Understanding Economic Dynamics Behind Growth-Inequality Relationships

Faculty of Business and Law School of Accounting, Economics and Finance ECONOMICS SERIES SWP 2008/21 Understanding Economic Dynamics Behind Growth-Inequality Relationships Xueli Tang Debasis Bandyopadhyay The working papers are a series of manuscripts in their draft form. Please do not quote without obtaining the author’s consent as these works are in their draft form. The views expressed in this paper are those of the author and not necessarily endorsed by the School or IBISWorld Pty Ltd. Understanding Economic Dynamics Behind Growth-Inequality Relationships BY XUELI TANG1 DEBASIS BANDYOPADHYAY2 We apply a stochastic dynamic general equilibrium model that jointly determines income inequality and the growth rate of income per capita to identify how different types of economic shocks and different states of the economy prior to the shock may imply different types of growth-inequality relationships over time series. In particular, we nd that an immigration shock that decreases diversity of labour types implies a positive relationship rst and then a negative relationship over time series while a scal policy shock designed to promote lower income inequality implies a positive relationship. The results may change depending upon the relative position of the economy with respect to its long run balanced growth state. We apply these results to explain Persson and Tabelini's nding of a negative growth-inequality relationship (reported in AER 1994) from the time series data of a selected OECD countries. In addition, we apply the model algorithm to identify relative strengths of different channels of interaction between income inequality and economic growth by examining data for the last 20 years on the New Zealand economy which experienced both types of shocks due to policy induced changes in immigration and redistribution. KEYWORDS: Heterogeneous agents, self-employment, externality, income distribution, inequality, growth, progressive redistribution. Contact address: School of Accounting Economics and Finance, Faculty of Business and Law, Deakin University; e-mail: xtang@deakin.edu.au 2 Contact address: Dept. of Economics, University of Auckland, 3A Symonds St, Auckland, New Zealand; e-mail: debasis@auckland.ac.nz. 1 1 1 Introduction The debate over the relationship between income inequality and economic growth is far from settled. Some literature nd positive theoretical relationships between inequality and growth, e.g., Kaldor (1956), Bourguignon (1981) and Li and Zou (1998), from different perspectives. But following Solow's (1992) hypothesis that more "equity" could promote more growth, a number of papers based on theoretical models started to question this positive relationship from the beginning of the 1990s (Bandyopadhyay 1993, Galor and Zeira 1993 and Benabou 1996, 2000 and 2002). They found that by giving a special role to human capital in production and ruling out a credit market for investing in human capital, the model environment can generate a negative relationship between income inequality and economic growth. Based on the model environment described by Bandyopadhyay (1993), Galor and Zeira (1993) and Benabou (1996, 2000, 2002), this paper re-examines the ambiguity in the inequalitygrowth relationship from two perspectives. First, by contrasting with Persson and Tabellini (1994), we examine whether past income inequality is harmful for future rates of growth. Persson and Tabellini (1994) ask the question: "is inequality harmful for growth?" and they report that it is. In particular, they nd that an increase in past inequality would lead the median voter to vote for higher redistributive income tax, which hinders future economic growth. That is why past inequality is harmful for future growth in their paper. In this model, this question is examined through simulations by choosing different levels of initial inequality relative to its steady state level. We nd that past inequality retards future growth no matter initial inequality is low or high relative to its steady state. It provides supports for Persson and Tabellini's (1994) conclusion. Secondly, the cross-country correlation between inequality and growth is discussed. Barro (2000) discusses the cross-country correlation between inequality and growth and reports that it is positive in rich countries but negative in poor countries. Dadkhah (2002), Castello and Domenech (2002) and Bandyopadhyay and Basu (2005), from different perspectives, provide alternative explanations for Barro's conclusion. The cross-country correlation is also discussed in this paper by changing some key institutional and scal policy parameters. The simulation results coincide with conclusions made in the literature, and provide explanations through a new model environment. In addition, we apply the model algorithm to identify relative strengths of different channels of interaction between income inequality and economic growth by examining data for the last 20 years on the New Zealand economy which experienced both types of shocks due to policy induced changes in immigration and redistribution. In this paper, we construct a Dynamic General Equilibrium (DGE) model to address the above issues and rationalize the con icting observations about the relationship between income inequality and economic growth from different perspectives. We nd that the relationship between income inequality and growth could vary signi cantly depending upon differences in scal policy regime, initial conditions, immigration or policy shocks. 2 2 2.1 The Model Preference, Technology and Endowments The model considers a continuum of in nitely lived dynasties i 2[0, 1]. Following Loury (1981), each dynasty is made of a sequence of families consisting of individuals who live for two periods or two generations, rst as a child and then as an adult. In each period t, the dynasty is represented by a family of an adult and a child. The adult, in period t represents the dynasty from that period onward and makes all decisions for that period subject to the constraint that she cannot pass on her debt to her child. We call this adult of the dynasty i in period t the dynastical agent i or simply agent i. The preference of the dynastical agent i at period t is given by: "1 # X n i (1) ln Uti = Et ln ci lt+n , > 1; t+n n=0 i where ci 0 and lt 2 [0; 1] denote, respectively, consumption and labor supply by the adult t of the dynasty i in period t; 2 (0; 1) is the discount factor. In the spirit of Benabou we assume that everyone operates a backyard technology but allow both physical and human capital to affect output as complementary inputs in the same way as Barro, Mankiw and Sala-i-Martin (1995) such that the output of the self-employed i i agent i as a function of her physical and human capital kt , hi and labor lt satis es3 t i i y t = kt (2) hi t i lt " , where, " = 1 . At each date, the disposable income yt of the agent i must equal the total expenditure on ^i i i consumption ct , consumption tax t ct , private education expenditure ei and bequest bi . In t t other words, (3) yt = (1 + ^i i t ) ct + ei + bi . t t The agent receives education subsidy at a rate dt 2 (0; 1) per unit of her expenditure ei t on the child's education such that in the following period her grown up child's human capital hi as a function of externality of human capital t , her innate ability i , external effects t+1 t+1 arising from neighborhood or family as proxied by parental human capital hi , and the sum t of private and public investment on her education (1 + dt ) ei , is given by, t (4) where, t 3 hi = t+1 i t t+1 hi t (1 + dt ) ei t = , Z 1 hi t di , 0. 