Income Distribution in Macroeconomic Models

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This book looks at the distribution of income and wealth and the effects that this has on the macroeconomy, and vice versa. Is a more equal distribution of income beneficial or harmful for macroeconomic growth, and how does the distribution of wealth evolve in a market economy? Taking stock of results and methods developed in the context of the 1990s revival of growth theory, the authors focus on capital accumulation and long-run growth. They show how rigorous, optimization-based technical tools can be applied, beyond the representative-agent framework of analysis, to account for realistic market imperfections and for political-economic interactions.

The treatment is thorough, yet accessible to students and nonspecialist economists, and it offers specialist readers a wide-ranging and innovative treatment of an increasingly important research field. The book follows a single analytical thread through a series of different growth models, allowing readers to appreciate their structure and crucial assumptions. This is particularly useful at a time when the literature on income distribution and growth has developed quickly and in several different directions, becoming difficult to overview.

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What People Are Saying

Kiminori Matsuyama
The interaction between the dynamics of economic growth and the evolution of economic inequality is an important and challenging problem. Recent advances in macroeconomics of heterogeneous agents have finally made it possible to investigate this question in a systematic manner. This timely book offers an excellent first broad overview in this area. The ideas in the book are so intuitive that they can be taught to advanced undergraduates. The exposition is so clear, simple and yet rigorous that the book is useful in a first-year graduate macro sequence. Its comprehensive coverage makes it an indispensable source of reference for the researcher in the field. A great achievement! I wish I had written this book.
Kiminori Matsuyama, Northwestern University
Alex Michaelides
Income distribution questions are becoming increasingly important in modern macroeconomic theory, and they will probably become even more so as computational techniques are utilized to move macroeconomics beyond the representative agent paradigm. This book does a good job in summarizing the current state of the literature in an interesting and hands-on way.
Alex Michaelides, London School of Economics
Frank Cowell
A well balanced, clearly argued, up-to-date, and informative account of the subject. The arguments that spin off from this book will interest not only macroeconomists but also others in the field.
Frank Cowell, Professor of Economics and Director of Distributional Analysis Research Programme, London School of Economics; author of "The Economics of Poverty and Inequality"
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Product Details

  • ISBN-13: 9780691121710
  • Publisher: Princeton University Press
  • Publication date: 12/5/2005
  • Edition description: New Edition
  • Pages: 440
  • Product dimensions: 6.20 (w) x 9.30 (h) x 1.50 (d)

Meet the Author

Giuseppe Bertola is Professor of Economics at the University of Turin. Reto Foellmi is Assistant Professor of Economics at the University of Zurich. Josef Zweimüller is Professor of Economics at the University of Zurich.
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Read an Excerpt

Income Distribution in Macroeconomic Models

By Giuseppe Bertola, Reto Foellmi, Josef Zweimüller


Copyright © 2006 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-6509-3


Production and Distribution of Income in a Market Economy

The aim of this book is to study the implications of economic interactions between heterogeneous individuals, both for macroeconomic outcomes and for the evolution of the income and wealth distribution. As these interactions are extremely complex, we organize our analysis around several key simplifications.

First, we will assume throughout that there are two factors of production: an "accumulated" factor and a "non-accumulated" factor. We will frequently refer to the former as "capital" and to the latter as "labor." As we discuss below, however, the important point is that the economy's (as well as the households') endowment with the former is endogenously determined by savings choices, whereas the economy's endowment with the latter is exogenously given.

Second, we will assume throughout that all individuals have the same attitude toward savings, i.e., that any two individuals would behave identically if their economic circumstances were identical. This is not to say that heterogeneity in preferences between present and future consumption is unimportant in reality. Allowing for systematic differences across individuals along this dimension, however, would tend to yield tautological results: one might, for example, find that the poor are and remain poor due to their low propensities to save. It is much more insightful to highlight other sources and effects of large differences in incomes across individuals: we will highlight the role of macroeconomic phenomena (such as capital accumulation and associated changes in factor prices, market imperfections, and economic policies) for the dynamics of the distribution of income and wealth and their feedback to the long-run process of economic development. Heterogeneous propensities to save are clearly of some importance in reality, but will not induce a systematic bias in our results if they are random and unrelated to economic circumstances.

Third, in many of our derivations we will assume that only one good is produced in the economy and can be used for either consumption or investment. Investment then coincides with forgone consumption, to be understood broadly as leisure choices are subsumed in consumption choices. The single-good assumption is adopted throughout part 1 (with the exception of the appendix to chapter 4) and part 2. In part 3, we relax it and consider the interrelation between distribution and growth when there are many goods and when the structure or consumption differs between rich and poor consumers.

As a further general principle, we will apply standard tools of modern macroeconomic analysis, formulating all models in formally precise and consistent terms. Even as we strive to take individual heterogeneity into account when studying macroeconomic phenomena, we will often find it useful to refer to situations where some or all of the implications of heterogeneity are eliminated by appropriate, carefully discussed assumptions, so that a representative agent perspective is appropriate for some or all aspects of the analysis. Specifying and carefully discussing deviations from these assumptions will make it possible to highlight clearly problems of heterogeneity and distribution, as well as their interaction with macroeconomic phenomena.

