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Optimization Letters (2019) 13:1803–1817https://doi.org/10.1007/s11590-018-1336-9
ORIG INAL PAPER
Monopolistic competition with investments in productivity
Igor Bykadorov1,2,3
Received: 27 February 2018 / Accepted: 2 October 2018 / Published online: 8 October 2018© Springer-Verlag GmbH Germany, part of Springer Nature 2018
AbstractWeconsider amonopolistic competitionmodelwith the endogenous choice of technol-ogy. We study the impact of technological innovation on the equilibrium and sociallyoptimal variables. We obtained the comparative statics of the equilibrium and sociallyoptimal solutions with respect to the technological innovation parameter and utilitylevel parameter.
Keywords Monopolistic competition · Investments in productivity · Relative love forvariety · Comparative statics
1 Introduction
We study a monopolistic competition model with endogenous choice of technologyin the closed economy case. We consider “technological innovation” non-negativeparameter α that influences on costs. Moreover, we consider “consumer utility level”non-negative parameter β that influences on utility. The aim is to make comparativestatistics of equilibrium and social optimal solutions with respect to parameters α andβ.
Our key findings are:
– When parameter α increases,
– consumption and investments in R&D both increase;
The work was supported by the program of fundamental scientific researches of the SB RAS No. I.5.1.,Project No. 0314-2016-0018. Supported in part by RFBR Grants 16-01-00108, 16-06-00101 and18-010-00728.
B Igor [email protected]
1 Sobolev Institute of Mathematics SB RAS, Acad. Koptyug avenue 4, Novosibirsk, Russia630090
2 Novosibirsk State University, Pirogova street 2, Novosibirsk, Russia 630090
3 Novosibirsk State University of Economics and Management, Kamenskaja street 56, Novosibirsk,Russia 630099
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1804 I. Bykadorov
– the behavior of the equilibrium and socially optimal variables does not dependon the properties of the costs as a function of investments in R&D;
– the behavior of the equilibrium variables depends on the elasticity of demandonly;
– the behavior of the socially optimal variables depends on the elasticity of utilityonly;
– the equilibrium variables depend on the elasticity of demand and the sociallyoptimal variables depend on the elasticity of utility in the identical way.
– When parameter β increases,
– the behavior of the equilibrium individual investments in R&D, individualconsumption, and mass of firms depend on the behavior of the demand elas-ticity;
– the behavior of the social optimal individual investments in R&D, individualconsumption, andmass of firms depend on the behavior of the utility elasticity;
– the behavior of the equilibrium total investments in R&D depends on thebehavior of the elasticities of both demand and marginal costs;
– the behavior of the social optimal total investments in R&D depends on thebehavior of the elasticities of both utility and marginal costs.
We discuss the generalization of the results to another monopolistic competitionmodels.
It looks natural to generalize our model to heterogeneous and/or dynamic versions.As to heterogeneity, in further research we plan to study heterogeneous firms a la [13,14]. As to dynamics, dynamic models (see, e.g., [1]) are useful in analyzing economicgrowth, but usually avoided within monopolistic competition framework, cf. [11,12].Here the typical research questions involve static market effects.
The paper is the extended and modified version of the paper [3] presented in the 8thInternational Conference “Optimization and Applications” (OPTIMA-2017) that washeld in Petrovac, Montenegro, during October 2–7, 2017. In this extended version, wemore carefully describe the model and present corresponding discussions. Moreoverwe present the main ideas (the sketch) of the proofs. The paper concerns with [2,9].Our research technique uses [15].
The paper is organized as follows.Section 2 presents the basic model of closed economy. Here we formulate main
assumptions of Monopolistic Competition concept (Sect. 2.1), the Consumer problem(Sect. 2.2), the Producer problem (Sect. 2.3), the concepts and the main results forthe equilibrium (Sect. 2.4, Proposition 1) and for the social optimality (Sect. 2.5,Proposition 2). In particular, the marginal cost function c( f ) is strictly convex as inequilibrium (Remark 1) as in social optimality (Remark 2).
In Sect. 3 we study a more complicated case where the cost function depends oninvestments and also on the parameter α showing how technological innovation affectsthe costs. In Sect. 3.1 we study the behavior of the equilibrium and socially optimalsolutions when the technological innovation parameter α increases, see Propositions3 and 4.
