Public capital and distributional dynamics in a two-sector growth model

Journal of Macroeconomics 32 (2010) 606–616

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Public capital and distributional dynamics in a two-sector growth model Yoseph Yilma Getachew * Maastricht University UNU-MERIT, Keizer Karelplein 19, 6211 TC, Maastricht, The Netherlands

a r t i c l e

i n f o

Article history: Received 9 May 2009 Accepted 30 December 2009 Available online 6 January 2010 JEL classification: D31 E25 H52 H54 O41 Keywords: Inequality Infrastructure and public services Growth

a b s t r a c t This paper mainly develops a joint theory of public capital, inequality, and growth, in a two-sector growth model that yields complete analytical solutions. Public capital plays an important role in long-run growth through enhancing productivity and complementing the accumulation of private inputs. Under certain conditions, it could also have important implications for income inequality dynamics. Inequality is bad for growth, when the credit market is imperfect and there is a diminishing marginal rate of return on private investment. Certain public services and investment may benefit the poor more than proportionally and thus improve the distribution of income, and hence, improve economic growth through an indirect channel. The key mechanism linking the distribution of income to public capital is its disproportional effect on the economy that affects factor shares of capital. The paper also studies the determination of optimal tax. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction The roles of public capital and income distribution in economic growth are well studied both analytically and empirically. However, the distributional effect of public capital, especially infrastructure, is usually ignored in the theoretical literature despite evidence and intuition to its disproportionate impact on household income (see, for e.g., Jacoby, 2000; Calderon and Chong, 2004). Provision of infrastructure or public services (such as public education, public health, and clean water) may benefit poor households more than proportionally due to their lack of access to their private substitutes. On the other hand, infrastructure (such as telecommunications and electricity) may tend to favor richer groups due to their greater access to their private complements. This disproportionate impact of public capital, once it is explicitly acknowledged, could be important for the dynamics of income distribution and hence the evolution of macroeconomic aggregates. This paper chiefly develops a joint theory of public capital, income distribution, and economic growth. Using a two-sector growth model, it extends the imperfect credit market theories in inequality and growth to public capital, inequality and growth. Inequality is bad for growth, when the credit market is imperfect and there is a diminishing marginal rate of return on private investment, because it prevents the poor from undertaking an efficient amount of investment (e.g., Loury, 1981; Benabou, 1996). The new theory suggests that certain public service or investment may benefit the poor more than proportionally, relax some of their resource constraints, and thus improve the distribution of income and hence economic growth through an indirect channel. The key mechanism linking the distribution of income to public capital is its disproportionate impact on the individual household, and, the subsequent effect on private factor (capital) shares. The basic idea and intuition behind the theory is described below.

* Tel.: +31 43 388 4450; fax: +31 43 388 4499. E-mail address: [email protected] 0164-0704/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2009.12.009

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Public capital plays an important role in long-run growth through enhancing productivity and complementing the accumulation of private inputs. Under certain conditions, it could also have important implications for income distribution dynamics. In an economy with heterogeneous agents and imperfect credit markets, the dynamics of aggregate variables such as public and private capital and the economy’s growth rate are determined jointly with those of income distribution. If there are diminishing returns to private factors in the economy, then income inequality is negatively related to the growth rate while the distributional dynamics is determined by the private factors shares. Therefore, if provisions of public capital affect the private factor shares (due to their disproportionate impact on the economy), they will also affect income distribution dynamics. In this case, public capital once more becomes an important determinant of long-run growth through its indirect effect on income distribution. The following example could help set the above theory in perspective. Consider an economy with heterogeneous agents in terms of their initial wealth but similar otherwise. If access to credit is limited, investment opportunities depend on individuals’ initial level of wealth. If production function faces diminishing returns to factor inputs, relatively poor individuals, who have relatively lower investment opportunities, would have high marginal productivity in production. This means initial wealth distribution also determines aggregate output that would be produced in this economy. Therefore, ceteris paribus, the more egalitarian (initial) wealth distribution is, the higher the aggregate production would be. What determines income distribution dynamics? When the credit market is imperfect, income distribution evolves according to private factor shares. When there are differences in initial endowment among households who are otherwise similar, and, in the presence of imperfect credit market, the dynamics of income distribution depends on the private factor shares or the degree to which households are able to exploit their relative initial advantage. The presence of a public goodtype input (e.g. an infrastructure or a public service) in production has no effect on income distribution dynamics unless it alters the private factor shares.1 If the provision of public capital as an additional input in production, however, could affect the private factor shares, then public capital becomes important for income inequality dynamics. This happens when the public or infrastructure services accrue heterogeneously among the individual households. In this case, public capital becomes important for income inequality dynamics and hence to growth (through an indirect channel). The next section presents the model. In the model, we suppose an economy, populated by heterogeneous agents, consists of two production sectors – human capital accumulation and goods production. In the former, human capital is generated using inputs from public and private resources while the production technology is characterized by inter-generational spillover. Production in the goods sector takes place also using private and public inputs. The benefit that accrues from using the public inputs is different among households. That is, depending on the type of the public good, a provision of public capital may benefit the poor (rich) more than proportionally due to their lesser (greater) access to their private substitutes (complements). Within such setup, we show that public capital not only affects growth but also inequality. That is, the dynamics of income inequality not only depends on the magnitude of the share of private capital but also public capital. The greater the output elasticity of public capital, the larger is its effect on inequality. Under some conditions, income inequality is negatively related to economic growth while certain infrastructure or public services could help to mitigate this effect, and hence could promote economic growth through an indirect channel. With respect to growth and public capital nexus, public capital in both sectors have a net positive effect on long-run growth while the magnitudes of optimal taxes (growth maximizing taxes) on both sectors depend on whether there is inter-generational spillover (in contrast to Barro-type findings). The study relates to strands of literature, which we extend along various dimensions. It relates to the large volume of literature dedicated to studying the relationship between public capital and economic growth (e.g., Barro, 1990; Futagami et al., 1993; Turnovsky and Fisher, 1995; Turnovsky, 2000; Agenor, 2008; Ziesemer, 1990, 1995, among many). This literature studies the relationship between public capital (stock or flow) and economic growth analytically in a representative agent framework. Barring a few exception, the literature has restricted public capital to a single sector (either in the goods production or human capital accumulation sector).2 In reality, the two public inputs coexist,3 and interact in their technological parameters, affecting each other’s macroeconomic performance. For instance, we show that Barro’s result holds only when there is no inter-generational spillover. Existence of inter-generational spillover affects the optimal tax rates in both sectors. The paper is also related to literature that studies the relationship between public capital and income inequality. Recently, a growing number of empirical studies try to assess the impact of infrastructure on income inequality. For instance, Calderon and Chong (2004) and Lopez (2003) show that infrastructure reduces income inequality and enhances economic growth at the same time. Jacoby (2000) argues that some infrastructure services could result in substantial benefits on average, much of it going to the poor. On the other hand, Garcia-Penalosa and Turnovsky (2007, 2008) and Chatterjee (2008) analytically studied the distributional impact of different ways of financing public good. They argue that growth-enhancing fiscal policies 1 This is simple to demonstrate. Suppose that the individual production function is taken place with two complementary inputs yt ¼ Aðkt Þa ðX t Þh where kt is ~ l, private capital and X t is public capital. Assume further that private capital, which initially differs among individuals, is distributed lognormally, i.e., ln kt Nð t r2t Þ. Then, an individual’s saving at t þ 1 is ktþ1 ¼ syt ¼ sAðkt Þa ðX t Þh , where s is an exogenous saving rate. Income distribution at t þ 1 is given by, a long story cut 2 2 2 short, v arðln ktþ1 Þ ¼ rtþ1 ¼ a rt . Therefore, in this economy, what matters for income distribution dynamics is neither X t nor its output elasticity h but the private capital income share a. 2 Rioja (2005) studies two public capitals, in education and goods sectors, with respect to growth, in a representative agent framework. But, he does not provide an analytical solution. 3 Public services on primary schooling, basic research and health, which are important for the accumulation of individuals’ human capital, essentially coexists with other infrastructure services such as roads, airports and energy, which are primarily crucial for the production of firms.

