- Email: [email protected]
Does the market deliver the right technology?
Economics Letters 150 (2017) 95–98
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Does the market deliver the right technology? Eric Toulemonde ∗ Université de Namur, CERPE, 8 Rempart de la Vierge, B-5000 Namur, Belgium
highlights • • • • •
Firms choose the technology that maximizes their profit. The size and the mass of firms/varieties vary with the technology choice. Firms do not internalize the effect of the technology choice on the mass of varieties. Under monopolistic competition, the mass of varieties affects welfare. Under conditions identified in the paper, firms choose the wrong technology.
article
info
Article history: Received 18 August 2016 Received in revised form 2 November 2016 Accepted 11 November 2016 Available online 17 November 2016
abstract We show that the market does not systematically deliver the right technology under monopolistic competition. (i) Firms might rush on large-scale technology, pushing to the exit many desirable varieties produced by small firms. (ii) Firms might shun large-scale technology, though that technology would benefit the society through lower prices. (iii) A bias towards small-scale technology in some stage of development, and a bias towards large-scale technology in another stage is also a possibility. © 2016 Elsevier B.V. All rights reserved.
JEL classification: D1 D2 D4 L1 Keywords: Welfare Monopolistic competition Productivity Technology
1. Introduction Consider two technologies: a large-scale technology with low marginal costs and high fixed costs, and a small-scale technology with high marginal costs and low fixed costs. Similarly, consider an old technology that can be upgraded through a fixed investment in R&D that reduces variable costs of production, as in Vives (2008). Is the technology chosen by firms the best for the society? This is a recurring question in our societies. Some argue that we should favor small-scale farming instead of industrial farming, artisanal products instead of industrial products, corner shops instead of supermarkets, independent booksellers instead of large online bookstore, independent taxi drivers instead of Uber, authors’
∗
Fax: +32 724840. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.econlet.2016.11.013 0165-1765/© 2016 Elsevier B.V. All rights reserved.
movies instead of Hollywood blockbusters, . . . . Others argue that policies that foster large-scale technology allow to enhance production and to bring more prosperity to citizens because of economies of scale. Citizens with some economic knowledge may think that firms will adopt the technology that is the most desired by the society, because otherwise they would be thrown out of the market. What to think of those presumptions? Since perfect competition is incompatible with fixed costs, a precise answer to this question requires considering imperfectly competitive markets. Therefore, we consider monopolistic competition in which firms sell differentiated varieties to consumers who love variety. On the one hand, lower marginal costs of the largescale technology push down prices, benefiting the consumers. On the other hand, its higher fixed costs reduce firms’ profitability and thus they contribute to decrease the number of varieties, which is detrimental to consumers. The balance between the benefits and the costs depends on the fixed and marginal costs and on the preferences of consumers. We show that under constant elasticity of
96
E. Toulemonde / Economics Letters 150 (2017) 95–98
substitution (CES) preferences, firms’ technological choice coincides with consumers preferred technology. By contrast, the market does not systematically deliver the right technology under nonCES preferences: (i) Firms might rush on large-scale technology, pushing to the exit many desirable varieties produced by small firms. (ii) Firms might shun large-scale technology, though that technology would benefit the society through lower prices. (iii) A bias towards small-scale technology in some stage of development, and a bias towards large-scale technology in another stage is also a possibility. Starting with Spence (1976) and Dixit and Stiglitz (1977), a large literature studies the (in)efficiency of the number of varieties delivered by the market. It is well known that the market achieves the optimum when the utility is CES. Recently, Parenti et al. (2016) prove that the market can also deliver the optimum under nonadditive and non-homothetic preferences. In general however, the market does not deliver the right number of varieties. We complement this literature by showing that the market might not only select the wrong number of varieties but also the wrong type of firms when preferences are additive. Ohkawa et al. (2005) is probably the paper with the closest focus to ours. They show that under asymmetric Cournot oligopoly the market also selects the wrong technology in the long run. They focus on homogeneous products whereas we emphasize the role played by the love for variety of diversified products. 2. The model We consider an economy endowed with L identical workers whose wage is chosen as the numeraire. There is a continuum N of firms, each producing a single variety indexed by i ∈ [0, N]. Each worker spends her entire income on the continuum of horizontally differentiated varieties. 2.1. Preferences As in Zhelobodko et al. (2012), we build on additive preferences.1 Each consumer chooses her consumption xi , to maximize her utility given her unit wage and the prices pi : N
u (xi ) di s.t.
