Bounded learning by doing, inequality, and multi-sector growth: A middle-class perspective

Journal Pre-proof Bounded learning by doing, inequality, and multi-sector growth: A middle-class perspective

Alain Desdoigts, Fernando Jaramillo

PII:

S1094-2025(18)30394-6

DOI:

https://doi.org/10.1016/j.red.2019.10.001

Reference:

YREDY 960

To appear in:

Review of Economic Dynamics

Received date:

18 July 2018

Revised date:

29 September 2019

Please cite this article as: Desdoigts, A., Jaramillo, F. Bounded learning by doing, inequality, and multi-sector growth: A middle-class perspective. Review of Economic Dynamics (2019), doi: https://doi.org/10.1016/j.red.2019.10.001.

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Bounded Learning by Doing, Inequality, and Multi-Sector Growth: A Middle-Class Perspective Alain Desdoigts† and Fernando Jaramillo‡ † Universit´ e ‡

Paris 1 Panth´eon Sorbonne, IEDES and UMR D&S. [email protected].

Corresponding author: Universidad del Rosario, Calle 12c # 4-69, Bogot´ a, Colombia. [email protected].

Abstract This paper develops a multi-sector model of middle-class-led economic growth, whereby (i) learning by doing interacts with scale economies and nonhomothetic preferences giving rise to endogenous growth, and (ii) the ”middle class” label is an endogenous outcome of the model, depending on past economic growth. The emphasis is placed on the entire income distribution, which affects the composition of demand - range of goods consumed - and in turn, the speed and extent of the learning process in the range of goods produced. Learning is sector specific, bounded from above, and constrained by a minimum scale restriction. It starts with a switch from traditional to modern technologies, understood as structural change. A higher share in the purchasing power of the middle class expands the market size for modern goods, and generates more learning in non-mature modern technologies, contributing to productivity gains, and under certain conditions (e.g. a significant learning potential), to sustained growth. Eventually, we make the case for a strong middle class; that is, redistributive policies towards the middle class and the poor. Key words: Learning by doing, Middle-class-led consumption, Inequality, Multi-sector growth, Income redistribution. JEL Classification: D31, L16, O33, O40.

1

Introduction

Wondering about the relative strength of the middle class and the prominent role it has to play in economic development has become increasingly common in the press, as well as in policy circles. The issue has also been widely discussed Preprint submitted to Elsevier Science

22 October 2019

by researchers in the social sciences, including in economics. 1 In this paper, we develop a multi-sector model of economic growth in which technological progress is driven by learning by doing in expanding emerging sectors that benefit in particular from middle-class-led consumption. Learning interacts with scale economies and nonhomothetic, hierarchical preferences, thus giving rise to endogenous growth. Specifically, low-income households consume only products that have a low income elasticity of demand. Thus, sectors that produce low income elasticity goods must cope with a large demand. As a consumer market grows, dependent on the distribution of income, the expanding sector may experience technology switching from smallto large-scale production or, in other words, from traditional (CRS) to modern (IRS) technology as formulated by Murphy, Shleifer and Vishny (1989). It is only after the technology has changed from CRS to IRS that learning can take place. Learning and productivity gains are therefore constrained by a minimum scale restriction. Learning is also assumed to be bounded from above so that the accumulated experience in a specific technology matures. Finally, the experience that the economy has accumulated across all sectors contributes to aggregate productivity growth. 2 Hence, the core mechanisms of our middle-class-led growth model are threefold. Firstly, learning takes place in an intermediate range of sectors, where 1

Development economists such as Adelman and Morris (1967) and the economic historian Landes (1998) exemplify the emergence of a middle class as a key feature of the industrial revolutions experienced by Western economies in the 18th and 19th centuries. Using post World War II cross-country data, Easterly (2001), among others, finds that relatively homogeneous middle-class countries have on average more income and growth. Partridge’s analysis (2005) of panel data from U.S. states also illustrates the importance of a vibrant middle class on long-run growth. Last but not least, it is now a well-established fact that the most important engine of growth for countries of the South is their domestic market, their middle class increasing both in size and share of income. See, for instance, Ravallion (2010) who claims that economic growth in developing countries typically comes with an expanding middle class, as well as the Human Development Report (The Rise of the South, UNDP 2013). 2 Kuznets (1955, 1966) provides an early analysis of the relationship between inequality and structural change as characterized by an increase in the scale of productive units, shifts in the structure of consumption, and the diversification of products. More recently, Jones and Romer (2010) emphasize increases in the extent of markets and IRS in their list of ”new” Kaldor facts as key features of modern economic growth, in particular through urbanization, which drives the emergence of mass consumers. See, for example, Ades and Glaeser (1999) for empirical evidence that supports the importance of the extent of the market and aggregate demand in having fostered growth in developing countries in the past century and nineteenth century U.S. states.

2

demand is high enough to allow a firm to take advantage of scale economies, and where the learning possibilities have not yet been exhausted. Secondly, how much a sector contributes to cost reduction is determined by its accumulated experience, which results from cumulative output driven by its market size evolution, and in turn depends on the shape of the income distribution. Thirdly, sustained growth requires that technology providing increasing returns be implemented in sectors one after the other. Knowledge exchange, resulting in productivity gains, must create sufficient additional demand for higher-ranking goods in the consumer’s hierarchy of needs for learning to never cease. Only then will the economy move forward. 3 Two original features of our model stand out. Firstly, both the size of the middle class and its total income share are simultaneously an input to our model and an outcome of past economic growth. Neither do we define the middle class in an ad hoc manner as is often the case in the theoretical and empirical literature addressing the issue, 4 nor do we consider it as a potentially influential actor in relation to political instability or changes in institutions (see Alesina and Perotti 1996, Birdsall 2010, and Halter et al. 2014, among others). Hence, the ”middle class” label is an endogenous, purely economic outcome of the model. 5 Secondly, our model shows how the distribution of income/skills across middle-class households determines the intensity of sector-specific learning by doing, and whether learning takes place across a broad range of sectors or not. In particular, we bring to light a mechanism through which an invertedU relationship between inequality and sustained growth emerges, which fits well with the empirical findings of Banerjee and Duflo (2003), Chen (2003), and Grigoli and Robles (2017). On the one hand, in a highly unequal economy, where an elite owns nearly all of the wealth, modern technologies are confined to a small range of sectors with limited opportunities for learning. On the other hand, an economy with an almost perfectly egalitarian income distribution needs to have a large population and/or a low minimum scale restriction to be able to ignite the learning process and achieve positive growth in the long run. The inverted U-shaped curve reflects the trade-off between labor allocation to a broad range of sectors in which learning may take place, and the speed of the learning process in each sector. Such a trade-off is, for the most part, influenced by the distribution of income/skills, which governs 3

Lucas in his seminal 1993 paper entitled ”Making a miracle” emphasizes the critical importance of such a continual process of change for achieving sustained growth. 4 For example, in the empirical literature: between 75% and 125% of median household income in Birdsall et al. (2000), ”the middle 60 percent” of the per capita distribution of consumption in Easterly (2001), or alternatively in developing countries, between $2 and $13 a day at 2005 PPP thresholds in Ravallion (2010). 5 For politico-economic models of growth that exhibit the endogenous emergence of a middle class, see Acemoglu and Robinson (2000) and Bourguignon and Verdier (2000), among others.

3

the duration and the extent of the learning process. Our model also has implications for growth-enhancing strategies by means of redistributions of income. 6 At the core of our results is that a higher share in the purchasing power of the middle class (i) expands the market size for modern products, and (ii) generates more learning in non-mature IRS technologies. This contributes to increased productivity and potentially to sustained growth. Nevertheless, if the middle class is too small, growth may only be short term and the economy will stagnate in the long run. More generally, we establish under what conditions income distribution and domestic market size help to promote sustained growth through a continuous process of structural changes involving all sectors of economic activity. Our study is thus related to the theoretical work carried out by Zeira and Zoabi (2015) who interpret structural changes as the transition between traditional and modern technologies in a growing economy where growth is driven by rising productivity in the modern sectors of economic activity only. However, they treat countries as small, open economies that benefit from new technologies invented elsewhere, and as such assume that productivity in modern sectors rises exogenously over time. Demand-led endogenous growth models with hierarchic preferences, which are closely related to our theoretical framework, have been developed by Matsuyama (2002), Foellmi and Zweim¨ uller (2006), and Foellmi, Wuergler and Zweim¨ uller (2014), among others. We tread in their footsteps here. On the supply side, all three models also exhibit increasing returns and feature both price and market-size effects. Matsuyama (2002) offers a model where there is only one possible production technology, namely CRS, on one hand. On the other hand, bounded sectorspecific learning by doing in perfectly competitive markets leads to external increasing returns to scale at the sector-wide level. This rules out market concentration dominated by large firms, as well as the role they play in modern economic development. 7 Firstly, a critical mass of rich households is needed for an industry to take off. This results in lower prices and triggers a sound trickle-down process, so that lower-income households may purchase additional goods, which further lowers prices, potentially giving rise thereafter to a trickle-up mechanism, and so on. Secondly, the extent of the trickle-down process depends on the income distribution. It starts at the top end of the distribution, but may be interrupted if the income gaps are too large. The au6

For example, Littrell et al. (2010) argue that the strength of the American middle class through the 1950s and 1960s can be ascribed to the New Deal policies that followed World War II. 7 It is now well established that learning opportunities in an industry have an impact on market structure with a tendency towards the emergence of large firms, which in turn leads to increased market concentration. See Thompson (2010) for an in-depth review of the theoretical and empirical literature on learning by doing.