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Monopolistic competition with investments in productivity 1805
In Sect. 4 we consider the situation when sub-utility function depends not only onindividual consumption, but also on parameter β. We can interpret this parameter asthe level of consumption utility (consumption quality). Analogously to Sect. 3.1, westudy in Sect. 4.1 the behavior of the equilibrium and socially optimal solutions whenthe technological innovation parameter β increases, see Proposition 5.
Section 5 (Conclusions) contains the review of the obtained results and the sugges-tions for further research.
2 The basic model of closed economy
In this section we set the basic monopolistic competition model for closed economy(one country case).
2.1 Main assumptions of monopolistic competition
Due to [11], the main assumptions of Monopolistic Competition are:
– consumers are identical, each endowed with one unit of labor;– labor is the only production factor; consumption, output, prices etc. are measuredin labor;
– firms are identical, but produce “varieties” (“almost the same”) of good;– each firm produces one variety as a price-maker, but its demand is influenced byother varieties;
– each variety is produced by one firm that produces a single variety;– each demand function results from additive utility function;– number of firms is big enough to ignore firm’s influence on the whole indus-try/economy;
– free entry drives all profits to zero;– labor supply/demand is balanced.
2.2 Consumer
Each consumer maximizes the total utility function under budget constraint by choos-ing an infinite-dimensional consumption vector X = (xi )i∈[0,N ] with coordinatesxi : [0, N ] → R+, where
– scalar xi is (individual) consumption of variety i by each consumer;– N is number (mass) of firms determined endogenously;– [0, N ] is the endogenous interval of the firms.
Since consumers are identical, we omit the index of a consumer. Denote by L thenumber of consumers. Thus the problem of consumer can be written as
⎧⎨
⎩
∫ N0 u (xi ) di → max
X∫ N0 pi xi di ≤ w +
∫ N0 πi diL = 1.
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1806 I. Bykadorov
We assume that sub-utility function u (·) satisfies the conditions
u(0) = 0, u′ (z) > 0, u′′ (z) < 0, (1)
i.e., it is strictly increasing and strictly concave.In the budget constraint, w is wage, pi is the unit price of the variety i , πi is the
profit of firm i . Due to the free entry condition, πi = 0 in the equilibrium. Since weconsider the general equilibrium model, wage can be normalized to w ≡ 1.
The First Order Condition (FOC) for the consumer’s problem entails the inversedemand for variety i :
p (xi , λ) = u′ (xi )λ
, (2)
where λ is the Lagrange multiplier associated with the budget constraint.
2.3 Producer
We assume that each variety is produced by one firm that produces a single variety.However, unlike the classical setting, each producer chooses the technology level.Namely, if producer i spends fi units of labor as fixed costs, then the total costs ofproducing yi = Lxi units of output are c( fi )yi + fi units of labor. It is natural tosuppose that the function c( f ) satisfies the condition
c′ ( f ) < 0. (3)
Using (2) the profit maximization problem of the producer i with respect to xi andfi can be formulated as
πi (xi , fi , λ) = (p (xi , λ) − c ( fi )) Lxi − fi
=(u′ (xi )
λ− c ( fi )
)
Lxi − fi → maxxi≥0, fi≥0
.
2.4 Equilibrium
The producers are assumed identical, and hence the producer’s problem acquires thesame form for each producer. Accordingly, further analysis focuses on the symmetricequilibria xi = x, fi = f for any i .
The FOC for the producer’s problem are
∂π
∂x= ∂π
∂ f= 0,
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Monopolistic competition with investments in productivity 1807
i.e.,
u′′(x)x + u′(x)λ
− c( f ) = 0, (4)
c′( f )Lx + 1 = 0. (5)
while the Second Order Conditions (SOC) are
∂2π
∂x2< 0,
∂2π
∂x2· ∂2π
∂ f 2−
(∂2π
∂x∂ f
)2
> 0,
i.e.,
u′′′(x)x + 2u′′(x)λ
< 0, (6)
−(u′′′(x)x + 2u′′(x)
)c′′( f )x
λ− (
c′( f ))2
> 0. (7)
Like in the standard monopolistic competition framework, the firms enter into themarket until their profit remains positive. Therefore, free entry implies the zero-profitcondition
u′(x)λ
− c( f ) = f
Lx. (8)
As to the labor supply/demand balance condition
N∫
0
(c ( fi ) xi L + fi ) di = L,
it can be written asN (c( f )xL + f ) = L. (9)
Summarizing, we define the symmetric equilibrium as a bundle
(x∗, p∗, λ∗, f ∗, N∗)
satisfying the following:
– the rational consumption condition (2);– the rational production conditions (4), (5) under (6), (7);– the free entry condition (8);– the labor supply/demand balance condition (9).
