Structural Change and Balanced Growth With A Hierarchy of Needs
Reto Foellmi Josef Zweimüller
¤May 2, 2001
Abstract
In this paper we study a model that generates, simultaneously, structural change and balanced growth. The pattern of structural change results from consumption along a hierarchy of needs which imply non-linear Engel-curves for the di¤erent goods. Despite this fact the model generatesbalanced growth in the aggregate.
In contrast to the standard monopolistic competition model of product in- novations we …nd that: (i) The model may generatemultiple equilibria due to a dynamic demand externality inherent in the innovation process. (ii) Con- sumers do not necessarily purchase all products available on the market, and innovators may have awaiting time until they can earn pro…ts.
The model implies that countries of di¤erent wealth levels di¤er in their demand and production structure. In a natural extension of the model, trade between these countries can be analyzed.
JEL classi…cation: O40, O31, L16
Keywords: balanced growth, structural change, innovation, hierarchic prefer- ences.
¤University of Zurich, Institute for Empirical Research in Economics, Bluemlisalpstrasse 10, CH- 8006 Zürich, Tel: ++41-1-634 37 26, Fax: ++41-1-634 49 07, e-mail: [email protected], [email protected]. We thank Josef Falkinger for helpful comments.
1 Introduction
The massive reallocation of labor from the agricultural sector to manufacturing and services is one of the most striking features in the growth experience of modern economies. Despite the large di¤erences in natural endowments and cultural heritage, the qualitative nature and the quantitative signi…cance of this reallocation process has been very similar across countries.1 Equally remarkable is the fact that, despite these large structural changes, the growth process of most economies has followed a relatively stable pattern with respect to macroeconomic aggregates. As stated in Kaldor’s (1961) famous stylized acts, ’balanced growth’ - a state where the growth rate, the interest rate, the capital output ratio, and the labor share are constant over time - is a reasonable …rst approximation of the actual long-run development of an economy.
Standard theory of economic growth has been predominantly concerned with models that exhibit balanced growth, but has largely ignored the issue of struc- tural change. In particular, there exists no uni…ed framework within which both of these phenomena can be studied. In this paper we study a model that generates, simultaneously, structural change and balanced growth.
Put in a general way, structural change results either from di¤erences in pro- ductivity growth or from di¤erences in the growth rate of product demand across sectors. In this paper we put particular emphasis on the demand side: We study structural change under the assumption that demand develops along a hierarchy of needs and wants. We abstract from the second possible source of structural change, namely technological shifts that a¤ect the various sectors di¤erently.
Structural change requires technical progress. Without productivity growth and the corresponding increase in income there is no change in demand and no need to reallocate resources between sectors of production. When there is economic growth, however, the corresponding changes in demand lead to sectoral changes in the de- mand for resources, unless demand expands uniformly for all products.
A hierarchy of needs implies that demand does not increase in proportion with income. Individuals purchase goods that satisfy more advanced wants only if the most basic needs have already been covered. This means that, given the income level and the prices for the various goods, it may not be optimal for a consumer to purchase all goods that are available on the market. A hierarchy of needs and wants thus implies non-homothetic preferences with the possibility that thenon-negativity constraints for some goods may become binding.
1The following …gures demonstrate the impressive size of the structural change: In 1870 the U.S.
employment share of the agricultural sector amounted to about 40 % and decreased to only 4 % a hundred years later; the service sector accounted for only 20 % in 1870 and absorbed about 40 % of total employment in 1970 ((Kongsamut et al. (1997)).
The idea that the consumption demand follows a hierarchy of needs has a long history in economics. Empirical research that supports this idea goes at least back to the 1850s when Ernst Engel (1857) showed that there exists a systematic relationship between the structure of consumption and a household’s budget. In particular, Engel (1857) found that the expenditure share for food decreases with income, the famous
’Engel’s law’.2 The empirical literature on household consumption has impressively con…rmed Engel’s conjectures. The vast majority of the studies rejects the hypothesis of homothetic preferences; and there is strong evidence that the income elasticity for food is smaller than the income elasticity for manufactures and for services.
This paper assumes that growth is driven by innovations. Each innovation in- creases the stock of public knowledge and raises the productivity level in the whole economy. This assumption, together with the assumption of hierarchic preferences leads to the following circular process: Innovations increase income and these in- creases in income create a desire to satisfy additional wants that makes new innova- tions pro…table.
It is intuitively clear that the sequence of innovations follows the hierarchy of needs and the most recent innovator always supplies the most ’luxurious’ good. As the economy grows new goods appear on the market and the relative position of the good of an innovator changes over time. Obviously, a good that was a luxury at the date of invention becomes a necessity once incomes have su¢ciently grown.
In other words, whether a good is a necessity or a luxury depends on the level of development.3
The assumption that the innovation process generates a uniform rate of produc- tivity growth disregards the second source of structural change, namely sector-speci…c changes in technology. One reason why we disregard uneven technical change is to keep the model tractable. A second reason is that, unlike for the demand side, it is less clear how the relative supply conditions in the various sectors change over time.4 Furthermore, abstracting from uneven technical change allows us to focus on the demand aspect.
The most interesting result of our analysis is that it allows us to discuss structural
2Engel’s analysis draws on the empirical work by Duceptiaux that is based on 199 individual household budgets from various Belgian provinces and by Le Play who analyses 39 budgets of working class families in various European countries. Engel used these data to estimate the structure of production and consumption in the kingdom of Saxony. Here he formulated his famous law:
”(...) je ärmer eine Familie ist, einen desto grösseren Antheil von der Gesammtausgabe muss zur Bescha¤ung der Nahrung aufgewendet werden (...).” (Engel, 1857, p. 28f., emphasis in original).
3There are lots of examples that come to mind: the automobile, the telefone, the radio, the TV, and so on. All these goods have been luxuries when they were invented but, in rich economies, have become necessities for everybody who wants to participate in the society.
4There is evidence that during the last century the highest rate of technical progress has taken place in agriculture, whereas the growth rate in the service sector has been lower (Fuchs, 1968).
change and balanced growth within a single model. Structural change means that there is a continuous change in the structure of consumption which in turn leads to a reallocation of labor across sectors. The choices of consumers imply market demand functions such that the goods with low priority (the luxuries) have a high income elasticity whereas the goods with high priority (the necessities) have a low income elasticity. Furthermore, not only the demanded quantities, but also the prices change over time as higher incomes imply a higher willingsness to pay for goods with a given position in the hierarchy. Finally, the expenditure share for the various products increases in the early stage, then reaches a maximum and …nally decreases in a more mature stage of the product cycle.