4

thor draws a useful analogy with a domino effect. The distance between two dominos, for example, the rich and the upper middle class, must be neither too large nor too small in order for the chain reaction not to be interrupted too early, even if this is at the expense of the poor, who may be used to narrow the income gap between the rich and the upper middle class. Foellmi and Zweim¨ uller (2006) study a two-class (rich and poor) society, and develop an innovation-based model, where goods are produced in monopolistic firms under IRS, and only rich households are initially willing to pay high prices for newly invented luxury goods. Thus, some degree of inequality defined by a high income ratio of rich to poor and/or a high concentration of income among the rich is needed to promote incentives for innovation. However, with too much inequality, the market for new goods is small, and these goods are unlikely to move into mass markets, which inhibits growth in the long run. As long as the price effect prevails over the market-size effect, a regressive transfer towards the rich (i.e. more inequality) fosters growth. If they were to introduce a middle class, either a transfer from middle-class to rich households or from poor to middle-class households would promote economic growth. Foellmi et al. (2014) distinguish between the invention of new high-quality products only accessible to the rich, and process innovations that transform luxury goods into low-quality mass products, thus becoming affordable to the poor. A household never consumes two qualities of the same product. Hence, either middle-class households consume only high-quality goods and are therefore almost as affluent as the rich, or they consume only low-quality goods and are almost as deprived as the poor. As long as the main driver of growth is product innovation, redistribution from the rich to the middle class reduces growth. It is only when process innovations become the driving force due to the income distribution that the positive effect of such a redistribution on growth becomes more likely. The results in Matsuyama (2002) and Foellmi and Zweim¨ uller (2006) stand in contrast with our core finding, outlined above. In these papers, the product life cycles and transmission channels through which inequality affects economic growth are different from ours. Their results emphasize both the importance of inequality at the top end of the income distribution and the distance between the different income groups in order to help an economy to form mass markets. Unlike these papers, we underline the importance of the economic situation of the middle class and the poor, and in particular the middle-class size and total income share, to account for the spread of modern sectors either to developing countries, where R&D and the scope for realizing scale economies remain limited, 8 or to any country where the trickle-down process may be ineffective 8

Ravallion (2012) finds, based on a sample of 90 developing countries, that those starting with a small middle class by these countries’ standards face a handicap, which hinders both growth and poverty alleviation.

5

due to too large an income gap between the wealthy elites and the middle class. We thus view our theory as complementary to theirs. It has a unique, special feature: the dynamic interplay between structural change (i.e. switch to modern technologies developed and implemented by large firms) and the share of the middle class’s purchasing power, both of which are interrelated endogenous outcomes of the model. The rest of the paper is organized as follows. In Section 2, we present the model. Section 3 discusses our set up regarding middle-class-led learning by doing and knowledge-based productivity growth. Section 4 characterizes the emergence of a middle class as an outcome of past economic growth and the steady-state growth rate. In Section 5, we examine the nexus between inequality and, in particular, the strength of the middle class and growth. Section 6 concludes.

2

The Model

Our model builds on the seminal static framework of Murphy, Shleifer, and Vishny (1989). It takes the ”high development theory” of early development economists such as Rosenstein-Rodan (1943) and Hirschman (1958) seriously, whose ideas stress the complementarity of demand between sectors on one hand, and market size and scale economies on the other hand as primary causes of economic growth (Krugman 1995). 9

2.1

Households’ Nonhomothetic Preferences, Distribution of Skills, and Budget Constraints

It is assumed that all households have the same preferences. They are modeled via a utility function that is defined over a continuum of indivisible goods 9

Desdoigts and Jaramillo (2009) set up a static two-country trade model, fairly similar in its premise. In a closed economy, all income distributed to the domestic middle class returns as demand addressed to the home industries. Under free trade, the domestic middle class becomes a relative notion. A middle class household in its own country may be rich enough to belong to the upper class of the world income distribution. Conversely, a household may be rich by national standards but belong to the global middle class. Their study is therefore more about how trade leads the global middle-class purchasing power to interact with scale economies, depending upon asymmetries in domestic income distribution, productivity, and market size, eventually leading to (de-)industrialization.

6

q ∈ (0, ∞) such that, at each date t, V (t) =

 ∞ 0

1 ı(q, t)dq, q

(1)

where ı(q, t) is an indicator function that takes a value of either one or zero according to:

ı(q, t) =

⎧ ⎪ ⎨1

if the agent consumes q

⎪ ⎩0

otherwise

.

Thus, a household’s utility increases with the range of goods (0, q) it consumes and not with the consumption of a single good q. Consumption is hierarchically structured, that is, needs are ordered so that the proportion of income that households spend on lower-indexed goods or, equivalently, on goods with lower income elasticities of demand, decreases with a household’s income. Different goods have different priorities in consumption and richer households can consume more than the bundle of goods available to poorer households (see Bertola, Foellmi, and Zweim¨ uller 2006, Chapter 12). Labor is the only input and households are organized by skill level, denoted by γL, where L is the labor force and γ ≤ γ < ∞. Each household is identified by its γ -type. The skills of the labor force are distributed according to the cumulative distribution function F (γ), which is assumed to be exogenous and constant over time. Thus, our framework allows us to distinguish between societies dominated by a highly educated elite and lacking semi-skilled and highly skilled workers, and societies that are endowed with a relatively large number of engineers, managers, and skilled workers. At each date t, the labor income of a γ -type household is proportional to its skill level, which is given by γw(t)L, where w(t) is the wage per unit of skill. Should there be firms making a profit in the economy, to simplify the analysis and without loss of generality, it is assumed that this profit will be redistributed to households up to their type γ. 10 The nominal income of a γ -type household is therefore defined as the sum of its labor income and income arising from the shares it holds in profit-making firms: Y (γ, t) = γ [w(t)L + Π(t)] , 10

All calculations below are based on this simplifying assumption. However, our results are qualitatively comparable when assuming that financial assets are more concentrated than labor earnings, i.e. a larger share of profits is assigned to highskilled workers. See the discussion in Section 5 below. We thank a reviewer for having invited us to check the robustness of our results to this stylized fact.

7

where Π(t) denotes corporate profits. Define (0, q(γ, t)) as the span of varieties purchased by a γ -type household at date t. The budget constraint that describes the consumption options available to this household with income Y (γ, t) can be written as:  q(γ,t) 0

p(q, t)ı(q, t)dq = γ [w(t)L + Π(t)] = Y (γ, t).

(2)

At the household level, income (wages and profits) affects the extensive margin of consumption: the higher the household’s income, the broader the span of varieties consumed.

2.2

Technology, Market Structure, and the Equilibrium Price

If market demand is low, a good is produced using a high-cost constantreturns-to-scale technology. Instead, if market demand is sufficiently large, firms may undertake an exogenous sunk cost investment allowing them to produce at lower marginal cost. Learning by doing implies dynamic scale economies, which typically involve a form of fixed sunk costs. This favors largescale over small-scale production, which in turn fosters market concentration; that is, the emergence of dominant firms (Dasgupta and Stiglitz 1988). For example, the automobile sector today is dominated by a small number of large firms with a clear tendency to merge over the last twenty years, whereas a century ago, the sector was dominated by a large number of small-scale innovative entrepreneurs. Concentration is also likely to be more pronounced in sectors where the learning curve substantially reduces the average cost of production of the original template (e.g. aeronautics and semi-conductor sectors). Specifically, each good q can be produced with two production function technologies. The first, loosely referred to as traditional, exhibits constant returns to scale (CRS). One unit of good q requires α/A(t) units of labor, where α > 1 and A(t) is knowledge-based productivity at time t. In our framework, such a technology is used to produce goods that possess the artisanal touch of luxuries. The alternative production technology, often described as modern, exhibits increasing returns to scale (IRS): 1/A(t) units of labor are required to produce one unit of good q. Nevertheless, to produce at such a marginal cost, a firm must also be able to cover a fixed sunk cost equal to C/A(t) units of labor, which is assumed to be independent of market size. On the one hand, each good may be produced by a competitive fringe of firms with CRS technology. Then, the free-entry equilibrium number of firms satisfies the zero-profit condition, and the equilibrium price is equal to the 8

average cost: p(q, t) = p(t) = α

w(t) . A(t)

(3)

On the other hand, we show that if the distribution function F (γ) is smooth enough, there is a standard sub-game perfect Nash equilibrium for a monopoly implementing IRS technology, which consists of setting the price at the same level as the competitive fringe (see Appendix A). Under the assumption of price competition (e.g. Bertrand competition), such a monopolistic structure may arise despite free entry, with the monopolist possibly reaping profits (see Motta 2004, pp 75-77). 11 Thus, the dynamics of the economy do not hinge on relative price effects, through either a trickle-down process or vertical product differentiation, but on the change over time in households’ income relative to the consumer price index.

2.3

Market Demand and the Static Output Multiplier

Let us define the demand for each good q(t) as x(q, t) ≡ [1 − F (γ(q, t))] L, where γ(q, t) is the share of income of households whose purchasing power is high enough to purchase exactly (0, q(t)), and F (γ(q, t)) = 0 for γ(q, t) < γ. Whenever the demand is high enough to cover the sunk costs, a good q is produced by a monopolist that implements IRS technology. Specifically, as soon as L > C/(α − 1), a good q is produced using the IRS technology if and only if the demand for this good is such that the following minimum efficiency scale is satisfied: (α − 1)

w(t) w(t) C x(q, t) − C ≥ 0 ⇔ x(q, t) ≥ . A(t) A(t) α−1

11

(4)

The reality is that the market structure must lie between these two extremes, a competitive fringe and a monopoly. Our framework allows us to simply take into consideration the fact that more mature technologies typically involve substantial economies of scale, and thus are most often implemented by a small number of large firms. See, for example, Autor et al. (2017) who find that (i) industry sales in the US increasingly concentrate in a small number of ”superstar” firms, with a ”winner take most” feature, and (ii) such a pattern can also be observed in 14 European countries and other OECD countries. Moreover, if large firms appear to have played a standout role in driving strong and sustained growth (i.e. GDP per capita annual growth of more than 3.5% and 5% for 50 and 20 years, respectively) in the so-called outperforming emerging economies (Woetzel et al. 2018), most firms in least developed countries remain at a microenterprise stage and tend to produce traditional goods with established CRS technologies, which furthermore require limited skills (The Least Developed Countries Report, UNCTAD 2018).

9

At each date t, there is a marginal good q ∗ (t) such that the break-even condition x(q ∗ , t) = C/(α − 1) holds true. Within our dynamic model, such a sector of activity can be said to have attained the take-off stage, which recalls the big push switching of technologies `a la Murphy et al. (1989). Note that x(q ∗ , t) is exogenous and constant over time. Then, sectors that produce goods q(t) ≤ q ∗ (t) and q(t) > q ∗ (t), use the IRS and CRS production technology, respectively. We define γ ∗ (t) as the share of income held by this marginal household whose purchasing power allows purchasing exactly the range of goods (0, q ∗ (t)), where 



A(t) ∗ A(t) w(t)γ ∗ (t) [L + Π(t)/w(t)] Π(t) w(t) q (t) = = γ (t) L + = . , and p(t) α w(t) p(t) α (5) ∗

We also define the upper class to be the set of households of type greater than γ ∗ (t). There is a number N ∗ (t) of such households, where N ∗ (t) = [1 − F (γ ∗ (t))] L.