Now let us characterize equilibrium conditions in more closed form.We use the concept of elasticity. The elasticity of a one-variable function g(x) is
εg(z) = g′(z)zg(z)
. (10)
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1808 I. Bykadorov
Moreover, we use the Arrow–Pratt measure of concavity defined for any function g (z)as
rg (z) = −g′′ (z) zg′ (z)
. (11)
Note that rg(z) = − εg′(z) and rg(z) + εg(z) = rln g(z). Note that for sub-utilityfunction u(·), Arrow–Pratt measure ru means the “relative love for variety”.
Proposition 1 (Cf. [2,9]) The equilibrium consumption/investment couple (x∗, f ∗) isthe solution of the equations
ru(x)x
1 − ru(x)= f
Lc( f ), (12)
(1 − εc( f )) · (1 − ru(x)) = 1 (13)
under the conditions
ru′(x) < 2, (14)
(2 − ru′(x)) rc ( f ) > 1. (15)
The equilibrium mass of firms N∗, price p∗ and markup are
N∗ = L
c ( f ∗) x∗L + f ∗ , (16)
p∗ = c ( f ∗)1 − ru (x∗)
, (17)
p∗ − c ( f ∗)p∗ = ru
(x∗) = N∗ f ∗
L. (18)
Proof of Proposition 1 (sketch) Using Arrow–Pratt measure (11), producer’s FOC (4)can be written as
(1 − ru(x)) · u′(x)λ
− c( f ) = 0, (19)
We obtain (12) by substituting (8) in (19); we obtain (13) by substituting (5) in (12)and using (10).
Further, due to (1) and (2), one has in equilibrium
λ > 0. (20)
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Monopolistic competition with investments in productivity 1809
Due to (1) and (20), producer’s SOC (6) can be written as (14). Let us show thatproducer’s SOC (7) can be written as (15). Indeed, due to (5), (8) and (12), we get
−(u′′′(x)x + 2u′′(x)
)c′′( f )x
λ− (
c′( f ))2
= u′′(x)xλ
· c′( f )f
·(
(2 − ru′(x)) rc( f ) + c′( f ) f · λ
u′(x)· 1
ru(x)
)
= u′′(x)xλ
· c′( f )f
·(
(2 − ru′(x)) rc( f ) − 1
Lx· f · 1 − ru(x)
c( f )· 1
ru(x)
)
= u′′(x)xλ
· c′( f )f
· ((2 − ru′(x)) rc( f ) − 1) .
Now, (15) holds due to (1), (3), (20). Further, (16) follows directly from (8). Due to(2) and (12), one has
p∗ = c(f ∗) ·
(
1 + f ∗
Lc ( f ∗) x∗
)
= c(f ∗) ·
(
1 + ru (x∗)1 − ru (x∗)
)
,
i.e., (17) holds. Finally, (18) holds due to (17) and (12).
Remark 1 Due to (1), Arrow–Pratt measure ru is positive. But in equilibrium, due to(12), one has ru (x∗) < 1. Moreover, due to (6), (15) and (3), one has c′′ ( f ∗) > 0,i.e., marginal cost function is strictly convex in equilibrium.
2.5 Social optimality
Let us consider another optimizationproblem:maximize the total (social) utility subjectto the labor balance condition.Within the framework of our model where xi = x, fi =f for any i , the problem is
⎧⎨
⎩
Nu(x) → maxN ,x, f
N (c( f )xL + f ) = L,
i.e.,Lu (x)
c ( f ) xL + f→ max
x, f.
A bundle(xopt , f opt , Nopt
)that solves this problem is called socially optimal.
Proposition 2 (Cf. [2]) The social optimal consumption/investment couple(xopt , f opt
)
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1810 I. Bykadorov
is the solution of the equations
(1 − εc( f )) · εu(x) = 1, (21)
c′( f )xL = −1 (22)
under the conditionεc( f ) + ru(x) · rc( f ) > 0. (23)
The social optimal mass of firms Nopt is
Nopt = L
c( f opt )xopt L + f opt. (24)
Proof of Proposition 2 (sketch) For the function
U (x, f ) = L · u(x)
c( f )xL + f
one has
∂U
∂x= − Lu(x) · ((−c′( f )xL − εc( f )
) · εu(x) + c′( f )xL)
(c( f )xL + f ) (c′( f )xL + εc( f )) x,
∂U
∂ f= − Lu(x)
(c( f )xL + f )2· (c′( f )xL + 1
).