0 In Benabou's technology = 0 and h can be interpreted as the sum of two types of capital (physical and human) that are perfect substitutes. 3 Following Lucas's (1988)4 basic idea of externality of human capital and considering the aggregate effect of ef ciency unit of human capital, (hi ) , as Benabou (2002), we introduce t an externality in the human capital accumulation. The idiosyncratic shocks i that arise t from discrepancies in innate ability or in ef ciency of human capital usage are i:i:d: with ln i s N ('; 2 ), where ' and 2 are constants.5 The parameter 2 (0; 1) measures the t child's human capital elasticity of "neighborhood externality," a phrase explored originally in Benabou (1996) in the context of human capital inequality and the parameter 2 (0; 1) measures the same elasticity of the education expenditure, which is primarily determined by the quality of the education system. Capital goods are complementary to human capital and become obsolete at the end of each generation. A tool loses value when its user dies. Parents buy new tools for their children at a subsidized rate set by the government and leave them as bequest. To capture this feature we assume that they depreciate completely in the production process. Consequently, i in the generation t+1, the agent i's physical capital kt+1 consists only of her parent's bequest i bt and a bequest subsidy from the government at the rate of vt 2 (0; 1) per unit of the bequest6 such that (5) i kt+1 = (1 + vt ) bi . t i Initial endowments of physical and human capital k0 and hi are jointly, lognormally distrib0 uted and the adult receives one unit of labor endowment in each period. 2.2 Redistribution with Progressive Income Tax The government has a scheme of progressive income taxation and transfer. By assumption, the government cannot detect individual innate ability i and neighborhood or family effects t i hi , but does observe individual incomes yt and their expenditure on education ei . Following t t Benabou (2002), the disposable income of a typical agent at a date t satis es (6) yt ^i i yt 1 t (~t ) t , y such that those with income higher than yt pay net tax while those with income below yt ~ ~ receive net transfers and the balanced-budget constraint is Z 1 i 1 t (7) yt (~t ) t di = yt , y 0 Lucas (1988) emphasizes the role of human capital externality, and nds that every one would be more productive and bene t from the average human capital if the level of other people's human capital is high. Agents communicate and interact with each other using their human capital. If, however, the gap of human capital between two individuals widens, then the communication exercise becomes harder. To model this, we assume that the ef ciency of communication increases with a diminishing rate with one's own human capital. The average productivity level in the economy depends on the average level of ef ciency in communication. 5 2 Benabou assumes ' = =2; but our results do not depend on that speci cation. 6 Bequest is usually taxed and not subsidized. We introduce bequest subsidy as a hypothetical tool for offsetting the distortionary effect of redistribution on physical capital accumulation. In principle, it could be zero or negative. 4 4 R1 i y di denotes per-capita income, yt represents the break-even level of income ~ where yt 0 t and 0 < t < 1 measures the average marginal tax rate and is identi ed as the degree of redistribution or progressivity in scal policy. Note that on a logarithmic scale t denotes the proportional tax rate on the log of personal income and we only focus on redistributive policies that transfer resources only from high income to low income people. The government nances education and bequest subsidies with consumption tax by setting the tax rate t such that Z 1 Z 1 Z 1 i i bi di. et di + t ct di = dt (8) t t 0 0 0 2.3 Individual Optimization To get our main points across easily we focus on the stationary policy sequence, T ( ; d; v; ).7 At each date t; let mht ; mkt denote the means and 2 , 2 denote the variht kt i ances of ln hi and ln kt , respectively, and let covt denote the covariance between ln hi and t t i ln kt . Suppose Mt (mht ; mkt ; 2 ; 2 ; covt ). Then for the agent's dynamic optimization ht kt i i problem, the state variables are (hi ; kt ; Mt ; T ), the control variables are (ci ; lt ; ei ; bi ) and t t t t the Bellman equation is as follows (9) i ln U hi ; kt ; Mt ; T = imax i t i i i (1 ) ln ci (lt ) t i + Et ln U (hi ; kt+1 ; Mt+1 ; T ) t+1 , ct ;lt ;et ;bt subject to (2), (3), (4), (5) and (6). i Lemma 1: The value function under scal redistribution is ln U (hi ; kt ; Mt ; T ) = Z1 (ln hi t t i Z2 (ln kt mkt ) + Wt , where mht )+ (10) Z1 = (1 (1 ) (1 (1 ) (1 R1 0 ) (1 ) (1 )) ) (1 (1 ) (1 )) ) (1 ) , (11) Z2 = (1 (1 ) , and aggregate welfare is Wt = i ln U (hi ; kt ; Mt ; T ) di. t The values of human capital and physical capital as expressed by their utility elasticities are respectively given by Z1 and Z2 . Note the tax rate can alter these values individually but does not alter the relative value of human to physical capital, (1 ) , which increases with output elasticity of human capital , neighborhood effect and patience but remains unaffected by the quality of education . A straightforward generalization of the analysis presented here could be done based on the arguments presented in page 488 of Benabou (2002). 7 5 2.4 Labor Supply and Savings Decisions The rst order conditions associated with the Bellman equation described by (9) yield complete solutions to the agent's problem. We rst discuss labor supply followed by investments in education and bequest. Lemma 2: The optimal labor supply remains invariant to time and personal characteristics and decreases with the average marginal income tax rate such that: (12) i lt =l ((1 (1 ) (1 ) = ) (1 (1 )) ) (1 (1 ) ) 1= . Next we consider investment propensities for the two forms of capital. We denote by si , jt j = 1, 2, respectively the fraction of disposable income that agent i invests in her children's i i education and for her bequest such that si ei =^t , si bi =^t . 