Fiscal Policy Puzzle and Intratemporal Substitution among Private Consumption, Government Spending, and Leisure. Masataka Eguchi Faculty of Economics, Keio University and Yuhki Hosoya Graduate School of Economics, Keio University Abstract This paper investigates how the response of private consumption to government spending can be changed by intratemporal substitution among private consumption, government spending, and leisure. We show that the response of private consumption to government spending can be positive even if private consumption and government spending are not complements and private consumption and leisure are not substitutes. In this case, substitution between leisure and government spending plays an important role. Previous works have overlooked this perspective. JEL codes: D11, E62, H31. Keywords: Fiscal policy, Government spending, Intratemporal substitution. We would like to thank Masao Ogaki, Yasuo Hirose, Eisei Otaki, and the other participants of the Public Economics Seminar at Keio University for their useful comments. Research Associate, Faculty of Economics, Keio University. Address: 2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan. E-mail: masataka.eguchi@gmail.com Ph.D. student, Graduate School of Economics, Keio University. 1
1 Introduction The aim of this paper is to provide a new solution to fiscal policy puzzle in recent macroeconomics. We show that the puzzle can be solved by considering the relationship between government spending and leisure. Empirical research indicates that private consumption rises in response to an increase in government spending ( e.g., Blanchard and Perroti (2002), Perotti (2005), Gali, López-Salido, and Vallés (2007)). As explained by Baxter and King (1993), however, an increase in government spending lowers the present value of after-tax income, thus generating a negative wealth effect that induces a fall in private consumption in the standard neoclassical model. This is called the Fiscal policy puzzle. Hence, several literatures introduce various assumptions to generate a positive effect of government spending on consumption. For example, Baxter and King (1993) introduces productive government spending to a pure Real Business Cycle model. Galí, López-Salido, and Vallés (2007) introduces a non-recardian household to the New Keynesian model with sticky price, creating a case in which the response of private consumption to government spending is positive. In contrast to these approaches, this paper applies a preference-based approach to solve fiscal policy puzzle. Particularly, we study how the response of private consumption to government spending can be changed whether they are substitutes or complements. 1 Bailey (1971), Barro (1981), Aschauer (1985), and, recently, Bouakez and Rebei (2007) and Ganelli and Tervara (2009) consider direct substitution between private consumption and government spending. They show that private consumption rises in response to an increase in government spending when their complementarity is strong enough. Kormendi (1983), Aschauer (1985), Karras (1994), Ni (1995), Amano and Wirjant (1998), Hamori and Asako (1999), Okubo (2003), and Bouakez and Rebei (2007) examine whether private consumption and government spending are complements or substitutes, but these empirical results are varying and inconclusive. On the other hand, Linnemann (2006), Monacelli and Perotti (2008), and Bilbiie (2009) argue that fiscal policy puzzle can be solved if private 1 In this context, the terms substitutes and complements are not used in the Hicks sense but Edgeworth s sense. Let the utility function be U(x 1, x 2 ). Then, we say that commodities i and j are substitutes in the sense of Edgeworth if U xix j < 0, complements if U xi x j > 0, and independent if U xi x j = 0. See, for example, Karras (1994), Ni (1995). 2
consumption and leisure are substitutes (or private consumption and labor are complements). But Bilbiie (2009) shows that the condition is equivalent to what consumption good is inferior. According to the previous work of the preference-based approach, (1) private consumption and government spending are complements and/or (2) private consumption and leisure are substitutes, is the necessary condition for the response of private consumption to government spending to be positive. We show that the response of private consumption to government spending can be positive even if private consumption and government spending are not complements and private consumption and leisure are not substitutes when government spending and leisure are substitutes. It is very important the substitution between leisure and government spending in this case. It is possible for consumption to respond positively to the increase of government spending if government spending and leisure are substitutes (or government spending and labor are complements). This perspective has been overlooked in previous work. 2 The Model The expected lifetime utility function of the representative household is E 0 t=0 β t U(C t, L t, G t ), (1) where, C t denotes private consumption, leisure is L t = 1 N t, N t denotes labor supply, and G t denotes government spending. Assume that U C > 0, U L > 0, U G > 0, U CC < 0, U LL < 0, and U G : (C, L) U(C, L, G) satisfies the strict bordered Hessian condition, that is, U CC U CL U C U CL U LL U L U C U L 0 > 0. The utility function (1) is non-separable in private consumption, government spending, and leisure. Bailey (1971), Barro (1981), Aschauer (1985), Bouakez and Rebei (2007), and Ganelli and Tervara (2009) consider the nonseparability of private consumption and government spending. On the other hand, Linnemann (2006), Monacelli and Perotti (2008), and Bilbiie (2009) 3
