- Email: [email protected]
On the monotonicity of balanced budget multiplier under imperfect competition
Economics Letters 59 (1998) 331–335
On the monotonicity of balanced budget multiplier under imperfect competition ´ J. Torregrosa* Ramon Universidad de Salamanca, ‘‘ Miguel de Unamuno’’, Ed FES Campus 37008 Salamanca, Spain Received 15 May 1997; accepted 11 February 1998
Abstract This paper presents a counterexample to the wellknown monotonicity result of Mankiw (1988) [Mankiw, N.G., 1988. Imperfect competition and the Keynesian cross. Economics Letters 26, 7–13]. In particular by considering taxes on labor income rather than lumpsum taxes it obtains a negative relation between the magnitude of the balanced budget multiplier and marketpower. 1998 Elsevier Science S.A. Keywords: Balanced budget multiplier; Imperfect competition; Distortionary taxation JEL classification: E12; L16; E62
1. Introduction Market-power foundations of macroeconomics point out that the inefficiency arising from imperfect competition allows us to explain some Keynesian features even in fully flexible price models.1 In this context Dixon (1987); Mankiw (1988) provide examples with Cobb-Douglas preferences, constant marginal costs, constant markup pricing and lumpsum taxes. In these examples the balanced budget multiplier rises monotonically in respect to market power. The question this paper seeks to answer is whether this monotonic behavior of the balanced budget multiplier remains unchanged under distortionary tax schemes. In an early paper Molana and Moutos (1992) without studying the monotonicity issue found that the balanced budget multiplier can be zero and even negative under tax rates on consumers’ income. Starting from here this paper studies the monotonicity of the balanced budget multiplier in relation with market power under a labor supply tax rate providing a counterexample to Mankiw’s monotonicity result. 2. The economy Following Mankiw (1988) I consider an economy with two goods: leisure which is used as *Tel.: 134 23294640; fax: 134 23294686; e-mail: [email protected] 1 See Silvestre (1993); Dixon and Rankin (1994). 0165-1765 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00055-X
R. J. Torregrosa / Economics Letters 59 (1998) 331 – 335
332
numeraire and a single produced commodity; m consumers that can be aggregated into a single representative consumer; an industry formed by n noncompetitive firms and a government.
2.1. Consumers The representative consumer maximizes a Cobb-Douglas utility function which depends on the amount C consumed of the single produced commodity and of the amount L of leisure consumed u(C, L) 5 a ln C 1 (1 2 a ) ln L.
(1)
Denote by w the endowment of time then w2L is the labor income. Considering a labor income tax rate t[1(0, 1) levied by the government, total after-tax income is (12t)(w2L)1 P where P are the firms’ profits. Thus the consumer’s budget constraint can be written as PC 5 (1 2 t)(w 2 L) 1 P where P is the price of the produced commodity. The consumer’s optimal choice allows us to express the consumption function as PC 5 a [(1 2 t)w 1 P ]
(2)
where a can be interpreted as the marginal propensity to consume. On the other hand labor supply is given by w 2 L 5 a w 2 (1 2 a )P /(1 2 t).
(3)
2.2. Government The tax revenue raised by the government is given by T5t(w2L) which is used to purchase the produced commodity and to hire government workers. Let us call as G the public spending and W the amount of labor demanded by the government. Then the government budget constraint requires: t(w 2 L) 5 G 1 W. Total expenditure on the produced commodity is given by Y 5 PC 1 G.
(4)
Substituting Eq. (2) into Eq. (4) we have Y 5 a ((1 2 t)w 1 P ) 1 G
(5)
i.e. total expenditure depends positively on profits and government purchases and negatively on taxes.
R. J. Torregrosa / Economics Letters 59 (1998) 331 – 335
333
2.3. Firms There are n firms producing an aggregated amount Q of the single commodity. The industry takes total expenditure as given.2 Hence it faces a unit elastic demand function Q 5 Y /P
(6)
Firms have the same constant returns to scale technology. The cost function of each firm in terms of the numeraire is C(q) 5 cq, where q is the quantity produced by the representative firm. We assume that firms are profit maximizers 3 and that play a oligopoly game. The result of the game determines the profit margin m as
m 5 (P 2 c) /P. We can interpret m as a measure of marketpower: when m 51 /n firms behave a la Cournot; when m 50 firms behave as Bertrand oligopolists (perfect competition); and finally when m falls to 1, firms behave as a monopolist (perfect collusion).4 Therefore m only depends on conjectural variations and n. The relationship between industry output and total expenditure is Q 5 [(1 2 m ) /c]Y, and aggregate profits can be written as a function of total expenditure and the profit margin as
P 5 m Y.
