Taxation and Fiscal Expenditure in a Growth Model with Endogenous Fertility

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1 Disussion Paper No May 08, Taxation and Fisal Expenditure in a Growth Model with Endogenous Fertility Norman Sedgley and Brue Elmslie bstrat Most growth theorists agree that understanding the eonomis of innovation and tehnologial hange is entral to understanding why some ountries are riher and/or grow faster than other ountries. The driving fore behind reent developments in endogenous innovation models of growth is a desire to eliminate population sale effets. In the semi endogenous growth model growth beomes proportional to the exogenous population growth rate but invariant to poliy. This paper makes population growth endogenous by modeling fertility along the lines of Barro and Beker (Fertility Choie in a Model of Eonomi Growth, 1989 and models an array of government poliies to demonstrate how some poliies an impat levels and growth rates in a sale free endogenous growth model. In the model government poliies are ategorized aording to whether they have level effets only, level and growth effets, or no impat on levels and/or growth. JEL H2 H0 O3 Keywords Poliy effets on growth; endogenous population; publi finane uthors Norman Sedgley, Sellinger Shool of Business and Management, Loyola University in Maryland Baltimore, US Brue Elmslie, Eonomis University of New Hampshire, US, brue.elmslie@unh.edu Citation Norman Sedgley and Brue Elmslie (2015. Taxation and Fisal Expenditure in a Growth Model with Endogenous Fertility. Eonomis Disussion Papers, No , Kiel Institute for the World Eonomy. Reeived pril 27, 2015 epted as Eonomis Disussion Paper pril 30, 2015 Published May 8, 2015 uthor(s Liensed under the Creative Commons Liense - ttribution 3.0

2 I. Introdution The literature on the impat of eonomi poliy variables on the steady state growth rate has been motivated, in part, by the attempt to understand why ross-ountry growth rates and levels of per apita inome differ so muh (Luas, 1988; Rebelo, 1991; Easterly, Models that allow poliy to impat steady state growth rates fit the data better sine there are very wide disrepanies in living standards and growth rates that appear to be very persistent. The neolassial model and semi-endogenous growth models share a ommon property in that these models typially predit that tax poliy impats levels only, in other words the effets are transitory. We address the question of the long run growth versus level effets of taxation in a relatively simple endogenous growth model that makes fertility hoie endogenous. dditionally, the model allows for a reasonably wide variety of poliy variables to determine the steady state growth rate. The model we generalize is diretly inspired by Jones s semi-endogenous model, but the tax struture of the eonomy is arefully speified to allow for taxes on wages, assets and onsumption, and for general government expenditures that do not enter the utility funtion or prodution funtion as well as government expenditures that support R&D. We find that an inrease in the asset tax lowers growth, while an inrease in the onsumption tax has level effets only and a wage tax has no impat on growth or levels. We an also look at pro family poliies that lower the ost of having hildren. Reent advanes in fully endogenous growth theory (Segerstrom, 1998; Young, 1998; Howitt, 1999; Zeng and Zhang, 2002; Peretto, 2007, 2011 generate growth and level effets from various forms of taxation. Empirially, links have long been established. Early researh (Barro, 1991; Kormendi and Meguire 1986; Islam,

3 finds, for example, that ross ountry growth rates are inversely related to the share of government onsumption in GDP and growth rates are positively orrelated with an index of property rights protetion. Reent empirial work has been diretly motivated by endogenous growth theory (Jones, 2002; Sedgley and Elmslie, 2010; Gemmell, Kneller, and Sanz, While fully aknowledging the limitations of the available time series data, Gemmell, Kneller, and Sanz (2011 show evidene of long run effets of taxation on a panel of OECD ountries overing 30 to 35 years. Given the results of Jones (2002 and Sedgley and Elmslie (2010, they also show a remarkably short transition period. While the urrent paper, along with Peretto (2007 and 2011 and Zeng and Zhang (2002 develop models implying differing growth versus level effets from various forms of taxation (e.g. onsumption and asset taxes, the empirial literature has not yet progressed to test for differential effets. 1 In a simpler framework, MGratten (2012 extends a standard neolassial growth model to allow for a wide range of taxes on property, apital, profits and sales. She finds strong level effets from data during the Great Depression. Durlauf, Kourtellos, and Ming Tan (2008 show that, in a ross setional regression framework, maroeonomi poliy and regional heterogeneity play a strong role in explaining growth. Romer and Romer (2010 find a strong relationship between hanges in taxes measured in terms of taxes as a perentage of GDP and subsequent growth over a three year period. Speifially they estimate that a 1% inrease in taxes dereases growth by 3% over the next 3 years. These results are onfirmed by Barro and Redlik (2011 who find a one-year tax rate multiplier of -1.1 using a new time series on 1 The pratial diffiulty arising from any empirial test surrounds the issue of differentiating between transitional and steady state effets in the models, Sedgley and Elmslie (