consider the non-separability of private consumption and government spending. In addition, We consider the non-separability of government spending and leisure. The existence of non-zero U LG is a feature of this paper. Let Y t denote output. the production function of the firm is given by Y t = F (N t ), (2) where F t denotes nonincreasing return to scale, that is, F (0) = 0, F N > 0, F NN 0. It is assumed for simplicity that the firm put only labor into production. Hence, the temporary resource constraint of this economy is given by 3 Results C t = F (N t ) G t. (3) In our model, the first welfare theorem holds, and thus, the competitive result must be the same as the social optimal solution. Hence, we solve only the social optimal solution under given G t and analyze it. Thus, our problem is to maximize (1) subject to (3) under given G t. We assume that the solution is not the corner solution. Then, the first order condition for this problem can be written as U L (C t, L t, G t ) = U C (C t, L t, G t )F N (1 L t ), (4) We redefine (4) and (5) as follows, F (1 L t ) = C t + G t. (5) H 1 (C t, L t, G t ) = U L (C t, L t, G t ) U C (C t, L t, G t )F N (1 L t ), H 2 (C t, L t, G t ) = F (1 L t ) C t G t, and check whether the implicit function theorem is applicable. Then, H t H1 C H1 L HC 2 H2 = U LL 2U CL F N + U CC (F N ) 2 + U C F NN L U CC U CL 1 = U C F NN U LC U LL F N 1 F N 0 U CC U CL U C = U C F NN (U C ) 2 U LC U LL U L U C U L 0 < 0, 4
where the last inequality comes from the strict bordered Hessian condition of U G. Therefore, the implicit function theorem is applicable, and both C t and L t can be represented as continuously differentiable functions of G t. We denote those functions as C t (G t ) and L t (G t ), respectively. Calculating C t(g t ) and L t(g t ) from the implicit function theorem, the following equation is obtained. C t(g t ) = U LL + U CL F N + U LG F N U CG (F N ) 2 U C F NN H t, (6) L t(g t ) = F (U CG U CC ) + U LC U LG H t. (7) Since H t is negative, we obtain the following proposition. proposition 1 C t > 0 if and only if or equivalently, U LL + U CL F N + U LG F N U CG (F N ) 2 U C F NN < 0, U CG U CL U C (U C ) 3 U LG U LL U L U C U L 0 < F NN. Since F NN 0, the necessary condition of C t > 0 (i.e., C t increases in response to G t ) is (a): U CG > 0 (private consumption and government spending are complements), (b): U CL = U LC < 0 (private consumption and leisure are substitutes), and/or (c): U LG < 0 (leisure and government spending are substitutes). Conditions (a) and (b) have already been shown in previous work. We show that the response of C t on G t can be positive when condition (c) holds even if condition (a) and (b) are not satisfied. That is, government spending reduces the marginal utility of leisure (or relieves marginal disutility of labor) and private consumption can rise in response to government spending. This is the main finding of this paper. 4 Discussion In this section, we consider the reason why private consumption can rise in response to government spending when U LG is negative. 5
Suppose that government spending increases. The increase in G t causes negative wealth effect, followed by a fall in leisure and rise in labor. Y t will increase as a result. The response of Y t on G t is Y t (G t ) = F N(U LG U LC ) + (F N ) 2 (U CC U CG ) H t. (8) To focus on U LG, we set U CG = U LC = 0. Then, If U LG = 0, Y t (G t ) = Y t (G t ) = F N U LG + (F N ) 2 U CC U LL + (F N ) 2 U CC + U C F NN. (9) (F N ) 2 U CC U LL + (F N ) 2 U CC + U C F NN. (10) Thus we get 0 < Y t (G t ) < 1. Hence, the increase in Y t is less than that in G t. This means that private consumption must fall. 2 A intuitive explanation of the result is the following. If government spending increases by 1 dollar, output must also increase by 1 dollar for maintaining the level of private consumption. However, marginal disutility of labor is higher and marginal productivity of production is lower than before. Thus, production increases by less than 1 dollar, and private consumption must fall because of resource constraint. It is essentially necessary for rise in private consumption that the government spending multiplier should be lager than 1. Howevwe, if U LG < 0, government spending relieves marginal disutility of labor, and it is possible that output increase is larger than 1 dollar, allowing for private consumption to rise. This result suggests that it will stimulate the economy if government spending improves the work environment and enhances the incentive to work. 5 Conclusion In this paper, we studied how the response of private consumption to government spending can be changed by intratemporal substitution among private consumption, government spending and leisure. As a result, considering 2 Christiano, Eichenbaum, and Rebero (2009) and Woodford (2010) also show that government spending multiplier is less than 1 if the utility function is additively separable in consumption and leisure and does not depend on government spending (i.e., U CG = U CL = U LG = 0.). 6
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