(7)
2.4. Feasibility condition in labor market Using Eqs. (4) and (6) the quantity of labor demanded (both firms and government) can be written as: W 1 cQ 5 [t(w 2 L) 2 G] 1 [Y 2 P ] and using Eqs. (3) and (5) we obtain W 1 cQ 5 t[a w 2 (1 2 a )P /(1 2 t)] 2 G 1 a ((1 2 t)w 1 P ) 1 G 2 P, W 1 cQ 5 a w(1 2 a )P /(1 2 t)
(8)
which is the labor supply function given by Eq. (3). Hence the feasibility condition in labor market 2
One might object this assumption is not reasonable because expenditure depends on industry profits. The model could be amended however to include a continuum of industries; the demand function of each industry would depend on aggregate profits. 3 Let us dispense the fact that if the price is the same for ownersconsumers and for consumers who are not owners then the goal of the firm might not be the profit maximization. 4 The demand function given by eq. (5) implies that the monopoly must be considered as a limiting case hence m [[0, 1).
R. J. Torregrosa / Economics Letters 59 (1998) 331 – 335
334
holds. Notice that from Eq. (8) the feasibility condition implies that in equilibrium a w2W 5 cQ 1 (12 a ) P /(12t).0.
3. The balanced budget multiplier (BBM) In order to display the shape of the BBM it is necessary to characterize the equilibrium total expenditure and the equilibrium public spending. Using Eqs. (3)–(5), (7) we obtain (1 2 t)[a w 2 W] Y 5 ]]]]], 1 2 t 1 m (t 2 a )
(9)
and G 5 t[a w 2 (1 2 a )P /(1 2 t)] 2 W.
(10)
By applying the implicit function theorem to the system of equations formed by Eq. (9) and Eq. (10) we can derive the BBM as a function of market-power m (1 2 a )m [a w 2 W] dY BBM( m ) ; ] 5 2 ]]]]]]]]]]]]]]]] . dG a w[1 2 t 1 m (t 2 a )] 2 2 (1 2 a )( m 2 am 2 )[a w 2 W]
(11)
It is easy to see that BBM(0)50 and limm →1 BBM( m )5 2[a w2W] /(12 a )W ,0. Also differentiating Eq. (11) with respect to m and manipulating yields dBBM( m ) (1 2 a )a [a w 2 W][w(1 2 am 2 )(1 2 2t) 1 wt 2 (1 2 m 2 ) 1 (1 2 a )m 2W] ]]]] 5 2 ]]]]]]]]]]]]]]]]]]]]] . dm [a w[1 2 t 1 m (t 2 a )] 2 2 m (1 2 a )(1 2 am )[a w 2 W]] 2
(12)
A sufficient (but far from necessary) condition for Eq. (12) to be negative is that t#1 / 2 which implies that the BBM is monotonically decreasing with market power.
4. Conclusion As we have seen Mankiw’s monotonicity result is not robust to different specifications of the tax scheme. In fact under a labor supply tax opposite results are reached. The reason for this is that an increase in this type of taxes distorts relative prices with the consequence of reducing output in greater proportion than the increase in demand due to government expenditure. Consequently total expenditure falls in relation to public expenditure. In the example this effect has also proved to be greater when market power increases. This is because the higher the degree of competitiveness the lower the pretax profits hence the lower the fall in private consumption.
R. J. Torregrosa / Economics Letters 59 (1998) 331 – 335
335
Acknowledgements ´ G. Fernandez ´ ´ ´ and an I am grateful to R. Caminal, L. Corchon, de Cordoba, T. Kehoe, B. Valdes, anonymous referee for their helpful comments and suggestions. I am also grateful for the financial support of the Spanish Ministry of Education (CICYT project PB93-0940) and the Instituto Valenciano de Investigaciones Economicas. The usual caveat applies.
References Dixon, H., 1987. A simple model of imperfect competition with walrasian features. Oxford Economics Papers 39, 134–179. Dixon, H., Rankin, N., 1994. Imperfect competition and macroeconomics: a survey. Oxford Economics Papers 46, 171–199. Mankiw, N.G., 1988. Imperfect competition and the Keynesian cross. Economics Letters 26, 7–13. Molana, H., Moutos, T., 1992. A note on taxation, imperfect competition and the balanced budget multiplier. Oxford Economic Papers 44, 68–74. Silvestre, J., 1993. The marketpower foundations of macroeconomic policy. Journal of Economic Literature 21, 105–141.