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are mostly related to greater pre-tax inequality although this might also depend on the type of financing the public good (Chatterjee, 2008). Taxation policy thus may determine existence of efficiency-equity trade-off. In contrast, the present paper argues that spending policies also determine the distributional dynamics while the existence of an efficiency-equity tradeoff depends on the existence of diminishing returns to private factors.4 The other strand of literature, may be, most closely, related to the present study deals with the dynamics of income inequality and long-run growth within an imperfect credit market scenario (e.g. Loury, 1981; Galor and Zeira, 1993; Banerjee and Newman, 1993; Aghion and Bolton, 1997; Aghion et al., 1999; Benabou, 1996, 2000, 2002). This literature in general states that when the credit market is imperfect, inequality negatively affects economic growth, because relatively more high-return investment opportunities would be forgone by resource-poor households in inegalitarian than egalitarian societies. We argue here that certain public provisions could come to the rescue due to their disproportional impact on the poor. This literature does not focus on public capital but on income distribution and growth. For example, Benabou (2000, 2002) argue that private factor income shares and family wealth determine the dynamics of the distribution of income and growth while we show here that public capital could also be an important determinant of the dynamics of inequality. The remainder of the paper is organized as follows. Section 2 provides the model. Section 3 is all about income distribution and public capital. Various macroeconomic aggregates that arise in the model and their dynamic behaviors are studied in Section 4. Section 5 concludes. 2. The model 2.1. Households and firms There is a continuum of heterogeneous households, i 2 ½0; 1. Each household i consists of an adult of generation t and a child of generation t þ 1. At the beginning, each adult of the initial generation is endowed with human capital h0 and a public infrastructure G0 which is shared among others. The distribution of income is assumed to take, initially, a known probability distribution of C0 ðÞ. Thus, the initial distribution is given and evolves over time at equilibrium.5 Agents care about their consumption level and the human capital stock of their children. When young, they accumulate human capital using both private and public input. When adult, they use their human capital for final goods production. The government taxes income with two fixed flat rate taxes, w and s, in order to finance infrastructure and public service, denoted by Gt and Mt , in the final goods production and human capital accumulation sectors, respectively. Individuals allocate after tax income between current consumption ct and children education et , while the latter represents private investment for human capital accumulation of the offspring. Preferences are logarithmic. Production functions are Cobb–Douglas. A utility of an individual is thus defined as

ln ct þ b ln htþ1

ð1Þ

subject to

ct þ et ¼ ð1  s  wÞyt

ð2Þ

where yt is income of an individual. Human capital accumulation function for the offspring htþ1 is a function of parental human capital ht , private educational investment et , and public service mt ,

htþ1 ¼ Bðht Þe ðmt Þt ðet Þg

ð3Þ

mt corresponds to a particular individual’s value of public services,

mt ¼

Mt ðht Þf

;