1t t y 2t t y Lemma 3: The education investment rate si and the bequest rate si are time invariant 1t 2t and decrease with the average marginal income tax rate : (13) si = s1 1t si = s2 2t 1 (1 1 (1 ) (1 ) s1 , (14) where s1 = and s2 = ) (1 ) s2 , denote the laissez-faire saving rates. From (13) and (14) we note that the relative propensity of investment between human and physical capital increases with the quality of education as well as all other factors that raise the relative utility valuation of human to physical capital. Lemmas 2 and 3 spell out explicitly the negative effect of redistribution on the incentives to supply labor and capital inputs8 . 2.5 Consumption Taxes, Education and Bequest Subsidies Benabou (2002) emphasizes that elected governments do use a wide range of instruments rather than mere income taxation to redistribute resources among people and across time. Note that saving rates for education and bequest are time invariant and constant across agents. The reason is because parents save a fraction of their income for their children whose ability or marginal returns are not observable. Therefore, every parent meets the same optimization problem and choose to save the same fraction of income. It's also constant because this situation happens in each generation. The reason why the amount of optimal labor supply across agents is the same is because the utility function agents are maximizing is the same across agents. Agents may have different income due to different levels of human or physical capital. But the optimal fraction of labor devoted to work is the same. 8 6 Typically governments attempt to offset some of the distortionary effects of income taxes with a package of redistributive policies. In particular, we assume that the government chooses the subsidy rates d and v such that: (15) (16) (1 + d) s1 = s1 , (1 + v) s2 = s2 . By the government budget constraint (8) and by Lemma 3, it follows that (17) (1 s1 1+ s2 ) = ds1 + vs2 , and by (15), (16) and (17), the subsidy rates d and v and the consumption tax rate satisfy, (18) d= 1 ,v= 1 and = s1 + s2 . 1 s1 s2 We can switch on the intertemporal distortions simply by setting either d or v or both equal to zero and by adjusting according to (17). By (18) the redistributive policy package can be represented by the parameter alone. To capture the breadth of redistribution, therefore, we refer to as the degree of redistribution under the income tax regime rather than just the average marginal tax rate. 3 (19) (20) (21) The Equilibrium Dynamics ln ci = ln (1 t s1 s2 ) ln (1 + ) + (1 i ) ln yt + ln yt , ~ The optimization problem (9) yields (12)–(14) and other decision rules as follows: ln ei = ln s1 + (1 t ln bi = ln s2 + (1 t i ) ln yt + ln yt , ~ i ) ln yt + ln yt . ~ Together with the government's budget constraints (7) and (8) the above decision rules imply a unique sequence of aggregate state variables fMt g that coincides with what the agent i takes as given in (9) such that at each date t = 0; 1; 2; ::; the following aggregate consistency condition holds: Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 i i i i i bi di. et di + d et di + bt di + v (22) yt di = ct di + t 0 0 0 0 0 0 7 3.1 Dynamic Path of Physical Capital, Human Capital and Income The logarithm of (5), combining with (12) and (14) yields the dynamics of physical capital for the dynasty i, (23) i ln kt+1 = ln s2 + (1 ) (1 ) ln l + (1 i ) ln kt + (1 ~ ) ln hi + ln yt . t The logarithm of (4), combining with (12) and (13) yields, (24) ln hi = ln t+1 + (1 t + ln s1 + (1 i ) ln kt + ( + ) (1 (1 ) ln l + ln ln yt . ~ i t+1 )) ln hi + t 2 i i where ln t = (mht + ht =2) since by (23) and (24), both kt and ht are lognormally distributed and the property of moment generating function on lognormal distribution could be applied9 . Substituting (23) and (24) into (2) yields the equilibrium path of income for agent i: (25) i ln yt+1 = + ln t + (1 ) (1 + ( + ) ln yt ~ ) (1 i )) ln yt ) ln l + ln ln yt ~ (1 1 i ) ln yt 1 , i t+1 +( + ( + where ln s1 + (1 ) ln s2 is constant. Note that the intergenerational persistence of human capital ph ( ) + (1 ) k and physical capital p ( ) (1 ) together imply the intergenerational persistence of y income p ( ) + ( + ) (1 ) between parents and children. It has a structural component re ecting the degree of segregations among the neighborhoods that cannot be lowered with redistribution alone. The other component of intergenerational persistence decreases with the degree of redistribution and through this channel a policy of redistribution enhances intergenerational social mobility.10 Next, we characterize the dynamic path of the aggregate state variables. 3.2 Dynamics of the Economy-wide State Variables Given the initial lognormal distribution, by (23) and (24), physical and human capital and income remain lognormally distributed over time such that at each date t, Mt satis es (26) 9 mkt+1 = ln s2 + (1 + (2 ) 2 2 kt ) ln l + mkt + mht + 2 2 ht +2 covt =2, Please nd the proof of ln t in appendix, the proof of Lemma 5. Note also that the income of the parents does not suf ciently determine children's income unlike Benabou but (25) is consistent with Becker and Tomes (1979). 10 8 (27) (28) 2 kt+1 = (1 )2 2 2 kt + 2 2 ht +2 covt , mkt covt =2, ))2 2 ht mht+1 = ' + ln s1 + (1 +( + + ) mht + + (2 2 ) ln l + 2 2 ht =2 2 ht + ) 2 2 kt + 2 (29) 2 ht+1 = 2+ 2 + 2 (1 (1 )2 2 + ( + (1 kt )( + (1 )) covt , (1 (30) )( + covt+1 = 2 (1 )2 2 + (1 kt + (1 ) ( + 2 (1 )) covt . )) 2 ht In line with Benabou (2002) we de ne for each date t an index of inequality logarithm of the ratio of mean to median income. t as the Lemma 4: At each date t, inequality index t equals the variance of logarithmic earnings of agents such that t = 2 2 + 2 2 + 2 covt =2 and the evolution of earnings of kt ht adults is governed by a lognormal distribution such that i ln yt N ( mkt + mht + (1 ) ln l, 2 t ). The break-even level of income yt at ~ which an agent's net tax obligation is zero satis es: (31) ln yt = ln yt + (1 ~ ) t = mkt + mht + (1 ) ln l + (2 ) t. By the de nition of inequality index shown in Lemma 4, we know (32) 0 = 2 2 k0 + 2 2 h0 +2 cov0 =2. By (32), we can get 1 . Along with (27), (29) and (30), the equation (32) provides a dynamic path of income inequality, i.e., (33) t+1 = f = g 2 2 kt+1 ; ht+1 ; covt+1 2 2 kt ; ht ; covt . The following Lemma describes the equilibrium dynamics of per capita income and inequality jointly. Lemma 5: At equilibrium, the growth rate of the per capita income, satis es (34) t t ln yt+1 ln yt , = (1 + + + ' + (1 + 2 kt =2 ) (1 ) ln yt +( + 2 2 ht =2 ) ln l ( + ) ln yt ) 2 t 1 t+1 ) (1 + (1 )2 t 1 + where ln s1 + (1 ) ln s2 and 9 t+1 is given by (33). 