max xi
N
pi xi di = 1 ∀i
0
q∗i /L u′′ q∗i /L + u′ q∗i /L
ru (x) ≡ −
xu′′ (x) u′ ( x )
> 0.
where λ =
− ci qi − fi .
cs
ru q∗s /L
⇐⇒ ru q∗s /L =
fs cs q∗s + fs
.
(5)
1 − ru q∗s /L
∗
cs
u′ q∗s /L
q∗ = s u′ q∗s /L ru q∗s /L . fs
(6)
Second, by (2), we find the optimal price and the number of firms:
Firm i faces a fixed cost fi and produces qi units of the variety i at constant marginal costs ci . It sells its output to the L identical consumers: qi = Lxi . Hence, pi = u′ (qi /L) /λ and firm i′ s profit is
λ
fs 1 − ru q∗s /L
q∗s =
λs =
2.2. Technology choice in the market solution
u (qi /L)
(4)
(2)
Under monopolistic competition, each firm is negligible to the market and treats the Lagrange multiplier λ as an exogenous aggregate variable.
′
At the equilibrium, the share of fixed costs in the total costs is thus equal to the relative love for variety. We also find the equilibrium value of the Lagrange multiplier:
0
where a star denotes the equilibrium value. Note that the second order condition requires that the left hand side of those expressions is decreasing in qi , that is, − (qi /L) ru′ < ru (1 − ru ). Therefore, firms with lower marginal costs ci must produce more in order to restore the equality in (4). We now limit the technology set to two technologies: the large-scale and the small-scale, denoted with subscripts l and s respectively. The large-scale technology entails high fixed costs fl and low marginal costs cl whereas the small-scale technology is characterized by smaller fixed costs fs < fl and higher marginal costs cs > cl . By symmetry, firms with the same technology sell the same quantities at the same prices. This allows us to replace the subscript i with the subscript t ∈ {l, s} that identifies the firm with its technology. To be clear, we consider two prices, pl and ps , and two possible output, ql and qs . All firms face the same aggregate conditions; they thus share the same Lagrange multiplier. At a free entry equilibrium, all firms will choose the same technology. This claim is easily proved by contradiction, as in general, there is no solution to a system of four equations with three unknowns. An equilibrium in which the two technologies are used would indeed require to find the three unknowns ql , qs and λ that simultaneously satisfy two zero-profit conditions (πs (qs ) = 0 and πl (ql ) = 0) and two first order conditions ((4) with i = l and (4) with i = s). Without loss of generality, let us characterize an equilibrium in which all firms adopt the small-scale technology. Let λ∗s denote the Lagrange multiplier faced by all firms when they all adopt technology s. First, the aggregate statistics λ∗s adjusts to make ∗ πs qs = 0 in (3). Equating this Lagrange multiplier with the value of the same multiplier found in (4) (with i = s), we find the output of a typical firm:
N
xi u′ (xi ) di.
(1)
The inverse demand function is pi = u′ (xi ) /λ
= λ∗ ci ⇔ 1 − ru q∗i /L u′ q∗i /L = λ∗ ci
0
where u (.) is thrice continuously differentiable, strictly increasing and strictly concave. The relative love for variety is
πi (qi ) =
Firm i chooses its production qi to maximize its profit. At the equilibrium,
(3)
1 See Parenti et al. (2016) for a discussion on the properties of additive and homothetic preferences.
p∗s =
λ∗s =
cs 1 − ru q∗s /L
and
Ns∗ q∗s ′ ∗ L u qs /L ⇐⇒ Ns∗ = ru q∗s /L . L fs
(7)
Third, by ensuring that a firm makes losses if it deviates from the equilibrium by choosing the large-scale technology, we find the following lemma, which will be further discussed in the next section.