1 < f < 1

ð4Þ

where Mt and ht denote the actual government spending in the sector, and the individual’s initial wealth (human capital), respectively. The use and efficiency of public capital thus vary among households. Depending on the type of the public good, productive government expenditure may benefit the poor (rich) more than proportionally due to their lack of (greater) access to its private substitutes (complements). The direction and magnitude of f, in (4), convey the type and the degree of the disproportionate impact of public services on the individual’s human capital accumulation, respectively. 0 < f 6 1 ð1 6 f < 0Þ corresponds to public service that benefits the poor (rich) more than proportionally. When f ¼ 0, which is usually (implicitly) assumed in the literature, public services benefit the poor proportionally. 4 The paper may also be related to literature in public education, redistribution and inequality (See, for instance, Glomm and Ravikumar, 1992, 2003b; SaintPaul and Verdier, 1993, among many). This literature focuses on the education sector with a particular emphasis of the distributional impact of public education (rather than the general infrastructure and public services). Glomm and Ravikumar (1992) compared public and private funding regimes and concluded that inequality more quickly declines in the public regime. But, in another paper (Glomm and Ravikumar, 2003b), they argued that equal access to public education may not yield income convergence. In contrast, this paper argues that the government’s specific public service choices are what determine income convergence (or divergence). 5 In all the text, small and capital letters represent individual and aggregate (average) variables, respectively.

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A number of practical instances can be mentioned where productive government spending could have a disproportionate impact on the economy. Construction of a new dam (or improved irrigation) may benefit more those who have access to fertilizer and high-yielding variety of seeds. The same goes for rural roads and internet use. They benefit more those with bicycles and motorbikes, and computers and better education, respectively (Songco, 2002; Estache et al., 2002). On the other hand, provision of public services such as public education, and/or, provision of infrastructure services such as clean water, sanitation and public transport, may benefit the poor more than proportionally. To a great extent, the poor are poor because they lack these basic inputs (such as clean water, sanitary surroundings, and mobility) (World Bank, 1994). While aggregating Eq. (4), interesting enough, the model captures congestion cost. In this case, f (where 0 < f < 1Þ denotes congestion factor.6 However, neither the distributional effect of public capital nor Eq. (4) shall be confused with congestion. At the individual level, Eq. (4) and its distributional parameter f do not reflect congestion but distribution.7 Combining (3) and (4), the human capital accumulation function becomes

htþ1 ¼ Bðht Þn ðM t Þt ðet Þg

ð5Þ

where 0 < e; g; t; < 1; 0 6 n  ðe  tfÞ < 1, and 1 6 f 6 1. 2.2. Firms We assume each household owns a firm.8 Aggregate output is thus the sum of the individuals’ productions. We also assume ~ l ; r2 Þ. individuals differ only in their initial human capital, which is lognormally distributed across agents: ln ht Nð t t 9 Thus, the income of an agent of generation t is

yt ¼ Aðht Þa ðg t Þh

ð6Þ

We define g t similar to (4)

gt ¼

Gt ; ðht Þj

1 < j < 1

ð7Þ

Thus, g t and j, in the goods production sector, are counterparts of mt and f, in the human capital accumulation sector, respectively; while (7) is a counterpart equation of (4). Therefore, g t – infrastructure service that corresponds to a particular individual’s value – depends on infrastructure stock Gt in the goods production sector and initial human capital of the agent. Combining (6) and (7), the production function for the goods production becomes

yt ¼ Aðht Þx ðGt Þh

ð8Þ

where 0 < a; h; x < 1; 1 6 j 6 1 and x  ða  hjÞ. While aggregate, production Y t is

Y t ¼ AðHt Þx ðGt Þh exp



r2t 2

10

 ðxðx  1ÞÞ

ð9Þ

r2

since E½ðht Þx  ¼ ðHt Þx exp 2t xðx  1Þ, where Ht is the aggregate human capital (see Appendix A). According to (9), aggregate income is smaller in heterogeneous economies than representative ones

r2t ¼ 0.

2.3. Government We assume that the government budget is at all times balanced:

Igt 

Z

Mt 

1

yt w dCt ðht Þ ¼ Y t w

ð10Þ

0

Z 0

1

yt s dCt ðht Þ ¼ Y t s

ð11Þ

Thus, the government collects proportional taxes w and s on output Y t to finance public investment and public service, Igt and Mt , respectively, while the accumulation of infrastructure follows the rule 6 Traditionally, congestion is modelled in the literature as M ct ¼ ðHMtÞf , where f represents the degree of congestion; Ht and M t are aggregate private capital and t public expenditure, respectively; and hence M ct represents public expenditure with congestion cost. See also Glomm and Ravikumar (1997). 7 First, congestion is an aggregate phenomenon. Congestion cost from an individual perspective (in light of equation (4)) is negligible. Moreover, if we had modelled (4) to reflect congestion, i.e., as ðHMtÞf , then public capital would not have had any effect on the individual private factor income share and hence income t distribution dynamics. 8 This assumption shuts off the input market, or it is another way of assuming that the credit market is imperfect (see, e.g., Benabou, 2000, 2002, for a similar specification). 9 yt shall be interpreted as income net of the cost of physical capital as in Barro et al. (1995) and Benabou (2002). See footnote 16 for a detailed discussion. 10 The case a ¼ hj is not included, in contrast to the human capital accumulation sector where a case of e ¼ tf is considered, because it is unlikely that goods would be produced using only infrastructure services. In fact, roads do not produce by themselves.