Furthermore, the balanced growth path is characterized by interesting features that are absent in the standard innovation driven growth model. (i) Complementar- ities in the innovation process. The expansion of the market of an innovator depends on the economy-wide growth rate. This generates a dynamic complementarity be- tween present and future innovations. The result is that there may be multiple equilibria: Optimistic (pessimistic) expections about the growth rate induce high (low) innovation activities that in turn sustain high (low) growth. (ii) Preemptive patenting. If non-negativity constraints are binding consumers do not buy the most luxurious good. Firms may nevertheless incur the R&D costs to get the patent and to preempt potential competitors trying to conquer the market. (iii) Stronger price distortions. Incumbent …rms have increasing market power as not only the income elasticities but also the price elasticities of demand for their products decrease over time. The growing mark-ups imply strong static price distortions. The socially optimal patent policy is characterized by a …nite patent length.
The previous literature that has dealt with the impact of non-homothetic pref- erences on macroeconomic outcomes has been primarily concerned with the role of income inequality. This issue arises naturally as such preferences imply that rich and poor consumers have a di¤erent structure of consumption.5
To the best of our knowledge, the only paper that aims explicitely at explaining both balanced growth and structural change within a uni…ed framework is the paper by Kongsamut, Rebelo, and Xie (1997). They show under which conditions a bal- anced growth path exists in a three-goods economy - where consumers’ preferences are characterized by a linear expenditure system with, respectively, a positive, zero, and negative, required consumption levels for food, manufactures and services. With such a framework balanced growth can only be reached if technology and taste pa-
5See, for instance, Murhpy, Shleifer and Vishny (1989), Eswaran and Kotwal (1993), Baland and Ray (1991), for static models, and Falkinger (1990, 1994), Chou and Talmain (1996), Matsuyama (2000) and Zweimüller (1996, 2000) for models that focus on the relationship between inequality and growth. Flam and Helpman (1987), Stockey (1991) and Matsuyama (2000) study international trade in the context of non-homothetic preferences.
rameters are linked in a very particular way. In constrast, no such link is necessary in our model. To generate balanced growth we need (i) that industries are homoge- neous with respect to production possibilities and (ii) technical progress a¤ects all sectors equally.
The paper is organized as follows. Section 2 presents the general set-up of the model and the particular assumptions about hierarchic preferences; solves the static problems of consumers and …rms; and discusses the resulting structure of demand and prices in the static equilibrium. Section 3 discusses the assumptions on technology and the allocation of ressources. Section 4 solves the the intertemporal problem of the consumer and Section 5 looks at the innovation choice of potential researchers.
Section 6 discusses the balanced growth path and Section 7 describes the struc- tural changes that occur along this path. Section 8 contains a discussion how the steady state changes with the interesting parameters that characterize the hierarchy of needs. Section 9 presents an application of the model to optimal patent policy.
Section 10 concludes.
2 The Static Equilibrium
2.1 Preferences and consumer demand
Consider a representative agent economy with in…nitely many potentially producable goods that are ranked by an indexi: We study the structure of consumption that is generated by preferences of the form
u(fc(i)g) =
Z 1
0 »(i)v(c(i))di (1)
wherev(c(i))is an indicator for the utility derived by consuming goodi in quantity cthat satis…es the usual assumptions v0 >0and v00<0. To capture the notion of a hierarchy of needs we introduce a weighting factor »(i) which ranks the potentially available goods. With»0(i)<0, lowi’s denote goods of higher priority:
A meaningful speci…cation of hierarchic preferences has to take account of two facts. First, some goods may not be consumed because the consumers cannot a¤ord it, since they are too poor. This implies that preferences must be such that thenon- negativity constraints may become binding and Engel-curves for the various goods are non-linear. Formally, the possibility that non-negativity constraints may become binding at some …nite, positive prices implies that the marginal utility of consuming good i, »(i)v0(0) is …nite. It is worth noting that non-negativity constraints never become binding in the standard monopolistic competition model that operates with constant elasticity of substitution preferences (Dixit and Stiglitz (1978)) and that is frequently used in the macroeconomic literature. With such a speci…cationv(c(i)) =
1
®c(i)®; with ® < 1; the condition »(i)v0(0) <1 is not satis…ed as long as »(i)> 0:
Thus in the standard monopolistic competition model all the currently available goods (i.e. all goods that are available at a …nite price) are always consumed in positive amounts.
Second, Engels’ law implies that additional income is spent primarily on low- priority goods. This feature is caught by the formulation that the utility of con- sumption of di¤erent goods di¤ers only in the multiplicative factor »(i): But this means that the marginal utility of a high priority good with»(i)high, falls quickly.
Optimal consumer behavior now implies that marginal utilities of di¤erent goods be equal. Hence, additional income is spent especially on the low-priority goods with slowly falling marginal utilities.
To keep the analysis tractable we make three assumptions that greatly simplify our analysis. First we assume that the weighting function is a power function so that »(i) = i¡° with ° 2 (0;1). We will see that this speci…cation guarantees thataggregate expenditures grow pari passu with income despite the fact that Engel- curves for particular goods are non-linear, and despite the fact that the expenditures on single goods do not grow pari passu with income.6 This will ultimately allow us to study equilibria in which the growth rate is constant. Second, we assume that v(c(i)) = ln³c(i)q + 1´ where q > 0 as some constant. This will allow us to …nd explicit solutions for the optimal quantities consumed by the households, for the prices charged, and the pro…ts made by monopolistic …rms, while at the same time
»(i)v0(0) = i¡°q <1for i >0. Third, it is convenient to model the goods space as a continuum.
With these assumptions, we can now speci…y the objective function of the con- sumer’s maximization problem. Assume that only goods with relatively high priority are acutally available on the market. In particular, assume that all goodsi2(0; N) are available at a …nite price, whereas alli > N have not yet been invented. In that case the consumers’ objective function is7
u(fcg) =
Z 1
0 i¡°ln
Ãc(i) q + 1
!
di: (2)
6Foellmi (1999) shows that this is the case for all continuous and twice di¤erentiable functions v(¢), withv0>0andv00<0.