4 overall average marginal tax rates inluding federal and state inome taxes and the soial seurity (FIC tax. Both of these results suggest possible growth effets from taxation and strong level effets. rin et al. (2013, using data on marginal tax rates from the U.S., U.K. and four Sandinavian ountries find strong growth effets from inreased average marginal rates. Reent work has pointed to a relationship between government poliies and eonomi growth. This is a major step forward in our understanding of the growth proess. The urrent paper represents an attempt to ath theory up with the developing empiris. Moreover, the model developed here generates new hypotheses to help guide future empirial work. Romer and Romer (2010 suggest that new work should investigate whether the output onsequenes of tax hanges depend not only on their size, but on their other features as well. (p. 800 Our model makes speifi preditions for differential effets of asset, wage, and onsumption taxation in terms of growth and level effets that are diretly testable. II. Theoretial Model The general endogenous innovation framework extended to inlude endogenous fertility has the tendeny to beome ompliated very quikly. It is our desire to show that adding endogenous demographis an be done in a relatively simple and well aepted framework. The analysis that follows borrows from Beker and Barro (1989 and Barro and Sala-i-Martin (1999 in assuming a linear ost relationship in raising hildren. Their analysis suggests that the fertility rate is related to the ratio of onsumption to the apital stok. n inrease in the level of onsumption inreases hild 3

5 quality and the demand for hildren. The inrease in the apital stok signals an inrease the opportunity ost of having hildren and lowers fertility. nother reent approah taken by Galor and Weil (2000 looks at a quantity/quality tradeoff in reprodutive hoie. Their model is based on the idea that greater future tehnologial hange auses families to shift from quantity to quality sine the reward to human apital investment is expeted to rise. This approah allows the atual transition from rising population growth rates to falling population growth rates to be modeled. The purpose of the urrent paper is not to predit the point of transition from inreasing population growth to dereasing population growth. The reason for our hoie of modeling strategy is simpliity. The Beker and Barro model, familiar to and aepted by most eonomists, allows for a fous on the relationship between apital aumulation, innovation, fertility, and the poliy impliations of the model as they would most likely apply to a modern R&D driven eonomy past the turning point of the demographi transition. Prodution Our speifiation of prodution is standard and follows a ommonly aepted formulation motivated by leading work in the field suh as Jones (1995a, 1995b, 1999 and Segerstrom (1998. Q 1, t xi, t 0 1 (1 Yt Ct I t Nt Gt Lt Qt i di, (2 x i, t xt K i, t i, t Equation (1 shows that final output, produed under the onditions of perfet ompetition, is produed using the available labor fore, and intermediate goods of 4

6 varying quality, i. Output an be used for onsumption, C, investment in apital, I, researh and development, N, and government spending on final goods and servies, G. We allow Q L where Q is the number of intermediate setors in the eonomy and 1. We follow ghion and Howitt (1998 in assuming that imitation happens as a onsequene of population growth and without the expenditure of additional resoures. We do not impose the ondition that 1 and, therefore, the expansion of Q over time does not eliminate the sale effet on the growth rate. We inlude the speifiation of Q for generality 2. Equation (2 is a prodution tehnology for intermediate goods supplied under monopolisti ompetition and demonstrates a fundamental omplimentary relationship between apital aumulation and tehnology as suggested by ghion and Howitt (1998. We agree with the position that the proess of apital aumulation and the proess of tehnologial hange are best viewed as omplementary. The total apital stok is: Q (3 Kt i, t x, tidi xt i, tdi 0 Q 0 We define the average level of produtivity to be t Q di / Q and the leading edge 0 i, t t tehnology to be MX. Define the produtivity adjusted apital stok per setor as (4 kˆ t Kt / t Qt xt i, tdi / Qt xi, t xt Q 0 2 The interested reader ould easily explore the model s impliations under a ghion Howitt type sale free growth model by setting 1 in Q L and 1(disussed below in the prodution funtion for new ideas. Sine our model follows Jones and builds a model with the least number of parameter restritions our model should produe robust results. 5