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Gtþ1 ¼ Igt þ Gt ð1  dg Þ

ð12Þ

g

where Gt and d denote infrastructure stock and depreciation, respectively. 2.4. Competitive equilibrium According to the above descriptions, an individual of generation t solves the following problem, which is derived by substituting (2) and (5) into (1),

Max lnðð1  s  wÞyt  et Þ þ b ln Bðht Þn ðM t Þt ðet Þg

ð13Þ

et

taking as given, s; w; Mt ; Igt and Gt . The first order condition gives

et ¼ að1  s  wÞyt

ð14Þ

bg where a ¼ 1þb g; (14) shows the agent’s optimal saving as the function of her income. Notice that the saving rate is identical

among individuals, due to logarithmic preferences, although the rate of return on investment is different.11 Individuals’ capital accumulation associated to their optimal behavior is derived by, first, substituting (14) and (11) into (5), and then, using (8) and (9)

htþ1 ¼ BAtþg st ðað1  s  wÞÞg ðht Þnþxg ðGt ÞhðtþgÞ ðHt Þtx exp



r2t 2

 ðtxðx  1ÞÞ

ð15Þ

From (15), an individual’s optimal human capital accumulation is determined by the human capital of her parent ht , the initial income distribution r2t , aggregate public and private capital (Ht and Gt respectively). The negative effect of income inequality in the individual human capital accumulation could not be a surprise. In fact, in the model, household human capital accumulation is a function of the provision of public capital M t , which, in turn, depends on the level of aggregate income Y t . But Y t has a negative relationship with income inequality r2t due to credit market imperfection and the existence of diminishing returns to private factors. Therefore, from (15), the following proposition can be established: Proposition 1. Income inequality bears additional cost to the individual household’s optimal human capital accumulation.

3. Income distribution and public capital From (15), we derive the following two difference equations that characterize the evolution of capital accumulation and income distribution in the economy12

ltþ1  E½ln htþ1  ¼ ðn þ xðg þ tÞÞlt þ ðt þ gÞ ln A þ ln B þ hðt þ gÞ ln Gt þ t ln s þ g ln að1  s  wÞ þ r2tþ1  v ar½ln htþ1  ¼ ðn þ xgÞ2 r2t

r2t 2

ðtx2 Þ

ð16Þ ð160 Þ

Eq. (160 ) has a solution, r2t ¼ ðn þ gxÞ2t r20 . Thus, steady state income distribution r2 takes a value of the initial distribution r20 , 0 or 1, depending on some conditions,

r2 ¼ 0 if n þ gx < 1 r2 ¼ r20 if n þ gx ¼ 1 r2 ! 1 if n þ gx > 1

ð17Þ ð18Þ ð19Þ

where 0 6 n  ðe  tfÞ < 1, 0 < e; g; t; a; h; x < 1; x  ða  hjÞ, and 1 6 f; j 6 1. Therefore, income inequality could decline through time and ultimately vanish for certain values on the parameters, n þ xg < 1.13 However, this should not be confused with a stylized fact. Heterogeneity, in this model, is only on individuals’ initial wealth; agents are similar otherwise, in their ability, technology, etc. Therefore, a diminishing return on net private accumulative factors in general implies that resource-poor households are more productive than rich ones that in turn leads to a declining income inequality. From (16) and (160 ), the model captures the intuition that differences in family wealth and the existence of public capital as an input for the production of goods and the accumulation of human capital play important role in the persistence of in11 In logarithmic utility function, inter-temporal elasticity of substitution is one, and consequently income effect exactly compensates substitution effect (see De la Croix and Michel, 2002, pp. 13–14). In this case, individuals’ saving rate is independent of the rate of return. r2 12 ~ l ; r2 Þ). We use the fact that E½ln ht  ¼ ln Ht  2t  lt in deriving (16) (see Appendix A, and recall that ln ht Nð t t 13 This result is comparable with that of Glomm and Ravikumar (1992). In a private education economy, they state that income inequality may decline, increase or remain constant depending on the sum of parameters in the sector.

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come inequality. Depending on the values of the distributional parameters, whether 0 < f; j 6 1 ð1 6 f; j < 0Þ, public capital would have a positive (negative) impact on inequality. Family wealth, however, similar to what is found by Benabou (2000, 2002), determinately exasperates income inequality. More important is the parent’s wealth, i.e., the larger is the inter-generational persistence e, for the accumulation of the offspring’s human capital, the more income inequality persists. But, more important are the public and infrastructure services for the accumulation of human capital and the production of goods, that is, the greater are h and t, also the larger their disproportionate impact on income, i.e., the greater are f and j, the faster (the slower) income inequality declines when 0 < f; j 6 1 (when 1 6 f; j < 0). In regard to the effect of public and infrastructure services on the economy, we thus have the following proposition: Proposition 2. Public and infrastructure services in human capital and goods production sectors speed up (slow down) income distribution convergence, in the short run, when 0 < f; j 6 1 ð1 6 f; j < 0Þ. Therefore, the dynamics of income distribution is governed by the private shares (such as e; g, and a). However, provision of public capital in both sectors could have important role in the distribution dynamics by altering these shares. The impact of public capital on income inequality dynamics depends on the degree of its importance to private production, which is reflected on the magnitude of h and t. Moreover, whether public and infrastructure services are pro-poor (to be determined on the magnitude and the direction f and j) is important to income distribution dynamics. We can have further insight on the importance of public capital, especially when it compares to private pre-existing conditions, to income distribution dynamics if we specify parameter values that seem reasonable for real economies. We set the inter-generational persistence e at .6. According to Solon (2002), the inter-generational persistence is about .4 or above for the United States. But, in general, it is expected to be higher in developing countries due to credit market imperfection (Grawe and Mulligan, 2002).14 We compute the public and private shares h and a respectively from a three-factors production function following Benabou (2002) and Barro et al., 1995. Following this literature, we use .3 and .5 for the shares of physical and human capital, respectively. We set the elasticity of output of public investment at .2.15 This gives us a ¼ :5=:7 ¼ :71, and h ¼ :2=:7 ¼ :29.16 We set ag ¼ :63; a value based on the estimate of Card and Krueger, 1992 for t, which is about .12. 17 But, there are no empirical estimates for the distributional parameters, which are assumed to be in ranges between 1 and 1. Calibrating Eq. (160 ) with these values yields a coefficient of income inequality dynamics ðr2tþ1 =r2t Þ about 1.5 for the private pre-existing condition (f ¼ 0 and j ¼ 0) implying a rising income inequality. A strong positive distributional effect of public capital (f ¼ 1 and j ¼ 1), however, could reduce the inequality coefficient significantly to .72. But, if f ¼ 1=2 and j ¼ 1=2, then the coefficient will be reduced to only 1.08 implying persistent inequality. Dollar (2005) argues that starting 1980 the trend in income inequality in general in developing countries is not rising, which may imply no insignificant values for the distributional parameters. Note that more than two third of the change in the coefficient is due to infrastructure services. For instance, j ¼ 1, (and f ¼ 0), only yields an inequality coefficient of .94. This is because infrastructure has a direct effect on the individuals’ income in contrast to public services that have an indirect effect through the accumulation of capital. Another reason is the assumed higher quality of infrastructure services, which is reflected by greater elasticity of output. This may have some implications for policy. First, public services (such as public education) may not be ‘‘the great equalizer” as expected to be (Glomm and Kaganovich, 2003a). Second, the government’s effort in using public capital as a distributional mechanism could be compromised by its quality. The expansion of public services (such as public education) in developing countries may not result in income convergence unless it is done together with improvement in quality. Both the quality and the quantity of public capital are important to income distribution dynamics. 4. Growth, inequality and public factors 4.1. Aggregate capitals To determine the remaining macro-variables, aggregate (15) to obtain the equation that characterizes the evolution of aggregate human capital