E. Toulemonde / Economics Letters 150 (2017) 95–98
Lemma 1. Let εu(x) denote the elasticity of u (x) with respect to x. Firms prefer the small-scale technology if and only if u q∗l /L ru q∗l /L εu(q∗l /L)
u q∗s /L ru q∗s /L εu(q∗s /L)
(ii)
fs
.
(8)
(i)
fl
<
(iii)
(iv)
The equilibrium output is larger when firms choose the large-scale technology, q∗l > q∗s . Proof. See a technical appendix available upon request (see Appendix A). 2.3. Market vs planner’s preferred technology In this section, we consider a benevolent planner who has a single instrument at its disposal: a regulation prohibiting the use of one technology. We keep the assumption that firms are free to enter and to leave the market; firms also choose their output and prices freely, but they must use the technology that is imposed by the planner. Free entry of firms drives profits to zero. Hence, the planner does not care about the profits and she seeks to maximize N the utility 0 u (xi ) di, which, by symmetry, is equal either to Ns u (qs /L) or Nl u (ql /L). We know that if firms adopt the smallscale technology, their number is given by (7). If instead firms adopt the large-scale technology, their number is given by a similar expression in which s is replaced by l. The technology that maximizes welfare is given in the next proposition. Lemma 2. A benevolent planner prefers the small-scale technology if and only if u q∗l /L ru q∗l /L
<
u q∗s /L ru q∗s /L
(i)
fl f
⇐⇒
s
q∗s cs + fs u q∗s /L
(iii)
(ii)
<
q∗l cl + fl u q∗l /L
.
(9)
The right-hand-side shows that the planner prefers the technology that minimizes the cost per unit of utility. As shown by the left-hand-side, three factors guide the planner. To understand those factors, recall that there are two components in the N utility 0 u (xi ) di. (i) First, u (xi ) is the utility derived from the consumption of xi units of variety i. Accordingly, the planner prefers the large-scale technology because it is the technology that allows firms to produce more. (ii) The preference for variety is taken into account by integrating u (xi ) over the N varieties. As shown in (9), the planner prefers the technology that delivers the highest value of the relative love for variety at the equilibrium. Note that under CES preferences, ru (x) is independent of x. (iii) The planner prefers the cheapest technology. It is the balance between those three factors that determines the planner’s optimal choice. Obviously, there might be a discrepancy between the firms’ preferred technology and the planner’s choice since (8) and (9) are not identical. Which are the factors that induce firms to adopt the small-scale technology? Lemma 1 shows that the same three factors (i)–(iii) that affect the planner’s choice, influence the firms’ choices. A fourth factor is at work here (iv): the elasticity of u (x) with respect to x. Suppose εu(q∗s /L) > εu(q∗ /L) . An increase of one percent of l the output has a higher relative impact on the utility of consuming variety i when all firms use the small-scale technology than when they use the large-scale technology. Firms can capture part of this
97
increase through higher prices, which induces them to adopt the small-scale technology. The opposite holds if εu(q∗s /L) < εu(q∗ /L) . l The discrepancy between the firms’ preferred technology and the planner’s choice is confirmed by the next proposition. Proposition 3. Consider the planner’s preferred technology as a benchmark.(i) The market is biased towards the large-scale technology for some values of fixed costs if εu(q∗ /L) > εu(q∗s /L) .(ii) By conl trast, the market is biased towards the small-scale technology for some values of fixed costs if εu(q∗ /L) < εu(q∗s /L) . l
Because consumers love variety, result (i) might be expectable. Firms adopt the large-scale technology because of its higher profitability. So doing, they push to the exit many small firms that are highly valued by the consumers. Result (ii) shows an opposite result in which the economy is trapped into a small-scale technology that delivers too many varieties. Note that those results do not appear under CES preferences, because εu(x) is independent of x. In this special case, the market outcome always delivers the right technology. Another new result shows that, depending on the specification of preferences, it is possible that the economy is trapped into a small-scale technology