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Taste for diversity and the optimality of economic growth
Economics Letters 58 (1998) 291–295
Taste for diversity and the optimality of economic growth Henri L.F. de Groot*, Richard Nahuis Tilburg University, Department of Economics and CenER, P.O. Box 90153, 5000 LE Tilburg, The Netherlands Received 3 July 1997; accepted 14 November 1997
Abstract We disentangle the taste for diversity in consumption present in endogenous growth models with expanding product variety from the elasticity of substitution between consumption goods. This seriously affects the welfare analysis of the model. If taste for diversity is relatively small, the market rate of growth may be suboptimally large. 1998 Elsevier Science S.A. Keywords: Taste for diversity; Social welfare; Endogenous growth JEL classification: D43; D60; O41
1. Introduction In their seminal work on endogenous growth and product variety, Grossman and Helpman (1991); (chapter 3) argue that the market provides insufficient incentives for investment in R&D from a social point of view. They derive this result, using the concept of ‘taste for diversity’ as has been developed by Dixit and Stiglitz (1977) and Spence (1976). The specific way of modelling the taste for diversity in this work has recently been shown to be crucial for a welfare evaluation (Benassy, 1996). The arbitrary way in which the elasticity of substitution and the taste for diversity are related yields results that do not generalize when this strict relation is broken down. In this paper, we will use a convenient way of disentangling the taste for diversity effect from the elasticity of substitution (capturing the market power of producers of the consumption goods). We thereby generalize the Dixit–Stiglitz specification of the consumption index as it is also used by Grossman and Helpman (1991). The rate of growth that results in the market is then no longer necessarily lower than the socially optimal rate of growth. An optimal R&D policy may hence require the taxation of R&D activities. This result will prevail if the taste for diversity is small relative to the elasticity of substitution between consumption goods.
2. A slight modification of the ‘expanding product variety’ model We take the expanding product variety model by Grossman and Helpman (1991), (chapter 3). *Corresponding author. Tel.: 0031 13 4662833; fax: 0031 13 4663042; e-mail: [email protected] 0165-1765 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0165-1765( 98 )00005-6
H.L.F. de Groot, R. Nahuis / Economics Letters 58 (1998) 291 – 295
292
Consumers maximize their intertemporal utility subject to a budget constraint. Intertemporal utility looks like `
E
U0 5 e
`
2r t
log D(t) dt, where D 5 n
0
s
3E 4 1 ] n
1 /a
a j
x dj
.
(1)
0
r is the subjective discount rate, n is the number of (unique) varieties / brands of the consumption good available in the economy (indexed j), and x j is the consumed quantity of the consumption good of brand j. The elasticity of substitution between any two consumption goods ´ equals 1 /(12 a ).1. The parameter s captures the taste for diversity. Under symmetry, we can write the consumption index D as n s 21 nx. So as long as s .1, consumers prefer diversity. Where we differ from Grossman and Helpman (1991) is in the specification of the consumption index D.1 In Grossman and Helpman, the parameter a both captures the elasticity of substitution and the taste for diversity. Our specification boils down to the Grossman and Helpman specification by taking s equal to 1 /a. Consumers maximize their intertemporal utility (Eq. (1)) subject to a dynamic budget constraint 2 n
E
~ 5 r(t)A(t) 1 w(t)L(t) 2 E(t) where E 5 x j pj dj, A(t)
(2)
0
in which A is financial wealth owned by the households, w is the wage rate, and L is (exogenous) labour supply. Standard optimization of this optimal control problem, taking consumption expendi´ tures as numeraire (E51) yields a Ramsey rule r(t) 5 r,
(3)
and a downward sloping demand curve for each brand of the consumption good 3 ´ p2 j x j 5 ]]] . n
Ep
12 ´ i
(4)
di
0
Producers of the differentiated goods produce one unit of the differentiated good with one unit of labour (x j 5l j ). Maximizing operating profits (pj 5x j pj 2wl j ) subject to the demand for variety j (Eq. (4)) gives rise to mark-up pricing w pj 5 ]. a
1
(5)
This specification is borrowed from Broer and Heijdra (1996). It was already present in Ethier (1982) and, according to Benassy (1996), in a working paper version of Dixit and Stiglitz (1977). The importance of this specification for welfare results in endogenous growth models has to our knowledge not been spelled out before. 2 ~ A dot above a variable indicates a derivative with respect to time (A;dA / dt). 3 Where it leads to no confusion, we drop time indices.