7 Substituting (4 into (1 and suppressing time subsripts gives: 1 ( 1 (5 Y L Q i ( Q 0 K Q di K ( L 1 showing that the prodution tehnology is Cobb Douglas., For innovation a Poisson proess is speified, with an arrival rate proportional to the level of resoures per setor devoted to R&D. (6 g MX N N, 1 1 MX MX Q MX L 0 1, 1. is the Poisson parameter and is a researh produtivity parameter. 0 1 aptures the potential for the stepping on toes effet or the possibility of dupliated researh effort. 1 aptures the balane between the fishing out effet where new disoveries beome harder to find as the tehnologial frontier advanes and the standing on shoulders effet where past disoveries aid in the searh for new disoveries. The exogenous arrival rate of horizontal innovations, whih do not enhane produtivity, is Q L. Q L Government One of the main motivations for this paper is to study the role of a realisti menu of government poliies in a sale free growth model with endogenous fertility. We keep the model tratable by making a number of simplifying assumptions. Government spending does not diretly impat utility or the prodution funtion. The government runs a balaned budget and always satisfies: (7 G T S N wl rk C R P w a 6

8 Where G is unprodutive government spending, T is the value of transfers (assumed lump sum, S R is an ad valorem subsidy to R&D, N P is private expenditures on R&D, w is the ad valorem inome tax, a is an ad valorem asset tax, and is an ad valorem onsumption tax. To simplify matters further we assume that agents do not expet tax rates to hange over time. Extensions of the model ould allow G to enter the prodution funtion and/or the utility funtion. Barro and Sala-I-Martin (1992 show how this an promote welfare and growth. We leave out these potentially important avenues of researh to fous on the role of poliy when fertility is endogenous in a sale free growth model. Utility Maximization The model of utility maximization follows the ontinuous time version of Barro s model developed fully in Barro and Sala-i-Martin (1999. Parents derive utility from onsumption, total family size, and the number of hildren. Eah adult inelastially supplies one unit of labor per time period. The problem is to maximize the present value of dynasti utility. The intertemporal utility funtion is: (8 U e 0 t ( ln L ln ln( b d dt, where 0 is the disount rate, is per apita onsumption, L represents family size, b is the birth rate and d is the mortality rate. 0 and 0 are elastiities. Utility is maximized subjet to the following onstraints: (9 k ( 1 w (1 rk t ( b d k Bbk (1, (10 L ( b d L, w a 7

9 Where w is the wage rate; is per apita firm profits; Bbk is a linear ost funtion for rearing hildren, suh that the present value of expenditures per hild is diretly proportional to the apital to labor ratio; r is the interest rate, t is per apita transfers, and n is the resoures devoted to R&D per apita. This type of dynami optimization problem is familiar to eonomists. The problem has two hoie variables, and b. The state variables are k and L. Denote the Hamiltonian as H. Define the multipliers for eah onstraint respetively as and. The first order onditions an be summarized as: a. H 0, b. H b 0,. H H, d., k L First order onditions a and provide the Euler equation. (11 g ( 1 r ( ( b d Bb. a Equations a, b, d, and g L b d defines the birth rate at eah point in time, equal to: (P L L b d ( / k. (1 B ( / k (1 Equation (11 is the familiar Euler equation where ( b d inreases the disount rate; this is due to the diminishing marginal utility of hildren. Bb is subtrated from r beause hild rearing osts, Bbk, inrease with k. s k rises, the return to apital falls. n asset tax lowers the return to foregoing onsumption and through an intertemporal substitution effet raises urrent onsumption and lowers the growth rate of onsumption. 8

10 s equation (P shows, the birth rate varies diretly with the death rate, and inversely with the ost of hild rearing, B. The relationship between / k and b is diret. Per apita onsumption,, represents an inome effet (a higher improves hild quality and raises b and k represents a substitution effet (a higher k inreases the ost of hild rearing and lowers b. Note that to be onsistent with optimizing behavior the rate of population growth, ( b d and, therefore / k, must be stritly positive (see Jones, onsumption tax has a diret impat on the fertility rate. Holding the / k ratio onstant an inrease in makes spending resoures on hild rearing relatively less expensive ausing substitution from other types of onsumption toward hild rearing and raising the birth rate. 3 Profit Maximization and Innovation Intermediate goods are supplied to the produers of final goods under the onditions of imperfet ompetition. Differentiation of equation (1 with respet to x yields the derived demand for an intermediate good. Profits, therefore, are: (1 (1 (12 il xi ( r i xi, where is the rate of apital depreiation. The first order ondition from maximizing equation (12 together with equation (4 implies that the demand for intermediates is symmetri (a fat expressed in equations (2 and (4 and verified here and: (13 ( kˆ / L 2 (1 1 r. Resoures flowing into innovation are determined by equating the marginal benefits of inreasing R&D resoures with the marginal osts of inreasing R&D 3 This an be thought of as a sort of hild tax redit. In our model the ost of hildren, bbk, is not subjet to the onsumption tax rate. More generous tax redits ould be modeled as a redution in B. 9

