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Growth and inequality: a demographic explanation


This paper investigates the relationship between growth and inequality from a demographic point of view. In an extended model of the accidental bequest with endogenous fertility, we analyze the effects of a decrease in old-age mortality rate on the equilibrium growth rate as well as on the income distribution. We show that the relationship between growth and inequality is at first positive and then may be negative in the process of population aging. The results are consistent with the empirical evidence in some developed countries.

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  1. The distribution of wealth is more concentrated and skewed to higher income groups than the distribution of earnings. See, for example, Castañeda et al. (2003) and De Nardi (2004).

  2. In an influential study, Kotlikoff and Summers (1981) estimate that a large fraction of the US capital stock was attributable to intergenerational transfers.

  3. Bowles and Gintis (2002) decompose a correlation between parent income and offspring income, which is estimated to be 0.32, into several causal channels. Wealth accounts for 0.12, which is much higher than schooling (0.07), race (0.07), and IQ (0.04).

  4. In an endogenous growth model à la Romer (1986), the growth rate of per capital income is related positively to the aggregate saving rate and negatively to the fertility rate. It can be shown below that the fertility rate decreases with population aging, which is well known as the demographic transition in the modern growth regime (Galor and Weil 1998, 2000). Combining the saving effect and the fertility effect, we can derive a hump-shaped pattern of the growth rate.

  5. See, for example, Bénabou (1996).

  6. Some papers conclude in the other contexts that the relationship between growth and inequality depends on the stage of economic development (see Aghion and Bolton 1992; Galor and Tsiddon 1997; and Banerjee and Duflo 2000).

  7. Precisely, the total number of types in period t is equal to t+1, which must be a finite number. Assume that the model economy starts at period 1. Assume also that a fraction (1−p) of individuals who are born at the beginning of period 1 do not receive bequests (type 0), and that the rest of the generation receive some bequests (say, type 1). In the next period, the population share of types 0, 1, and 2 is, respectively, given by 1−p, (1−p)p, and p 2. In the same way, the population share of type i (i=0, 1,...t−1) in period t is (1−p)p i, and that of type t is given by p t. However, the economic impact of type t in period t may be measured zero if t is sufficiently large.

  8. If the assumptions are relaxed, the population distribution becomes complex, since the fertility rate depends on the wealth accumulation. One can incorporate fertility decision into a computational general-equilibrium method, such as Huggett’s (1996) and Huggett and Ventura’s (2000), although it is beyond the scope of the paper.

  9. We assume f′>0, f″<0, f(0)=0, limk→+0 f′(k)=+∞, and limk→∞ f′(k)=0.

  10. The definition of the saving rate follows Zhang and Zhang’s (2001a,b). From Eqs. 20 and 22, the saving rate is proportional to the aggregate saving rate, K t+1/Y t. Note that \(\widehat{s}_{{\text{t}}} \) may exceed one if the share of bequest in the household income is relatively large.

  11. β corresponds to z in Zhang and Zhang (2001a,b).

  12. Zhang et al. (2003) derives a similar hump-shaped pattern in a Lucas-type endogenous growth model. The reason is that it is the aggregate saving rate that determines the growth rate if the public schooling is not controlled (p. 91).

  13. It is assumed that α 1=0.5, α 2=0.2, v=0.1, η=10, and β=0.05. From Eq. 35, the critical value is β g=0.122.


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I would like to thank Junsen Zhang, the editor of this journal, and three anonymous referees for their useful suggestions and comments. I am grateful to Frank Cowell, Akira Yakita, Hikaru Ogawa, Makoto Hirazawa, Yuji Nakayama, Akira Momota, Yoshinao Sahashi, and workshop participants at Chukyo University and Osaka Prefecture University for helpful comments on an earlier version of the paper. The hospitality at STICERD, LSE is gratefully acknowledged. The research is supported by grants from the Japan Society for the Promotion of Science (No. 13630023) and the Pache Research Subsidy I-A-2 (Nanzan University, 2006). All errors are the author’s.

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Correspondence to Kazutoshi Miyazawa.

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Proof of Proposition 2

Differentiating Eq. 32 with respect to p, we have

$$\frac{{d\widehat{s}}}{{dp}} = \frac{{\alpha _{1} {\left\{ {{\left( {1 - p} \right)}{\left[ {1 + \alpha _{1} {\left( {1 - p} \right)}} \right]} - {\left( {\beta + p} \right)}} \right\}}}}{{\beta {\left[ {1 + \alpha _{1} {\left( {1 - p} \right)}} \right]}^{2} }}$$

We know

$$\begin{array}{*{20}c} {\frac{{d\widehat{s}}}{{dp}}\left| {_{{p = 1}} } \right. = - \frac{{{\left( {1 + \beta } \right)}\alpha _{1} }}{\beta } < 0} \\ {\frac{{d\widehat{s}}}{{dp}}\left| {_{{p = 0}} } \right. = \frac{{\alpha _{1} {\left( {1 + \alpha _{1} - \beta } \right)}}}{{\beta {\left( {1 + \alpha _{1} } \right)}^{2} }}} \\ \end{array} $$