7v(c(i)) = ln (c(i) +q)¡lnq = ln³c(i)
q + 1´
has been normalized such that v(0) = 0. This normalization is necessary to prevent divergence of the utility integral because the consumer’s preferences are de…ned over an in…nite number of goods. Since only goods in the interval i 2 (0; N) can be consumed in positive amounts the consumer’s objective can be written as u(fcg) = RN
0 i¡°ln³c(i)
q + 1´
di+R1
N i¡°ln(1)di: To prevent divergence of the …rst integral we must have ° < 1: By the normalization of v(:) above, the second integral is zero and does not diverge. Without loss of generality we can then restrict our attention to the utility function u(fcg) =RN
0 i¡°ln³c(i)
q + 1´ di:
which will be maximized subject to the budget constraint
Z N
0 p(i)c(i)di =E (3)
and the non-negativity constraints
c(i)¸0 8i: (4)
The optimality conditions require that the above constraint and the …rst order con- ditions
c(i)
à i¡°
c(i) +q ¡¸p(i)
!
= 0 8i (5)
i¡°
c(i) +q ¡¸p(i) · 0 8i:
be satis…ed, where ¸ denotes the Lagrangian multiplier. The …rst order condition speci…es the demand for goodi.
2.2 Prices
Next we consider the determination of prices. Assume that all goods are produced at the constant marginal cost, equal for all goods. We take these marginal costs as the numéraire. (When we discuss intertemporal issues below, we will see that marginal cost are constant over time, since wages grow pari passu with with the inverse of the unit labor requirement to produce output. Thus marginal cost will not be only equal for all goods but also constant over time).
In what follows we consider a situation where goods with lower priority (high-i goods) are supplied on monopolistic markets, whereas the goods with higher priority (low-i goods) are supplied by competitive producers. While this is an assumption at this stage, we will see below that this will be the equilibrium outcome of the model. Our goal is to study long-run growth when new goods are continuously introduced. It is intuitively obvious that the chronological sequence in which these new products appear on the market follows the hierarchy of wants: there will be demand for products with lower priority only if goods with high priority have already been satis…ed. In other words, the goods that have been introduced most recently are the goods with the least priority.
So why are the goods with highest priority priced at marginal cost whereas the remaining products are sold at the monopoly price? The reason is patent protection.
Because monopoly pro…ts are necessary to provide …rms with the necessary incen- tives, we assume that innovators are protected by patents of some …nite duration.
At any given point of time we thus have a fraction of available products a which are priced at marginal cost and a fraction of products1¡a for which the monopoly price can be charged. Since, at a given point of time, N goods are available in the economy, the above arguments imply that all goodsi2[0; aN]have a pricep(i) = 1 whereas the producers of the remaining available goodsi2(aN; N]are protected by patents and charge the monopoly price.
The market demand faced by this monopolist is given by the representative house- hold’s demand function (5) so that the monopolist maximizes the objective function
¼(p(i)) = [p(i)¡1]
"
max
Ã
0; i¡° 1 p(i)¸ ¡q
!#
;
the solution to which is given by:
p(i) = max
2 41;
à 1
qi°¸
!1=23
5: (6)
2.3 Equilibrium composition of demand and the structure of prices
We can now characterize the composition of demand and the structure of prices in the static equilibrium, given the representative agent’s budgetE and the measure of currently producable goodsN. We will do this separately for the two scenarios which can occur in equilibrium. In the …rst case, the consumers do not …nd it optimal to consume all goods that are available on the market. Alternatively, it may be the case that the consumer can a¤ord all goods that are supplied. We discuss the two cases in turn.
Consider the …rst case where the consumer may …nd it optimal to consume not all available goods so that the measure of products consumed in positive amountsn falls short of the measure of available goodsN. If the goods are consumed in positive amounts and the goods are supplied at the monopoly price we know from (5) and (6) that the consumed quantity of good i equals c(i) = ³i°q¸
´12
¡q which clearly is decreasing in i. This means that there is a good, call it n; that for goods i > n it is optimal to consume quantity zero. All goods i < n are consumed in positive amounts, and all goods i > n are not consumed.
It turns out convenient to express the endogenous variablesc(i)andp(i)in terms of the endogenous variable n rather than ¸: From c(n) = ³nq°¸
´12
¡q = 0 it is straightforward to calculate¸= qn1°:Substituting this into equations (5) and (6) we get
c(i) =
8>
>>
<
>>
>:
qh³ni´°¡1i; i2[0; aN];
q·³ni´°=2¡1
¸
; i2(aN; n]; and
0; i2(n; N]
(7) and
p(i) =
8>
><
>>
:
1; i2[0; aN];
³n i
´°=2
; i2(aN; n]; and 1; i2(n; N]:
(8) Note that in the above expressions forp(i)andc(i) the variablenis endogenous, chosen by the consumer on the basis of the variables N and E which are taken as given. We can easily express n as a function of E using the consumer’s budget constraint
E =
Z n
0 p(i)c(i)di (9)
= qn
0 B@ 1
1¡° +1¡³aNn ´1¡°=2 1¡°=2 ¡ aN
n
1 CA
= qn¢B(1;aN n ;°):
We see that the optimal measure of consumed goods is proportional to the consumer’s budget since aNn will be constant over time. The …rst argument in the B(¢)-function stands forp= 1(see below).
Now consider the alternative scenario that the consumer chooses to consume all available goods in positive amounts. Obviously, this is the case if c(N) =³Nq°¸
´12
¡ q > 0. Also here it is convenient to replace ¸: However, we cannot express it in terms of the optimal bundle of consumed goods n which is trivially determined by the number of available goods N: Instead we express ¸ in terms of the price of the consumed good that has least priority in consumption, that is by the endogenous variable p(N) ´ p. From (6) it is straightforward to express the marginal utility of income as ¸= qN1°p2: The same expression (6) can be used to express the monopoly prices for the goods i2(aN; N]in terms of p asp(i) =p¢³Ni´°=2. The structure of prices and the equilibrium composition of demand can now be expressed as
c(i) =
8>
<
>:
qhp2¢³Ni ´°¡1i; i2[0; aN] and q
·
p¢³Ni´°=2¡1
¸
; i2(aN; N] (10)
and
p(i) =
8<
:
1; i2[0; aN] and
p¢³Ni ´°=2; i2(aN; N]: (11) Now the variablesc(i)andp(i)are determined by the endogenous variablepthat is chosen by the monopolist who supplies the good that has the least priority. p depends on the consumer’s budget E, and the measure of available goods N both of which are taken as given in the monopolist’s choice of p. To see the relationship between p, E; andN; we insert equations (10) and (11) into the consumer’s budget constraint
E =
Z N
0 p(i)c(i)di (12)
= qN
Ã
p2 1
1¡° +p1¡a1¡°=2 1¡°=2 ¡a
!