11 resoures. llow V to represent the present value of innovating. Equations (4, (12, and (13 allow the present value of profits to be expressed as: (1 (1 MX L (1 ( kˆ (14 V N r 1 L MX. For an individual investing in researh in the leading edge tehnology resulting in a marginal benefit of V 1 the marginal ost is MX (1 S R. Setting marginal benefits equal to marginal osts gives a familiar researh arbitrage equation, 1 (1 (1 (15 L (1 ( kˆ MX MX ( 1 S R. N r 1 L MX Equilibrium and analysis Equations (1 through (15 ompletely define the dynamis of the model eonomy in the steady state 4. Note that the model is no longer a semi-endogenous growth model. It is now a fully endogenous growth model where g (. (1 k (1 (1 B (. (1 k and (. expresses the ratio of onsumption to apital as a funtion of poliy variables k 4 We fous on a positive analysis of poliy effets on growth rates. In terms of soial welfare the deentralized equilibrium will generally differ from the equilibrium hosen by a soial planner. Our model inludes all of the externalities disussed in Jones and Williams (2000. They outline four externalities where two externalities lead to underinvestment in R&D in a deentralized equilibrium and two externalities lead to over investment in R&D in a deentralized equilibrium. fter alibrating the model they find that the eonomy is likely to under invest in R&D. Our model inludes one additional externality assoiated with the marginal soial benefit of raising hildren that is not inorporated in the private utility funtion (see Jones This externality reinfores the Jones and Williams onlusion that soiety is likely to under invest in R&D. In our model a Pareto optimal equilibrium an be obtained with a lump sum tax and an R&D subsidy or a diret subsidy to hild prodution. Sine the welfare impliations are standard we leave a full exploration to the reader. 10

12 and parameters. Equation (5 speifies onstant returns to sale in rival inputs and inreasing returns in rival inputs and knowledge in goods prodution. These properties together with dereasing returns in knowledge prodution, equation (6, and a linear differential equation desribing population growth in equation (P guarantee the existene of a positive stable steady state growth rate. Our fous is on the steady state. To understand the nature of equilibrium we derive a graphial representation of the determination of the steady state growth rate similar to that found in ghion and Howitt (1998. Begin by expressing the model in terms of the average produt of apital, Z k ~ 1 ~ (where k K / L and the ratio / k. system of graphial representations of the equilibrium is derived, so that taken together, it allows for an easy exposition of omparative steady state analysis under alternative value of poliy parameters. Proeed by deriving two unique relationships between Z and. In the steady state g g. Using the Euler equation this implies r g ( b d Bb. Combine (1 a this equation with equation (13 and note that k ˆ kl ~ / Q : (Z g 2 (1 Z ( (1 d (1 B b. a a This Z equation implies a positive relationship between Z and sine the partial derivative is g Z 2 ( 1, is greater than zero. This positive relationship is a interpreted as the impat of a hange in g on the Z. n inrease in g inreases the growth rate of onsumption via the Euler equation. This implies an inrease in the rate of interest and an inrease in the rental ost of apital. The demand for apital falls. lower level of apital implies that the average produt of apital is greater. 11

13 The seond relationship between Z and equation. Equation (15 is rewritten g is derived from the researh arbitrage as r g / ( 1 MX (1 S R (1 L (1 (1 an be used to express the arbitrage equation as: kˆ MX. Sine ~ 1 k Z equation (13 (R g (1 L (1 S 1 R 1 1 MX Z 1 2 Z Where L MX is onstant in the steady state. 5 The R equation defines a negative relationship between Z and as seen from the partial derivative, g Z (1 L 1 (1 S 1 R 1 1 MX Z 1 1 2, is less than zero. The intuition is, again, straight forward. higher Z implies a lower apital stok. This diretly raises the rental rate of apital and diminishes the present value of an innovation (see equation (14. With the marginal benefits of raising R&D resoures falling relative to the marginal osts of raising R&D resoures, resoures to R&D are lowered and lowered. is The model is now expressed in the form of the three equations: (Z, (R, and (P. It is onvient to re-write the (R and (Z equations in two alternative forms to use together to understand the properties of balaned growth. One transformation, labeled (R and MX 5 Log differentiating L with respet to time gives 1 ( 1 g (1 g L (1 g L (1 g L 0 (1 equation (6, it must be true in a steady state that g (1. where we use the result that, aording to L 12