3.3 Balanced Growth Path ) (1 ) = 0, 1 A In the balanced growth path, under the parametric condition (1 (34) becomes (35) 0 + ' + (1 ) (1 ) ln l 1 @ + t+1 + + ( + ) (1 )2 t = 1 ( + ) + ( 2 =2 + 2 2 =2) kt ht If (36) and (35) becomes = 0 and > 0, by Lemma 4 and (29), we have 2 2 t+1 t + (1 )2 t 1 = 2 +( + (1 ))2 t, 2 2 (37) where lower t . If (38) where t = (1 ln s1 + ' + (1 ))2 + (1 ) (1 ) ln l + 2 + t. ( + )2 < 011 . It implies that higher t leads to > 0 and = 0, (35) becomes t = 1 1 ln s1 + (1 + ' + (1 ) (1 + t+1 +( + ) ln s2 . ) (1 ) ln l )2 t + (1 )2 t 1 = At steady state, we get the following Proposition. PROPOSITION 1: The per capita income grows in the balanced growth path at the rate as follows (39) = a5 a1 + a2 ln l + a3 2 k =2 + 2 2 h =2 a4 , (1 (1 1 )2 + where a1 (1 ) ln s1 + 1 ln s2 + ' (1 ), a2 2 (1 ), a4 (1 ) (1 )( 2 ) + (1 )2 , a5 expressions of l, s1 , and s2 are given by (12), (13), (14), and (40) 11 ), a3 , analytical 2 = (1 (1 (1 + )) (1 + (1 2 (1 ))2 )) (( + ) (1 = ( ) + )2 2 1) < 0. , By setting = 0, and substituting 1 = 0, it's easy to get 10 (41) 2 h = (1 (1 (1 )( + 2 (1 (1 ))) 1 (1 ))2 2 (1 )2 ) (1 2 3 (1 )4 2 )) (1 + (1 )) (1 + 2 (( + ) + )2 , (42) 2 k = (1 (1 )2 (1 + (1 )) 2 (1 )) (( + ) (1 2 )+ ) 2 . Proof: See Appendix. The above Proposition gives us an equation of balanced growth rate which shows the relation between inequality and balanced growth rate. To show the relation clearly, we set = 0, and by (37), we have 2 2 (43) where = ln s1 + ' + (1 ) (1 ) ln l + 2 + . 2 2 (44) = 1 ( (1 ) + )2 2 . Since < 0, the above equation shows that in the long run, high inequality leads to low balanced growth rate if the high inequality is not due to high 2 . If we substitute (44) into (43) and rearrange, then we get (45) = ln s1 + ' + (1 ) (1 ) ln l + , 2 where 1 +2 > 0. From the above equation, we can see that high inequality leads to high growth rate. This conclusion is opposite to the conclusion made in (43). This is because steady state inequality is a function of 2 . Equation (45) embraces effects of 2 on the balanced growth rate. Hence, by (45), the balanced growth rate increases with high inequality which arises due to high 2 . If, however, inequality varies due to other parameters, inequality is still harmful for the balanced growth rate. The above discussion gives us an implication, like Partridge (1997, page 1030) argued, that "the overall negative inequality-economic growth relationship may only apply to developing or newly industrialized nations, but not for advanced nations." This conclusion is supported by (43) which gives us a clue that the negative relationship between inequality and growth may be due to the differences of institutional or technology factors. In contrast, in Partridge (1997), he emphasizes that advanced countries, like U.S. and UK, have experienced high income inequality and economic growth rate since 1980. This fact could be explained by (45) which shows that countries which experience high inequality due to high 2 would have high economic growth. The reason is, like Partridge (1997) mentioned, that low-income families migrate from high inequality states to low inequality states while 11 high-income families which have high levels of skills and abilities would be attracted to high inequality states because of lower tax rate. Hence, high inequality state would have more talent people and hence have high mean and variance of innate ability which are positively related to economic growth. Therefore, we can nd that advanced countries which have high inequality may have high growth rate. 4 Quantitative Analysis In the paper, we discuss time series correlation between inequality and growth using (35) while we do calibrations based on NZ data using (39). In this section, we rst estimate parameter values for NZ economy. And then, we discuss the time-series correlation between income inequality and economic growth. In that subsection, we rst present NZ's inequality and growth during the last decade. Then, we compare it with the model result. Using (35), we discuss effects of immigration and tax policies on the correlation between inequality and growth through impulse response analysis. At the end, we discuss the effects of past income inequality and future growth rate in the transition path. It provides explanations for Persson and Tabellini (1994). 4.1 Parameter Values The key parameters we need are the output elasticity of unskilled labor " and that of physical and human capital and , respectively, the `neighborhood externality' parameter , the quality of education system , the mean and variance of logarithm of innate ability, ' and 2 , and intertemporal elasticity of labor supply which identi es the preference parameter and agent's attitude towards patience . Production Parameters Following Gollin (2002), the labour share is equal to the ratio of the labour income to GDP. Statistics New Zealand provide values of income of self-employed, wage and salary and investment. The sum of these values is regarded as the total labour income while government transfer, bene ts, business and rent losses are excluded. The ratio of the labour income to GDP was 0.45 in 1996. After calculating the labour share, we use the measure which is introduced by Mankiw, Romer and Weil (1992) to specify shares of human capital and raw labour ". In MRW (1992), they regard the labour who earn minimum wage as having no human capital. The share of human capital is calculated by multiplying one minus the ratio of minimum wage to average wage by the labour share, i.e., = (1 ratio) labour share. The ratio of minimum wage to average wage, according to the data from New Zealand Statistics in 1996, is 0.42. Therefore, we get = 0:55, = 0:26 and " = 0:19. Accumulation 12 The intergenerational persistence of income between parents and children is de ned in (25) as follows (46) py ( ) +( + ) (1 ). In Maloney, Maani and Pacheco (2003), they analyzed whether bene t dependency is passed on from parents to children and found the correlation coef cient on the bene t propensity of the parents is 0.372 in New Zealand. Moreover, this transmission of welfare dependency from one generation to the next may be somewhat stronger among females and Maori youth. So, we set = 0:2, and = 0:4, which allows py ( ) to range from 0.26 to 0.85 which is consistent with Maloney, Maani and Pacheco (2003). Inequality In Benabou (2002), he measures the family income inequality by using the logarithm of the ratio of mean to median income. In NZ, according to the data from Statistics New Zealand, Household Economic Surveys, the logarithm of the ratio of mean to median income decreases from 0.31 to 0.23 from 1998 to 2006. By (40), I set 2 = 4, so that the feasible range is [0.15, 0.34]. Balanced Growth Rate According to Statistics New Zealand12 , the average GDP growth rate, based on 1995/96 prices per capita, from 1994 to 2004 is 2.4% while according to New Zealand