= qN¢B(p; a;°)
This expression de…nespas a function ofEandN;and other parameters of the model.
We also observe that, for a givenp, consumption expenditures E are proportional to the measure of available goodsN:
3 Technical Progress and the Labor Market
Until now we have not been speci…c about the supply side of the economy. In this section we will describe and discuss our assumptions about technology and technical progress in more detail. To highlight the role of hierarchic preferences we assume that the technology is equal for all goods so that all heterogeneity across goods comes from preferences rather than technology.8 Talking about technical progress means that the dynamic dimension comes into play. We capture this by the continuous time indext. While goods are heterogenous with respect to preferences, we assume that the supply conditions are equal for all goods. To keep things as simple as possible we assume that labor is the only production factor and that the labor market is competitive. The market clearing wage at datet is denoted byw(t):
Before a good can be produced with this technology there has to be an innovation.
Just like in other innovation-driven growth models, we assume that a …xed cost has to be incurred to acquire the blueprint for a new good. More precisely, we assume that a
…xed labor inputF(t)is necessary, so the …xed cost arew(t)F(t), equal for all goods.
8Note that this is perhaps somewhat unrealistic since one could argue that in the industries where demand and production is very high, there is much learning by doing. Assuming symmetry of good with respect to technology therefore means that we abstract from such learning e¤ects.
It is assumed that F(t) decreases over time as a result of technical progress. We model technical progress by assuming thatF(t)is inversely related to the aggregate knowledge stockA(t)that re‡ects the economy-wide productivity at datet and that is growing over time. Thus we can write F(t) = A(t)F where F > 0 is an exogenous parameter.
Once an innovation has taken place the corresponding output good can be pro- duced with the linear technology
l(i; t) =b(t)c(i; t) (13)
wherel(i; t)is labor employed to produce goodi at datet, andb(t)is the unit labor requirement. Marginal cost at date t is w(t)b(t), equal for all goods, where w(t) is the wage rate that applies to the whole economy.
We assume that - as a result of technical progress - not only F(t) but also b(t) decreases over time. Again we model this by assuming thatb(t) = A(t)b , where b >0 is an exogenously given parameter.
At this stage we want to point out an important di¤erence between the hierarchy model of product innovations and the standard ’love-for-variety’ model by Grossman and Helpman (1992). In the hierarchy model there has to be technical progress otherwise innovations comes to a halt. To see this recall that, as a result of growing incomes, consumers have always an incentive to consume more low-priority goods but do not necessarily consume the good of the most recent innovator. Without technical progress in production this means that sooner or later the whole labor force will be employed to satisfy the demand of consumers on the already existing goods. Higher-priority goods have no demand and therefore there are no incentives for further innovations.
In the love-for-variety model this is di¤erent. Here new goods are always de- manded by the household and, even if there is no technical progress in production as in Grossman and Helpman (1992), innovation incentives are still there. The al- ready existing goods are consumed in ever lower amounts which releases the resources necessary to produce the new goods. The consumption expenditures are allocated equally among the existing goods and the new ones and therefore the expenditures allocated to a particular good become smaller and smaller. While this means that revenues and pro…ts of an innovator decrease over time, innovation incentives con- tinue to exist, if there is technical progress in R&D. In that case not only pro…ts but also the innovation costs decrease.
In sum, the love-for-variety model of product innovations needs the assumption of productivity progress only in the research sector, but not in the production sector.
In the hierarchy model of product innovations, however, there has to be technical progress both in the R&D and in the production of …nal output, otherwise the incentive to innovate disappears.
Young (1993) justi…es the assumption that not only the research productivity increases as a result of innovation but also the productivity in production of all in- dustries. By introducing an intermediate goods sector he assumes that the invention of a new goodi leads as a by-product to the discovery of a new intermediate input.
According to Romer (1990) a higher variety of inputs makes the …nal output tech- nology more productive. In particular, as Young (1993) notes, if the …nal goods are produced by combining these inputs using a constant returns to scale CES technol- ogy, the productivity of the output sector rises linearly in the number of inputs.
In the following we will assume that the economy-wide knowledge stock equals the measure of innovations that have taken place up to datet, so we have A(t) =N(t);
and the technology parameters can be written as b(t) = b
N(t) andF(t) = F
N(t) (14)
Moreover, on a balanced growth path it must be that wages grow at the same rate as productivity which means that wages grow pari passu with N(t): Thus we can write
w(t) =w¢N(t) (15)
wherew is an exogenous parameter.
The marginal cost in production at datet are then w(t)b(t) =wb; constant over time. Without loss of generality, we take marginal cost as the numeraire thus, for all t; we have wb= 1. Note that also the …xed innovation cost w(t)F(t) = wF are constant over time.
We can now describe the resource constraint of the economy. The labor supply in the economy isL which we normalize to 1 and, in equilibrium, this labor force is either employed in the R&D sector to develop new blueprints or in the production of consumer goods. At datet,N_(t)new goods are introduced and each such innovation requires a unit labor input N(t)F :This means, at datet,N_(t)NF(t) workers are employed in the R&D sector. All other employees work in …rms that produce consumer goods.
Employment in production is the sum of all workers that are employed to produced the various goods which is Nb(t)R0n(t)c(i; t)di: The resource constraint can then be written as
1 = N_(t)
N(t)F + b N(t)
Z n(t)
0 c(i; t)di: (16)
To calculate the labor market equilibrium condition we again have to distinguish (i) the case, where the consumer purchases only a fraction of the goods that are actually available on the market, so that n(t) < N(t) and (ii) the case where the consumer purchases all available goods, so that n(t) =N(t):On a balanced growth path, n(t) grows at the same rate as N(t). We denote the growth rate by n(t)n(t)_ =
N(t)_
N(t) =g: So if it is optimal to consume only a subset of the available goods at date t this is also optimal at all later dates; in that case equation (7) is relevant. If it is optimal to consume all available goods at datet, it is optimal to consume all available goods also at later dates; then (10) is relevant.