14 (Z expresses relationships between the fertility rate, b, and the average produt of apital, Z. The seond transformation, labeled (R and (Z provides for a unique relationship between Z and /k on a balaned growth path. (R and (Z are derived by noting that, aording to equation (6, the steady state growth rate must satisfy: (G (1 g 1 L where it is assumed that 1. With g L b d equation (G is used with equations (Z and (R to express a relationship between b and Z. (Z 2 (1 (1 a (1 b Z [ d (1 a ] (1 (1 (1 B (1 (1 (1 B (1 d (1 (1 (1 B (R (1 b (1 1 1 MX 1 L (1 Z (1 S R 1 2 (1 (1 (1 Z d (1 (Z defines a positive relationship between b and Z and (R defines an inverse relationship between b and Z. The intuition for the slopes is the same as the intuition for the slopes of (R and (Z reognizing that b and are positively related. The (R and (Z equations are represented in the top left graph in Figure 1. (R and (Z are derived by taking the equation (P for population growth and substituting out b in (R and (Z. Given the positive relationship between b and the ratio / k it should be lear that (Z defines a positive relationship between Z and 13

15 / k while (R implies a negative or inverse relationship between Z and / k. Defining the right hand side of (Z as F and the right hand side of (R as G we have: (Z (1 B ( F( Z, a, B d (1 C / k ( F( Z,, B d a (1 B ( G( Z, S R d (1 C (R / k ( G( Z, S d R (Z and (R appear in the bottom graph in Figure 1. Equation (G appears in the right hand graph in Figure 1. Consider what is gained from this extension of the endogenous innovation model. This is a model of fully endogenous growth that is onsistent with the lak of empirial evidene of sale effets. Furthermore, it reintrodues poliy variable effetiveness in levels and growth rates. ny poliy variation, therefore, has a muh greater potential to explain ross ountry differenes in growth rates and levels of per apita inome. The latter must be the ase sine some poliy variations have growth rate effets and these differenes ompound into greater and greater differenes in levels over time. This only strengthens this approah to modeling endogenous growth sine the model an more easily explain the empirial regularity of great variane in levels and growth rates aross eonomies at a point in time and within some ountries over time. This model onsiders a number of poliy variables in a fully endogenous new growth model. The government ollets taxes from wage inome, onsumption expenditures, and asset inome. The government uses these funds to support R&D subsidies, lump sum transfers to households, and unprodutive government spending. w 14

16 does not appear in any of the equilibrium equations. Beause labor supply is inelasti w does not influene equilibrium. This is probably a reasonable approximation for most industrialized eonomies given the empirial evidene that individual labor supply is very inelasti. a appears in the Euler equation. n inrease in a dereases the return of assets and lifts urrent onsumption: lowering the growth rate of onsumption intertemporally. The equation desribing the birth rate inludes the tax rate. This is a novel result given that agents don t expet the onsumption tax rate to hange over time. If fertility where exogenous then would not impat the eonomies balaned growth path (Barro and Sala-i-Martin The researh subsidy appears in the researh arbitrage equation with an inrease in S R promoting private R&D spending. The general equilibrium effets of different poliies is easily understood using the (R, (Z, (R, (Z and (G equations. These equations are summarized together here. (Z b F( Z,, B (R b G( Z, S a R (Z (R (G k k (1 B ( F( Z, a, B d (1 ( ( F( Z,, B d (1 B ( G( Z, S (1 ( ( G( Z, S (1 g 1 L a R R d d 15

17 The signs over the poliy variables refer to the partial derivative of the funtion with respet to the assoiated variable. The signs of the partial derivatives are easily derived by referring to the (R and (Z equations. Refer to Figure 1. Consider two eonomies that differ only in the level of the onsumption tax. Sine does not appear in (Z, (R or (G the two eonomies share the same fertility rate and rate of eonomi growth. The tax rate does appear in the (R and (Z equations. The eonomy with the higher tax rate, eonomy two( (1< below the (R and (Z shedules of the eonomy with a lower (2, has (R and (Z equations that lie. In equilibrium G(.=F(. whih is enough to establish that the equilibrium for the two eonomies exists at the same value of Z but different values of the / k ratio. The onsumption tax has a level effet but no growth effet. This result is similar to the results for all poliy variables in a semi-endogenous growth model when population growth is exogenous. Higher onsumption taxes are onsidered permanent and have strong permanent inome effets on onsumption. This lowers the / k ratio by just enough to offset the diret impat of the higher tax rate on fertility in equation (P. The net result is a lower level of per apita onsumption at all points in time with no effets in first differenes. Figure 2 demonstrates the differene between two eonomies that differ only in the size of the R&D subsidy. Eonomy two has a higher subsidy ( S (1< eonomy with the higher R&D subsidy, R S R (2. The S R, has an (R and (R shedule that are to 16