(6)

Their purchasing power allows them to buy goods produced with the IRS technology, as well as goods with higher income elasticity of demand that are produced using the CRS technology. We therefore have the following breakeven condition, which is time-independent. (We thus get rid of the t notation in both variables, N ∗ and γ ∗ .) x(q ∗ , t) = N ∗ = [1 − F (γ ∗ )] L =

C . α−1

(7)

Learning is constrained by a minimum scale restriction, which depends on the distribution of income, the size of the labor force, and technical parameters (i.e. fixed sunk costs and markup). The aggregate profit in the economy is the sum of profits realized by those sectors of activity that produce goods q in the range (0, q ∗ (t)):  q∗ (t)

Π(t) = p(t)

0

x(q, t)dq −

 q∗ (t) w(t) 0

A(t)

[x(q, t) + C] dq.

Combining the above profit expression with (5) and (6) yields Π(t) α − 1 w(t)L + Π(t) = T, p(t) α p(t)

∗

where T = L γγ γdF (γ) is defined as the share of income held by those households of type smaller than γ ∗ whose income is entirely devoted to purchasing 10

mass-consumption goods in the range (0, q ∗ (t)). From this definition, we deduce: Π(t) α−1 1 w(t)L = T, p(t) α 1 − α−1 p(t) T α where the output multiplier, which is independent of time, is defined by M=

1 1−

α−1 T α

.

As a consequence, the higher the real income per capita, denoted by y(t), the lower the share of income held by the upper class, or, equivalently, the higher the share of income held by those households of type smaller than γ ∗ . This increases monotonically with both the output multiplier and knowledge-based productivity, and takes the form y(t) =

Y (t) w(t)L + Π(t) 1 A(t) A(t) = = =M . α−1 p(t)L p(t)L α 1− α T α

(8)

For now, the setting of the model is very much inspired by the static framework in Murphy, Shleifer and Vishny (1989). We now turn to the two following questions: (i) what about the extent and speed of the learning process? and (ii) how do they depend on the income distribution and, in particular, the middle-class purchasing power?

2.4

Sectors of Activity Featuring Learning by Doing, Learning Spillovers, and the Middle Class

Similar to Stokey (1988), Young (1993) and Lucas (1993), technological progress is the by-product of economic activity in which learning by doing is the main source of accumulated experience, which in turn yields knowledge-based productivity gains. 12 We also allow for transfers of knowledge across sectors. Therefore, learning may not necessarily be bounded at the aggregate level and all sectors of activity have a chance to benefit from productivity gains. 13 In our dynamic framework, sectors grow at different rates. At each date t, learning leads to an accumulation of experience at the sector level, denoted 12

Argote and Epple (1990), Bahk and Gort (1993) and Thompson (2001) among many others provide evidence of a significant impact of learning by doing on productivity. 13 For example, the primary sector that produces necessary products is nowadays benefiting most from the knowledge accumulated in other sectors, such as mechanization of agriculture, chemistry, agriculture v3.0 and geolocation, etc.

11

by E(q, t). The level of knowledge-based productivity in the economy, A(t), then equals the sum of experiences gathered over time by all sectors that have implemented IRS technologies. Specifically, the accumulated stock of knowledge at time t is defined by A(t) =

 q∗ (t) 0

E(q, t)dq.

(9)

Experience increases with cumulative production levels over a time interval (t∗ (q) , t), where t∗ (q) denotes the date at which the IRS technology was adopted for the first time by the sector that produces good q(t). Learning in a sector of activity is constrained by a minimal scale restriction. At each date t, among those sectors of activity that use the IRS technology, we can distinguish between two groups of sectors. The first group includes sectors where learning is depleted and has reached the upper bound being defined by

whereas the second group includes those in which the accumulated experiE, ence has not yet reached the upper bound. This group of non-mature sectors



lies within the range of activities (q(t), q ∗ (t)), where q(t) is defined by

t) = E(q,

E

⇔

 t q) t∗ (

v)dv = 1/λ, x(q,

 t

where λ is the learning rate. As soon as

t∗ (q)

(10)

x(q, v)dv ≥ 1/λ, learning in sector

that is, the sector of activity has reached q has reached the upper bound E; ∗

maturity. In addition to q (t), there exists another marginal sector q(t), which is defined as the most recent sector of activity at time t where learning has

declined to zero. Solving (10) yields a solution for q(t). It follows that, sim∗ ilar to the γ -type household, there is another key marginal household of type γ (t) whose purchasing power allows buying exactly the range of goods

(0, q(t)). Thus, our framework allows us to provide an economic definition of who belongs to the middle class, although we acknowledge that the term may mean different things depending on one’s perspective.

Definition 1 If households of type γ ∗ and above constitute the upper-income class, as long as γ < γ (t) ≤ γ ∗ , mass consumers (the rest) may now be divided into a low-income class, which includes households of types between γ and γ (t), and a middle-income class where we find households of types ranging from γ (t) to γ ∗ . If γ (t) ≤ γ, the economy is composed of only two classes: the upper-income class and the rest. Learning eventually occurs only in an intermediate range of modern sectors of

activity (q(t), q ∗ (t)), that is, those for which there is enough demand to prompt the learning process but that have not yet exhausted the learning possibilities. In contrast to the empirical literature on this subject, which identifies the middle class by setting limits either in the people space (e.g. the middle 60 12

percent) or the income space (e.g. 75%–125% of the median income), we stress the importance of the size of the middle class in conjunction with its income share as key determinants of sustained economic growth (see, discussions by Atkinson and Brandolini 2013, and Dallinger 2013, among others).

3

Middle-Class-Led Consumption, Learning Curve, and Growth

3.1

Distribution of Income, Production Diversification, and Aggregate Economic Growth

Knowledge-based productivity A(t) is a function of the experience accumulated by all sectors of the economy in which learning occurs and has already taken place at time t. We thus rewrite (9) as A(t) =

 q∗ (t) 0

E(q, t)dq = E q(t) +

 q∗ (t)

q (t)

E(q, t)dq.

Changes in A(t) are the result of the experience accumulated in the economy at time t:

˙ = A(t) and

 q∗ (t)

q (t)

⎧ ⎪

⎨ λEx(q, t)

˙ t) = E(q, ⎪

˙ t)dq = λE

E(q,

q (t)

x(q, t)dq,

if E(q, t) < E and q(t) < q ≤ q ∗ (t)

⎩0

otherwise

 ⎧ t ⎪ ⎪ ⎨ x(q, t)/

˙ t) E(q, = E(q, t) ⎪ ⎪ ⎩

 q∗ (t)

t∗ (q)

⇔



x(q, v)dv if E(q, t) < E and q(t) < q ≤ q ∗ (t)

0

.

otherwise

Accumulated experience is a by-product of the economic activity in those sectors where there is mass consumption. It is worth noting here that given x(q, t) = [1 − F (γ(q, t))] L, the evolution of demand in sector q is at the origin of its learning curve, which explicitly depends on the survival function of the distribution of income. This yields the following productivity rate of growth arising from economy-wide learning by 13

doing: g(t) =

 q∗ (t)  q∗ (t) ˙ x(q, t) [1 − F (γ(q, t))] L A(t) = λE

dq = λE

dq. A(t) A(t) A(t)

q (t)

q (t)

(11)

At each time t, the growth rate of knowledge-based productivity equals the amount of labor (excluding fixed sunk costs) required to produce quantity x(q, t) in sectors using the IRS technology and in which learning takes place. Thus, unit cost reductions as a result of increased knowledge arise from increased cumulative output brought about by both learning and economies of scale. Whereas q ∗ (t) evolves over time, recall that γ ∗ is constant over time and equal to q ∗ (t)/y(t)L (see (2) and (8)). We use the change in variables γ(q, t) = q/y(t)L to rewrite (11): g(t) = λE

y(t)L  γ ∗ [1 − F (γ)] Ldγ. A(t) γ (t)

(12)

This discussion allows us to establish the following first important result. Proposition 1 The rate of growth of A(t) is positively associated with the output multiplier, i.e. the share of income held by those households of type smaller than γ ∗ . It increases with the proportion of income held by the middle class T − T (t) and, more specifically, with the share of total income spent in the

intermediate range of modern sectors (q(t), q ∗ (t)) where learning takes place. It is equal to: g(t) = λE

where M/α = y(t)/A(t),



M  ∗ ∗ (t) + T (t) , L γ N + T − γ (t)N α (t) N

= [1 − F (γ (t))] L, and

T (t)

=

(13)



γ (t)

γf (γ)Ldγ.

γ

Whenever q(t) < q(t), which implies that γ (t) < γ, we have F (γ (t)) = (t) = L and T (t) = 0. f (γ (t)) = 0, which yields N  γ∗

Proof. Replacing (8) in (12) and integrating by parts

γ (t)

[1 − F (γ)] Ldγ,

we obtain (13). Figure 1, which is inspired by Atkinson and Brandolini (2013), is comprised of two graphical representations: on the left, the Lorenz curve associated with the cumulative income distribution F (γ), which is shown on the right. Similar to the model of Murphy, Shleifer, and Vishny (1989) without learning, γ ∗ N ∗ + T is the proportion of total income spent in sectors that have implemented the 14

Fig. 1. Middle-class identification and growth. Right quadrant: cumulative incone distribution F (γ) and lower and upper cutoffs γ

(t) and γ ∗ , respectively. Left quad

rant: Lorenz curve, and middle-class size N (t) − N ∗ and income share T − T (t).

IRS technology (see the left-hand panel on Figure 1). The growth rate at time t depends on both the output multiplier and the proportion of income that is spent in sectors where learning takes place and has not yet been exhausted;

that is, given (8), on total income spent in the range of sectors (q(t), q ∗ (t)). Therefore, (13) provides us with a relationship between growth and middleclass-led consumption, where growth depends on the share of aggregate real income held by those households of type between γ (t) and γ ∗ . The proportion

of total income spent in the intermediate range of sectors (q(t), q ∗ (t)) where (t) − N ∗ learning takes place depends on both the size of the middle class N and the share of total income held by the middle class T − T (t). Intuitively, a given middle-class share of total income may be high, but concentrated among either a large or a small number of middle-class consumers, depending on the shape of the income distribution. Thus, the entire income distribution and, more specifically, the middle class identified by its lower and upper bounds, γ (t) and γ ∗ , respectively, determine the extent of the learning process. The right-hand panel on Figure 1 shows where the expenditure of various consumers goes, and represents the sectors of activity where learning takes place. Indeed, as alluded to earlier, we have γ(q, t) = q/y(t)L. This also allows us to determine the product life cycle, which goes through three stages. In the first stage, the demand is too low for a firm to be able to implement IRS technology, i.e. to incur fixed sunk costs. At this stage, no learning has taken place yet, and firms remain at a small or microenterprise stage. In the second phase of the product life cycle, the product is produced with increasing returns and a middle-class-led learning phase takes place, accompanied by high market 15

concentration. Finally, learning stops and the good becomes affordable to a large population of consumers, including the most deprived.