H.L.F. de Groot, R. Nahuis / Economics Letters 58 (1998) 291 – 295
293
Using the fact that in symmetry E5nxp51, we can write operating profits as 12a p 5 ]]. n
(6)
In the R&D sector of the model, new blueprints for consumption goods are created according to nl n n~ 5 ], a
(7)
where l n is labour employed in R&D, and n /a is labour productivity of R&D labour. As labour productivity increases with the number of patents invented in the past, growth will not be exhausted. Perfect competition prevails in the R&D sector so the price of a blueprint (v) will equal wa v 5 ]. n
(8)
The model is closed by imposing labour market equilibrium n
E
L 5 l j dj 1 l n ,
(9)
0
and capital market equilibrium according to which
p 1 v~ 5 rv.
(10)
This is the no-arbitrage condition. The return on investing in R&D which equals profits in the consumption goods sector plus the change in the value of the patent has to equal the return on a riskless loan. We can now solve for the steady state rate of growth (the rate of innovation, g5n~ /n) which equals 4 L g 5 (1 2 a ) ] 2 ar. a
(11)
The growth rate in the market equilibrium exactly equals the growth rate derived by Grossman and Helpman. The parameter measuring the taste for diversity does not show up in this solution. This reflects the fact that the market does not take into account the externality resulting from diversity. In the next section, we derive the social optimum and compare it with the market equilibrium. 4
~ /n51 /p1na ~ /n. So g;n~ /n5L /a2 a /vn (using Eqs. (5) and (8)). The According to labour market equilibrium, L5nl j 1na no-arbitrage condition can be written as v~ /v 5 r 2 p /v. Using Eqs. (3) and (6), we can then write v~ /v 5 r 2 (1 2 a ) /nv. In a steady state with a constant rate of growth, it holds that v~ /v 5 2 n~ /n. Solving this system of three equations yields the steady state rate of growth.
H.L.F. de Groot, R. Nahuis / Economics Letters 58 (1998) 291 – 295
294
3. The market versus the social optimum In the social optimum, the planner maximizes `
E
U0 5 e 2 r t [(s 2 1) log n(t) 1 log X(t)] dt,
(12)
0
subject to the resource constraint an~ 1 X 5 L.
(13)
Here an~ is labour employed for the development of new patents and X(5nl j ) is labour employed for the production of consumption goods. The current value Hamiltonian for this problem is
F
G
(L 2 X)n H 5 (s 2 1) log n 1 log X 1 u ]]] . a
(14)
Here u is the shadow value of variety. Solving this optimal control problem gives the optimal rate of growth 5 L r g* 5 ] 2 ]]. a s 21
(15)
An increase in the taste for diversity raises the optimal rate of growth. Note that the elasticity of substitution does not show up in the socially optimal rate of growth. Fig. 1 depicts the market rate of growth and the socially optimal rate of growth as a function of the taste for diversity. Whether the market generates the socially optimal rate of growth crucially depends on the taste for diversity. In the specific case considered by Grossman and Helpman, the market provides too little incentive for investments in R&D ( g,g* at s 51 /a ). To explain these results we consider the three effects associated with increased diversity. First, there is a consumer surplus effect which is external to the firm and relates to the taste for diversity. Secondly, there is a profit destruction effect deriving from the fact that the introduction of a new brand of a consumption good diminishes the market share of existing producers. Finally, there is an intertemporal knowledge spill-over effect. By introducing a new brand, a firm attributes to the common stock of knowledge and thereby reduces the future cost of inventing a new brand. In the specific case studied by Grossman and Helpman, the profit destruction effect and the consumer surplus effect exactly cancel. What remains is the intertemporal spill-over effect. The market consequently provides too little growth. When generalizing the utility function, the profit destruction effect and the consumer surplus effect no longer need to cancel. If the taste for diversity effect is very small (s is small) relative to the mark-up, the profit destruction effect exceeds the consumer surplus and the spill-over effect (to the left of point q). Growth is then suboptimally large in the market economy. R and D activities should in this case be taxed. 5
The first order conditions corresponding to this problem are 1 /X5u n /a and (s 21) /nu 1(L2X) /a5 r 2 u~ /u. In a steady state with a constant allocation of labour, it holds that u~ /u 5 2 n~ /n. The combination of these equations yields the socially optimal rate of growth.