On the monotonicity of balanced budget multiplier under imperfect competition ´ J. Torregrosa* Ramon Universidad de Salamanca, ‘‘ Miguel de Unamuno’’, Ed FES Campus 37008 Salamanca, Spain Received 15 May 1997; accepted 11 February 1998
Abstract This paper presents a counterexample to the wellknown monotonicity result of Mankiw (1988) [Mankiw, N.G., 1988. Imperfect competition and the Keynesian cross. Economics Letters 26, 7–13]. In particular by considering taxes on labor income rather than lumpsum taxes it obtains a negative relation between the magnitude of the balanced budget multiplier and marketpower. 1998 Elsevier Science S.A. Keywords: Balanced budget multiplier; Imperfect competition; Distortionary taxation JEL classification: E12; L16; E62
1. Introduction Market-power foundations of macroeconomics point out that the inefficiency arising from imperfect competition allows us to explain some Keynesian features even in fully flexible price models.1 In this context Dixon (1987); Mankiw (1988) provide examples with Cobb-Douglas preferences, constant marginal costs, constant markup pricing and lumpsum taxes. In these examples the balanced budget multiplier rises monotonically in respect to market power. The question this paper seeks to answer is whether this monotonic behavior of the balanced budget multiplier remains unchanged under distortionary tax schemes. In an early paper Molana and Moutos (1992) without studying the monotonicity issue found that the balanced budget multiplier can be zero and even negative under tax rates on consumers’ income. Starting from here this paper studies the monotonicity of the balanced budget multiplier in relation with market power under a labor supply tax rate providing a counterexample to Mankiw’s monotonicity result. 2. The economy Following Mankiw (1988) I consider an economy with two goods: leisure which is used as *Tel.: 134 23294640; fax: 134 23294686; e-mail: [email protected] 1 See Silvestre (1993); Dixon and Rankin (1994). 0165-1765 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00055-X
R. J. Torregrosa / Economics Letters 59 (1998) 331 – 335
332
numeraire and a single produced commodity; m consumers that can be aggregated into a single representative consumer; an industry formed by n noncompetitive firms and a government.
2.1. Consumers The representative consumer maximizes a Cobb-Douglas utility function which depends on the amount C consumed of the single produced commodity and of the amount L of leisure consumed u(C, L) 5 a ln C 1 (1 2 a ) ln L.
(1)
Denote by w the endowment of time then w2L is the labor income. Considering a labor income tax rate t[1(0, 1) levied by the government, total after-tax income is (12t)(w2L)1 P where P are the firms’ profits. Thus the consumer’s budget constraint can be written as PC 5 (1 2 t)(w 2 L) 1 P where P is the price of the produced commodity. The consumer’s optimal choice allows us to express the consumption function as PC 5 a [(1 2 t)w 1 P ]
(2)
where a can be interpreted as the marginal propensity to consume. On the other hand labor supply is given by w 2 L 5 a w 2 (1 2 a )P /(1 2 t).
(3)
2.2. Government The tax revenue raised by the government is given by T5t(w2L) which is used to purchase the produced commodity and to hire government workers. Let us call as G the public spending and W the amount of labor demanded by the government. Then the government budget constraint requires: t(w 2 L) 5 G 1 W. Total expenditure on the produced commodity is given by Y 5 PC 1 G.
(4)
Substituting Eq. (2) into Eq. (4) we have Y 5 a ((1 2 t)w 1 P ) 1 G
(5)
i.e. total expenditure depends positively on profits and government purchases and negatively on taxes.
R. J. Torregrosa / Economics Letters 59 (1998) 331 – 335
333
2.3. Firms There are n firms producing an aggregated amount Q of the single commodity. The industry takes total expenditure as given.2 Hence it faces a unit elastic demand function Q 5 Y /P
(6)
Firms have the same constant returns to scale technology. The cost function of each firm in terms of the numeraire is C(q) 5 cq, where q is the quantity produced by the representative firm. We assume that firms are profit maximizers 3 and that play a oligopoly game. The result of the game determines the profit margin m as
m 5 (P 2 c) /P. We can interpret m as a measure of marketpower: when m 51 /n firms behave a la Cournot; when m 50 firms behave as Bertrand oligopolists (perfect competition); and finally when m falls to 1, firms behave as a monopolist (perfect collusion).4 Therefore m only depends on conjectural variations and n. The relationship between industry output and total expenditure is Q 5 [(1 2 m ) /c]Y, and aggregate profits can be written as a function of total expenditure and the profit margin as
P 5 m Y.