3.2

Distribution of Income and the Speed of Learning

In the long run, as long as there is positive growth, a sector of activity q begins producing good q with the CRS technology. At this stage, q can be considered a luxury good. It is produced for a minority of consumers, namely the highestincome households N ∗ of type greater than γ ∗ . Then, it may go through a stage where it becomes, at some point in time, the marginal sector q ∗ (t). It produces q in the amount x(q ∗ , t) = [1 − F (γ ∗ )] L. From then on, it starts to accumulate experience, the magnitude of which depends on aggregate economic growth and the distribution of income. Its apprenticeship may continue until the point

When the of exhaustion, that is, until it has reached the upper bound E. sector has exhausted its potential for learning, it becomes a marginal sector

t) = [1 − F (γ

(t))] L of activity q(t) whose good is purchased in quantity x(q, (see the right-hand panel on Figure 1).

that is, to reach How long does it take for a sector to reach the upper bound E; maturity? What is the duration of the learning period? The analysis below can be seen as a survival analysis in which we are interested in the length of time before the upper bound is reached by a sector once it has implemented the IRS technology.

Note first that the experience accumulated by one sector of activity depends on its market size and growth rates over the period prior to the point at which

Thus, changes in the demand structure depend on aggregate it reaches E. economic growth and the income distribution. As long as aggregate economic growth is taking place, the growing demand in a sector of activity depends on the hazard rate of the income distribution: if y(t) increases by 1% during the learning process, the demand for good q increases by γ(q, t) times the hazard rate f (γ(q, t))/ [1 − F (γ(q, t))]. More specifically, the income elasticity of demand for any good q at time t depends on the hazard rate of the income distribution f (γ(q, t)) / [1 − F (γ (q, t))], which is the number of households of type γ(q, t) relative to the number of households whose income is greater 16

than γ(q, t). Specifically, we have 14 .

.

x(q, t) y (t) f (γ(q, t)) / = γ(q, t) . x(q, t) y(t) 1 − F (γ (q, t))

(14)

Second, recall from (10) that for all q ∈ (q (t) , q ∗ (t)), the experience accumulated at time t by the sector of activity indexed by q is described using the survival function 1 − F (γ(q, t)):  t  t E(q, t) x(q, v)dv = λ [1 − F (γ (q, v))] Ldv. = λ t∗ (q) t∗ (q) E

Learning duration is determined by the law of motion of demand for q over the learning period. Specifically, the learning process in the sector of activity

where q that has reached maturity E at time t has a duration of t − t∗ (q), ∗

t (q) is such that:  t  t

t) E(q, 1

v) dv =

(v))] Ldv = , x ( q, [1 − F ( γ = 1 ⇔ λ q) q) t∗ (

t∗ (

E

 

with γ (v) = γ ∗ exp −

 v

q) t∗ (

m) dm , g (q,

(15) (16)

.

m) = x(q,

m)/x(q,

m) is the rate of change that the sector of activity and g (q, q experiences at time m. Recall that when γ (v) < γ, then F (γ (v)) = 0.

Firstly, learning duration is determined by the law of motion of demand for

t), where changes in demand depend on the q over the time interval (t∗ (q),

which itself evolves over the learning process income elasticity of demand for q, . according to both aggregate economic growth y (v) /y(v) and the hazard rate f (γ (v))/ [1 − F (γ (v))]. Secondly, implicit differentiation of (15) leads to the following inequality, where the higher prior economic growth in the q sector, the shorter the duration of the learning process:

d(t − t∗ (q))   ≤ 0, ∀t∗ (q)

≤ tˇ ≤ t.

tˇ dg q, 14

(17)

Note that .

x(q, t) =

∂γ (q, t) ∂ [1 − F (γ (q, t))] L = −f (γ(q, t)) L , ∂t ∂t

where γ (q, t) = q/y(t)L implies that ∂γ (q, t) /∂t = −(q/y 2 (t)L)(∂y(t)/∂t). We get .

.

x(q, t) = f (γ(q, t)) γ (q, t) L

.

.

y (t) x(q, t) f (γ(q, t)) y (t) ⇒ = γ(q, t) . y(t) x(q, t) 1 − F (γ (q, t)) y(t)

17

Finally, the shape of the income distribution in the range (γ (t), γ ∗ ) or, equiva

lently, (q(t)/y(t)L, q ∗ (t)/y(t)L), determines the duration of the lifelong learning process in a sector of activity: the higher the growth rate of demand, the shorter the apprenticeship period and the sooner the sector reaches maturity.

4

The Middle Class as both an Input and an Outcome of Growth

It is now clear that the combined composition of aggregate demand and the learning process as determined by the entire income distribution implies that mass consumption at time t of the intermediate range of modern goods

(q(t), q ∗ (t)) is the engine of growth that enables an economy to keep moving forward. We now turn to the identification of the middle class (i.e. households of type ranking from γ (t) to γ ∗ ), which is derived from past economic growth.

4.1

The Size of the Middle Class Increases with Past Economic Growth

Recall that the upper cut-off, which demarcates the middle class from the upper class γ ∗ , is determined regardless of the state of economic growth. It depends on the income distribution, the size of the labor force, and both CRS and IRS technology cost parameters. Only the lower cut-off of the middle class depends on prior economic growth. In fact, the higher the past economic growth, the lower the lower bound of the middle class γ (t) or, equivalently, the larger the size of the middle class. (t) − N ∗ is an More specifically, at time t, the size of the middle class N increasing function of compound growth for good q over its learning period and, as a consequence, of past aggregate economic growth. To see this, note that implicit differentiation of (16) and use of (17) for γ (t) > γ yields: (t) d(γ ∗ − γ (t)) dN   >0⇒   > 0, with t∗ (q)

≤ tˇ ≤ t.

tˇ

tˇ dg q, dg q,

(18)

Hence, the delimitation of the size of the middle class and its relative position in the income hierarchy are explicitly derived from past economic growth. 18

4.2

The Steady-State Growth Rate and the Size and Share of Total Income of the Middle Class

We are now in a position to determine the aggregate growth rate and both the size and share of total income of the middle class in the steady-state equilibrium. First, it must be noted that in the steady state, we have (see (8)):

g=

A˙ (t) y˙ (t) q˙∗ (t) = = ∗ = constant. A(t) y(t) q (t)

Once it is recognized that the middle class is not only an input, but also an outcome of economic growth, we can establish the following proposition. Proposition 2 The size and share of total income of the middle class in the steady-state equilibrium depend on aggregate economic growth g, such that γ

is time-independent and satisfies the following equality:  γ∗ g  γ ∗ [1 − F (γ)] L ∗ ∗ = dγ = N ln(γ ) − N ln(γ ) + ln(γ)f (γ)Ldγ, λ γ

γ

γ = [1 − F (γ

)] L and T = with N

(19)

 γ∗

γf (γ)Ldγ.

γ

∗

= q(t)/y(t

exp(−g(v− (q))L, with y(v) = y(t∗ (q)) Proof. Note that γ ∗ = γ (t∗ (q)) ∗

∗

t (q)) for t (q) ≤ v ≤ t. This implies that γ (v) = γ ∗ exp(−g(v − t∗ (q)). Total differentiation of the latter leads to

dγ (v) = −γ ∗ exp(−g(v − t∗ (q))gdv = −γ (v)gdv ⇔ dv = −

dγ (v) . γ (v)g

By replacing in (15), we obtain  γ∗ [1 − F

γ  γ∗

Integrating by parts

γ

(γ)] L

γ

dγ =

g . λ

([1 − F (γ)] L/γ) dγ then yields (19).

The model therefore provides an example of an economic relationship between middle-class-led consumption and aggregate economic growth, where the former plays a key role in the latter, while past economic growth determines both the size and the income share of the middle class. In the steady-state, 19

we obtain the following solutions (g, γ ) to equations (13) and (19): ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ g/λ





M γ ∗N ∗ + T − γ + T

N g/λ = EL α ∗

= N ln(γ

∗

ln(γ

) + )−N



 γ∗

γ

.

(20)

ln(γ)f (γ)Ldγ

In the next section, we study these conditions in terms of inequality and the values of the various parameters for which there will be positive long-term economic growth, as well as the trade-off underlying the inverted-U relationship between inequality and growth.

5

Bounded Learning by Doing, Inequality, and Sustained Growth in the Long Run

From that point on, conditions regarding the parameters of both the CRS and IRS production functions, the size of the labor force, the income distribution, and last, but not least, the potential for learning in each sector are crucial to ensuring that there is sustained growth in the long run. This requires that certain conditions be met.

5.1

Necessary and Sufficient Conditions for Sustained Growth

The system of equations (20) can be written as follows:

g = Λ (g) = λEL



M ∗ ∗ (g) + T (g) γ N + T − γ (g) N for γ > γ, α

(21)

(g), and T (g) are implicitly defined by equation (19). where γ (g), N

Firstly, differentiation of equation (21) implies that 



(g) dΛ (g) λEL dT (g) dγ (g) dN (g) + γ

(g) =− M N + . dg α dγ (g) dγ (g) dg

Using equation (19) and both

N

=

[1 − F (γ )] L

and

T

=



γ

(22)

γf (γ)Ldγ, we

γ

(g)/dγ

(g), and dT (g)/dγ

(g), respectively, which are reobtain dγ (g)/dg, dN placed in (22) to obtain

dΛ (g) d2 Λ (g) > 0 and < 0. dg dg 2 20

Secondly, note from (19) that γ = γ ∗ ⇔ g = 0 ⇒ Λ (0) = 0,

and g→∞ lim γ (g) = 0 ⇒ g→∞ lim dΛ (g) /dg = 0 .

Hence, there exists one fixed point that exhibits positive long-run growth if 



dΛ (g)  dΛ (g)    > 1 and < 1 ⇔ E > E c (α, C, L, F ) , dg g=0 dg g=0,g=Λ(g) where E c (α, C, L, F ) ≡

5.1.1



(23)



α α−1 1 α T = ∗ . 1− ∗ γ L α γ LM

(24)

Economies of Scale and Learning Possibilities for Sustained Growth

Recall that learning may only occur in those sectors of economic activity using the IRS technology, and that it is bounded from above. The analysis then leads to the following proposition. Proposition 3 For there to be a fixed point with positive growth in the long run, E must be greater than a critical level E c (.), which decreases with the output multiplier M , and where E c (.) is a decreasing function of α and L and an increasing function of C. Proof. See Appendix B. The above proposition shows that relaxing the minimal-scale restriction (see the break-even condition (7)) will yield sustained growth for a lower upper

The higher the fixed cost and the lower the markup and bound for learning E. the size of the labor force, the smaller the ability of many sectors to exploit economies of scale, which in turn yields both a low output multiplier process and aggregate demand spillovers that are limited in scope. In such a situation, a higher upper bound for learning is necessary to offset the impact of either a fall in α and L or a rise in C.