at early stage of development whereas it is trapped into a large-scale technology at a later stage . . . . But the opposite result can also be found with different preferences. Note that Vives (1999) and Kühn and Vives (1999) define the degree of preference for variety as 1 − εu(x) .2 In their opinion, an increasing degree of preference for variety can be considered as the normal case, meaning that εu(x) would be a decreasing function. Bykadorov et al. (2015) go in the same direction. Hence, the bias towards the small-scale technology might be more plausible. We now illustrate our results with four examples (a technical appendix is available upon request for the detailed computations, Appendix A). As a first example, consider the utility u (x) = γ x + xβ where γ ≥ 0 and β ∈ (0, 1). CES preferences arise as a special case, if γ = 0. It is readily checked that εu(q∗ /L) ≥ εu(q∗s /L) . That l is, for some values of fixed costs, prohibiting the use of the largescale technology would improve welfare. As a second example, consider the constant average risk aversion (CARA) utility with u (0) = 0: u (x) = 1 − exp (−α x) where α > 0. It is readily checked that εu(q∗ /L) < εu(q∗s /L) . Hence, there always exists some l values of fixed costs such that the market selects the small-scale technology whereas it should select the large-scale technology. Consider now a CARA utility with u (0) > 0: u (x) = β−exp (−α x), where β > 1 and α > 0. It can be shown that the preferred technology of the planner depends on the value of β and on the market size L. For a small value of β and for a small market size L, there exists values of fixed costs such that the economy is trapped into the small-scale technology whereas the planner prefers the large-scale. The opposite holds for a larger values of β and L. Finally, consider the utility u (x) = x/ ln (x + 2). It can be shown that there exists values of fixed costs such that the market selects the largescale technology whereas it should ideally choose the small-scale technology when the market is small whereas the opposite holds when the market is large. Back to some examples provided at the beginning of the introduction, we conclude that the market sometimes promotes excessively some type of firms: supermarkets instead of corner shops, Hollywood blockbusters instead of authors’ movies, . . . or the opposite.
2 When introducing a new variety, the proportion of social benefits not captured by revenues is the degree of preference for variety.
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E. Toulemonde / Economics Letters 150 (2017) 95–98
Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2016.11.013. References Bykadorov, I., Gorn, A., Kokovin, S., Zhelobodko, E., 2015. Why are losses from trade unlikely? Econom. Lett. 129, 35–38. Dixit, A.K., Stiglitz, J.E., 1977. Monopolistic competition and optimum product diversity. Amer. Econ. Rev. 67 (3), 297–308. Kühn, K.-U., Vives, X., 1999. Excess entry, vertical integration, and welfare. Rand J. Econ. 30 (4), 575–603.
Ohkawa, T., Makoto, O., Noritsugu, N., Kazuharu, K., 2005. The market selects the wrong firms in the long run. Internat. Econom. Rev. 46 (4), 1143–1165. Parenti, M., Ushchev, P., Thisse, J.-F., 2016. Toward a theory of monopolistic competition. Higher School of Economics Research Paper WP BRP 121/EC/2016. Spence, M., 1976. Product selection, fixed costs and monopolistic competition. Rev. Econom. Stud. 43 (2), 217–235. Vives, X., 2008. Innovation and competitive pressure. J. Ind. Econ. 56 (3), 419–469. Vives, X., 1999. Oligopoly Pricing - Old Ideas and New Tools. The MIT Press, Cambridge, Massachusetts. Zhelobodko, E., Kokovin, S., Parenti, M., Thisse, J.-F., 2012. Monopolistic competition: Beyond the constant elasticity of substitution. Econometrica 80 (6), 2765–2784.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Does the market deliver the right technology? Eric Toulemonde ∗ Université de Namur, CERPE, 8 Rempart de la Vierge, B-5000 Namur, Belgium
highlights • • • • •
Firms choose the technology that maximizes their profit. The size and the mass of firms/varieties vary with the technology choice. Firms do not internalize the effect of the technology choice on the mass of varieties. Under monopolistic competition, the mass of varieties affects welfare. Under conditions identified in the paper, firms choose the wrong technology.