(7)
2.4. Feasibility condition in labor market Using Eqs. (4) and (6) the quantity of labor demanded (both firms and government) can be written as: W 1 cQ 5 [t(w 2 L) 2 G] 1 [Y 2 P ] and using Eqs. (3) and (5) we obtain W 1 cQ 5 t[a w 2 (1 2 a )P /(1 2 t)] 2 G 1 a ((1 2 t)w 1 P ) 1 G 2 P, W 1 cQ 5 a w(1 2 a )P /(1 2 t)
(8)
which is the labor supply function given by Eq. (3). Hence the feasibility condition in labor market 2
One might object this assumption is not reasonable because expenditure depends on industry profits. The model could be amended however to include a continuum of industries; the demand function of each industry would depend on aggregate profits. 3 Let us dispense the fact that if the price is the same for ownersconsumers and for consumers who are not owners then the goal of the firm might not be the profit maximization. 4 The demand function given by eq. (5) implies that the monopoly must be considered as a limiting case hence m [[0, 1).
R. J. Torregrosa / Economics Letters 59 (1998) 331 – 335
334
holds. Notice that from Eq. (8) the feasibility condition implies that in equilibrium a w2W 5 cQ 1 (12 a ) P /(12t).0.
3. The balanced budget multiplier (BBM) In order to display the shape of the BBM it is necessary to characterize the equilibrium total expenditure and the equilibrium public spending. Using Eqs. (3)–(5), (7) we obtain (1 2 t)[a w 2 W] Y 5 ]]]]], 1 2 t 1 m (t 2 a )
(9)
and G 5 t[a w 2 (1 2 a )P /(1 2 t)] 2 W.
(10)
By applying the implicit function theorem to the system of equations formed by Eq. (9) and Eq. (10) we can derive the BBM as a function of market-power m (1 2 a )m [a w 2 W] dY BBM( m ) ; ] 5 2 ]]]]]]]]]]]]]]]] . dG a w[1 2 t 1 m (t 2 a )] 2 2 (1 2 a )( m 2 am 2 )[a w 2 W]
(11)
It is easy to see that BBM(0)50 and limm →1 BBM( m )5 2[a w2W] /(12 a )W ,0. Also differentiating Eq. (11) with respect to m and manipulating yields dBBM( m ) (1 2 a )a [a w 2 W][w(1 2 am 2 )(1 2 2t) 1 wt 2 (1 2 m 2 ) 1 (1 2 a )m 2W] ]]]] 5 2 ]]]]]]]]]]]]]]]]]]]]] . dm [a w[1 2 t 1 m (t 2 a )] 2 2 m (1 2 a )(1 2 am )[a w 2 W]] 2
(12)
A sufficient (but far from necessary) condition for Eq. (12) to be negative is that t#1 / 2 which implies that the BBM is monotonically decreasing with market power.
4. Conclusion As we have seen Mankiw’s monotonicity result is not robust to different specifications of the tax scheme. In fact under a labor supply tax opposite results are reached. The reason for this is that an increase in this type of taxes distorts relative prices with the consequence of reducing output in greater proportion than the increase in demand due to government expenditure. Consequently total expenditure falls in relation to public expenditure. In the example this effect has also proved to be greater when market power increases. This is because the higher the degree of competitiveness the lower the pretax profits hence the lower the fall in private consumption.
R. J. Torregrosa / Economics Letters 59 (1998) 331 – 335
335
Acknowledgements ´ G. Fernandez ´ ´ ´ and an I am grateful to R. Caminal, L. Corchon, de Cordoba, T. Kehoe, B. Valdes, anonymous referee for their helpful comments and suggestions. I am also grateful for the financial support of the Spanish Ministry of Education (CICYT project PB93-0940) and the Instituto Valenciano de Investigaciones Economicas. The usual caveat applies.
References Dixon, H., 1987. A simple model of imperfect competition with walrasian features. Oxford Economics Papers 39, 134–179. Dixon, H., Rankin, N., 1994. Imperfect competition and macroeconomics: a survey. Oxford Economics Papers 46, 171–199. Mankiw, N.G., 1988. Imperfect competition and the Keynesian cross. Economics Letters 26, 7–13. Molana, H., Moutos, T., 1992. A note on taxation, imperfect competition and the balanced budget multiplier. Oxford Economic Papers 44, 68–74. Silvestre, J., 1993. The marketpower foundations of macroeconomic policy. Journal of Economic Literature 21, 105–141.