5.1.2

Conditions of Inequality and Learning for Sustained Growth

As shown in (24), the critical level E c (α, C, L, F ) also depends on the distribution of income. More specifically, it depends on both the lower and upper bounds of the middle class; that is γ and γ ∗ , respectively.

5.1.2.1 Very unequal versus perfectly egalitarian distribution of income and growth Let us consider the limiting cases of either a very unequal or almost perfectly egalitarian economy. 21

Proposition 4 On the one hand, there can be no sustained growth in a very unequal economy. On the other hand, an economy that tends toward perfect equality exhibits a positive growth rate in the long run only if E > E c (.) = 1 + C/L. Proof. See Appendix C. On the one hand, in a highly unequal economy where a small number of households owns nearly all of the wealth, modern technologies are confined to a small range of sectors with limited opportunities for learning. Meanwhile, the upper class consumes a wide range of goods, many of which are produced with a traditional, artisanal method (CRS), plus a touch of technology A(t). Thus, aggregate demand spillovers only have a limited scope, and the ability to exploit economies of scale remains moderate. This results in a low multiplier effect. Furthermore, middle-class-led consumption determines the steady-state growth rate of the economy, which also depends on the share of income spent in sectors where learning takes place in each period. In a very unequal economy, there are no middle-market consumers, which explains the no-growth scenario in the long run. 15 On the other hand, an economy with an almost perfectly egalitarian distribution of income needs to have a large enough population and/or a low minimum scale restriction to be able to prompt the learning process and exhibit positive long-run growth. What happens in between these two extremes is ambiguous.

5.1.2.2 Inequality, income transformation and growth How income transformation influences the size and income position of the middle class is crucial to achieve sustained growth. The model has specific implications for growth-enhancing strategies by means of income transformations. 16 For example, let us assume the following income transformation based on Hemming 15

Note that Matsuyama (2002) also shows that either very unequal or very egalitarian societies may fail to generate growth. The rationale for why this occurs in his study are however qualitatively different from ours. On the one hand, with too much inequality, neither the trickle-down nor the trickle-up processes work because of too large income gaps. On the other hand, if there is too much equality, the economy is unable to boost the trickle-down process of economic take-off and stagnates in a poverty trap. In Foellmi and Zweim¨ uller (2006), recall that more inequality affects the value of an innovation through two competing effects: (i) a positive price effect that provides incentives for innovation, and (ii) a negative market-size effect, which makes the market for newly invented luxury goods small, and the move into mass markets for these goods unlikely. As a consequence, a very unequal society may also fail to generate growth, should the market-size effect prevail over the price effect. 16 Traditional tax distortions as raised in the optimum tax theory (e.g. labor-leisure choice and/or the consumption-savings trade-off) are not dealt with in our framework since the only input is the inelastic households’ labor supply.

22

and Keen (1983), where the purchasing power of the pre-tax middle class increases to the detriment of the initial upper class:

γ r ≡ ψ (γ, τ ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

γ

γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩



∀γ ∈ γ, γ



+ ϕ (γ, τ ) ∀γ ∈ [γ , γ ∗ ] ,

γ − ρ (γ, τ ) ∀γ ∈ ]γ ∗ , ∞[

where (i) ψ(.) is positive, monotone-increasing, and continuous with γ, that is ∂ϕ (γ, τ ) /∂γ > −1 and ∂ρ (γ, τ ) /∂γ < 1, and (ii) τ > 0 reflecting the intensity of income redistribution with ∂ϕ (γ, τ ) /∂τ ≥ 0 and ∂ρ (γ, τ ) /∂τ ≥ 0. In addition, we assume γ r = γ and γ ∗r = γ ∗ , which implies that ϕ (γ , τ ) = ϕ (γ ∗ , τ ) = ρ (γ ∗ , τ ) = 0, and

 γ∗

γ

ϕ (γ, τ ) f (γ) dγ =

 ∞ γ∗

ρ (γ, τ ) f (γ) dγ.

Such an transformation verifies the following single-crossing condition:  income  ∀γ ∈ γ, ∞ , we have γ  γ ∗ ⇒ ψ (γ, τ )  γ. As a consequence, the posttransfer income distribution is continuous and Lorenz dominates the initial distribution of income. The above transformation also satisfies the principle of rank-preserving income transfers from high to low households’ earnings capacities. Proposition 5 Let us assume that the single-crossing condition is satisfied and we allow only rank-preserving transfers. Firstly, as long as additional income is to the benefit of pre-tax lower- and middle-class households at the expense of the initial upper class, the output multiplier increases by raising demand in modern sectors of activity. A higher share in the purchasing power of the middle class also generates more learning in those non-mature modern technologies, which fosters economic growth. Secondly, additional income to pre-tax poorer households at the expense of the initial middle class leaves the multiplier unchanged. Demand shifts from sectors where learning is taking place to mature sectors where it has been exhausted, which weakens economic growth. Proof. See Appendix D for details. There is an improvement in economic growth as long as income transformation expands the total income initially held by either the middle class or the lower class at the expense of the pre-tax upper class. In both instances, the amount of output produced with IRS technologies increases with the decline in inequality. The post-transfer share of total income spent in modern sectors increases. This leads to a higher transformed output multiplier, which in turn yields greater economic efficiency thanks to larger scale economies. Eventually, this has a positive effect on economic growth. When the income transformation increases the purchasing power of the pre-tax 23

middle class at the expense of the initial upper class, there is also a positive middle-class learning effect. It turns out that the purchasing power of the ”transformed” middle class that contributes to the rate of growth increases. Not only does the multiplier increase, but we also observe an improvement in the proportion of income spent in sectors of activity where there is learning potential. In the end, what matters for growth is also the relative change + T . Thus, an economy in which a in γ ∗ N ∗ + T compared with that in γ N large share of income is initially concentrated in the hands of the upper class, such that there is no growth, may experience sustainable growth through redistribution to lower classes. This growth will be driven by the positive effect of redistribution on both the output multiplier that in turn leads to a decline in E c (.) and learning by doing. 17 Let us now consider that the income transformation is at the expense of the pre-tax middle class and that it is to the benefit of the initial lower class. Even though such a transformation tends to make the income distribution more equal, it has no impact on the output multiplier. Recall that lowerclass households spend all of their income in sectors where learning has been exhausted. It is interesting to note that the total share of post-transfer income held by the ”transformed” middle class declines, which results in a smaller steady-state rate of growth. It is thus entirely possible for an economy with a Lorenz-dominating distribution to be characterized by both a lower growth rate and a weaker middle class. These results allow us to gather a better understanding of the inverted Ushaped relationship between inequality and growth, which is further discussed below.

17

In neoclassical economics, higher income inequality is considered to positively affect savings, capital accumulation, and economic growth, with the rich having a higher marginal savings rate than the poor. Given that (skilled) labor is the only production factor in the models with nonhomothetic preferences outlined above. They thus differ from another class of supply-side growth models, which emphasize the pivotal function of the middle class, by using the median-voter paradigm. In particular, more pressure for redistributive policies is expected in a more unequal society (i.e. the greater the distance between the median and mean income). This in turn reduces both the incentives and the resources available for private investment, slowing down the growth process. See, for instance, Persson and Tabellini (1994), and Perotti (1996) for empirical evidence, which however does not support this argument. The collection of articles compiled in Inequality and economic development: The modern perspective (2009), with its introduction by Galor, provides an overview of this ”new” class of supply-side models, which emphasize the role of both human and physical investments in economic growth.

24

5.1.2.3 The distribution of profit is skewed towards high-skilled workers Let us specify a wealth distribution rule that assigns a disproportionately larger share of profits to high-skilled workers; that is, F (γ) Lorenz dominates the distribution of profit among households. More specifically, the share of aggregate corporate profits of a household of type γ is defined by ⎧ ⎪ ⎨γ

− φ (γ, ) ∀γ ∈ γ, γ c

⎩γ

+ ζ (γ, ) ∀γ ∈ ]γ c , ∞[

γ π ≡ ψ (γ, ) = ⎪



,

where γ ∗ < γ c , ψ(.) is assumed to be positive, monotone increasing, and continuous with γ, and ≥ 0 reflects the degree of inequality in the profit distribution with φ (γ, 0) = 0, ∂φ (γ, ) /∂ > 0 and ∂ζ (γ, ) /∂ > 0. Moreover, c

c

c

when γ π = γ , we have φ (γ , ) = ζ (γ , ) = 0, and  ∞ γc

 γc

φ (γ, ) dF (γ) = γ



ζ (γ, ) dF (γ). We also assume that ∂ψ (γ, ) /∂γ ≥ 0 ∀γ ∈ γ, ∞ . As a

consequence, γ π  γ π if and only if γ   γ. It is expected that such a transformation of the income distribution should lead to a smaller multiplier effect. Indeed, the demand complementarity between sectors and associated cumulative processes brought about through the market size of the modern sectors has become, by definition, lower. Firstly, we obtain

∗

α−1 L γγ φ (γ, ) dF (γ) Ty ( ) α =1− ⇒ Ty ( ) < T. ∗ T 1 + α−1 L γγ φ (γ, ) dF (γ) α

Secondly, as long as we assume that the skill distribution Lorenz dominates the distribution of profit (i.e. > 0), we have, on one hand M ( ) < M ⇔ y( , t) < y(t); and, on the other hand, ∂Ty ( )/∂ < 0, ∂M ( )/∂ < 0, and ∂y( , t)/∂ < 0. Thirdly, similar to (13), the growth rate can be rewritten as a function of M ( ) and the new share of total income spent in the range of sectors where learning takes place (see Proposition 1 above):

g( , t) = λEL



M ( )  ∗ ( , t) + T ( , t) , γ y ( )N ∗ ( ) + Ty ( ) − γ y ( , t)N y α

where T y ( , t) = L



γ y (,t) γ (,t) y

Π(,t)/w(t)L γ y ( , t)dF (γ y ) with γ y ( , t) = γ−φ (γ, ) 1+Π(,t)/w(t)L ,

which implies γ y ( , t)  γ for households of type γ  γ c . Given that M ( ) < M , what about the income share spent in modern sectors where learning takes place? Recall that the size and share of total income of the middle class depends on past economic growth. Using (19) adjusted to reflect the skewed profit distribution, the steady-state growth rate g( ) can 25

be rewritten as in (21):

g( ) = λEL



M ( )  ∗ (g( ), ) + T (g( ), ) , γ y ( )N ∗ ( ) + Ty ( ) − γ y (g( ), )N y α

where γ ∗y ( ) = γ ∗ − φ (γ ∗ , ) J( ) with J( ) = Π( )/ (wL + Π( )), i.e. the share of corporate profits in total income, γ y (g( ), ) = γ (g( ), ) − 



φ (γ (g( ), ), ) J( ), and T y (g( ), ) = T − L γ γ (g(),) φ (γ, ) dF (γ) J( ). As with the procedure used in the proof of Proposition 5, we show that a more unequal distribution of profit relative to the skill distribution reduces economic growth in the long run: 

lim

φ(γ,)→0

dg( )/d

g( )



< 0.