Both (7) and (10) tell us that the …rstaN(t)goods are available at the competitive market price p(i; t) = 1, whereas for all other goods the monopoly price is charged.
The latter …rms are still protected by patents whereas for the innovators of the former products patent protection has expired and this markets are served by competitive producers. If the length of the patent is¢, it is clear that the share of competitively supplied products a depends on the growth rate g: To see the relationship between a, g; and ¢; note that between the date of innovation of a certain good and the expiration of the patent the number of goods in the economy has risen by the factor eg¢. Therefore a fractiona= 1=eg¢ =e¡g¢goods are competitive and the remaining fraction 1¡a = 1¡e¡g¢ is monopolistic.
When it is optimal for the consumer to consume only a subset of the available goods,n(t)< N(t);we can express the resource constraint in terms of the endogenous variablesg and±;where±is the time it takes until a product that is innovated today is purchased by the consumer in positive amounts. ± is de…ned by the equation n(t)e±g =N(t): Substituting this and equation (7) into the resource constraint (16) we get
1 = gF +bqe¡±g
"
e¡(1¡°)g(¢¡±)
1¡° +1¡e¡(1¡°=2)g(¢¡±) 1¡°=2 ¡1
#
(17) In the case where all goods are always consumed, n(t) = N(t); we express the labor market equilibrium condition in terms of the endogenous variables g and the price of the good with least priority p substituting equation (10) into the resource constraint (16) to get
1 = gF +bq
"
p2e¡(1¡°)g¢
1¡° +p1¡e¡(1¡°=2)g¢
1¡°=2 ¡1
#
(18)
4 Consumption Expenditures and Savings
The next step of our analysis is to ask the question how consumption expenditures are allocated over time. Note that we follow a two-stage budgeting procedure. We have already studied how a given amount of expenditures is allocated across goods at a given point of time. It remains to study the time path of these expenditures.
We study the situation of a representative consumer who maximizes utility over an in…nite horizon. To study the intertemporal allocation of expenditures we insert, respectively, the optimal quantities (7) (if n < N) and (10) if (n=N) of each good i into the utility function (2), we get the instantaneous utility function int
u(t) = n(t)1¡°
1¡° K(ae±g; p;°) (19)
whereK(ae±g; p;°) =
µ
1 +³ae±g´1¡°
¶
lnp¡° 2
³ae±g´1¡°lnae±g+ ° 2
1 +³ae±g´1¡° 1¡° The formulation encompasses both the casen < N when we have± >0andp= 1 and the casen=N when we havep >1and±= 0:The utility value has a constant elasticity of substitution not only in the number of consumed goodsn(t) but also in the consumption expendituresE(t), because of the proportionality between n(t)and E(t); see equations (9) and (12).9
The instantaneous utility index (19) depends also on the variables p; the price of the consumed good that has least priority in consumption, and the fraction of all goods a that is supplied under competitive conditions. Both of these variables are endogenous in the model, but taken as given by the consumer when deciding about the optimal allocation of consumption expenditures over time. Nevertheless, it is interesting to ask how the instantaneous utility u(t) is a¤ected by a and p:
Clearly, a p > 1 means that we are in the situation where the consumer decides to purchase all available goods, so n(t) =N(t): With a larger p, for a given N(t); the consumed quantities for all goodsi < N(t)are larger, and therefore the utility index takes a higher value. Furthermore, the utility index also increases in the fraction of competitively supplied goods a. As these goods have lower prices, their consumed quantities are higher, which ceteris paribus, leads to a higher level of the utility index.
We can now turn to the intertemporal allocation of consumption expenditures.
The additively separable intertemporal utility function is assumed to have the con- stant elasticity form with the intertemporal elasticity of substitution 1=¾: As u(s) is the instantaneous utility received in period ¿ with the intratemporal elasticity of substitution1=° we can write the intertemporal utility as follows
U(t) =
Z 1
t
(u(s))1¡¾
1¡¾ e¡½(s¡t)ds (20) where ½ denotes the rate of time preference. Consumers choose the time path of E(s)such that the the objective function (20) is maximized under the intertemporal budget constraint
Z 1
t E(s)e¡r(s¡t)ds ·
Z 1
t w(s)e¡r(s¡t)ds+V(t)
9This is the result of our choice of the hierarchy weighting factor i¡°. It can be shown (see Foellmi, 1999) that, no matter what the particular form ofv(¢)is, if and only if the hierarchy factor takes the formi¡°, the number of goodsn(t)and the incomeE(t)are proportional, and the utility function is constant elasticity in these two variables.
wherer is the interest rate, which is constant over time since we only consider steady states and V(t) denotes the assets that the consumer owns at datet.
Recognizing thatK in equation (19) is a constant from individual’s point of view and that, by the proportionality ofn(t)andE(t), we have E(t)E(t)_ =
n(t):
n(t) =g we get the following Euler equation for the growth rate of expenditures
E(t)_
E(t) =g = r¡½
¾(1¡°) +° (21)
In the symmetric case (° = 0) we get the usual form g = r¡½¾ . Note that the e¤ect of ° on the growth rate of consumption is ambigous (remember that ° < 1).
A higher ° raises g when ¾ >1 and it decreases g if ¾ <1: The intuition is subtle:
With ° > 0, the expenditures E(t) enter themselves as a concave function in the utility function. Remember that the growth rate of consumption depends on how fast marginal utility falls. In the symmetric case marginal utility declines at rate ¾:
In comparison to the symmetric case the asymmetry now has two e¤ects. On the one side, the intertemporal substitution e¤ect causes marginal utility to fall only at the rate ¾(1¡°) but on the other side the intratemporal substitution additionaly implies that marginal utility falls at rate °.10 In total, marginal utility falls at rate
¾(1¡°) +° which is less than ¾ if ¾ >1 and bigger than ¾ if¾ < 1:
5 Research and Development
To study growth and structural change we have to give an answer to the question what determines the incentives to innovate. Above we have already seen that the costs of an innovation are equal to wF. As long as the value of an innovation does not fall short of these set-up costs, …rms have an incentive to conduct R&D and incur these set-up costs.