18 the right of the (R and (R of the eonomy with the lower R&D subsidy 6. The eonomy with the higher R&D subsidy experienes a higher steady state growth rate. Thus the level and growth rate must be higher for the eonomy with the higher R&D subsidy. Inreased innovation boosts the average produt of apital and the / k ratio. The inrease in the / k inreases the population growth rate and the long run rate of growth. While the result that an R&D subsidy boosts innovation is familiar and seems reasonable, Jones's (2003 omes to the opposite onlusion after endogenizing the fertility rate. s outlined earlier, our paper assumes an inelasti supply of labor in the output setor of the eonomy, whih is enough to overturn Jones's onlusion that subsidies will lower growth rates in an endogenous fertility framework. Jones ignores any expliit goods ost of hild rearing, assumes no substitution between labor and leisure, and assumes that only labor is used in the prodution of new ideas. 7 researh 6 The diretion of the shift in (R and (Z is very easy to verify. To establish the diretion of shift in (R and (Z note that from R / k F 2 (1 B (1 ( ( F d / k G 2 2 (1 B (1 ( ( G d 2 >0 and from Z >0. These results follow from noting that the fertility rate is a number between zero and one and F=G=b in the steady state. Therefore the diretion of shift in the (R and (Z is in the same diretion as the shifts in (R and (Z for all poliy variables with the exeption of and B. 7 truly integrated model would speify a household s hoie between labor, leisure, and hild rearing where the three are distint ativities. Furthermore it would allow apital to be used in knowledge prodution as in the present paper. This generality would ome at a ost in tratability, but now the birth rate would be a funtion of the time ost of raising hildren, as in Jones, and the goods ost, as in the present model. subsidy would raise the wage rate in the formal setor and this would tend to lower the birth rate as in Jones, but this effet would be offset by an inrease in the /k ratio and a quality to goods ost ratio, whih would raise the birth rate as in this model. The net effet is likely to be ambiguous and would depend on the elastiity of the birth rate with respet to the goods ost relative to the elastiity of labor supply relative to the wage rate. 17

19 subsidy then inreases the opportunity ost of spending time having hildren and workers take time away from hild rearing and spend more time in R&D. This lowers the population growth rate, and, subsequently the steady state growth rate in his model. Figure 3 ompares two eonomies where the asset tax rate, lower than the asset tax rate in eonomy two( a (1< a a, for eonomy one is (2. The (Z and (Z shedules are unambiguously lower for the eonomy with the higher tax rate. The higher tax rate lowers the effetive rate of return on assets and, through standard intertemporal inome and substitution effets, auses a rise in urrent onsumption and the growth rate of onsumption falls. In the steady state g g. In other words the higher tax auses a lower / k ratio onsistent with lower fertility and lower long run growth. The last poliy experiment omes from a hange in the ost of having hildren, B. This is viewed as a poliy sine governments an promote larger families in a number of important ways. Providing quality publi eduation, medial insurane overing pregnany and subsequent expenses 8, tax breaks for firms who provide on site day are, and tax redits related to the number of hildren are all examples of government poliies that lower the ost of rearing hildren. These varied poliies an be investigated by omparing the steady states of two eonomies where the seond eonomy has a higher ost of rearing hildren(b(1<b(2. Refer to Figure 4. In the enter graph only (Z is different, and is learly lower for eonomy two. The higher B lowers the hoie of the number of hildren to have in eonomy two, lowers b, for any value of the average produt of apital along the (Z shedule implying a (Z that is lower and to the right of 8 This may also impat growth through an exogenous derease in d, whih Barro and Sala-i-Martin (1999 argue an be interpreted as a hild mortality rate in the infinite time horizon setup used here. llowing for this effet would only strengthen the argument that improved health are for mothers and hildren inreases the long run growth rate. 18