This first chapter sets the stage for our analysis. We introduce notation and set out basic relationships both at the level of the family and at the aggregate, making the important distinction between accumulated and non-accumulated income sources. Then, we analyze the relationship between distribution and the efficiency of production in a "neoclassical" setting of perfect and complete markets. Firms maximize profits and take prices as given, all factors of production are mobile, there is complete information, and all economic interactions are appropriately accounted for by prices (there are no externalities). In that setting we discuss in some detail the conditions under which macroeconomic aggregates do not depend on income distribution and on technological heterogeneity, so that production and accumulation can be studied as if they were generated by decisions of "representative" consumers and producers. As is often the case in economics, the model's assumptions are quite stringent, so we discuss briefly conceptual problems arising when certain tractability conditions are not met. In particular, if factors of production cannot be reallocated, aggregation becomes very problematic unless stringent conditions are met regarding the character of technological heterogeneity. This qualifies, but certainly does not eliminate, the usefulness of stylized models as a benchmark when assessing the practical relevance of deviations from the neoclassical assumptions.

1.1 Accounting

Consider an economy with many households endowed with two types of production factors: accumulated and non-accumulated. By definition, accumulated factors are inputs whose dynamics are determined by microeconomic savings decisions. At the aggregate level, these decisions affect both the distribution of accumulated factors across individuals and the dynamics of macroeconomic accumulation. In contrast, non-accumulated factors are, by definition, production factors that evolve exogenously (or, for simplicity, remain constant) in the aggregate. We will frequently refer to the accumulated factor as "capital" and to the non-accumulated factor as "labor." However, the simple capital/labor distinction may be misleading. For instance, an individual's human capital is essential for the efficiency of its "labor" but clearly affected by an individual's savings choices. In contrast, incomes from real estate ("land") as well as non-contestable monopolies are often counted as part of capital income but are, according to our definition, part of non-accumulated factors' rewards.

While here we take the evolution of non-accumulated factors as given, it is important to note that, in reality, the economy's supply with these factors is subject to households' supply choices. Here we abstract from the endogeneity of the supply of their non-accumulated factors and from endogenous fertility behavior. We subsume labor/leisure choices under the consumption choice.

A family or household i is endowed with k(i) units of an accumulated factor and l(i) units of a non-accumulated factor. In general, households differ in endowments k and l. Moreover, factor rewards may also differ between households, hence r = r(i) and w = w(i). However, when there are perfect factor markets, all households get the same returns and r and w no longer depend on individual endowment levels but are determined by their aggregate counterparts.

The models reviewed below can be organized around a simple accounting framework. The income flow y accruing to a family also depends on endowments k and l and equals

y(i) = w(i) x l(i) + r(i) x k(i).

The dynamic budget constraint, at the household level, is given by

Δk(i) = y(i) - c(i), or Δ k(i) = r(i)k(i) + w(i)l(i) - c(i), (1.1)

where c(i) denotes the consumption flow of a household who owns accumulated factor k and non-accumulated factor l in the current period. The change in the family's stock of the accumulated factor, denoted Δk(i), coincides with forgone consumption (income not consumed). Income y(i) is measured net of depreciation of the accumulated factor, and r(i) is the net return of this factor. Consumption c, income y, and savings Δk are, in general, heterogeneous across individuals. This heterogeneity may be due to two sources: households own different baskets of factors (k(i), l(i)), and they may earn different rewards r(i) and/or w(i).

There are two important assumptions implicit in the above formulation. The first is that there is only one consumption good, and the second is that consumption is convertible one to one into the accumulated factor. We will stick to these assumptions throughout most of parts 1 and 2 of this book. In part 3 we will relax the first assumption: we will study conditions under which differentiating output by different consumption purposes becomes relevant for distribution and growth. In appendix 4.6 we will address the latter assumption. There a model with two sectors is presented where the accumulated factors and consumption goods are produced with different technologies.

Any of the variables on the right-hand sides of the expressions in (1.1) may be given a time index, and may be random in models with uncertainty. In (1.1), Δk(i) = kt+1 (i) - kt(i) is the increment of the individual family's wealth over a discrete time period. In continuous time, the same accounting relationship would read

[??] = y - c = rk + wl - c, (1.2)

where [??]k(t) [equivalent to] dk(t)/dt = limΔt -> 0 [(k(t + Δt) - k(t))/Δt] is the rate of change per unit time of the family's wealth.

The advantage of a continuous-time formulation is that it frequently yields simple analytic solutions, and it is not necessary to specify whether stocks are measured at the beginning or the end of the period. The advantage of discrete time models is that empirical aspects and the role of uncertainty are discussed more easily in a discrete-time framework. We will use the continuous-time formulation in some chapters, the discrete-time formulation in others.