Treasury, the average marginal income tax rate in the corresponding period is about 26.3%. To match 2.4% growth rate in the case without subsidy when = 26:3%, we set ' = 2:04213 . Labor Supply Kalb and Scutella (2003), by using four separately estimated sets of discrete choice labour supply models, found that in New Zealand, the average wage elasticities are 0.24, 0.40, 0.63 and 0.82 for married men and women, single men and single women respectively. From the New Zealand Statistics 2001 Household Economic Survey, we found that married men, married women, single men and single women account for more than 90% of total labour in the market. Therefore, after multiplying the wage elasticities with its corresponding percentage of labour population and taking sum, the average labour supply elasticity of New Zealand is around 0.45. It equivalently means that = 3:2. Discount Factor Following Benabou (2002), we set 12 13 = 0:4 for New Zealand. Data source: Statistics New Zealand, Table 6.1, Series SNCA.S6RB01NZ. If education subsidy is considered, then to match 2.4% growth rate with = 26:3%, ' = 1:92. 13 The following Table gives the benchmark parameter values14 . TABLE 1 BENCHMARK PARAMETERS 0.55 0.26 " 0.19 0.20 0.40 ' 2.042 2 4 3.20 0.40 4.2 Time-series Correlation In this subsection, we rst presents some NZ data and describe how the frenquency of NZ residents and immigrants which is proxied as the change of 2 , average marginal tax rate and GDP growth rate change during the last decade. Secondly, we calibrate the model to match NZ data to see whether the model's result is consistent with the real data. Thirdly, we do impulse response analysis about the correlation between inequality and growth when there is an immigration or scal policy shock. Fourthly, we discuss effects of past inequality on future growth by changing the initial level of income inequality. 4.2.1 NZ data In this subsection, we present four Figures to show New Zealand income inequality, distribution of skilled and unskilled residents and immigrants, average marginal tax rate and GDP growth rate during the last decade. 14 Note that =1 1 . 14 Figure 1—New Zealand income inequality from 1991 to 2007. Figure 1 shows that income inequality15 increases from 1991 to 2001 and then decreases until 2004. To explain the change of income inequality over time, by (27), (29), and (30) and then (33), we need to focus on the change of the variance of innate ability 2 and income tax . The proxy for the change of 2 is the change of the variance of NZ residents and immigrants. First, we look at how the percentage of skilled and unskilled residents changes over time from 1993 to 2007. 15 Note that the income inequality is the logarithm of the ratio of mean to median income. 15 Frequency: NZ Residents 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 Frequency: Skilled Frequency: Unskilled Figure 2—New Zealand distribution of skilled and unskilled residents from 1993 to 2007. Figure 2 shows that from 1993 to 2002, the gap between skilled and unskilled residents becomes smaller and smaller. It means that people becomes more homogeneous, i.e., 2 decreases. After 2002, the gap becomes larger. This is not consistent with Figure 1. There must be some reasons. New Zealand is a small and immigration country. The composing of immgrants has signi cant in uence on the distribution of income of New Zealand residents. In this paper, therefore, we provide some data about the frenquency immigrants to explain Figure 2. 16 Frequency: Immigration 1.2 1 0.8 0.6 0.4 0.2 0 1992199319941995199619971998199920002001200220032,0042,0052,0062,007 Frequency: Skilled Frequency: Unskilled year -0.2 Figure 3—New Zealand distribution of skilled and unskilled immigrants from 1992 to 2007. The above Figure presents the identical trend of variance of skilled and unskilled immigrants. It can be seen that the gap of frequency between skilled and unskilled immigrants becomes smaller from 1992 to 2001 and then increase until 2007. Therefore, we can say that the distribution of skilled and unskilled immigrants has important in uences on the distribution of NZ residents. From Figure 2 and 3, we can say that 2 should decrease from 1992 to 2002. By (27), (29), and (30), or by (40), we know that income inequality should decrease as 2 decreases. However, Figure 1 shows that income inequality does not decrease but increase during that period. The reason could be due to the change of income tax during that period which has direct negative effect on inequality. It is shown in the following graph. 17 Figure 4—New Zealand average marginal tax rate from 1990 to 2008. The above graph shows how NZ marginal tax rate (MTR) changes from 1990 to 2008. It can be seen that from 1997 to 1999, MTR decreases greatly. It then, leads to high levels of income inequality. This is consistent with Figure 1. It implies that effect of income tax on inequality is more important than the change of residents on inequality during that period. After 2002 until 2004, income inequality decreases even though the gap between skilled and unskilled residents becomes larger during that period. It shows again that effect of income tax is more important to determine the change of inequality when we look at the increase of income tax from 2003 to 2004. After describing and explaining the change of income inequality, next Figure shows how the growth rate of GDP changes from 1994 to 2006. 18 Figure 5—New Zealand GDP growth rate from 1992 to 2006. The above graph shows that the growth rate of per capita GDP increases from 1992 to 1994, decreases from 1994 to 2001 and then increases until 2003. The signi cant increase of growth rate from 1992 to 1994 may be attribute to the decrease of the gap of skilled and unskilled immigrants, shown in Figure 3. Because New Zealand is a small and immigration country, more immigrants coming with high skills would help to promote growth greatly. If we compare Figure 5 with Figure 1, we can see that the correlation between growth rate and income inequality is positive from 1992 to 1994, negative from 1994 to 2004, and positive from 2004 to 2006. In order to explain the change of the sign of the correlation using the model, in the following subsection, we do calibrations to match the growth rate and income inequality by calibrating the mean and variance of logarithm of the innate ability, ' and 2 to see whether the change of ' and 2 could provide some explainations for the real data. 4.2.2 Calibration We have data for , t and t , but the values of the mean and variance of innate ability are not available. By (40), we calibrate 2 to match income inequality and by (39) calibrate ' to match the growth rate from 1998 to 2005. The following Table gives calibration results for ' and 2 , and , t and t in the corresponding year. 19 TABLE 2 Year 1998 1999 2000 2001 2002 2003 2004 