To make clear what the value of an innovation is we have to look at the life cycle of an innovator. Consider a …rm that, at datet;incurs the set-up costs and is granted a patent until datet+ ¢for the new product. This new …rm has initially no demand if it is not optimal for the consumer to purchase all goods (the casen < N) but has to wait a period of length± until consumers start to purchase the new good in positive amounts. Note that, even if there is initially no demand, the …rm has an incentive to
10Ignoring the constant, the marginal utility of expenditures is given by
@
@E
³E1¡°
1¡°
´1¡¾
1¡¾ = µE1¡°
1¡°
¶¡¾
E¡° .
incur the cost and apply for a patent already at date t, to preempt other potential innovators from entering the market. We will call this scenariopreemptive patenting.
After datet+±, the demand for this product starts to increase as consumers become richer and demand the good in increasing amounts. Due to this higher demand the innovator can charge higher prices. As a result, pro…ts continuously increase after datet+±when consumers start to buy until datet+¢when the patent expires. From t+ ¢onwards, the pro…t ‡ow from the innovation stops because competition drives the price down to marginal cost and the innovating …rm is displaced by competitive producers.
The situation is only slightly di¤erent in the case when the innovator has already demand from the beginning (the case ifn=N). At datet;when the good is invented and the patent is granted, the innovator can already sell the good in positive amount.
At date t this good is the one that has the least priority in consumption and has price p ¸1: From date t onwards, the pro…t ‡ow increases as a result of increasing demand, and the pro…t ‡ow stops at date t+ ¢ when the innovator gets displaced by competitive …rms as a result of the expired patent.
To calculate the value of an innovation, we …rst consider the pro…ts that an innovator can make at a given point of time. Substituting, respectively, equations (7) and (8) (when n < N) or (10) and (11) (when n= N) into the pro…t equation
¼(p(i)) = [p(i)¡1]c(i) yields the pro…ts a …rm, that introduces a new good N(t) at date t; can make at some dates
¼(N(t); s) =
8>
>>
<
>>
>:
0 s2[t; t+±)
q
·
p³Nn(s)(t)´°=2¡1
¸2
s2[t+±; t+ ¢]
0 s > t+ ¢
(22) where we note that, if not all goods are consumed (n < N), the price of the good with least priorityp = 1 and the waiting time ± >0 ; alternatively, when all goods are consumed (n = N) we have p ¸ 1 and ± = 0: Using this and (22) it is now straightforward to calculate the value of an innovation ¦(t)as
¦(t) =
Z t+¢
t+± q
2 4p
Ãn(s) N(t)
!°=2
¡1
3 5
2
e¡r(s¡t)ds
Usingn(s) =e¡±gN(s) andN(s) =N(t)eg(s¡t) we receive
¦(t) = q
Z t+¢
t+±
hpeg°2 (s¡[t+±])¡1i2e¡r(s¡t)ds
Evaluating this integral we get for the case where not all goods are consumed and innovators have waiting time± until there is demand (n < N)
Ặ(t) =q
ẳ1âeâ(đâẸ)ạ
ạ ::19è19::eâ(đâẸ)[ạ+g°2]
ạ+g°2 +1âeâ(đâẸ)[ạ+g°]
ạ+g°
!
đeâẸ[ạ+g°] (23) where we used the de…nitionạ =râg° and the fact that from (21)r =½+g(¾(1â
°) +°):
For the case where innovators have demand from the beginning (n = N) the value of an innovation is
Ặ(t) =q
ẳ
p21âeâđạ
ạ â2p1âeâđ[ạ+g°2 ]
ạ+ g°2 +1âeâđ[ạ+g°]
ạ+g°
!
: (24)
We assume that there is free access into the R&D sector, meaning that in equi- librium, the value of an innovation must not be larger than the cost of an innovation, so we must have
Ặ(t)·wF: (25)
If the inequality is strict, no innovations take place in equilibrium. If the value of an innovation equals the innovation cost, we have an equilibrium where new products are introduced on a continuous basis.
6 Balanced Growth
In this section we describe the balanced growth equilibrium of the model which is characterized by structural change. The equilibrium exhibitsbalanced growthbecause aggregate consumption expenditures (or, alternatively, the value of production) and investment expenditures (the aggregate costs of innovation) grow at a common con- stant rate. Moreover, in this equilibrium the allocation of labor between production and research remains constant.
In the following we will …rst discuss under which conditions an equilibrium with balanced growth and structural change exists. We will then turn to the question un- der which conditions the equilibrium is unique. It turns out that, for some parameter values, the model exhibits multiple equilibria.
6.1 The Balanced Growth Equilibrium
The task of this section is to …nd the equilibrium growth rate in this economy. It can be determined by focusing on the resource constraint and the zero-pro…t condition.
When the consumer does not buy all goods supplied on the market, n < N; the relevant equation is (17) for the resource constraint and (25) with the value of an
innovation ¦(t) given by (23) for the zero-pro…t condition. The two endogenous variables are the growth rateg and the innovator’s waiting time ±:
Under the alternative scenario when the consumer buys all available goods, n= N, the resource constraint is given by (18) and the zero pro…t conditions is given by (25) where ¦(t) is now given by (24). The two endogenous variables are g and the price of the good with the least priorityp.
To examine under which conditions an equilibrium exists, and when it is unique, we have to scrutinize the behavior of these equations. This will be done by a graphical exposition. We denote the curve that represents the resource constraint byRand the curve that shows the zero pro…t condition by¦. It is convenient to drawR and¦ in a(±; g; p)-diagram as this allows us to discuss the above two scenarios simultaneously (Figure 1). The left part of in Figure 1 measures the innovator’s waiting time±(from right to left, starting at ± = 0) whereas the right part of in Figure 1 measures the price of the good with least priority p (starting at p = 1). Observe that ± = 0 and p = 1 is the limiting case where the innovator has neither a waiting time nor an incentive to charge a price above marginal cost. We now discuss the shape of the two curves in turn.
Figure 1
The shape of the ¦-curve We …rst consider the case where where the consumer does not buy all goods n < N so that innovators have to wait for an interval ± >0 until they can sell their product. To get the slope of the ¦-curve we make use of the implicit function theorem and calculatedg=d± =¡¦±=¦g (where¦x denotes the partial derivative of the right-hand-side of equation (23) with respect to x).