In other words, a mismatch between labor earnings and income, such that lower income households receive an income/profit that is disproportionately low relative to their skills leads to a lower steady-state rate of growth. The more unequal the distribution of financial assets relative to wage inequality, the greater the need to target redistributive policies towards the low- and middle-income class.

5.2

Inverted-U Relationship Between Inequality and Growth

At this stage, it is necessary to specify the distribution of income F (γ). We adopt the Pareto distribution, which exhibits useful properties as a functional form for income distribution: F (γ) = 1 − (γ/γ)β with β > 1 as the shape parameter, and γ ≥ γ > 0 as the scale parameter. The larger the value of parameter β, the more equal the distribution of income; dispersion increases monotonically as β decreases. Note that β = γ(q, t)f (γ(q, t)) / [1 − F (γ (q, t))] implies that . . x(q, t) y (t) =β . x(q, t) y(t) The income elasticity of demand addressed to each sector is equal to the Pareto shape parameter; it is constant across all sectors of activity q > q(t). In the Pareto case, the system of two equations (20) becomes ⎧ ⎪ ⎨g ⎪ ⎩



[M/α] T − T

= β −1 λEL 

− N∗ g = β −1 λ N



,

which leads to a solution for γ and g and, consequently, for both the size − N ∗ and its share of total income T − T . We now of the middle class N show that there is an inverted-U-shaped relationship between T − T and the 26

degree of inequality in the distribution of income β. When we use the Pareto distribution and take advantage of both changes in variables, n = N/L and ˘ = 1/(2β − 1), which denotes the Gini index of the Pareto distribution, we G have: ˘ ˘ 1−G 1−G ˘ − T (G) ˘ = (n

) 1+G˘ − (n∗ ) 1+G˘ ⇒ T (G) 

˘ 1−G

˘ 1−G



˘=

) 1+G˘ − (n∗ ) 1+G˘ /∂ G ∂ (n

2

 2

˘ 1+G

˘ 1−G



˘ 1−G

) 1+G˘ ln n

. (n∗ ) 1+G˘ ln n∗ − (n

˘ as the unique solution to Denote G 

) ∂ (n

˘ 1−G ˘ 1+G

− (n∗ )

˘ 1−G ˘ 1+G



∗

˘ 1−G ˘ 1+G

⎛



n

2 (n ) ⎜ ∗ ˘=0⇔  /∂ G 2 ⎝ln n − n∗ ˘ 1+G

⎞

 1−G˘ ˘ 1+G

⎟

⎠ = 0. ln n

In addition, we have: ∂

2



) (n

˘ 1−G ˘ 1+G

∗

− (n )

˘ 1−G ˘ 1+G



  ˘ ˘ G=G

˘ 2

/∂ G

=

2 ˘ 1+G

 2

n n∗

 1−G˘ ˘ 1+G

ln ln n

n ≤ 0. n∗

˘ − T (G) ˘ has a global maximum. Interestingly enough, BanerTherefore, T (G) jee and Duflo (2003) find a non-linear empirical relationship between inequality and growth, where a shift in the income distribution may negatively affect economic growth whether the distribution becomes more unequal or more egalitarian. Furthermore, using cross-country data, Chen (2003) finds an inverted-U relationship between initial income distribution and long-run economic growth. In order to illustrate the inverted-U relationship between inequality and growth in the steady state, we provide a small numerical example where the values of the parameters are set as follows: the learning rate λ is set to 0.1, the labor force L equals 1, and the fixed sunk costs C and α are set to 0.03 and 1.25, respectively. Thus, we have L > C/(α − 1) and the calculations are performed ˘ ≤ 1 (see (A.3) in Appendix A). This for a Gini index such that 0.11 < G numerical example depicted in Figure 2 shows the inverted-U relationship between inequality as measured by the Gini index and the steady-state growth rate g. An interesting result in this respect is documented in Grigoli and Robles (2017) who find a relationship between income inequality and growth where the slope switches from positive to negative at a net Gini of about 27 percent. It also shows that given the same level of equality, the growth rate increases with the potential for learning, which must be significant enough so as to achieve sustained growth (see Proposition 3). 27

and the steady-state growth Fig. 2. Gini index, upper bound value of learning E, rate g.

5.3

Learning by Doing: The Range–Speed Trade-Off

The inverted-U-shaped relationship represented in Figure 2 also corroborates the idea that beyond a certain level of (in)equality, growth starts to decrease. Too much equality slows down economic growth. This is also the case for a polarized income distribution. Here, it is useful to think in terms of the way the distribution of income affects the allocation of labor across sectors where some potential for learning exists. As discussed above, a sector can learn and reach the upper bound E more or less rapidly depending on the rate of growth of demand for its good, which in turn depends on the income distribution. Therefore, there is a trade-off between the learning speed within a sector, which depends on the income elasticity and the range of sectors in

which learning takes place (q(t), q ∗ (t)). An economy cannot learn both quickly and simultaneously in a wide range of sectors. It is constrained in this respect by its income/skill distribution and the size of its labor force. Imagine that a peak arises in the distribution of income. That is, a high proportion of households in the economy now share the same γ. Furthermore, imagine that this leads to a high demand for some goods whose learning potential has not yet been exhausted. The high demand for these goods requires the allocation of a significant amount of labor to their sectors. This may be so great that not only will learning rapidly cease, but a residual part of the workforce or, equivalently, skills allocated to these sectors may generate no extra productivity; learning is indeed bounded from above. It would then be more efficient in terms of cumulative learning experience to reallocate these skills 28

to other sectors which have not exhausted their learning potential. However, given how goods are arranged in order of decreasing hierarchy of needs, the demand for such goods may be insufficient to carry out this reallocation of labor to a downstream sector, whilst learning may have been exhausted in upstream sectors. Thus, for particular values of the technical parameters, a distribution of income which either deteriorates into a very unequal distribution, or approaches an almost perfectly egalitarian distribution may produce a decline in the economy’s steady-state growth rate. Eventually, the trade-off consists in learning quickly in a small number of sectors versus learning more slowly in a broad range of sectors. Such a trade-off is for the most part influenced by the entire distribution of income, which governs the product cycle. On the one hand, the lack of economies of scale hurts non-egalitarian societies with a small elite, which in turn prevents technology maturation through learning by doing. On the other hand, in a society characterized by an egalitarian distribution of income, the learning process occurs rapidly; however, it is confined to a small number of sectors which attract most (too much) of the labor force, eventually preventing the economy from moving forward. This refers primarily to what we call a range–speed of learning trade-off.

6

Conclusion

The purpose of this paper is to offer a compelling theory of middle-classled economic growth. In a stylized way, our framework provides an economic relationship between middle-class-led consumption and multi-sector growth, in which the latter depends on both the size of the middle class and its share of total income, the delimitation of the size of the middle class and its relative position in the income hierarchy being explicitly derived from past economic growth. In brief, we make the case for a strong middle class, i.e. redistributive policies towards the poor and the middle class. The model relies on the capacity of income (re)distribution to drive a dynamic switching of technologies from small- to large-scale production thanks to middle-class-led learning by doing in the modern sectors of economic activity. Our assumption about the product life cycle differs from the existing related endogenous growth models with nonhomothetic preferences like innovation-based or trickle-down models, which underline the importance of the top end of the income distribution. We argue that it is more appropriate to account for economic development and in particular the spread of modern technologies in countries where either innovation is limited, like developing countries, or where a trickle-down process is unlikely to take place. In light of the available empirical evidence, we would like, however, to emphasize that the demand-side models of growth as discussed above are not necessarily in 29

competition, but should rather be seen as complementary in order to better understand the complexity of the inequality-growth nexus. After all, this also applies to supply-side models which examine this nexus, relying either on the investment capacity of households at the top end of the income distribution, or on the credit constraints encountered by households at the bottom end of the distribution. We also show that there is a labor/skill allocation trade-off concerning the range of sectors in which learning may occur simultaneously versus the speed at which learning takes place in each sector of activity. This range–speed of learning trade-off depends on the entire distribution of income, which determines the pace of learning in a sector of activity. Thus, there is a level of inequality, measured here by the shape parameter of the survival function, namely the income elasticity of demand, which maximizes economic growth. This trade-off yields an inverted-U-shaped relationship between inequality and growth in which beyond a certain level of (in)equality, growth starts to decrease. We anticipate that this analysis will stimulate empirical research to further investigate the relationships between middle-class-led consumption/learning, market structure and multi-sector growth, particularly in the developing world where R&D expenditures remain scarce, and where the income gap between the rich and the rest of the population is still high.

Acknowledgements

This work has benefitted from the generous support of our institutions. We are grateful to three anonymous referees for their useful comments. We also would like to thank audiences at the 8th Annual Conference on Economic Growth and Development (Indian Statistical Institute at New Delhi), International Workshop on Economic Growth, Macroeconomic Dynamics and Agents’ Heterogeneity (European University at St. Petersburg), 14th EUROFRAME Conference on Growth and Inequality: Challenges for EU Countries (German Institute for Economic Research at Berlin), 8th ECINEQ Meeting (Paris School of Economics), universit´e de Rennes 1, Aix-Marseille School of Economics, Banco de la Republica de Colombia, Universidad de Antioquia (Medell´ın), and Universidad de Los Andes (Bogot´a).