2005 2006 ' 1.815 1.835 2.21 1.914 1.944 2.12 2.183 2.042 2.053 2 t t 5.5 5 2.3 4.5 4.5 2.8 3.3 4.8 4.4 24% 22% 21% 21% 21% 21% 33% 33% 33% 0.3% -0.3% 4.7% 1.5% 2.8% 2.9% 1.7% 2.5% 1.1% 0.31 0.30 0.14 0.27 0.27 0.17 0.17 0.25 0.23 The following Figure shows the calibrated mean and variance of the logarithm of innate ability. Figure 6: Calibrated mean and variance of innate ability. The above Figure shows the calibrated mean and variance of innate ability such that the balanced growth rate and income inequality are matched with data. From Figure 6, it can be seen that the variance of innate ability 2 decreases while the mean ' increases from 1998 to 2000. The reason is due to the decrease of the gap of frequency of NZ residents between skilled and unskilled which is shown in Figure 2. From 2003 to 2005, 2 increases. This is consistent with Figure 2 where the gap of frequency of NZ residents between skilled and unskilled becomes larger relative to previous year. From 2000 to 2003, Figures 2 and 3 show uctuations of frequency of skilled and unskilled workers. Accordingly, Figure 6 also 20 presents big jumps of the variance of innate ability. By comparing Figure 2, 3 with 6, it can be seen that Figure 6 could match the real data very well. It implies that the model could be used to predict the trend of income inequality and economic growth. Therefore, in the following subsection, impulse response of growth and inequality to immigration or scal policy shock will be discussed in order to see whether these two shocks could have effects on the correlation between growth and inequality. 4.2.3 Impulse response analysis We consider two shocks. One is immigration shock and the other one is scal policy shock. They have been chosen because these two shocks occur commonly in many modern societies, such as U.S., New Zealand and Australia. First, we discuss effects of an immigration shock on the correlation through impulse response analysis using equation (35). Suppose that before period t, the economy is in the balanced growth path where there is no subsidy and = 26:3% such that = 2:4%. In period t, the government announces an immigration policy which restricts people to immigrate easily. Then the number of immigrants will decrease and then the variance of innate ability 2 will decrease as a result of less people with different culture, educational background, and abilities living in the country16 . The drop of 2 is interpreted as the immigration shock and leads to a decrease of income inequality in the following period. Using benchmark values, Figure 7 shows the impulse response of growth rate and income inequality to the immigration shock in the transition path when the redistributive income tax rate is set equal to its actual level 26.3%. Note that the mean of innate ability would change as well. However, the change of the mean of innate ability has no effects on the income inequality both in the transitional path and steady state. In addition, the change of the mean of innate ability can only shift up or down the balanced growth rate horizontally, shown in (35). The correlation between inequality and growth does not change with the change of the mean of innate ability. Therefore, we do not discuss it in the paper. 16 21 Figure 7—Impulse response of growth and inequality to a change of diversi cation of innate ability. The above Figure shows that in period three, both growth and income inequality decrease from steady state as the government puts more restrictions on immigration policy so that less people immigrate into the country. The variance of innate ability decreases correspondingly17 . It then leads to the decrease of income inequality and growth rate. This is because the term t+1 in (35) decreases18 . It drags down the t . But after period three, the growth rate increases because the negative effects of inequality on growth, shown by the term + + ( + ) (1 )2 t . In the long run, inequality and growth rate will converge to its new steady state. Before the shock, inequality and growth rate are two paralleled lines. But from period two to period three, the correlation between inequality and growth is positive as both decrease. After period three, however, the correlation becomes negative as inequality decreases while growth rate increases to a new steady state. Therefore, we can see the correlation between inequality and growth rate changes signi cantly because of the immigration shock. Figure 3 shows that the gap between skilled and unskilled immigrants decreases signi cantly from 1992 to 1994. It means that the variance of ability should decrease. From Figure 7, we can see that the sudden decrease of 2 would lead to a positive correlation between inequality and growth temporarily. Compared Figure 5 with Figure 1, we can see it is true. The correlation between growth rate and income inequality is positive from 1992 to 1994. After 1994, the correlation becomes negative. This is also supported by Figure 7. Therefore, In the simulation, we decrease 2 from 4 to 3 from period two. Note that by (27), (29) and (30) and then (33), we know income inequality is a state variable and determined by the last period variables. Therefore, the change of immigration policy in period t can only affect t+1 . 18 17 22 we can say that the impulse response of growth and inequality to immigration shock could help to explain the real data. The above discussion also gives econometricians an implication that when they do empirical analysis, they may need to pay more attention to the data in order to determine whether any policy shocks occurred around the period of their study. This concern also appears in Banerjee and Du o (2003) in which they emphasize that the quality of data may affect the estimation of the relationship between inequality and growth. The above simulation results also give a message to policy makers that immigration may cause higher income inequality but lead to higher growth rate because of skill spillover. After discussing how immigration shocks affect the correlation between inequality and growth, next, we discuss whether a scal policy shock could have the same effect or not. A sudden change of redistributive income tax in a speci c period is interpreted as a scal policy shock in the model. If the government increases (decreases) a redistributed income tax in a period, (27), (29) and (30) and then (33) show that inequality will decrease (increase), and so does the growth rate shown in (35). Consequently, the correlation between inequality and growth will change and it is shown through the impulse response of the correlation to the scal policy shock in the following Figure. Figure 8—Impulse response of growth and inequality to a change of scal policy. In period 2, the government increases redistributive income tax from 26.3% to 33%. The above Figure shows that the balanced growth