Htþ1 ¼ BAst ðGt Þh ðHt Þnþx ðað1  s  wÞÞg Xt

ð20Þ

where g þ t ¼ 1,18 14 For instance, Hertz (2001) and Dunn (2007) estimated .6 and .69 for South Africa and Brazil respectively. Grawe and Coraks (2004) estimates for Malaysia and Peru (.54 and .67 respectively) are also larger than the US. 15 Fay (2001) and Kamps (2005) estimated .2 for Latin America and the majority of the OECD countries respectively while Miller and Tsoukis (2001) calibrated their model to estimate about 0.18 for a wide range of low and middle income countries. 16 Following Benabou (2002), yt , in (6), is interpreted as income net of the cost of physical capital kt . If e y t is gross output with three-factors production 0 00 e e @yt k0 k00 y t  rkt ¼ e y t ð1  kÞ where @@kytt ¼ r. Since @@hytt ¼ @h (or k0 e and h ¼ 1k . y t ¼ ayt ), it follows that a ¼ 1k function: e y t ¼ Aðkt Þk ðht Þk ðg t Þk , then yt  e t 17 Rioja (2005) also calibrated his model based on the estimate of Card and Krueger (1992) for Latin America countries ‘‘given the absence of systematic estimates” for developing countries. r2 18 While computing (20), we use the fact that E½ðht Þnþxg  ¼ ðHt Þnþxg expð 2t ðn þ xgÞðn þ xg  1ÞÞ (see Appendix A).

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r2t

Xt ¼ exp

2

 ððn þ xgÞðn þ xg  1Þ þ txðx  1ÞÞ

The difference equation for infrastructure service is easily derived by substituting the last term of (10) into (12), using (9), and assuming a complete depreciation ðdg ¼ 1Þ

Gtþ1 ¼ wAðHt Þx ðGt Þh F t

ð21Þ

where

F t ¼ exp



r2t 2

 ðxðx  1ÞÞ

Eqs. (16), (20), and (21) determine the dynamics of the economy including the growth rate. From (20) and (21), income inequality could undermine the accumulation of aggregate capital under certain conditions. If n þ xg < 1 and x < 1, then Xt ; Ft 6 1. The maximum values Ft ; Xt ¼ 1 are thus reached when r2t ¼ 0, which implies that the highest capital accumulation is realized when the society is perfectly egalitarian. This implies that policies that decrease income inequality may also enhance growth. However, if n þ xg > 1 (e.g., due to a large inter-generational persistence), then inequality could be good for growth – creating an efficiency-equity trade-off – and hence, policies that enhance growth might also aggravate income inequality. 4.2. Dynamics and steady state The value that nð e  tfÞ assumes is important in determining the long-run behavior of the system. First of all, when 0 < f < 1, recall that e and tf have opposite roles on income distribution dynamics. The former (the greater) makes income inequality persistent whereas the later (the greater) reduces it through time. We analyze two different cases below, when n ¼ 0; and n > 0.19 When n ¼ 0, the system behaves in the long run similar to a standard AK model. In steady state, the ratio HG is constant. To see this, divide (20) by (21) and r2 ¼ 0:

H ¼ Bag st w1 ð1  s  wÞg G

ð22Þ

In the long run, the system is thus characterized by a continuum of steady state equilibria while each can be reached only if the system starts at equilibrium. Moreover, aggregate variables will be in a balanced growth path, where H; G and Y grow at the same rate.20 However, in the short run, it exhibits transitional dynamics, unlike the textbook AK model, which arises from the existence of income inequality dynamics r20 –0 in the model. The growth rate of the economy ctþ1 is analytically determined at any point in time:

ctþ1 ¼ x ln v þ xg lnð1  s  wÞ þ xt ln s þ h ln w þ Dt

ð23Þ

where21

Dt ¼ x 3

r2t 2

1

gðxg  1Þ < 0 and v ¼ Bag Ax

According to (23), r2t is important for the economy’s growth rate. The term Dt captures the extent to which inequality hampers economic growth during transition in a heterogeneous economy with a production function that exhibits diminishing returns to factors, and imperfect credit market. Public capital could mitigate (exasperate) this effect under the condition 0 < j 6 1 ð1 6 j < 0Þ. Note that if Dt ¼ 0 ðr2t ¼ 0Þ, then the growth rate of output c > 0. But, for greater Dt (due to greater r2t ), the growth rate of output could be zero and even negative. The relationship between the taxes used to finance public capital and long-run growth is non-linear, in line with the lit@c ¼ @@sc ¼ 0: erature. The growth maximizing taxes (wgmax and sgmax ), for the case n ¼ 0, are derived by @w

wgmax ¼ h

ð24Þ

sgmax ¼ txð tða  hjÞÞ

ð25Þ

The optimal tax for infrastructure service wgmax is equal to the share of public capital in the sector (similar to what is found by Barro, 1990) whereas the growth maximizing tax for public service sgmax is equal to the share of the public capital t in that sector times the net output elasticity of human capital xð a  hjÞ (in line with Ziesemer, 1990, 1995).22 The reason that technological parameters from the goods production sector (such as a; h and j) are related to sgmax is because the tax used to finance the public service s affects growth via its positive role in the accumulation of human capital, which in turn will be used for the final goods production. 19 20 21 22

We exclude the case n < 0 because of its unlikeliness. The variables without time subscript (H; G; Y and r2 ) denote steady state values. See Appendix B for details on the derivation. But, as we see below, these results hold only when there is no inter-generational spillover.

Y.Y. Getachew / Journal of Macroeconomics 32 (2010) 606–616

613

Fig. 1. Phase diagram for a case h > n or 0 < hn < 1. Together with the eigenvalues the steady state is globally stable. h

When n > 0, there exists a stable and unique global steady state where H; G and Y converge. The steady state is saddle point stable. The local stability is established in Appendix C. Whereas, the global analysis is done here using phase diagrams near the set of points where r2t ¼ 0; n – 0. Thus, from (20) and (21) we have

Htþ1 ¼ BAðHt Þnþx ðGt Þh ðað1  s  wÞÞg st

ð26Þ

Gtþ1 ¼ wAðHt Þx ðGt Þh

ð27Þ

To build the phase diagram, first we need to characterize the set of points where there is no change on the variables, for (26) and (27). That is, for (26) we solve Htþ1 ¼ Ht for Gt and for (27) we solve Gtþ1 ¼ Gt for Ht , to get hn h

1

Gt ¼ ðBAðað1  s  wÞÞg st Þh ðHt Þ 1

1h

ð28Þ

1

Ht ¼ ðwAÞ x ðGt Þ x ¼ ðwAÞ x Gt

ð29Þ

The slope of the phase line (28) depends on the relative values of h and n. If h > n, then 0 < hn < 1 and hence Gt is increash < 0, and Gt is decreasing at an increasing rate in Ht ing at a decreasing rate in Ht , in space ðHt ; Gt Þ (Fig. 1). If h < n, then hn h 1 (Fig. 2). The curve (29) is easy to characterize. The phase line is a diagonal line, with slope ðwAÞx . By combining (28) into (29), we obtain the equilibrium values where the two phase lines meet 1

nh

1 n

xn

G ¼ ðBAðað1  s  wÞÞg st Þn ðwAÞ xn H ¼ ðBAðað1  s  wÞÞ

g t

s Þ ðwAÞ

ð30Þ

h

ð31Þ

Figs. 1 and 2 capture the qualitative feature of the model. Notice that although the slopes of the phase lines for Ht are different for the two cases, (h < n and h > n), the steady state equilibrium loci Z remain the same. Moreover, the saddle path has a negative slope. Note that (in the case of n > 0) aggregate variables exhibit imbalance growth. While human capital grows faster than output, public capital grows at the same rate with the latter. If we log-linearize the system, (20) and (21), near a local steady state point ðH; GÞ,23 we can characterize the growth rates,

ðln Htþ1  ln HÞ ¼ ðn þ xÞðln Ht  ln HÞ þ hðln Gt  ln GÞ

ð32Þ

ðln Gtþ1  ln GÞ ¼ xðln Ht  ln HÞ þ hðln Gt  ln GÞ

ð33Þ

The economy’s growth rate near the steady state is defined, c ¼ ln Y t  ln Y,

c ¼ xðln Ht  ln HÞ þ hðln Gt  ln GÞ

ð34Þ

Combining (33) and (34), we obtain

c ¼ ln Gtþ1  ln G 23

The loglinearization can be done near equilibrium points of the distribution which exist,

ð35Þ

r2 ¼ r20 or r2 ¼ 0.

614

Y.Y. Getachew / Journal of Macroeconomics 32 (2010) 606–616

Fig. 2. Phase diagram for a case h < n or

hn h

< 0. this result is the same with that of Fig. 1 except here the phase line for Eq. (28) is downward slopping.