Aggregating across individuals leaves us with the macroeconomic counterparts of income, consumption, and the capital stock. We allow the distribution to be of discrete or continuous nature. In the former case, p(i) denotes the population share of group i, with n different groups in the population, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] p(i)= 1. If distribution is continuous, p(i) denotes the density, and with a population distributed over the interval we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] p(i)di = 1. For the sake of compact notation we use the Stjelties integral, which encompasses both the discrete and the continuous case. The measure P(•), where ∫NdP(i) = 1, assigns weights to subsets of N, the set of individuals in the aggregate economy of interest. To gain more intuition with the weight function P(•) consider the special case where N has n elements (of equal population size). Then, the weight function P(i) = 1/n defines Y as the arithmetic mean of individual income levels y(i).

With continuous distribution, the relative size or weight P(A) of a set A [subset] N of individuals is arbitrarily small, and conveniently lets the idiosyncratic uncertainty introduced in chapter 8 average to zero in the aggregate.

We use the convention to write uppercase letters for the aggregate counterpart of the corresponding lowercase letter. Hence aggregate income is denoted by Y and equals

Y [equivalent to] ∫N y(i)dP(i), (1.3)

where N denotes the set of families. For the most part, we take N as fixed. However, when we want to study issues like population growth, finite lives, or immigration, we will allow N to be variable over time.

Recall that heterogeneity of the non-accumulated income flow wl may be accounted for by differences in w and/or l across individuals. We take l as exogenously given. Hence we sum up and get

L [equivalent to] ∫N l(i)dP(i), (1.4)

where L denotes the amount of non-accumulated factors available to the aggregate economy.

Recall from (1.1) that we assumed the relative price of c and Δk to be unitary. This allows us to aggregate families' endowments with the accumulated factor. The aggregate stock of the accumulated factor K is measured in terms off or gone consumption

K [equivalent to] ∫N k(i)dP(i), (1.5)

The definitions in (1.3), (1.4), and (1.5) readily yield a standard aggregate counterpart of the individual accumulation equation (1.1):


Corresponding to its individual counterpart we define Y = RK + WL, where R and W denote the aggregate rate of return on the accumulated and non-accumulated factor, respectively. The definition directly implies that R and W are weighted (by factor ownership) averages of their heterogeneous microeconomic counterparts,


Interestingly, the economic interpretation of these aggregate factor prices is not straightforward in a world where inequality plays a role. In the models discussed in part 1, all units of each factor are rewarded at the same rate. In this case r(i) = R and w(i) = W, which denotes an economy-wide interest rate and wage rate (or land rent), respectively. In the more complex models of part 2, however, unit factor incomes may be heterogeneous across individuals. This introduces interesting channels of interaction between distribution and macroeconomic dynamics. At the same time, such heterogeneity also makes it difficult to give an economic interpretation to aggregate factor supplies and remuneration rates.

Finally, note that the individual-level budget constraint (1.1) features net income flows, and so does (1.6). Hence, the aggregate Y flow is obtained subtracting capital depreciation, say δK, from every period's gross output flow, say [??], and (1.6) may equivalently be written

ΔK = [??] - δK - C.

In order to economize on notation and obtain cleaner typographical expressions, from now on we abstain from making explicit the indexing of (lowercase) individual-level variables. A convention we adopt throughout the book is the use of lowercase letters to denote variables relating to individuals and capital letters for variables relating to the aggregate economy.

Before proceeding it is important to note that we use the term "inequality" as a relative concept. More inequality can therefore be characterized by a shift in the Lorenz curve, which clearly is measured in relative terms. For example, the Lorenz curve for income depicts the relative share of total income of the poorest x percent of the population where the population percentages are on the horizontal axis. Obviously, we could also be interested in absolute differences in income. However, most of our discussions will not depend on details of such definitions. The interested reader is referred to Cowell (2000).

1.2 The Neoclassical Theory of Distribution

Let production take place in firms that rent factors of production from households, and use these factors in (possibly heterogeneous) production functions. (Now lowercase letters refer to a particular firm rather than a household.) A firm produces y = f(k, l) units of output, takes as given the (possibly heterogeneous) rental prices r and w of the factors it employs, and maximizes profits as in


If technology is convex, i.e., f(•, •) is a concave function, the first-order conditions


are necessary and sufficient for solution of the problem (1.8). Note that f(•,•), r, and w may, in general, be different by firms.


Excerpted from Income Distribution in Macroeconomic Models by Giuseppe Bertola, Reto Foellmi, Josef Zweimüller. Copyright © 2006 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Ch. 1 Production and distribution of income in a market economy 3
Ch. 2 Exogenous savings propensities 14
Ch. 3 Optimal savings 28
Ch. 4 Factor income distribution 51
Ch. 5 Savings and distribution with finite horizons 88
Ch. 6 Factors shares and taxation in the OLG model 127
Ch. 7 Investment opportunities and the allocation of savings 147
Ch. 8 Risk and financial markets 180
Ch. 9 Uninsurable income shocks 204
Ch. 10 Distribution and market power 241
Ch. 11 Indivisible goods and the composition of demand 266
Ch. 12 Hierarchic preferences 302
Ch. 13 Dynamic interactions of demand and supply 321
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