rate decreases correspondingly because of less labor supply and savings while income inequality starts decreasing from period three. This is because inequality is a state variable and determined by the last period variables. Hence, 23 the decrease of in period t affects inequality in period t+1. Then, the correlation between inequality and growth becomes positive from period three. After a few periods, both income inequality and growth rate will go to its respective steady states. Therefore, we can say that the correlation between inequality and growth rate changes signi cantly but temporarily because of the scal policy shock. From Figures 7 and 8, it can be seen that both immigration shock and policy shock could have signi cant effects on the correlation between inequality and growth. This gives econometricians an implication that when they do empirical analysis, they may need to pay more attention to the data in order to determine whether any policy shocks occurred around the period of their study. This concern also appears in Banerjee and Du o (2003) in which they emphasize that the quality of data may affect the estimation of the relationship between inequality and growth. The above simulation results also give a message to policy makers that immigration may cause higher income inequality but lead to higher growth rate because of skill spillovers. 4.2.4 Effects of past inequality on future growth Persson and Tabellini (1994) nd that past inequality is negatively related to future growth by using Ordinary Least Square (OLS) regressions over a cross-section of nations. Aghion, Caroli and Garcia-Penalosa (1999) argue that when there are credit barriers, inequality may limit the ability of low-income households to make investments in human or physical capital, thus reducing the subsequent economic growth. They analyze the effect of inequality and economic growth in economies in which wealth is distributed heterogeneously. Because of credit constraints, dynasties with high initial wealth can invest in human capital and remain rich; the poor, with low initial wealth, cannot borrow to invest, and remain poor for ever. Banerjee and Newman (1993) argue that occupational structure depends on the distribution of wealth because of capital market imperfections. Poor agents work and earn wage income while wealthy agents become entrepreneurs and monitor workers. They nd that if the income inequality in the initial period is low, the economy will converge to a high-wage, high-employment steady state. Otherwise, the economy will end up with low wage and low employment. Bandyopadhyay (1997) nds similar conclusions to Banerjee and Newman (1993) that the initial conditions play an important role in wealth distribution in the short and long run: a country with a more equal distribution of initial wealth will grow more rapidly and have a higher level of average income in the long run. In section 3.3 or by (37), we already see that inequality is harmful for growth in the transition path. In this section, using simulation, we discuss the time-series correlation between past income inequality and future economic growth in order to examine Persson and Tabellini's (1994) hypothesis by plotting graphs. First, we use equation (35) to generate a time-series growth rate and use (27), (29) and (30) to generate a time-series income inequality. To check whether past income inequality is harmful for the future growth rate, we plot a graph using past income inequality as horizontal axis and future growth rate as vertical axis. Since, in the long run, both balanced growth 24 rate and income inequality converge to its respective steady states which is a function of the parameters, we examine the time-series correlation by changing the initial levels of the variances of the logarithm of physical and human capital, 2 and 2 , and the covariance k0 h0 between these two variables cov0 such that the initial level of income inequality is lower or higher than the steady state level. Note that parameters are xed at benchmark values with = 0. Figure 9—Balanced growth rate against income inequality in the transition path when initial inequality is low. 25 Figure 10—Balanced growth rate against income inequality in the transition path when initial inequality is high. Figure 9 and 1019 show how past inequality affects future growth rate when initial inequality is lower or higher than its steady state level. Both Figures show that the future growth rate decreases with higher past income inequality no matter initial inequality is low or high relative to its steady state level. It proves the conclusion made in (37) and it is also consistent with Persson and Tabellini's (1994) nding but in a non-political environment. It implies that for the country where the composing of agents based on skills and abilities does not change over time, higher inequality is always harmful for growth. It means that if the economy wants to have a higher growth rate, the government needs to keep income inequality at a low level. 5 Conclusion In this paper, the relationship between income inequality and growth is discussed. First, by calibration, we nd that the model result are consistent with data very well. It implies that our model could be used to predict the trend of income inequality and economic growth. Second, the time-series correlation between past income inequality and the future growth rate is examined. Following Bandyopadhyay's (1993) idea to change levels of initial inequality, the simulation results show that the time-series correlation is negative. This is consistent with Persson and Tabellini's (1994) hypothesis that past income inequality is harmful for future growth. Finally, simulations show that immigration and policy shocks may have signi cant effects on the correlation between inequality and growth in the transition path. This shows that paying more attention on the data and understanding its backgrounds in some particular periods may be necessary for interpreting estimation results correctly. Banerjee and Du o (2003) also address this concern. Thus, we can say that the above discussion gives econometricians and empirical analysts guidances for differentiating between countries according to different institutions, scal policies, initial conditions, immigration or policy shocks. Appendix PROOFS OF LEMMAS 1, 2 AND 3: Note that based on the NZ benchmark values, the steady state 2 = 0:67; 2 = 4:53; cov = 0:57. We kt ht choose 2 = 0; 2 = 0; cov = 0 for Figure 9 and 2 = 0:7; 2 = 4:6; cov = 0:6. for Figure 10. kt ht kt ht 19 26 By (3), (4) and (5), we rewrite (9) as follows: (A.1) (1 ) ln (1 si si ) ln (1 + ) + ln yt ^i i 1t 2t ln U hi ; kt ; Mt ; T = imax i i i t + Et ln U ht+1 ; kt+1 ; Mt+1 ; T s1t ;si ;lt 2t Agent solves (A.1) subject to (2), (6) and (A.2) hi = t+1 (A.3) t i (lt ) . (1 + d) si 1t i t+1 i kt (1 ) hi t (1 + (1 ) i lt (1 )(1 ) (~t ) y , and i i kt+1 = (1 + v) si kt 2t (1 ) hi t ) i lt (1 )(1 ) (~t ) . y i i We guess the value function as: ln U (hi ; kt ; Mt ; T ) = Z1 ln hi + Z2 ln kt + Bt . Then by t t substituting this value function into (A.1), we get (A.4) si ) = (1 + ) ln (1 si i 2t 1t Z1 ln hi +Z2 ln kt +Bt = (1 ) i i t ~ +(1 ) (1 ) ln lt + ln yt (lt ) i + (1 + Z1 + Z2 ) (1 ) ln kt + ((1 + Z1 + Z2 ) (1 ) + Z1 ) ln hi t 1 0 i ln t + ln(1 + d)s1t + ' Z i A. ln yt ~ ) (1 ) ln lt + + @ 1 + (1 i i ~ ) (1 ) ln lt + ln yt ) + Bt+1 +Z2 (ln (1 + v) s2t + (1 i Taking partial differentials with respect to ln kt and ln hi yield t (A.5) (A.6) Z1 = (1 Z2 = (1 + Z1 + Z2 ) (1 + Z1 + Z2 ) (1 )+ ). Z1 , Rearranging (A.5) and (A.6), we verify the guess and con rm the existence of (A.4) and thus Lemma 1 is established. The rst-order conditions of (A.1) with respect to the saving rates and labour supply are (A.7) 1 1 si 1t 1 1 si 1t (1 ) si 2t = i i i @ ln Ut+1 @ ln hi @ ln Ut+1 @ ln kt+1 t+1 + i @ ln hi @si @ ln kt+1 @si t+1 1t 1t , (A.8) si 2t i lt = i i i @ ln Ut+1 @ ln hi @ ln Ut+1 @ ln kt+1 t+1 + i @ ln hi @si @ ln kt+1 @si t+1 2t 2t , (A.9) 1 = (1 + i ) (1 ) (1 ) =l1t i i i @ ln Ut+1 @ ln hi @ ln Ut+1 @ ln kt+1 t+1 + i i i @ ln hi @lt @ ln kt+1 @lt t+1 , 27 i i where @ ln kt+1 =@si = 0, @ ln kt+1 =@si = 1=si , @ ln hi =@si = =si , @ ln hi =@si = 1t 2t 2t t+1 1t 1t t+1 2t i i i i i 0, @ ln kt+1 =@l1t = (1 ) (1 ) =l1t and @ ln hi =@l1t = (1 ) (1 ) =l1t . t+1 The above optimization problem (A.4) is strictly concave. Consequently, (A.7)—(A.9) are suf cient for the optimization exercise and the Lemmas 2 and 3 follow immediately after we substitute (10) and (11) into (A.7)—(A.9). PROOF OF LEMMA 4: By assumption, at the initial date t = 0, physical and human i capitals are lognormally distributed. By (23) and (24), it follows that kt and hi remain t i lognormally distributed over time and hence by (2) yt is lognormal and is given by, (A.10) i ln yt = i ln kt + ln hi + (1 t i ) ln lt . i By (12), it follows that the mean of the lognormal distribution of yt is given by, Z 1 i ln yt di = mkt + mht + (1 (A.11) ) ln l. 0 i i The variance of ln yt is the sum of variances of ln kt , ln hi plus the covariance of these two t variables (A.12) i var ln yt = 2 2 kt + 2 2 ht +2 covt . The income per capita yt , following Crow and Shimizu's (1988) description about properties of moment generating function on lognormal distribution on page 9, is Z 1 Z 1 1 i i i ln yt di + var ln yt . yt di = exp (A.13) yt = 2 0 0 The median income is (A.14) Therefore, inequality index is (A.15) t yt;median = exp Z 1 i ln yt di . 0 log yt yt;median = 1 i var ln yt = 2 2 2 kt + 2 2 ht +2 covt =2. i To derive the expression for the break-even point given by (7), we note the mean of yt in logarithm, by (A.13), satis es (A.16) 1 ln yt = mkt + mht + (1 ) ln l + t, i and the mean of (yt ) in logarithm is Z 1 i 1 (A.17) ln yt di = (1 ) ( mkt + mht + (1 0 ) ln l) + (1 )2 t. 28 Taking the difference between before and after tax income yields Z 1 i 1 (A.18) ln yt ln yt di = mkt + mht + (1 ) ln l + (2 0 ) t. It means that ln yt = mkt + ~ (31). This proves Lemma 4. mht + (1 ) ln l + (2 ) t, then we can get PROOF OF LEMMA 5: Integrating both sides of (25) across all agents i we get, (A.19) Z 1 i ln yt+1 di = 0 + ln t + (1 ) (1 +( +( + From (A.13), we know (A.20) Z 1 ) (1 Z )) 0 ) ln l + ' + ( + ) ln yt ~ Z 1 1 i i ln yt di ln yt 1 di. (1 ) 0 1 i yt di ln yt ~ 1 i ln yt di = ln 0 Z 0 1 i var ln yt . 2 Combining (A.20) with (A.19) yields Z 1 1 i i (A.21) ln yt+1 di var ln yt+1 = + ln t + ' + (1 ) (1 2 0 + ( + ) ln yt ~ ln yt 1 ~ Z 1 1 i i + ( + ( + ) (1 )) ln yt di var ln yt 2 0 Z 1 1 i i yt 1 di (1 ) ln var ln yt 1 . 2 0 ) ln l i From the proof of Lemma 4, we know that kt and hi are lognormally distributed. By the t property of moment generating function on the lognormal distribution, we can get (A.54) ln kt = mkt + t 2 kt =2 and ln ht = mht + 2 ht =2. Similary, by the de nition of (47) t we can get mht + ln ht + ( 2 ht =2 2 ht =2 = exp = exp 1) . Substituting (A.54) into (A.16), we get (48) ln yt = ln kt + ln ht + (1 ) ln l + ( 29 1) 2 kt +( 1) 2 ht +2 covt =2. Substitute (48) into (47), we get 0 ln yt @ ((( (49) ln t = + ( (50) ln kt (1 2 1) kt + ( 2 1) ht =2 ) ln l 2 1) ht + 2 Integrating (5), and taking logarithm, we get ln kt = ln s2 + ln yt 1 . covt ) =2) A . 1 Substitute (50) into (49), and rearrange, we get (51) ln t = ln yt (1 ) ln l + (ln s2 + ln yt 1 ) . 2 kt =2 + 2 2 ht =2 t Substituting (51) and (31) into (A.21), and by (A.15) yield (34). PROOF OF PROPOSITION 1: Writing the system of linear equations (27), (29) and (30) in a matrix form, we get (52) where Nt+1 Nt+1 = A0 + A1 Nt , 2 4 2 2 kt+1 2 ht+1 covt+1 3 5 , A0 2 4 0 2 0 3 5, 2 3 A1 2 4 2 2 (1 2 (1 2 (1 )2 )2 )2 (1 )2 ))2 (1 ( + (1 (1 )( + (1 )2 2 (1 )( + (1 )) (1 ) ( + 2 (1 )) 5 . )) The sequence Nt converges to a steady state if all eigenvalues of A1 , denoted by S (A1 ) (Ej ), j = 1; 2; 3, are less than one20 . By setting = 0 to avoid unnecessary details, we solve det jA1 S (A1 ) Ij = 0, where I is identity matrix, to get E1 < 1 since ; 2 (0; 1). 3 3 Note that when = 0, jA1 j = . It implies that E1 E2 E3 = 3 3 , since A1 is 2 2 symmetric. We know E1 . The symmetry of A1 implies P, it means that E2 E3 = also that the trace fA1 g = Ej . Or, equivalently, j=1;:::3 (A.43) 20 E2 + E3 = 2 +( + )2 + ( + 2 ). For detailed discussion about this property, please see Reich (1949), Lorenz (1993) and Young (2003). 30 By the assumption (1 ) (1 ) = , it follows that both E2 < 1 and E3 < 1. Thus, S (A1 ) < 1. Consequently, Nt converges to a unique steady state which we denote as N . By (52), N satis es the following xed point problem (A.44) N = A0 + A 1 N , and has a unique solution, since I A1 is nonsingular. It follows, therefore, a unique steady state exists and the equilibrium sequence of Nt converges to it. Moreover, 0 < S (A1 ) < 1 implies that fNt g constitutes a monotone sequence21 . 2 2 2 2 2 Then, on the balanced growth path, 2 h and ht = ht+1 = k, kt = kt+1 = covt+1 = covt = cov. By (27), (29) and (30), we can get (41), (42) and steady state cov as follows 2 2 (53) cov = (1 (1 )4 + (1 )) (1 + )( + (1 ))2 (1 )) 1 ) (1 2 (1 )2 2 (1 (( + ) + )2 . Following Lemma 4, inequality index is denoted by t = 2 2 + 2 2 + 2 covt =2. kt ht Then we get (40). We write the system of linear equations (26) and (28) in a matrix form, and get (54) where22 N Nt+1 mkt+1 mht+1 , AA0 ln s2 + (1 ) ln l + (2 ) ' + ln s1 + (1 ) ln l + AA1 + + . 2 h =2 N Nt+1 = AA0 + AA1 N Nt , + (2 ) , The sequence N Nt converges to a steady state if all eigenvalues of AA1 , denoted by S (AA1 ) (EEj ), j = 1; 2, are less than one. If (1 ) (1 ) = 0, however, we solve det jAA1 S (AA1 ) Ij = 0, where I is identity matrix, and get EE1 ( + ) < 1 and EE2 = 1. 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