Alternatively, from (32) and (34), we can get

ln Htþ1  ln H ¼ nðln Ht  ln HÞ þ c

ð36Þ

From (35) and (36), we see that there is an imbalance in the growth rates of the macro-variables. While Y t grows at the same rate with Gt ; Ht grows faster. The growth rate of the economy is derived by substituting (27) and (30) in (35): 1

nh

c ¼ ln wAðHt Þx ðGt Þh  lnðBAðað1  s  wÞÞg st Þn ðwAÞ xn

Then, the growth maximizing tax rates for infrastructure @c ¼ @@sc ¼ 0:24 puted @w

hð1  nÞ 1  hn

wgmax ¼

sgmax ¼

xt

1  nð1  xÞ

ð37Þ wgmax

and public services

s

 gmax ,

for the case of n > 0, are com-

ð38Þ ð39Þ

Eqs. (38) and (39) are equivalent to (24) and (25) (not surprisingly), respectively, if n ¼ 0. sg max is increasing at n but wg max is decreasing at it. Therefore, the optimal tax rate for infrastructure service is lower than that of predicted by Barro (1990) and other similar studies while the optimal tax for public service is higher than that of Ziesemers (1990, 1995), in the presence of inter-generational spillover.25 The existence of (net) inter-generational spillover in the human capital accumulation sector, i.e., n > 0, increases the role of human capital in the economy, which is reflected by a positive relation between sg max and n. On the other hand wg max and n are inversely related while both have a similar role in the economy – spillover effect. According to Barro and Sala-i Martin, 1992, the tax rate w raises growth since the private rate of return on investment falls behind the social return, which, in turn, invites some forms of stimulus (such as public investment) to investment. 5. Conclusion We studied public spending, in a two-sector economy populated with heterogeneous agents, as a factor that both enhances productivity and promotes the accumulation of human capital. We showed that public and infrastructure services in both the human capital accumulation and the goods production sectors have a net positive effect on long-run growth while the magnitudes of the growth maximizing taxes on the sectors depend on whether there is inter-generational spillover in the human capital accumulation sector. We disclosed the effect of income inequality on the individual and aggregate production and accumulation of capital. That is, we captured the negative effect of income inequality on economic growth, when the credit market is imperfect and there are diminishing returns to private investment. 24

We use g þ t ¼ 1, and x þ h ¼ 1. Using parameter values from Section 3, (f ¼ 0; j ¼ 0, a ¼ :71; h ¼ :29, and e ¼ :6), the growth maximizing tax rates for infrastructure and public service are about .14 and .1, respectively. Note that when r2t ¼ 0; f and j may not be interpreted as distributional parameters but congestion factors (see Section 2.1, and footnote 7). They may have important role in determining the optimal taxes. For instance, f lowers sg max but it raises wg max . 25

Y.Y. Getachew / Journal of Macroeconomics 32 (2010) 606–616

615

More importantly, we showed that certain public and infrastructure services could improve income inequality dynamics, and hence could promote economic growth, once more, through an indirect effect of mitigating the negative influence of income inequality on economic growth. We calibrated the model and showed how important public interventions (compare to ‘‘private pre-existing conditions”) could be to the distribution of income. Therefore, we conclude that under certain conditions, public capital could promote pro-poor growth (i.e., loosely defined as an increase in growth and reduction in income inequality simultaneously). In particular, with public investment which is pro-poor, not only could the economic pie grow but also a larger slice could pass to the poor. That makes a wise investment on productive public good an area that belongs to the win–win type of policies. Acknowledgements I would like to thank Thomas Ziesemer, Abbi Kedir, and anonymous referees for providing helpful comments at various stages of the paper. Appendix A. Aggregation The logarithm of a variable with lognormal distribution will have a normal distribution (and vice versa). A normal distribution preserves under linear transformation (Greene, 2003, Appendix B). We use these facts and other important relations between lognormal and normal distribution to study the evolution of income distribution in our model. ~ l ; r2 Þ, we have the following Since we assume a lognormal distribution for individual’s initial human capital, i.e., ln ht Nð t t relation

ln E½ht  ¼ E½ln ht  þ

r2t 2

() E½ln ht  ¼ ln E½ht  

r2t 2

 ln Ht 

r2t 2

ðA1Þ

since Eðht Þ  Ht  2  r We derive E½ðht Þx  ¼ ðHt Þx exp 2t xðx  1Þ , for example, in Eq. (9), using the above facts. If ht is a lognormal distribution then ðht Þx is also a lognormal distribution, thus, according to Eq. (A1),

  1 1 r2 r2 ln E½ðht Þx  ¼ E½lnðht Þx  þ v ar½lnðht Þx  ¼ E½x ln ht  þ v ar½x ln ht  ¼ x ln Ht  t þ x2 t 2 2 2 2  2  rt x x xðx  1Þ E½ðht Þ  ¼ ðHt Þ exp 2 

ðA2Þ ðA3Þ

 r To derive E½ðht Þnþxg  ¼ ðHt Þnþxg exp 2 ðn þ xgÞðn þ xg  1Þ for Eq. (20), follow similar steps as above. 2 t

Appendix B. The growth rate For the case n ¼ 0, growth rate ctþ1 can be derived as follows. Since

ctþ1 ¼ ln Y tþ1  ln Y t

ðB1Þ

From (9) and (B1), we have

ctþ1 ¼ xðln Htþ1  ln Ht Þ þ hðln Gtþ1  ln Gt Þ þ

ðr2tþ1  r2t Þ ðxðx  1ÞÞ 2

ðB2Þ

0

By substituting (20) and (21), and using (16 ), in (B2), we obtain

þ

!

ln BAst ðHt Þx ðGt Þh ðað1  s  wÞÞg

ctþ1 ¼ x

2 t

r

expð 2 xgðxg  1Þ þ txðx  1ÞÞ  ln Ht

r2t 2

 2    rt þ h ln wAðHt Þx ðGt Þh exp ðxðx  1ÞÞ  ln Gt 2

ðxðx  1ÞÞððxgÞ2  1Þ

ðB3Þ

Then, simplifying (B3), while applying x þ h ¼ 1 and

t þ g ¼ 1 repeatedly, we get (23).