H.L.F. de Groot, R. Nahuis / Economics Letters 58 (1998) 291 – 295
295
Fig. 1. The market versus the socially optimal rate of growth.
4. Conclusion We generalized the expanding product variety model developed by Grossman and Helpman by disentangling the taste for diversity from the elasticity of substitution between any pair of consumption goods. The welfare results derived by Grossman and Helpman turned out not to generalize. It may well be the case that the rate of growth sustained by the market is too high from a social point of view.
Acknowledgements Valuable suggestions were made by Sjak Smulders. The usual disclaimer applies.
References Benassy, J.P., 1996. Taste for variety and optimum production patterns in monopolistic competition. Economics Letters 52, 41–47. Broer, D.P., Heijdra, B.J., 1996. The intergenerational distribution effects of the investment tax credit under monopolistic competition. OCfEB Research Memorandum 9603, Rotterdam. Dixit, A.K., Stiglitz, J.E., 1977. Monopolistic competition and optimum product diversity. American Economic Review 67, 297–308. Ethier, W.J., 1982. National and international returns to scale in the modern theory of international trade. American Economic Review 72, 389–405. Grossman, G.M., Helpman, E., 1991. Innovation and Growth in the Global Economy. MIT Press, Cambridge, MA. Spence, M., 1976. Product selection, fixed costs, and monopolistic competition. Review of Economic Studies 43, 217–235.
Taste for diversity and the optimality of economic growth Henri L.F. de Groot*, Richard Nahuis Tilburg University, Department of Economics and CenER, P.O. Box 90153, 5000 LE Tilburg, The Netherlands Received 3 July 1997; accepted 14 November 1997
Abstract We disentangle the taste for diversity in consumption present in endogenous growth models with expanding product variety from the elasticity of substitution between consumption goods. This seriously affects the welfare analysis of the model. If taste for diversity is relatively small, the market rate of growth may be suboptimally large. 1998 Elsevier Science S.A. Keywords: Taste for diversity; Social welfare; Endogenous growth JEL classification: D43; D60; O41
1. Introduction In their seminal work on endogenous growth and product variety, Grossman and Helpman (1991); (chapter 3) argue that the market provides insufficient incentives for investment in R&D from a social point of view. They derive this result, using the concept of ‘taste for diversity’ as has been developed by Dixit and Stiglitz (1977) and Spence (1976). The specific way of modelling the taste for diversity in this work has recently been shown to be crucial for a welfare evaluation (Benassy, 1996). The arbitrary way in which the elasticity of substitution and the taste for diversity are related yields results that do not generalize when this strict relation is broken down. In this paper, we will use a convenient way of disentangling the taste for diversity effect from the elasticity of substitution (capturing the market power of producers of the consumption goods). We thereby generalize the Dixit–Stiglitz specification of the consumption index as it is also used by Grossman and Helpman (1991). The rate of growth that results in the market is then no longer necessarily lower than the socially optimal rate of growth. An optimal R&D policy may hence require the taxation of R&D activities. This result will prevail if the taste for diversity is small relative to the elasticity of substitution between consumption goods.