Hence, (21) and (22) hold. Moreover, under (21) and (22),
∂2U
∂x2= Lu′′(x)
c( f )xL + f< 0,
∂2U
∂x2· ∂2U
∂ f 2−
(∂2U
∂x∂ f
)2
= −(
L2 · u(x)
(c( f )xL + f )2
)2
· c( f )c′( f )f
· (εc( f ) + ru(x)rc( f )) .
Due to (3), the SOC is (23).
Remark 2 Due to (3), the elasticity εc is negative. Hence, due to (23), one hasc′′ ( f opt
)> 0, i.e., marginal cost function is strictly convex in social optimality.
3 Generalization 1: the case c = c(f ,˛)
Let us study a more complicated case where the cost function depends on investmentsand also on the parameter α showing how technological innovation affects the costs.
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Monopolistic competition with investments in productivity 1811
Let c = c( f , α) with
∂c
∂ f< 0,
∂2c
∂ f 2> 0,
∂c
∂α< 0,
∂2c
∂ f ∂α< 0.
The solution is the same as in the case c = c( f ), Propositions 1 and 2 remain validunder the notation
rc := rc( f , α) := − ∂2c
∂ f 2· f
∂c
∂ f
> 0, εc := εc/ f := ∂c
∂ f· f
c.
We study the elasticities Ex/α = dxdα
· αx , E f /α, EN/α, EN f /α, Ep/α with respect to
the parameter α. Note that
εu > 0, εc/ f < 0, εc/α := ∂c
∂α· α
c< 0,
εc′f /α
:= ∂
∂α
(∂c
∂ f
)
· α
∂c
∂ f
= ∂2c
∂ f ∂α· α
∂c
∂ f
> 0.
3.1 Comparative statics w.r.t.˛
Here we study the behavior of the equilibrium and socially optimal solutions when thetechnological innovation parameter α increases. More precisely, we study the signs ofthe derivatives w.r.t. α. We present the results in terms of the elasticities w.r.t. α.
Proposition 3 The elasticities of the equilibrium variables x∗, f ∗, N∗, N∗ f ∗ and p∗w.r.t. α are
Ex∗/α =εc′
f /α− (1 − ru) rcεc/α
(2 − ru′) rc − 1> 0, (25)
E f ∗/α =(2 − ru′) εc′
f /α− (1 − ru) εc/α
(2 − ru′) rc − 1> 0, (26)
EN∗/α = εru · Ex∗/α − E f ∗/α, (27)
EN∗ f ∗/α = E p∗−c( f ∗)p∗ /α
= εru · Ex∗/α, (28)
Ep∗/α =−ruεc′
f /α+ (1 − ru) ((2 − ru′) rc − 1 + rurc) εc/α
(2 − ru′) rc − 1< 0. (29)
The signs of the elasticities can be found in Table 1, where the symbol “?” means thatthe sign of corresponding elasticity is not uniquely determined.
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1812 I. Bykadorov
Table 1 Equilibrium:comparative statics w.r.t. α
r ′u < 0 r ′
u = 0 r ′u > 0
Ex∗/α > 0 > 0 > 0
E f ∗/α > 0 > 0 > 0
EN∗/α < 0 < 0 ?
EN∗ f ∗/α < 0 = 0 > 0
Ep∗/α < 0 < 0 < 0
Proof of Proposition 3 (sketch) Let us totally differentiate (12) and (13) w.r.t. α. Using(12) and (13) and the identity for Arrow–Pratt measure
r ′g(z) · z = (
1 + rg(z) − rg′(z)) · rg(z), (30)
we get
(2 − ru′(x)) · Ex/α − E f /α = − (1 − ru(x)) εc/α
1 + ru(x) − ru′(x)
1 − ru(x)· Ex/α − 1 − (1 − ru(x)) · rc
1 − ru(x)· E f /α = εc′
f /α− εc/α,
i.e., (25) and (26) hold. Further, due to (18),
EN∗ f ∗/α = E N∗ f ∗L /α
= Eru(x∗)/α = εru(x∗) · Ex∗/α,
i.e., (28) holds. Moreover,
EN∗/α = EN∗ f ∗/α − E f ∗/α.