Since the numerator of \(d\widehat{s}/dp\) is a quadratic function of p with a positive coefficient of the quadratic term, the saving rate has an interior maximum if and only if \(d\widehat{s}/dp\left| {_{{p = 0}} } \right. > 0\), that is,

$$\beta < 1 + \alpha _{1} $$

Proof of Proposition 3

From Eq. 33, we know

$$1 + g = G{\left( p \right)} = \frac{{\alpha _{1} }}{{\alpha _{2} }}\eta v\phi {\left( p \right)}$$


$$\begin{array}{*{20}c} {\phi {\left( p \right)} = {\left( {1 - p} \right)}\varphi {\left( p \right)}} \\ {\varphi {\left( p \right)} = 1 + \frac{{\alpha _{2} {\left( {\beta + p} \right)}}}{{\beta {\left[ {1 + \alpha _{1} {\left( {1 - p} \right)}} \right]}}}} \\ \end{array} $$

Note that G(0) > 0, which contrasts with Fuster (1999) and Cipriani (2000). First, we show that the growth rate is concave with respect to p, that is, G′′ < 0. Then, we show that G′(0) > 0 is equivalent to Eq. 35. Since G(1) = 0 and G(0) > 0, g has an interior maximum if and only if the condition (Eq. 35) is satisfied because of the concavity.

We know

$$\begin{array}{*{20}c} {{\phi \prime {\left( p \right)} = - \varphi {\left( p \right)} + {\left( {1 - p} \right)}\varphi \prime {\left( p \right)}}} \\ {{\phi \prime \prime {\left( p \right)} = - 2\varphi \prime {\left( p \right)} + {\left( {1 - p} \right)}\varphi \prime \prime {\left( p \right)}}} \\ \end{array} $$


$$\begin{array}{*{20}c} {{\varphi \prime {\left( p \right)} = \frac{{\alpha _{2} {\left( {1 + \alpha _{1} + \beta \alpha _{1} } \right)}}}{{\beta {\left[ {1 + \alpha _{1} {\left( {1 - p} \right)}} \right]}^{2} }} > 0}} \\ {{\varphi \prime \prime {\left( p \right)} = \frac{{2\alpha _{1} \varphi \prime {\left( p \right)}}}{{1 + \alpha _{1} {\left( {1 - p} \right)}}}.}} \\ \end{array} $$

First, we have

$$\phi \prime \prime {\left( p \right)} = - \frac{{2\varphi \prime {\left( p \right)}}}{{1 + \alpha _{1} {\left( {1 - p} \right)}}} < 0$$

which implies that G is concave.

Second, we have

$$\phi \prime {\left( 0 \right)} = \frac{{\alpha _{2} }}{{\beta {\left( {1 + \alpha _{1} } \right)}}} + \frac{{\alpha _{1} \alpha _{2} }}{{{\left( {1 + \alpha _{1} } \right)}^{2} }} - {\left( {1 + \frac{{\alpha _{2} }}{{1 + \alpha _{1} }}} \right)}$$

Thus, ϕ′(0) > 0 is equivalent to Eq. 35.

Proof of Proposition 4

From Eq. 36, we have γ = (1−p)f 1(p)f 2(p), where

$$\begin{array}{*{20}c} {f_{1} {\left( p \right)} = 1 - \frac{\beta }{{\beta + p}}} \\ {f_{2} {\left( p \right)} = 1 - \frac{{\alpha _{2} }}{{1 + \alpha _{1} {\left( {1 - p} \right)} + \alpha _{2} }}} \\ \end{array} $$

First, we have γ(0) = γ(1) = 0.

Second, we have f 1′ > 0, f 1″ < 0, f 2′ < 0, and f 2″ < 0 for p∈[0, 1], and

$$\begin{array}{*{20}c} {\gamma \prime {\left( p \right)} = - f_{1} {\kern 1pt} f_{2} + {\left( {1 - p} \right)}{\left( {f^{\prime }_{1} {\kern 1pt} f_{2} + f_{1} {\kern 1pt} f^{\prime }_{2} } \right)}} \\ {\gamma \prime \prime {\left( p \right)} = - 2{\left( {f^{\prime }_{1} f_{2} + f_{1} {\kern 1pt} f^{\prime }_{2} } \right)} + {\left( {1 - p} \right)}{\left( {f^{{\prime \prime }}_{1} f_{2} + 2{\kern 1pt} {\kern 1pt} f^{\prime }_{1} {\kern 1pt} f^{\prime }_{2} + f_{1} {\kern 1pt} f^{{\prime \prime }}_{2} } \right)}} \\ \end{array} $$

Since γ(p) is continuous in [0,1] and γ(p) > 0 for ∀p∈(0, 1), there exists at least one solution p*∈(0, 1), such that it maximizes γ . The necessary condition, γ′(p*) = 0, requires

$$f^{\prime }_{1} {\kern 1pt} f_{2} + f_{1} {\kern 1pt} f^{\prime }_{2} = \frac{{f_{1} {\kern 1pt} f_{2} }}{{1 - p*}} > 0$$

which implies γ″(p*) < 0. Therefore, we know that γ does not have any local minimums in (0,1). It proves the uniqueness of p*.

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Miyazawa, K. Growth and inequality: a demographic explanation. J Popul Econ 19, 559–578 (2006).

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  • Inequality
  • Growth
  • Fertility
  • Accidental bequest

JEL Classification

  • D31
  • J13
  • O41