Appendix C. Local stability In matrix form, (32) and (33) become



ln Htþ1  ln H





¼

ln Gtþ1  ln G   ðn þ xÞ h A¼ x h

ðn þ xÞ h x h

where A is the Jacobian matrix.



ln Ht  ln H ln Gt  ln G

 ðC1Þ

616

Y.Y. Getachew / Journal of Macroeconomics 32 (2010) 606–616

Then, if n ¼ 0, the system is non-hyperbolic, i.e., one of the characteristic roots is a unit. The characteristic polynomial PðkÞ for the linear system, which is given by (32) and (33), is (recall x þ h ¼ 1)

PðkÞ ¼ k2  TrðAÞk þ DetðAÞ ¼ k2  ðn þ 1Þk þ nh

ðC2Þ

2

If n ¼ 0, then k  k ¼ 0 and hence k1 ¼ 1. But if n > 0; PðkÞ admits two positive roots, where only one root is stable. Given G0 and H0 , there exists a unique solution to (32) and (33), which converges to ðH; GÞ. The path is monotonic and the steady state is saddle point stable. We can show this following De la Croix and Michel (2002), A.3.4. When j 1 þ DetðAÞ j. De la Croix, D., Michel, P., 2002. A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations. Cambridge University Press, Cambridge. Dollar, D., 2005. Globalization, poverty, and inequality since 1980. World Bank Research Observer 20 (2), 145–175. Dunn, C.E., 2007. The intergenerational transmission of lifetime earnings: evidence from brazil. B.E. Journal of Economic Analysis and Policy: Contributions to Economic Analysis and Policy 7 (2). Estache, A., Foster, V., Wodon, Q., 2002. Accounting for poverty in infrastructure reform: learning from latin america’s experience. The World Bank, Washington DC. Fay, M., 2001. Financing the future: infrastructure needs in latin america, 2000–05. The World Bank, Policy Research Working Paper Series, 2545. Futagami, K., Morita, Y., Shibata, A., 1993. Dynamic analysis of an endogenous growth model with public capital. Scandinavian Journal of Economics 95 (4), 607–625. Galor, O., Zeira, J., 1993. Income distribution and macroeconomics. The Review of Economic Studies 60 (1), 35–52. Garcia-Penalosa, C., Turnovsky, S.J., 2007. Growth, income inequality, and fiscal policy: what are the relevant trade-offs? Journal of Money, Credit, and Banking 39 (2-3), 369–394. Garcia-Penalosa, C., Turnovsky, S.J., 2008. Taxation and income distribution dynamics in a neoclassical growth model. Memo, University of Washington. Glomm, G., Kaganovich, M., 2003a. Distributional effects of public education in an economy with public pensions. International Economic Review 44 (3), 917–937. Glomm, G., Ravikumar, B., 1992. Public versus private investment in human capital: Endogenous growth and income inequality. The Journal of Political Economy 100 (4), 818–834. Glomm, G., Ravikumar, B., 1997. Productive government expenditures and long-run growth. Journal of Economic Dynamics and Control 21 (1), 183–204. Glomm, G., Ravikumar, B., 2003b. Public education and income inequality. European Journal of Political Economy 19 (2), 289–300. Grawe, N.D., Corak, M., 2004. Intergenerational mobility for whom? the experience of high- and low-earning sons in international perspective. In: Generational Income Mobility in North America and Europe. Cambridge University Press, Cambridge, New York and Melbourne, pp. 58–89. Grawe, N.D., Mulligan, C.B., 2002. Economic interpretations of intergenerational correlations. Journal of Economic Perspectives 16 (3), 45–58. Greene, W., 2003. Econometric Analysis, fifth ed. Prentice-Hall. Hertz, T., 2001. Education, inequality and economic mobility in south africa. Ph.D. Dissertation, University of Massachusetts. Jacoby, H.C., 2000. Access to markets and the benefits of rural roads. Economic Journal 110 (465), 713–737. Kamps, C., 2005. Is there a lack of public capital in the european union? EIB Papers 10 (1), 72–93. Lopez, H., 2003. Macroeconomics and inequality. World Bank, Washington DC. Loury, G.C., 1981. Intergenerational transfers and the distribution of earnings. Econometrica 49 (4), 843–867. Miller, N.J., Tsoukis, C., 2001. On the optimality of public capital for long-run economic growth: evidence from panel data. Applied Economics 33 (9), 1117– 1129. Rioja, F.K., 2005. Roads versus schooling: growth effects of government choices. B.E. Journals in Macroeconomics: Topics in Macroeconomics 5 (1), 1–22. Saint-Paul, G., Verdier, T., 1993. Education, democracy and growth. Journal of Development Economics 42, 399–407. Solon, G., 2002. Cross-country differences in intergenerational earnings mobility. Journal of Economic Perspectives 16 (3), 59–66. Songco, J.A., 2002. Do rural infrastructure investments benefit the poor? evaluating linkages: a global view, a focus on vietnam. The World Bank, Policy Research Working Paper Series, 2796. Turnovsky, S.J., 2000. Government policy in a stochastic growth model with elastic labor supply. Journal of Public Economic Theory 2 (4), 389–433. Turnovsky, S.J., Fisher, W.H., 1995. The composition of government expenditure and its consequences for macroeconomic performance. Journal of Economic Dynamics and Control 19 (4), 747–786. World Bank, 1994. Infrastructure for Development. World Bank/Oxford University Press, Washington DC/Oxford and New York. Ziesemer, T., 1990. Public factors and democracy in poverty analysis. Oxford Economic Papers 42 (1), 268–280. Ziesemer, T., 1995. Endogenous growth with public factors and heterogeneous human capital producers. FinanzArchiv 52 (1), 1–20.