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A

Price Equilibrium

On the one hand, at time t, a monopolist entering the market for a particular good q cannot set a price higher than the competitive price without giving way to a competitive fringe of firms. On the other hand, could he seriously consider increasing profits by lowering the price unilaterally below p(t) = αw(t)/A(t) 33

(i.e. while all other firms keep their price unchanged)? The answer is no, as long as the marginal profit satisfies the following condition: 



∂π(q, t) ∂ x(q, t) w(t)  t) − = p(q, + x(q, t) > 0  t)  t) ∂ p(q, ∂ p(q, A(t)    t) − w(t)/A(t)  t) p(q, ∂ x(q, t) p(q, ⇔− < 1,  t) x (q, t)  t) ∂ p(q, p(q,

(A.1)

 t) ≤ p(t), and where x (q, t) = [1 − F (γ (q,  t))] L is the effective with p(q,  t). In other words, the price elasticity of demand for good q produced at p(q, demand multiplied by the price-cost margin should not exceed unity.

Let us define q ≤ q such that  t) p(q, 1 1 1 1 = ⇒ q = q.  t) q p(q, q p(t) p(t)

Customers for the variety of good q include all those that are rich enough  t) where γ(q,  t) = q(t)/y(t)L  to buy q (i.e. households of type γ ≥ γ(q, =  t)q) / (p(t)y(t)L)). Let f (γ) be the density of γ -type households. We (p(q, also define β(γ) ≡ γf (γ)/ [1 − F (γ)]. The price elasticity of demand for good q can be written as follows −

 t)  t))q  t) p(q, ∂ x(q, t) p(q, f (γ(q, = (A.2)  t) x  t))] L (q, t) ∂ p(q, p(t)y(t) [1 − F (γ (q,  t))γ(q,  t) p(t)y(t)L  t)f (γ(q,  t)) f (γ(q, γ(q,  t)). = = = β(γ(q,  t))])L p(t)y(t)  t)) [1 − F (γ (q, 1 − F (γ(q,

We can show that using (A.1) and (A.2), we have: 







 t) − w (t) /A (t)  t) − w(t)/A(t) p(q, ∂ x (q, t) p (q, t) p(q,  t)) = β (γ(q, −  t) ∂ p (q, t) x (q, t) p (q, t) p(q,   p(q, t) − w(t)/A(t) α−1   t)) . β (γ (q, < β (γ(q, t)) = p(q, t) α

Therefore, the following inequality provides a sufficient condition for ruling out price-cutting equilibria: 



 t) − w(t)/A(t)  t) p(q, α−1 ∂ x(q, t) p(q,  t)) < 1 ⇒ − β (γ(q, < 1.    t) α ∂ p(q, t) x(q, t) p(q,

(A.3)

Indeed, as long as (A.1) is satisfied, when a firm with access to IRS technology in sector q aims to cut the price below αw(t)/A(t), it is unable to expand its customer base to such an extent as to compensate for the loss in profit per customer, thus discouraging price-cutting. In our framework, such a condition results in (A.3). The income distribution should not degenerate around any γ -type. 34

B

Proof of Proposition 3: Scale Economies, Learning Potential, and Growth

Using the break-even condition (7), it is easy to verify that dγ ∗ dα

=

1−F (γ ∗ ) (α−1)f (γ ∗ )

> 0,

dγ ∗ dL

=

1−F (γ ∗ ) f (γ ∗ )L

> 0,

dγ ∗ dC

= − (α−1)f1 (γ ∗ )L < 0,

which implies that 







1 − F (γ ∗ ) 1 α 1 − F (γ ∗ ) dE c (.) = ∗ 1− + − 1 T − [1 − F (γ ∗ )] γ ∗ < 0 dα γ L α − 1 f (γ ∗ ) γ ∗ f (γ ∗ ) γ ∗ α − 1 f (γ ∗ ) γ ∗ < 1 (see the price equilibrium section above); ⇔ α 1 − F (γ ∗ ) 



dE c (.) E c (.) (1 − F (γ ∗ )) α ∗ =− − ∗ + (α − 1) f (γ ) < 0; dL γ∗ f (γ ∗ ) L γ (L)2 



dE c (.) E c (.) 1 = > 0. + (α − 1) f (γ ∗ ) ∗ dC γ (α − 1) f (γ ∗ ) L

C

Proof of Proposition 4: Very Unequal Versus Almost Perfectly Egalitarian Income Distribution and Growth

First, for a very unequal distribution of income, the Lorenz curve is such that its slope tends to be zero for γ ≤ γ ≤ γ ∗ . Thus, we have γ ≈ γ ≈ γ ∗ ≈ 0 which implies both T ≈ T ≈ 0 and M ≈ 1. Replacing in (13) immediately leads to g = 0. Second, for an almost perfectly egalitarian distribution of income, the ◦ Lorenz curve approaches the 45 line, and we have γ ≈ γ ∗ ≈ 1/L which implies that T ≈ 1 − N ∗ /L, T ≈ 0, and M ≈ α/(1 + C/L). Again, replacing in (13) we obtain: 1

g = λEL [1 − γ L] . 1 + C/L Using (19) and F (γ) ≈ 0 for γ < γ ∗ , we get 



 γ∗ g 1 1 =L dγ = L ln , λ γ L

γ γ

which implies that γ L = exp(−g/λL). Eventually, if we substitute this result into the above expression, the steady-state growth rate becomes 35



g = Λ(g) = λEL 

1 [1 − exp(−g/λL)] 1 + C/L

E

dΛ(g)  C ⇒ > 1 ⇔ E > E c (.) = 1 + .  = dg g=0 1 + C/L L

D

Proof of Proposition 5: Inequality, Transformation of Income, and Growth

The proof proceeds in two parts. The first part consists of demonstrating that an income transformation that increases the purchasing power of either the pre-tax middle class or the pre-tax lower class to the detriment of the initial upper class will produce an upward shift of Λ(g), which eventually leads to a new equilibrium with a higher steady-state rate of growth. In the second part, we show that an income transformation that benefits the pre-tax lower class at the expense of the initial middle class lowers economic growth. 1.a/ Let us assume the following income transformation so that the posttransfer income distribution denoted by Fr (γ) Lorenz dominates the pre-tax distribution of income F (γ) (see Figure D.1): ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

γ



∀γ ∈ γ, γ



γ r ≡ ψ (γ, τ ) = γ + ϕ (γ, τ ) ∀γ ∈ [γ , γ ∗ ] , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩γ

− ρ (γ, τ ) ∀γ ∈ ]γ ∗ , ∞[

where ψ(.) is positive, monotone-increasing, and continuous with γ (i.e. ∂ϕ (γ, τ ) /∂γ > −1, ∂ρ (γ, τ ) /∂γ < 1), and τ > 0 reflecting the intensity of income redistribution with ∂ϕ (γ, τ ) /∂τ ≥ 0 and ∂ρ (γ, τ ) /∂τ ≥ 0. In addition, we have γ r = γ and γ ∗r = γ ∗ , which implies that ϕ (γ , τ ) = ϕ (γ ∗ , τ ) = ρ (γ ∗ , τ ) = 0,  γ∗

and

γ

ϕ (γ, τ ) f (γ) dγ =

 ∞ γ∗

ρ (γ, τ ) f (γ) dγ.

First, such an income transformation satisfies both the single-crossing condition and the principle of rank-preserving transfers. Second, note that it implies ) that Fr (γ r ) = F (γ). Then, using dγ r = ∂ψ(γ,τ dγ, we get ∂γ 

∂ψ (γ, τ ) dFr (γ r ) = dF (γ) ⇒ fr (γ r ) = f (γ) ∂γ

−1

.

The middle-class share of post-tax income becomes: Tr (γ ∗r , τ ) =

 γ∗ r γ

r

 γ∗

γ r fr (γ r ) dγ r =

ψ (γ, τ ) f (γ) dγ = T (γ ∗ )+

γ

36

 γ∗

γ

ϕ (γ, τ ) f (γ) dγ.

Fig. D.1. Income transformation from the upper to the middle class.

Let γ ∗r (τ ) be the implicit solution to 1 − Fr (γ ∗r (τ )) = C/ ((α − 1) L), where γ ∗r (τ ) is the post-transfer income of household of type γ ∗ (τ ), which is equal to ψ −1 (γ ∗r (τ )) and defined as the upper cut-off for the middle class after transformation. Given that 1 − F (γ ∗ ) = C/ ((α − 1) L) and γ ∗r (τ ) = ψ (γ ∗ (τ )) = γ ∗ (τ ), we get γ ∗r (τ ) = γ ∗ (τ ) = γ ∗ . We now analyze the effect of an income transformation on Λ (g) (see (21)). The system of equations (20) can be rewritten in terms of both the transformed multiplier Mr (τ ) and the proportion of total income spent in the intermediate range of sectors where learning takes place, which is denoted by N (γ (τ ) , τ ):

g = Λ (γ (τ ) , τ ) = λEL

Mr (τ ) N (γ (τ ) , τ ) α

(D.1)

g/λ = Γ (γ (τ ) , τ ) = N ∗ ln (γ ∗ ) − [1 − F (γ (τ ))] L ln (γ (τ ) + ϕ (γ (τ ) , τ ))

(D.2)

 γ∗

+

γ (τ )

ln (γ + ϕ (γ, τ )) f (γ) Ldγ

where γ (τ ) = γ (g, τ ) is the implicit solution to (D.2) so that the household of γ (τ ) -type is the new lower cut-off for the middle class after transformation, N (γ (τ ) , τ )

≡

 γ∗

γ r (τ )

[1 − Fr (γ r )] Ldγ r , and γ r (τ ) = γ (τ ) + ϕ (γ (τ ) , τ ) is the

post-transfer income of household of type γ (τ ). The righ-hand side of (D.2)  γ∗

is the result of the integration by part of

γ r (τ )

1−Fr (γ r ) dγ r . γr

Total differentiation of (D.1) leads to

dg =

⎡

λEL ⎢ ⎢ α ⎣

⎤

∂N(

γ (τ ),τ ) ∂Mr (τ ) N (γ (τ ) , τ ) + Mr (τ ) dτ ⎥ ∂τ  ∂τ ⎥. ⎦ ∂N(

γ (τ ),τ ) ∂

∂N

γ (τ ),τ ( ) γ (g,τ ) (g,τ ) dg + ∂ γ (τ ) ∂ γ∂τ dτ +Mr (τ ) ∂g ∂

γ (τ )

37

Therefore, we have 

−1



dg/dτ

Mr (τ ) ∂N (γ (τ ) , τ ) ∂ γ (g, τ ) = 1 − λEL

g α ∂ γ (τ ) ∂g ⎡ ⎤ ∂N(

γ (τ ),τ ) ∂N(

γ (τ ),τ ) ∂

γ (g,τ ) ∂Mr (τ ) + ∂ γ (τ ) ⎢ ∂τ ⎥ ∂τ ∂τ ⎢ ⎥. + ⎣ M (τ ) ⎦

N (γ (τ ) , τ ) r

(D.3)

First, note that

λEL

Mr (τ ) ∂N (γ (τ ) , τ ) ∂ γ (g, τ )

Mr (τ ) (γ

(τ ) + ϕ(γ

(τ ) , τ )) , = λEL α ∂ γ (τ ) ∂g α

where ∂ γ (g, τ ) /∂g is the implicit derivative of γ (g, τ ) using (D.2): 

∂ γ (g, τ ) 1 ∂Γ (γ (τ ) , τ ) = ∂g λ ∂ γ (τ )

−1

.