2. A slight modification of the ‘expanding product variety’ model We take the expanding product variety model by Grossman and Helpman (1991), (chapter 3). *Corresponding author. Tel.: 0031 13 4662833; fax: 0031 13 4663042; e-mail: [email protected] 0165-1765 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0165-1765( 98 )00005-6
H.L.F. de Groot, R. Nahuis / Economics Letters 58 (1998) 291 – 295
292
Consumers maximize their intertemporal utility subject to a budget constraint. Intertemporal utility looks like `
E
U0 5 e
`
2r t
log D(t) dt, where D 5 n
0
s
3E 4 1 ] n
1 /a
a j
x dj
.
(1)
0
r is the subjective discount rate, n is the number of (unique) varieties / brands of the consumption good available in the economy (indexed j), and x j is the consumed quantity of the consumption good of brand j. The elasticity of substitution between any two consumption goods ´ equals 1 /(12 a ).1. The parameter s captures the taste for diversity. Under symmetry, we can write the consumption index D as n s 21 nx. So as long as s .1, consumers prefer diversity. Where we differ from Grossman and Helpman (1991) is in the specification of the consumption index D.1 In Grossman and Helpman, the parameter a both captures the elasticity of substitution and the taste for diversity. Our specification boils down to the Grossman and Helpman specification by taking s equal to 1 /a. Consumers maximize their intertemporal utility (Eq. (1)) subject to a dynamic budget constraint 2 n
E
~ 5 r(t)A(t) 1 w(t)L(t) 2 E(t) where E 5 x j pj dj, A(t)
(2)
0
in which A is financial wealth owned by the households, w is the wage rate, and L is (exogenous) labour supply. Standard optimization of this optimal control problem, taking consumption expendi´ tures as numeraire (E51) yields a Ramsey rule r(t) 5 r,
(3)
and a downward sloping demand curve for each brand of the consumption good 3 ´ p2 j x j 5 ]]] . n
Ep
12 ´ i
(4)
di
0
Producers of the differentiated goods produce one unit of the differentiated good with one unit of labour (x j 5l j ). Maximizing operating profits (pj 5x j pj 2wl j ) subject to the demand for variety j (Eq. (4)) gives rise to mark-up pricing w pj 5 ]. a
1
(5)
This specification is borrowed from Broer and Heijdra (1996). It was already present in Ethier (1982) and, according to Benassy (1996), in a working paper version of Dixit and Stiglitz (1977). The importance of this specification for welfare results in endogenous growth models has to our knowledge not been spelled out before. 2 ~ A dot above a variable indicates a derivative with respect to time (A;dA / dt). 3 Where it leads to no confusion, we drop time indices.
H.L.F. de Groot, R. Nahuis / Economics Letters 58 (1998) 291 – 295
293
Using the fact that in symmetry E5nxp51, we can write operating profits as 12a p 5 ]]. n
(6)
In the R&D sector of the model, new blueprints for consumption goods are created according to nl n n~ 5 ], a
(7)
where l n is labour employed in R&D, and n /a is labour productivity of R&D labour. As labour productivity increases with the number of patents invented in the past, growth will not be exhausted. Perfect competition prevails in the R&D sector so the price of a blueprint (v) will equal wa v 5 ]. n
(8)
The model is closed by imposing labour market equilibrium n
E
L 5 l j dj 1 l n ,
(9)
0
and capital market equilibrium according to which
p 1 v~ 5 rv.
(10)
This is the no-arbitrage condition. The return on investing in R&D which equals profits in the consumption goods sector plus the change in the value of the patent has to equal the return on a riskless loan. We can now solve for the steady state rate of growth (the rate of innovation, g5n~ /n) which equals 4 L g 5 (1 2 a ) ] 2 ar. a
(11)
The growth rate in the market equilibrium exactly equals the growth rate derived by Grossman and Helpman. The parameter measuring the taste for diversity does not show up in this solution. This reflects the fact that the market does not take into account the externality resulting from diversity. In the next section, we derive the social optimum and compare it with the market equilibrium. 4
~ /n51 /p1na ~ /n. So g;n~ /n5L /a2 a /vn (using Eqs. (5) and (8)). The According to labour market equilibrium, L5nl j 1na no-arbitrage condition can be written as v~ /v 5 r 2 p /v. Using Eqs. (3) and (6), we can then write v~ /v 5 r 2 (1 2 a ) /nv. In a steady state with a constant rate of growth, it holds that v~ /v 5 2 n~ /n. Solving this system of three equations yields the steady state rate of growth.
