose some theoretical mech-anisms of interaction between economic and demographic processes.

2. The dynamic economic-demographic models presented in this articleare based on two streams of literature: about economic growth and economic-demographic interaction. It is argued that high income disparity has a neg-ative effect on fertility, and the dynamics of TFR is CIS countries confirmthis. In one of previous author’s papers it has been shown that low popu-lation density has negative effect on economic growth. Hence, by boostingfertility, Russia can weaken this effect and reach higher growth level. Prepa-ration for this today is especially importance since Russia has oil resourcesonly for 20 years, and need to diversify its economy anyway.

3. Several optimization problems for economic-demographic interactionare suggested. There are two key problems: specification of fertility as thefunction of consumption and income distribution, and objective to maximize.The general and special cases are studied, both by analytical and numericalmethods.

4. in numerical simulations, it is shown that the policy of constant sav-ing rate may have catastrophic consequences, if income disparity is above acertain threshold. Hence, Russia needs to make its income distribution moreequal (like in EU) in order to escape this scenario.

6 Literature

1. Atkinson A., Micklewright J. (1992) Economic Transformation in EasternEurope and the Distribution of Income. - Cambridge Univ. Press.2. Barro R., Sala-i-Martin X. (1995) Economic growth.3. Yegorov Y. (2009) Socio-economic influences of population density, Chi-nese Business Review, vol.8, No. 7, p.1-12.4. Yudaeva (2002) Globalization and Inequality in CIS Countries: Role of In-stitutions. - CEFIR, Moscow, WP24. (http://www.cefir.ru/papers/WP25.pdf)

17

5. http://en.wikipedia.org/wiki/Total fertility rates by federal subjects of Russia6. http://en.wikipedia.org/wiki/Fertility-development controversy

18

1

Figures for DEGIT 2011, Yegorov

Fig. 1. Dynamics of TFR in European CIS countries (both within and out of EU) shows the similar pattern:

rapid decline between 1990 and 1995 and then stabilization.

Fig.2. Dynamics of population for G=0.4. Decline is due to overconsumption (C>1)

1

1,2

1,4

1,6

1,8

2

2,2

1990 1995 2000 2005

TFR

Year

Dynamics of Total Fertility Rates

Belarus

Latvia

Lithua

Russia

Ukraine

Estonia

0

0,2

0,4

0,6

0,8

1

1,2

0 5 10 15 20 25 30 35 40

Dynamics pf population growth L(t) for A=1,del=0.1, s=0.3, K(0)=0.5, L(0)=0.5, b=0.4, G=0.4, eps=0.05

2

Fig.3. Simulation of co-dynamics of capital and population in the case of underpopulated economy,

for constant saving rate policy, I(t)=sY(t). In the case of not high inequality (G=0.4) both capital and

population grow, with further stabilization of population due to overconsumption (C>1).

Fig.4. Simulation for higher G=0.6 results in lower population growth, after initial decline.

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40

K(t

), L

(t),

C(t

)

Time

Dynamics of capital and population growth. A=1, =0.1, s=0.3, K(0)=0.5, L(0)=0.5, b=0.4, G=0.4, =0.05

C(t)

K(t)

L(t)

0

0,5

1

1,5

2

2,5

3

0 5 10 15 20 25 30 35 40

K(t

), L

(t),

C(t

)

Time

Dynamics of capital and population growth. A=1, =0.1, s=0.3, K(0)=0.5, L(0)=0.5, b=0.4, G=0.6, =0.05

C(t)

K(t)

L(t)

3

Fig. 5. For G=0.75 there is a catastrophe: economic growth goes along with population decline.

Consumption first grow, but then decline catastrophically. The reason is too low population for

production. This can correspond to Russian case at present.

Fig.6. Simulation for initially overpopulated economy (L>Kb). In this case we have rapid

accumulation of capital, leading to overconsumption and population decline. At some stage model

no longer valid, since current L(t)<Kb (t).

-1

-0,5

0

0,5

1

1,5

0 5 10 15 20 25 30 35 40

K(t

), L

(t),

C(t

)

Time

Dynamics of capital and population growth. A=1, =0.1, s=0.3, K(0)=0.5, L(0)=0.5, b=0.4, G=0.75, =0.05

C(t)

K(t)

L(t)

00,5

11,5

22,5

33,5

44,5

5

0 5 10 15 20 25

K,L

,C

Time

Dynamics of capital and population growth. L>K^b. A=1, del=0.1, s=0.3, K(0)=0.5, L(0)=0.5, b=0.4, G=0.5,

eps=0.05

C(t)

K(t)

L(t)