A longer waiting time ± reduces the value of an innovation. This simply results from discounting: the longer I have to wait for a given pro…t ‡ow the lower is the present value of this ‡ow. This e¤ect is still enhanced by the fact that, due to a …xed patent duration ¢; the period over which the innovator earns positive pro…ts does not only start later but also becomes shorter. Hence we have ¦± <0:
The impact of the growth rate g on the value of an innovation is ambiguous.
In a world with homothetic preferences where all goods enter the utility function in a symmetric way, a higher growth rate always lowers the value of an innovation.
This is because, in equilibrium, a higher growth rate is always associated with a higher interest rate that discounts future revenues more strongly (see discussion in Romer (1990)). With a hierarchic structure of preferences instead, we have a second e¤ect: a higher growth rate means that the demand for an innovator’s product rises more quickly. Hence the innovator can charge higher future prices and make higher future pro…ts which raises the value of an innovation. If the hierarchy parameter ° is large enough,¦g >(<)0wheng is low (high): the demand e¤ect of higher growth
dominates the interest rate e¤ect at low level of g and vice versa. Taken together, this implies thatdg=d± >0 wheng is low and dg=d± <0 when g is high (recall that in Figure 1 we measure± from left to right).
When the consumers buy the good of the most recent innovator immediately, n = N; we have p ¸ 1 and ± = 0: In that case; the picture is very similar. Just like a lower ±; a higher p increases pro…ts so ¦~p > 0 (to avoid confusion the tilde indicates the case p >1) and the sign of ¦~g remains ambigous. Just like before the demand e¤ect of a higher g is dominant at low growth rates if hierarchy is steep enough,11 whereas the interest e¤ect dominates at high growth rates. This implies that the ¦-curve has a negative slope at low g and a positive one at high g: If ° is close to zero, however, the interest e¤ect will always dominate and ¦ will always have a positive slope.
We summarize our discussion of the ¦-curve in the following
Lemma 1 If the slope of ¦ is positive at g = 0 the ¦- curve in monotonically increasing in (g; p)-space. If the slope of ¦ is negative at g = 0, the curve bends leftwards for low g and turns to the right for higher g.
Proof. See Appendix.
The shape of the R-curve The slope of the resource constraint R (18) can be derived in an analogous way as before by calculating, respectively,dg=d±=¡R±=Rg
anddg=dp=¡R~p=R~g for the two alternative regimes. (R andR~ now denote, respec- tively, the right-hand-sides of equations (17) as a function of g and ± and (18) as a function ofg andp).
A higher waiting time± means that a lower number of goods are produced in the economy which saves resources for other research, soR±<0:A higherpis associated with higher consumption levels for all goods (see equation 10), hence a larger amount of ressources in the production of …nal output is needed so thatR~p >0.
A higher growth rateghas an ambiguous e¤ect on the demand for labor resources.
On the one hand, there is the direct e¤ect of a higherg that increases the demand for workers in the research sector. On the other hand, there is an indirect e¤ect which
11Whether¦ bends leftwards at g = 0can be seen when we calculate which ° implies ¦~g(g = 0) > 0. Note that ¦ hits the p-axis at p = 1 +q
F bq
½
1¡e¡¢½: After some manipulations, we get (e¡z(1 +z)<1forz >0)
¦~g(g = 0) =¡1¡(¢½+ 1)e¡¢½
½2 (p¡1) [(p¡1)¾(1¡°)¡°]>0
< => ° > (p¡1)¾
1 + (p¡1)¾ wherep¡1 = sF
bq
½ 1¡e¡¢½ The latter condition is always ful…lled if°is su¢ciently close to1.
is due to the increase in the size of the monopolistic sector. (Recall from Section 3 above that, with a given patent duration ¢; a fraction e¡g¢ of all goods is supplied by competitive producers and a fraction 1¡e¡g¢ by monopolistic …rms.) As the monopolists charge a higher price than the competitive …rms there are more sectors in which demand is lower. This leads to a lower demand for production workers due to an increase ing. We summarize our discussion of theR-curve in the following
Lemma 2 At low levels of g the resource constraint can be positively sloped in (g; p)-space. This slope eventually becomes negative for larger g, and then approaches the highest feasible growth rate g = F1:
Proof. See Appendix.
Having discussed the shapes of the two curves we can now consider the general equilibrium of this model. In this equilibrium both the resource constraint and the zero pro…t condition have to be satis…ed which is the case at the point of intersection E in Figure 1. That …gure suggests that the equilibrium is unique if R hits the p-axis more to the right than ¦ does. Indeed, this is the condition that a unique equilibrium exists.
Proposition 1 If and only if the exogenous parameters satisfy 1 +qbqF 1¡e½¡¢½ <
p1¡°qbq1 + 1, there exists a unique balanced growth equilibrium.
Proof. See Appendix
6.2 Stagnation and Multiple Equilibria
Proposition 1 implies that a necessary condition for multiple equilibria is that 1 +
qF bq
½
1¡e¡¢½ ¸ p
1¡°qbqL + 1 . The condition is not su¢cient because it might be that the curves do not cross. In that case an equilibrium exists, but the economy exhibits no growth.
Stagnation The zero-pro…t condition above states that in equilibrium the value of an innovation has to be either equal or smaller than the costs of an innovation, the latter being an equilibrium because there is no incentive to enter the research sector. In other words, in this equilibrium no innovations take place and the economy stagnates. In terms of Figure 1, the stagnation point lies o¤ (more precisely, to the left of) the ¦-curve. As the stagnation equilibrium satis…es the full employment condition, the equilibrium point lieson theR-curve. Taken together this means that the stagnation equilibrium is located where theR-curve intersects the horizontal axis (atg = 0).
Ifn=N, the R-curve hits the horizontal axis in the right part of Figure 1. This occurs atp=p
1¡°qbq1 + 1which must be larger than1otherwise the casen=N is not feasible. In the latter situation, n < N, the waiting time ± can no longer
be used as the endogenous variable, since Ẹ will be necessarily in…nity as there is no growth. Instead, the resource constraint has to be solved for n=N, the share of available products that is acutally consumed. In such an equilibrium there are …rms that know how to produce the goodsi 2 (n; N] but no production ever takes place since demand given the (stagnant) income level is too small.