20 the (Z shedule for eonomy one. This auses b to be lower in the right hand graph as eonomy two operates at a point to the left of eonomy one along the (G shedule. In the bottom graph things are only slightly more omplex. Sine F(.=G(. in the steady state the diret impat of a higher B is an equal move up in both the (Z and (R shedules in a manner similar to our analysis of a onsumption tax variation. Stopping here would ignore the indiret impat of a higher B on Z, whih partially offsets the higher plaement of the Z shedule for eonomy two. The net effet is a higher average produt of apital and / k ratio for eonomy two 9. The higher / k ratio is not enough to offset the inrease in B in the (P equation and eonomy two experienes a lower fertility rate and a lower growth rate in the right hand graph. Poliies that promote a lower ost of having hildren have level and growth effets on the steady state equilibrium. III Conlusion It is our desire to show that adding endogenous demographis an be done in a relatively simple and well aepted framework. This paper develops a fully endogenous non-sale growth model by allowing families to make hoies about family size, thereby making the fertility rate endogenous. In order to allow for fertility hoie we follow the well aepted approah developed by Barro and Beker (1989. Government is inluded in a more realisti formulation ompared to the previous literature. This paper fills two long standing gaps in the growth literature. First, a model is developed that is sale free and does not rely on the equality between the growth of setors and the growth of population, but reintrodues a poliy impat on the steady state growth rate. Seond, the 9 This does not imply that steady state per apita onsumption is higher in eonomy 2. In fat and k are lower in eonomy two. Comparing one and two the differene in the apital labor ratio is relatively larger than is the differene in per apita onsumption ausing a higher /k ratio in eonomy two. 19

21 model inorporates a more realisti government setor and an be used to understand a wide array of government spending and taxation poliies. In our model government poliies an be ategorized aording to whether they have level effets only, level and growth effets, or no impat on levels and/or growth. This makes a generalized Jones model a more powerful tool for understanding long standing differenes in growth rates and inome levels aross ountries at a point in time and within some ountries over time. Wage taxes are found not to impat levels or growth rates beause we assume that labor supply is inelasti. This result may seem ounter intuitive and inonsistent with the available empirial findings of Romer and Romer (2010 and Barro and Redlik (2011 who find strong level effets from hanges in average marginal tax rates. However, Barro and Redlik persuasively argue that these results are oming mostly from labor-leisure substitution effets from a hange in inome. The mehanism that we are modeling of an inelasti labor supply regarding fertility deisions is independent of the onsideration of the elastiity with regard to inome. Thus, we find that there are no additional empirially relevant growth or level effets from wage taxation arising from hanges in fertility deisions that the traditional fous of steady-state growth models. Consumption taxes have a level effet but no growth effet. This result is novel from a neolassial perspetive but is onsistent with the endogenous growth perspetive of Zeng and Zhang (2002 and Peretto (2007. In a neolassial setting with exogenous population growth the onsumption tax would have no impat on levels or growth rates. This result is the diret onsequene of making population growth endogenous and does 20

22 not depend on the speifiation of tehnologial hange. Our model predits that an asset tax disourages the long run growth rate. This paper also predits that a researh subsidy promotes long run growth. This result is in ontrast to Jones (2003 who finds that a subsidy to researh lowers long run growth in a new growth model with endogenous fertility. The assumption of inelasti labor supply in the output produing setor of the eonomy is enough to hange the diretion of the impat of a subsidy on steady state growth. We also find level and growth effets for poliies that diretly lower the ost of having hildren. We argue that these poliies an inlude providing quality publi eduation, medial insurane overing pregnany and subsequent expenses, tax breaks for firms who provide on site day are, and tax redits related to the number of hildren. These are all examples of government poliies that lower the ost of rearing hildren. Certainly there are many more examples of related government poliies that influene both levels of per apita inome and long run growth rates. 21

23 Referenes ghion, P. and P. Howitt (1998 Endogenous Growth Theory, MIT Press, London, England. rin, K. Peren, M. Berlemann, F. Koray andt. Kuhlenkasper (2013 Nonlinear Growth Effets of Taxation: Semi-Parametri pproah Usinverage Marginal Tax Rates, Journal of pplied Eonometris, 28, Barro, R. (1991 Eonomi Growth in a Cross Setion of Countries, Quarterly Journal of Eonomis, 106, Barro R. and G. Beker (1989 Fertility Choie in a Model of Eonomi Growth, Eonometria, 57, Marh, Barro R. and C. J. Redlik (2011 Maroeonomi Effets from Government Purhases and Taxes, Quarterly Journal of Eonomis, 126, 1, Barro, R. and X. Sala-I-Martin (1992 Publi Finane in Models of Eonomi Growth, Review of Eonomi Studies, 59, Otober, Barro, R. and X. Sala-I-Martin (1999 Eonomi Growth, MIT Press, Cambridge, Massahusetts. Durlauf, S.,. Kourtellos, and C. Ming Tan (2008 re ny Growth Theories Robust? Eonomi Journal, 118, Easterly, W. (1993 How Muh Do Distortions ffet Growth? Journal of Monetary Eonomis, 32, Galor, O. and D. Weil (2000 Population, Tehnology, and Growth, merian Eonomi Review, 90, 4, Gemmell, N., R. Kneller and I. Sanz (2011 The Timing and Persistene of Fisal poliy impats on Growth: Evidene from OECD Countries, Eonomi Journal, 121, F33-F58. Howitt, P. (1999 Steady Endogenous Growth with Population and R&D Inputs Growing, Journal of Politial Eonomy, 107, 4, Islam, N. (1995 Growth Empiris: Panel Data pproah, Quarterly Journal of Eonomis, 110, 4, Jones, C. (1995a R&D Based Models of Eonomi Growth, Journal of Politial Eonomy, 103,