Using properties of Λ (g) as discussed in Section 5.1, we infer that for g such that Λ (g, τ ) = g, we have ∂Λ (g, τ ) /∂g < 1, where ∂Λ (g, τ )

Mr (τ ) (γ

(τ ) + ϕ(γ

(τ ) , τ )) , = λEL ∂g α which implies that 

λEL ∂N (γ (τ ) , τ ) ∂ γ (g, τ ) Mr (τ ) 1− α ∂ γ (τ ) ∂g

−1

> 0.

Second, the transformed multiplier is given by

1− ⇒

1



Mr (τ ) = α−1 α

T (γ ∗ ) +



 γ∗

γ (τ )

ϕ (γ, τ ) f (γ) dγ

α − 1  γ ∗ dϕ (γ, τ ) ∂Mr (τ ) /∂τ = Mr f (γ) dγ > 0. Mr (τ ) α γ (τ ) dτ

Finally, the share of total income spent in the ”transformed” range of sectors where learning takes place can be written as N (γ (τ ) , τ ) = γ ∗ N ∗ − [γ (τ ) + ϕ (γ (τ ) , τ )] [1 − F (γ (τ ))] L ∗

+T (γ ) − T (γ (τ )) +

38

 γ∗

γ (τ )

ϕ (γ, τ ) f (γ) Ldγ.

Given that

∂ γ (g, τ ) ∂Γ (γ (τ ) , τ ) /∂τ =− , ∂τ ∂Γ (γ (τ ) , τ ) /∂ γ (τ )

we get ∂N (γ (τ ) , τ ) ∂N (γ (τ ) , τ ) ∂ γ (g, τ ) + ∂τ ∂ γ (τ ) ∂τ    γ∗



γ (τ ) + ϕ (γ (τ ) , τ ) ∂ϕ (γ, τ ) = f (γ) Ldγ > 0 1− γ + ϕ (γ, τ ) ∂τ

γ (τ ) ⇔ γ + ϕ (γ, τ ) > γ (τ ) + ϕ (γ (τ ) , τ ) , for γ (τ ) < γ < γ ∗ . 1.b/ We use the same approach to prove that an income transformation, which increases the purchasing power of the pre-tax lower class at the expense of the initial upper class will lead to a rise in the steady-state growth rate. Consider the following income transformation: ⎧ ⎪ ⎪ ⎪ γ ⎪ ⎪ ⎨



+ ϕ (γ, τ ) ∀γ ∈ γ, γ

γ r ≡ ψ (γ, τ ) = ⎪

⎪ ⎪ ⎪ ⎪ ⎩γ



∀γ ∈ [γ , γ ∗ ] ,

γ

− ρ (γ, τ ) ∀γ ∈ ]γ ∗ , ∞[

where ψ(.) is again positive, monotone-increasing and continuous with γ, both γ r = γ

 ∞ γ∗

and

γ ∗r

=γ

∗

imply that ϕ (γ , τ )

∗

= ρ (γ , τ ) = 0, and



γ

ϕ (γ, τ ) f (γ) dγ =

γ

ρ (γ, τ ) f (γ) dγ. Such an income transformation also satisfies the single-

crossing condition and the principle of rank-preserving income transfers, and implies that Fr (γ r ) = F (γ). The middle-class share of post-tax income becomes Tr (γ ∗ , τ ) =

 γ∗

ψ (γ, τ ) f (γ) dγ = T (γ ∗ ) +

γ



γ

ϕ (γ, τ ) f (γ) dγ.

γ

In this case, (D.2) is given by Γ (γ (τ ) , τ ) = N ∗ ln (γ ∗ ) − [1 − F (γ (τ ))] L ln (γ (τ ) + ϕ (γ (τ ) , τ )) +



γ

γ (τ )

 γ∗

ln (γ + ϕ (γ, τ )) f (γ) Ldγ +

γ

ln (γ) f (γ) Ldγ.

We show that: ∂Mr (τ ) ∂τ

> 0 and

∂N(

γ (τ ),τ ) ∂τ

+

∂N(

γ (τ ),τ ) ∂

γ (g,τ ) ∂τ ∂

γ (τ )

> 0,

which given (D.3) implies that dg/g > 0. On the one hand, we have 39





−1



γ α−1 Mr (τ ) = 1 − T (γ ∗ ) + ϕ (γ, τ ) f (γ) dγ α γ ∂Mr (τ ) /∂τ α − 1  γ ∂ϕ (γ, τ ) = Mr (τ ) f (γ) Ldγ > 0. ⇒ Mr (τ ) α γ ∂τ

On the other hand, the post-transfer proportion of total income spent in the range of sectors where learning now takes place is given by 18 N (γ (τ ) , τ ) = γ ∗ N ∗ − [1 − F (γ (τ ))] L (γ (τ ) + ϕ (γ (τ ) , τ )) ∗

+T (γ ) − T

(γ (τ )) +



γ

γ (τ )

ϕ (γ, τ ) f (γ) Ldγ

∂N (γ (τ ) , τ ) ∂N (γ (τ ) , τ ) ∂ γ (g, τ ) + ∂τ ∂ γ (τ ) ∂τ   

γ γ (τ ) + ϕ (γ (τ ) , τ ) ∂ϕ (γ (τ ) , τ ) = f (γ) Ldγ > 0 1− γ + ϕ (γ, τ ) ∂τ

γ (τ ) ⇔ γ + ϕ (γ, τ ) > γ (τ ) + ϕ (γ (τ ) , τ ) , for γ (τ ) < γ < γ . ⇒

2/ Let us assume ⎧ ⎪ ⎪ ⎪ γ ⎪ ⎪ ⎨



+ ϕ (γ, τ ) ∀γ ∈ γ, γ



γ r ≡ ψ (γ, τ ) = γ − ρ (γ, τ ) ∀γ ∈ [γ , γ ∗ ] , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

γ

∀γ ∈ ]γ ∗ , ∞[

where ψ(.) has the same properties as above, and



γ

 γ∗

ϕ (γ, τ ) f (γ) dγ =

γ

γ

ρ (γ, τ ) f (γ) dγ.

The middle-class share of post-tax income is equal to ∗

∗

Tr (γ , τ ) = T (γ ) +



γ γ

ϕ (γ, τ ) f (γ) dγ −

 γ∗

γ

ρ (γ, τ ) f (γ) dγ = T (γ ∗ ) .

Note that (D.2) is given by 18

We make use of the inequality γ

(τ ) < γ

, which can be shown using d

γ (τ ) ∂

γ (g, τ ) dg ∂

γ (g, τ ) = + < 0. dτ ∂g dτ ∂τ

40

Γ (γ (τ ) , τ ) = N ∗ ln (γ ∗ ) − [1 − F (γ (τ ))] L ln (γ (τ ) + ϕ (γ (τ ) , τ )) +



γ

 γ∗

ln (γ + ϕ (γ, τ )) f (γ) Ldγ +

γ (τ )

γ

ln (γ − ρ (γ, τ )) f (γ) Ldγ.

Here, we show that: ∂Mr (τ ) ∂τ

= 0 and

∂N(

γ (τ ),τ ) ∂τ

+

∂N(

γ (τ ),τ ) ∂

γ (g,τ ) ∂τ ∂

γ (τ )

> 0,

which implies that dg/g > 0. We focus on the post-transfer share of total income spent in the intermediate range of sectors where learning takes place: N (γ (τ ) , τ ) = N ∗ γ ∗ − [1 − F (γ (τ ))] L (γ (τ ) + ϕ (γ (τ ) , τ )) + T (γ ∗ ) − T (γ (τ )) 

γ

+

γ (τ )

ϕ (γ, τ ) f (γ) Ldγ −

 γ∗

γ

ρ (γ, τ ) f (γ) Ldγ

∂N (γ (τ ) , τ ) ∂N (γ (τ ) , τ ) ∂ γ (g, τ ) + ∂τ ∂ γ (τ ) ∂τ   

γ γ r (τ ) ∂ϕ (γ (τ ) , τ ) f (γ) Ldγ 1− = γ + ϕ (γ, τ ) ∂τ

γ (τ )   γ∗  γ r (τ ) ∂ρ (γ (τ ) , τ ) − f (γ) Ldγ < 0, 1− γ − ρ (γ, τ ) ∂τ

γ ⇒

recalling that γ r (τ ) = γ (τ ) + ϕ (γ (τ ) , τ ) . Note first that the following two inequalities hold:  

γ

γ (τ )  γ∗ 

γ

γ r (τ ) 1− γ + ϕ (γ, τ )

γ (τ ) 1− r γ









γ ∂ϕ (γ (τ ) , τ ) γ (τ ) f (γ) dγ < 1− r ∂τ γ

γ (τ ) 

 γ∗ ∂ρ (γ (τ ) , τ ) γ r (τ ) f (γ) dγ < 1− ∂τ γ − ρ (γ, τ )

γ





∂ϕ (γ (τ ) , τ ) f (γ) dγ, ∂τ

∂ρ (γ (τ ) , τ ) f (γ) dγ. ∂τ

Second, the above budget constraint leads to:  

γ

γ (τ )

γ (τ ) 1− r γ





 γ∗ ∂ϕ (γ (τ ) , τ ) γ (τ ) f (γ) dγ < 1− r ∂τ γ

γ



∂ρ (γ (τ ) , τ ) f (γ) dγ. ∂τ

Finally, the following inequality is obtained using the principles of transitivity:  

γ

γ (τ )

γ r (τ ) 1− γ + ϕ (γ, τ )





 γ∗ ∂ϕ (γ (τ ) , τ ) γ r (τ ) f (γ) dγ < 1− ∂τ γ − ρ (γ, τ )

γ

41



∂ρ (γ (τ ) , τ ) f (γ) dγ. ∂τ