article
info
Article history: Received 18 August 2016 Received in revised form 2 November 2016 Accepted 11 November 2016 Available online 17 November 2016
abstract We show that the market does not systematically deliver the right technology under monopolistic competition. (i) Firms might rush on large-scale technology, pushing to the exit many desirable varieties produced by small firms. (ii) Firms might shun large-scale technology, though that technology would benefit the society through lower prices. (iii) A bias towards small-scale technology in some stage of development, and a bias towards large-scale technology in another stage is also a possibility. © 2016 Elsevier B.V. All rights reserved.
JEL classification: D1 D2 D4 L1 Keywords: Welfare Monopolistic competition Productivity Technology
1. Introduction Consider two technologies: a large-scale technology with low marginal costs and high fixed costs, and a small-scale technology with high marginal costs and low fixed costs. Similarly, consider an old technology that can be upgraded through a fixed investment in R&D that reduces variable costs of production, as in Vives (2008). Is the technology chosen by firms the best for the society? This is a recurring question in our societies. Some argue that we should favor small-scale farming instead of industrial farming, artisanal products instead of industrial products, corner shops instead of supermarkets, independent booksellers instead of large online bookstore, independent taxi drivers instead of Uber, authors’
∗
Fax: +32 724840. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.econlet.2016.11.013 0165-1765/© 2016 Elsevier B.V. All rights reserved.
movies instead of Hollywood blockbusters, . . . . Others argue that policies that foster large-scale technology allow to enhance production and to bring more prosperity to citizens because of economies of scale. Citizens with some economic knowledge may think that firms will adopt the technology that is the most desired by the society, because otherwise they would be thrown out of the market. What to think of those presumptions? Since perfect competition is incompatible with fixed costs, a precise answer to this question requires considering imperfectly competitive markets. Therefore, we consider monopolistic competition in which firms sell differentiated varieties to consumers who love variety. On the one hand, lower marginal costs of the largescale technology push down prices, benefiting the consumers. On the other hand, its higher fixed costs reduce firms’ profitability and thus they contribute to decrease the number of varieties, which is detrimental to consumers. The balance between the benefits and the costs depends on the fixed and marginal costs and on the preferences of consumers. We show that under constant elasticity of
96
E. Toulemonde / Economics Letters 150 (2017) 95–98
substitution (CES) preferences, firms’ technological choice coincides with consumers preferred technology. By contrast, the market does not systematically deliver the right technology under nonCES preferences: (i) Firms might rush on large-scale technology, pushing to the exit many desirable varieties produced by small firms. (ii) Firms might shun large-scale technology, though that technology would benefit the society through lower prices. (iii) A bias towards small-scale technology in some stage of development, and a bias towards large-scale technology in another stage is also a possibility. Starting with Spence (1976) and Dixit and Stiglitz (1977), a large literature studies the (in)efficiency of the number of varieties delivered by the market. It is well known that the market achieves the optimum when the utility is CES. Recently, Parenti et al. (2016) prove that the market can also deliver the optimum under nonadditive and non-homothetic preferences. In general however, the market does not deliver the right number of varieties. We complement this literature by showing that the market might not only select the wrong number of varieties but also the wrong type of firms when preferences are additive. Ohkawa et al. (2005) is probably the paper with the closest focus to ours. They show that under asymmetric Cournot oligopoly the market also selects the wrong technology in the long run. They focus on homogeneous products whereas we emphasize the role played by the love for variety of diversified products. 2. The model We consider an economy endowed with L identical workers whose wage is chosen as the numeraire. There is a continuum N of firms, each producing a single variety indexed by i ∈ [0, N]. Each worker spends her entire income on the continuum of horizontally differentiated varieties. 2.1. Preferences As in Zhelobodko et al. (2012), we build on additive preferences.1 Each consumer chooses her consumption xi , to maximize her utility given her unit wage and the prices pi : N
u (xi ) di s.t.