24 Jones, C. (1995b Time Series Tests of Endogenous Growth Models, Quarterly Journal of Eonomis, 110, Jones, C. (1999 Growth: With or Without Sale Effets? merian Eonomi Review, Papers and Proeedings, Jones, C. (2001 Was the Industrial Revolution Inevitable? Eonomi Growth Over the Very Long Run, BEP dvanes in Maroeonomis, 1, Number 2, rtile 1. Jones, C. (2002 "Soures of U.S. Eonomi Growth in a World of Ideas," merian Eonomi Review, 92, 1, Jones, C. (2003 Population and Ideas: Theory of Endogenous Growth, ghion, Frydman, Stiglitz, and Woodford (eds. Knowledge, Information, and Expetations in Modern Maroeonomis: In Honor of Edmund S. Phelps, Prineton University Press. Jones C. and J. C. Williams (2000 Too Muh of a Good Thing? The Eonomis of Investment in R&D, Journal of Eonomi Growth, 5, 1, Killingsworth, M. R. (1983 Labor Supply, Cambridge University Press, Cambridge. Kormendi, R. and P. Meguire (1986 Maroeonomi Determinants of Growth: Cross- Country Evidene, Journal of Monetary Eonomis, 16, 2, Luus, R. E. (1988 On the Mehanis of Eonomi Development, Journal of Monetary Eonomis, 22, MGrattan, E. R. (2012 Capital Taxation During the U.S. Great Depression, Quarterly Journal of Eonomis, Peretto, P. (2007 Shumpeterian Growth with Produtive Publi Spending and Distortionary Taxation, Review of Development Eonomis, 11, 4, Peretto, P. (2011 The Growth and Welfare Effets of Defiit-Finaned Dividend Tax Cuts, Journal of Money, Credit and Banking, 43, 5, Rebelo, S. (1991 Long-Run Poliy nalysis and Long-Run Growth, Journal of Politial Eonomy, 99, Romer, C. and D. Romer (2010 The Maroeonomi Effets of Tax Changes: Estimates Based on a New Measure of Fisal Shoks, merian Eonomi Review, 100, 3, Segerstrom, P. (1998 Endogenous Growth without Sale Effets, merian Eonomi Review, 88, 5,

25 Sedgley, N. and B. Elmslie (2010 Reinterpreting the Jones Critique: Time Series pproah to Testing and Understanding Idea Driven Growth Models with Transition Dynamis, Journal of Maroeonomis, 32, Sedgley, N. and B. Elmslie (2013 The Dynami Properties of Endogenous Growth Models, Maroeonomi Dynamis, 17, Young,. (1998 Growth without Sale Effets, Journal of Politial Eonomy, 106, 1, Zeng, J. and J. Zhang (2002 Long-Run Growth Effets of Taxation in a Non-Sale Growth model with Innovation, Eonomis Letters, 75,

26 Figure 1. Consumption Tax and the Steady State b b G Z b* b* R Z* Z * Z (1 Z (2 / k / k(1 / k(2 R (2 R (1 Z* Z 25

27 Figure 2. Researh Subsidy and the Steady State b b G Z b*(2 b*(2 b*(1 R (2 b*(1 R (1 Z* Z (1* (2* Z / k / k(2 / k(1 R (1 R (2 Z(1* Z(2* Z 26

28 Figure 3. n sset Tax and the Steady State b Z (1 b G Z (2 b*(1 b*(2 R b*(1 b*(2 / k Z* Z Z (1 Z (2 (2* (1* / k(1 / k(2 R Z(1* Z(2* Z 27

29 Figure 4. The Cost of Child Rearing and the Steady State b Z (1 Z (2 b G b*(1 b*(2 R b*(1 b*(2 / k Z* Z Z (2 Z (1 (2* (1* / k(2 / k(1 R (1 R (2 Z(1* Z(2* Z 28

30 Please note: You are most sinerely enouraged to partiipate in the open assessment of this disussion paper. You an do so by either reommending the paper or by posting your omments. Please go to: The Editor uthor(s Liensed under the Creative Commons ttribution 3.0.