H.L.F. de Groot, R. Nahuis / Economics Letters 58 (1998) 291 – 295
294
3. The market versus the social optimum In the social optimum, the planner maximizes `
E
U0 5 e 2 r t [(s 2 1) log n(t) 1 log X(t)] dt,
(12)
0
subject to the resource constraint an~ 1 X 5 L.
(13)
Here an~ is labour employed for the development of new patents and X(5nl j ) is labour employed for the production of consumption goods. The current value Hamiltonian for this problem is
F
G
(L 2 X)n H 5 (s 2 1) log n 1 log X 1 u ]]] . a
(14)
Here u is the shadow value of variety. Solving this optimal control problem gives the optimal rate of growth 5 L r g* 5 ] 2 ]]. a s 21
(15)
An increase in the taste for diversity raises the optimal rate of growth. Note that the elasticity of substitution does not show up in the socially optimal rate of growth. Fig. 1 depicts the market rate of growth and the socially optimal rate of growth as a function of the taste for diversity. Whether the market generates the socially optimal rate of growth crucially depends on the taste for diversity. In the specific case considered by Grossman and Helpman, the market provides too little incentive for investments in R&D ( g,g* at s 51 /a ). To explain these results we consider the three effects associated with increased diversity. First, there is a consumer surplus effect which is external to the firm and relates to the taste for diversity. Secondly, there is a profit destruction effect deriving from the fact that the introduction of a new brand of a consumption good diminishes the market share of existing producers. Finally, there is an intertemporal knowledge spill-over effect. By introducing a new brand, a firm attributes to the common stock of knowledge and thereby reduces the future cost of inventing a new brand. In the specific case studied by Grossman and Helpman, the profit destruction effect and the consumer surplus effect exactly cancel. What remains is the intertemporal spill-over effect. The market consequently provides too little growth. When generalizing the utility function, the profit destruction effect and the consumer surplus effect no longer need to cancel. If the taste for diversity effect is very small (s is small) relative to the mark-up, the profit destruction effect exceeds the consumer surplus and the spill-over effect (to the left of point q). Growth is then suboptimally large in the market economy. R and D activities should in this case be taxed. 5
The first order conditions corresponding to this problem are 1 /X5u n /a and (s 21) /nu 1(L2X) /a5 r 2 u~ /u. In a steady state with a constant allocation of labour, it holds that u~ /u 5 2 n~ /n. The combination of these equations yields the socially optimal rate of growth.
H.L.F. de Groot, R. Nahuis / Economics Letters 58 (1998) 291 – 295
295
Fig. 1. The market versus the socially optimal rate of growth.
4. Conclusion We generalized the expanding product variety model developed by Grossman and Helpman by disentangling the taste for diversity from the elasticity of substitution between any pair of consumption goods. The welfare results derived by Grossman and Helpman turned out not to generalize. It may well be the case that the rate of growth sustained by the market is too high from a social point of view.
Acknowledgements Valuable suggestions were made by Sjak Smulders. The usual disclaimer applies.
References Benassy, J.P., 1996. Taste for variety and optimum production patterns in monopolistic competition. Economics Letters 52, 41–47. Broer, D.P., Heijdra, B.J., 1996. The intergenerational distribution effects of the investment tax credit under monopolistic competition. OCfEB Research Memorandum 9603, Rotterdam. Dixit, A.K., Stiglitz, J.E., 1977. Monopolistic competition and optimum product diversity. American Economic Review 67, 297–308. Ethier, W.J., 1982. National and international returns to scale in the modern theory of international trade. American Economic Review 72, 389–405. Grossman, G.M., Helpman, E., 1991. Innovation and Growth in the Global Economy. MIT Press, Cambridge, MA. Spence, M., 1976. Product selection, fixed costs, and monopolistic competition. Review of Economic Studies 43, 217–235.