max xi
N
pi xi di = 1 ∀i
0
q∗i /L u′′ q∗i /L + u′ q∗i /L
ru (x) ≡ −
xu′′ (x) u′ ( x )
> 0.
where λ =
− ci qi − fi .
cs
ru q∗s /L
⇐⇒ ru q∗s /L =
fs cs q∗s + fs
.
(5)
1 − ru q∗s /L
∗
cs
u′ q∗s /L
q∗ = s u′ q∗s /L ru q∗s /L . fs
(6)
Second, by (2), we find the optimal price and the number of firms:
Firm i faces a fixed cost fi and produces qi units of the variety i at constant marginal costs ci . It sells its output to the L identical consumers: qi = Lxi . Hence, pi = u′ (qi /L) /λ and firm i′ s profit is
λ
fs 1 − ru q∗s /L
q∗s =
λs =
2.2. Technology choice in the market solution
u (qi /L)
(4)
(2)
Under monopolistic competition, each firm is negligible to the market and treats the Lagrange multiplier λ as an exogenous aggregate variable.
′
At the equilibrium, the share of fixed costs in the total costs is thus equal to the relative love for variety. We also find the equilibrium value of the Lagrange multiplier:
0
where a star denotes the equilibrium value. Note that the second order condition requires that the left hand side of those expressions is decreasing in qi , that is, − (qi /L) ru′ < ru (1 − ru ). Therefore, firms with lower marginal costs ci must produce more in order to restore the equality in (4). We now limit the technology set to two technologies: the large-scale and the small-scale, denoted with subscripts l and s respectively. The large-scale technology entails high fixed costs fl and low marginal costs cl whereas the small-scale technology is characterized by smaller fixed costs fs < fl and higher marginal costs cs > cl . By symmetry, firms with the same technology sell the same quantities at the same prices. This allows us to replace the subscript i with the subscript t ∈ {l, s} that identifies the firm with its technology. To be clear, we consider two prices, pl and ps , and two possible output, ql and qs . All firms face the same aggregate conditions; they thus share the same Lagrange multiplier. At a free entry equilibrium, all firms will choose the same technology. This claim is easily proved by contradiction, as in general, there is no solution to a system of four equations with three unknowns. An equilibrium in which the two technologies are used would indeed require to find the three unknowns ql , qs and λ that simultaneously satisfy two zero-profit conditions (πs (qs ) = 0 and πl (ql ) = 0) and two first order conditions ((4) with i = l and (4) with i = s). Without loss of generality, let us characterize an equilibrium in which all firms adopt the small-scale technology. Let λ∗s denote the Lagrange multiplier faced by all firms when they all adopt technology s. First, the aggregate statistics λ∗s adjusts to make ∗ πs qs = 0 in (3). Equating this Lagrange multiplier with the value of the same multiplier found in (4) (with i = s), we find the output of a typical firm:
N
xi u′ (xi ) di.
(1)
The inverse demand function is pi = u′ (xi ) /λ
= λ∗ ci ⇔ 1 − ru q∗i /L u′ q∗i /L = λ∗ ci
0
where u (.) is thrice continuously differentiable, strictly increasing and strictly concave. The relative love for variety is
πi (qi ) =
Firm i chooses its production qi to maximize its profit. At the equilibrium,
(3)
1 See Parenti et al. (2016) for a discussion on the properties of additive and homothetic preferences.
p∗s =
λ∗s =
cs 1 − ru q∗s /L
and
Ns∗ q∗s ′ ∗ L u qs /L ⇐⇒ Ns∗ = ru q∗s /L . L fs
(7)
Third, by ensuring that a firm makes losses if it deviates from the equilibrium by choosing the large-scale technology, we find the following lemma, which will be further discussed in the next section.