Multiple equilibria Multiple equilibria arise when the condition in Proposition 1 is violated and the two curves cross. We then have three equilibria: the stagnation equilibrium and the two points of intersection of the Ặ- and theR-curve. There are two potential sources of multiplicity in this model that give rise to complementarities in the innovation process. The …rst is…nite patent length. The second is thedynamic demand externality due to hierarchic preferences.
To study the sources of multiplicity in more detail, it is convenient to compare the behavior of an economy where consumers have homothetic preferences (that is when ° = 0) to the more interesting case when we allow for hierarchic preferences which are non-homothetic (° >0).
Figure 2
Multiple equilibria with symmetry (° = 0). With symmetry each good faces the same demand, hence all monopolistic prices are equal top >1:Note that a situation where p = 1 andẸ > 0 cannot arise in an equilibrium with positive growth since a new good isimmediatelypurchased in the same amounts as all other goods supplied by the monopolists.
When° = 0, the zero pro…t condition and the resource constraint read F
bq = (pâ1)21âeâđạ
ạ ; and (26)
1 =gF +bqhp2eâgđ+p(1âeâgđ)â1i: (27) The slope of the zero pro…t condition Ặ is now unambigously positive because the above demand externality does not arise: higher growth is not associated with higher future demand; instead demand jumps from zero to a positive level and stays there until the patent has expired. (Thereafter demand makes a further jump due to the fall in prices that occurs as soon as the competitive producers take over the market;
needless to say, this demand increase is irrelevant for the innovations incentives). In other words, whenever there are higher prices to charge, new innovators are attracted that raise the growth rateg:
The resource constraintR, however, still has an ambigous slope. A higher growth g not only raises the demand for labor in research but also decreases the demand for
production labor. The larger fraction of monopolistic markets imply high prices on more markets leading to lower aggregate consumption demand. When patent length is in…nite, this complementary vanishes. In that case changes in the growth rate cannot a¤ect the market structures because all markets are monopolized. This point has been explored by Laussel and Nyssen (1999) who showed that multiple equilibria can arise in a standard endogenous growth model with quality improvements and
…nite patent length.
Multiple equilibria with hierarchic preferences (° > 0). With hierarchic prefer- ences the situation is quite di¤erent. Multiple equilibria can arise even when there are in…nitely lived patents because the ¦-curve is not necessarily monotonic. With symmetry (° = 0), the ¦-curve is monotonically increasing, but with a steeper hi- erarchy (an increase in °) the ¦-curve eventually bends backwards for low levels of g. The reason is the above mentioned demand externality that characterizes the innovation process. With hierarchic preferences the demand of an innovator grows over time as consumers get richer and can satisfy wants of lower priority to larger degree. If innovators expect high growth they expect that the demand for their prod- ucts expands more quickly so that future prices, quantities, and pro…ts are larger.
This increases the incentives to innovate. If innovators expect low growth, pro…t expectations and the resulting incentives to innovate are correspondingly low. Obvi- ously, thisdemand externality is at work even in a fully monopolized economy where patents last in…nitely long.12
It is worth noting that the intercept of the resource constraint with the horizon- tal axis, which is given byp
1¡°qbq1 + 1 shifts to the left with an increase in °:
According to Proposition 1, this implies that multiple equilibria become more likely.
Unlike to the symmetric case,preemptive patenting, i.e. an equilibrium with± >0 (and hence p = 1) is possible. If the costs to innovate are so low and hierarchy is steep enough, it is eventually the case, that with higher growth,¦reaches the region with ± >0:
We summarize our discussion in this section in the following
Proposition 2 If and only ° > 0; that is with a hierachic structure of preferences, (i) multiple equilibria can arise even when patent length is in…nite; (ii) an equilibrium with ± >0, that is preemptive patenting, is possible:
Moreover, multiple equilibria are more likely to arise when ° >0:
12A similar mechanism is present in the model of Zweimüller (2000). This paper studies the impact of inequality on the aggregate innovation rate. In that model, consumers buy one unit of each product and a demand e¤ect of higher growth rates arises because the waiting time of the innovator for the demand of the poor becomes shorter. The assumptions in that model are more restrictive than in the present model: consumers buy only one unit of each good, and the mark-up of an innovator is assumed to be exogenous.
7 Structural Change
We have now seen that the balanced growth equilibrium in the hierarchic demand model of product innovations di¤ers signi…cantly from the standard monopolisitic competition model. In this section we will focus on thestructural change that takes place along the balanced growth path. We will see that the hierarchic demand model allows us to discuss the notion of structural change in a meaningful and non-trivial way.
For a given product, the consumption levels, the consumption expenditures, and the expenditure shares do not change in proportion with income. For instance, we will show that the ratio of consumption expenditures of an arbitrarily chosen range of products relative to total consumption expenditures rises initially, reaches a maximum, and then starts to decrease. Moreover, there is a tendency to devote a disproportionate share of theadditional expenditures towards goods that have lower consumption priorities (’services’) and only a small share to the most basic goods (’food’). Whether or not the expenditure share of intermediate priority (’manufac- tures’) increases or decreases depends on the stage of development. This pattern of change in the expenditure shares is also re‡ected in the allocation of labor between the food, the manufacturing, and service sector.
Structural change means there is continuous change in the composition of demand and production and to make these changes explicit we are primarily interested in the income elasticities of demand for each good i. Moreover we want to make explicit the corresponding changes in the price, expenditures for good i and the production labor necessary to produce goodi. We will calculate these elasiticities in turn. The results in Section 2.3 can be used to derive these elasticities. Consider …rst the income elasticity of consumption demand for goodiwhich is at the heart of the problem. We are interested in the change of theequilibrium quantityc(i; t)as a result of a change in income. This incorporates both the direct and the indirect e¤ects of a change in income on the demand for goodi. The direct e¤ect is the change in demand holding the price of goodiand all other prices constant; and the range of the supplied goods constant. Clearly, the direct e¤ect of an increase in income leads to more demand for good i. This is simply the result of our speci…cation of preferences which rules out the possibility of inferior goods. There are two further, more indirect e¤ects of a change in income: the …rst works via the range of available products and the second works via the prices. First, in equilibrium a higher income level is associated with a larger menu of purchased products. Ceteris paribus, a larger menu of purchased goods tends to decrease the demand for good i because expenditures are spread out over more varieties. The second e¤ect captures the fact that the willingness to pay increases as incomes grow. This implies higher prices as long as good i is still supplied by a monopolist. After patents have expired, the monopolist gets displaced