E. Toulemonde / Economics Letters 150 (2017) 95–98
Lemma 1. Let εu(x) denote the elasticity of u (x) with respect to x. Firms prefer the small-scale technology if and only if u q∗l /L ru q∗l /L εu(q∗l /L)
u q∗s /L ru q∗s /L εu(q∗s /L)
(ii)
fs
.
(8)
(i)
fl
<
(iii)
(iv)
The equilibrium output is larger when firms choose the large-scale technology, q∗l > q∗s . Proof. See a technical appendix available upon request (see Appendix A). 2.3. Market vs planner’s preferred technology In this section, we consider a benevolent planner who has a single instrument at its disposal: a regulation prohibiting the use of one technology. We keep the assumption that firms are free to enter and to leave the market; firms also choose their output and prices freely, but they must use the technology that is imposed by the planner. Free entry of firms drives profits to zero. Hence, the planner does not care about the profits and she seeks to maximize N the utility 0 u (xi ) di, which, by symmetry, is equal either to Ns u (qs /L) or Nl u (ql /L). We know that if firms adopt the smallscale technology, their number is given by (7). If instead firms adopt the large-scale technology, their number is given by a similar expression in which s is replaced by l. The technology that maximizes welfare is given in the next proposition. Lemma 2. A benevolent planner prefers the small-scale technology if and only if u q∗l /L ru q∗l /L
<
u q∗s /L ru q∗s /L
(i)
fl f
⇐⇒
s
q∗s cs + fs u q∗s /L
(iii)
(ii)
<
q∗l cl + fl u q∗l /L
.
(9)
The right-hand-side shows that the planner prefers the technology that minimizes the cost per unit of utility. As shown by the left-hand-side, three factors guide the planner. To understand those factors, recall that there are two components in the N utility 0 u (xi ) di. (i) First, u (xi ) is the utility derived from the consumption of xi units of variety i. Accordingly, the planner prefers the large-scale technology because it is the technology that allows firms to produce more. (ii) The preference for variety is taken into account by integrating u (xi ) over the N varieties. As shown in (9), the planner prefers the technology that delivers the highest value of the relative love for variety at the equilibrium. Note that under CES preferences, ru (x) is independent of x. (iii) The planner prefers the cheapest technology. It is the balance between those three factors that determines the planner’s optimal choice. Obviously, there might be a discrepancy between the firms’ preferred technology and the planner’s choice since (8) and (9) are not identical. Which are the factors that induce firms to adopt the small-scale technology? Lemma 1 shows that the same three factors (i)–(iii) that affect the planner’s choice, influence the firms’ choices. A fourth factor is at work here (iv): the elasticity of u (x) with respect to x. Suppose εu(q∗s /L) > εu(q∗ /L) . An increase of one percent of l the output has a higher relative impact on the utility of consuming variety i when all firms use the small-scale technology than when they use the large-scale technology. Firms can capture part of this
97
increase through higher prices, which induces them to adopt the small-scale technology. The opposite holds if εu(q∗s /L) < εu(q∗ /L) . l The discrepancy between the firms’ preferred technology and the planner’s choice is confirmed by the next proposition. Proposition 3. Consider the planner’s preferred technology as a benchmark.(i) The market is biased towards the large-scale technology for some values of fixed costs if εu(q∗ /L) > εu(q∗s /L) .(ii) By conl trast, the market is biased towards the small-scale technology for some values of fixed costs if εu(q∗ /L) < εu(q∗s /L) . l
Because consumers love variety, result (i) might be expectable. Firms adopt the large-scale technology because of its higher profitability. So doing, they push to the exit many small firms that are highly valued by the consumers. Result (ii) shows an opposite result in which the economy is trapped into a small-scale technology that delivers too many varieties. Note that those results do not appear under CES preferences, because εu(x) is independent of x. In this special case, the market outcome always delivers the right technology. Another new result shows that, depending on the specification of preferences, it is possible that the economy is trapped into a small-scale technology at early stage of development whereas it is trapped into a large-scale technology at a later stage . . . . But the opposite result can also be found with different preferences. Note that Vives (1999) and Kühn and Vives (1999) define the degree of preference for variety as 1 − εu(x) .2 In their opinion, an increasing degree of preference for variety can be considered as the normal case, meaning that εu(x) would be a decreasing function. Bykadorov et al. (2015) go in the same direction. Hence, the bias towards the small-scale technology might be more plausible. We now illustrate our results with four examples (a technical appendix is available upon request for the detailed computations, Appendix A). As a first example, consider the utility u (x) = γ x + xβ where γ ≥ 0 and β ∈ (0, 1). CES preferences arise as a special case, if γ = 0. It is readily checked that εu(q∗ /L) ≥ εu(q∗s /L) . That l is, for some values of fixed costs, prohibiting the use of the largescale technology would improve welfare. As a second example, consider the constant average risk aversion (CARA) utility with u (0) = 0: u (x) = 1 − exp (−α x) where α > 0. It is readily checked that εu(q∗ /L) < εu(q∗s /L) . Hence, there always exists some l values of fixed costs such that the market selects the small-scale technology whereas it should select the large-scale technology. Consider now a CARA utility with u (0) > 0: u (x) = β−exp (−α x), where β > 1 and α > 0. It can be shown that the preferred technology of the planner depends on the value of β and on the market size L. For a small value of β and for a small market size L, there exists values of fixed costs such that the economy is trapped into the small-scale technology whereas the planner prefers the large-scale. The opposite holds for a larger values of β and L. Finally, consider the utility u (x) = x/ ln (x + 2). It can be shown that there exists values of fixed costs such that the market selects the largescale technology whereas it should ideally choose the small-scale technology when the market is small whereas the opposite holds when the market is large. Back to some examples provided at the beginning of the introduction, we conclude that the market sometimes promotes excessively some type of firms: supermarkets instead of corner shops, Hollywood blockbusters instead of authors’ movies, . . . or the opposite.
2 When introducing a new variety, the proportion of social benefits not captured by revenues is the degree of preference for variety.
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Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2016.11.013. References Bykadorov, I., Gorn, A., Kokovin, S., Zhelobodko, E., 2015. Why are losses from trade unlikely? Econom. Lett. 129, 35–38. Dixit, A.K., Stiglitz, J.E., 1977. Monopolistic competition and optimum product diversity. Amer. Econ. Rev. 67 (3), 297–308. Kühn, K.-U., Vives, X., 1999. Excess entry, vertical integration, and welfare. Rand J. Econ. 30 (4), 575–603.
Ohkawa, T., Makoto, O., Noritsugu, N., Kazuharu, K., 2005. The market selects the wrong firms in the long run. Internat. Econom. Rev. 46 (4), 1143–1165. Parenti, M., Ushchev, P., Thisse, J.-F., 2016. Toward a theory of monopolistic competition. Higher School of Economics Research Paper WP BRP 121/EC/2016. Spence, M., 1976. Product selection, fixed costs and monopolistic competition. Rev. Econom. Stud. 43 (2), 217–235. Vives, X., 2008. Innovation and competitive pressure. J. Ind. Econ. 56 (3), 419–469. Vives, X., 1999. Oligopoly Pricing - Old Ideas and New Tools. The MIT Press, Cambridge, Massachusetts. Zhelobodko, E., Kokovin, S., Parenti, M., Thisse, J.-F., 2012. Monopolistic competition: Beyond the constant elasticity of substitution. Econometrica 80 (6), 2765–2784.