Hence, (27) holds. Finally, due to (17),
Ep∗/α = 1
p∗ · c( f ∗, α)
1 − ru(x∗)·(
r ′u(x
∗)x∗
1 − ru(x∗)· Ex∗/α + εc/ f · E f ∗/α + εc/α
)
.
Using (17), (25) and (26), we get (29).
Proposition 4 The elasticities of the socially optimal variables xopt , f opt , Nopt andNopt f opt w.r.t. α are
Exopt/α =εc/ f ·
(εu · εc/α · rc − εc′
f /α
)
ru · rc + εc/ f> 0, (31)
E f opt/α =εc′
f /α+ Exopt/α
rc> 0, (32)
ENopt/α = − εu · (εc/α + Exopt/α
), (33)
ENopt f opt/α = εεu · rcru · rc + εc/ f
·(
εu · εc/α −εc′
f /α
rc
)
. (34)
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Monopolistic competition with investments in productivity 1813
Table 2 Social optimality:comparative statics w.r.t. α
ε′u > 0 ε′
u = 0 ε′u < 0
Exopt /α > 0 > 0 > 0
E f opt /α > 0 > 0 > 0
ENopt /α < 0 < 0 ?
ENopt f opt /α < 0 = 0 > 0
The signs of the elasticities can be found in Table 2, where the symbol “?” means thatthe sign of corresponding elasticity is not uniquely determined.
Proof of Proposition 4 (sketch) Let us totally differentiate (21) and (22) w.r.t. α, weget
1 − εc/ f
εc/ f· εεu · Ex/α − (
1 − εc/ f − rc) · E f /α = εc′
f /α− εc/α, (35)
Ex/α − rc · E f /α = − εc′f /α
. (36)
Using (21), (22) and the identity for elasticity
ε′g(z) · z = (
1 − εg(z) − rg(z)) · εg(z) (37)
(cf. (30)), we get (31) and (32). Now let us totally differentiate (24) w.r.t. α. Using(22), we get
L2c
(cLx + f )2· x
N· (Ex/α + εc/α
) + EN/α = 0.
Due to (24), (22) and (22), we get (33). Finally,
ENopt f opt/α = ENopt/α + E f opt/α.
Hence, due to (32) and (33),
ENopt f opt/α =(−εc/ f · εu − ru
) · rcru · rc + εc/ f
·εu · εc/α · rc − εc′
f /α
rc.
Using (21) and (37), we get (34).
Remark 3 Tables 1 and 2 show that the equilibrium variables depend on the elasticityof demand in a similar way as the socially optimal variables depend on the elasticityof utility.
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1814 I. Bykadorov
4 Generalization 2: the case u = u(x,ˇ)
Now let us consider the situation when sub-utility function u depends not only onindividual consumption x , but also on parameter β. We can interpret this parameter asthe level of consumption utility (consumption quality). Thus, u = u(x, β). Of course,
it is natural to assume that∂u(x, β)
∂β> 0. But under comparative statics with respect
to β, as we will see, the signs of equilibrium variables depends essentially on thepartial elasticity w.r.t. β of the relative love for variety ru ,
εru/β := ∂
∂β(ru(x, β)) · β
ru(x, β)
≡ ∂
∂β
⎛
⎜⎝
∂2u(x, β)
∂x2· 1
∂u(x, β)
∂x
⎞
⎟⎠ · ∂u(x, β)
∂x· 1
∂2u(x, β)
∂2x
· β,
while the signs of socially optimal variables depends essentially on the partial elasticityw.r.t. β of the elasticity of sub-utility u,
εεu/β := ∂
∂β(εu(x, β)) · β
εu(x, β)
≡ ∂
∂β
(∂u(x, β)
∂x· 1
u(x, β)
)
· u(x, β) · 1∂u(x, β)
∂x
· β.
4.1 Comparative statics w.r.t.ˇ
Analogously to Sect. 3.1, here we study the behavior of the equilibrium and sociallyoptimal solutions when the technological innovation parameter β increases. Moreprecisely, we study the signs of the derivatives w.r.t. β. We present the results interms of the elasticities w.r.t. β. By total differentiation w.r.t. β the equations for theequilibrium (see Proposition 1) and social optimality (see Proposition 2), we obtain
Proposition 5 The elasticities of the equilibrium variables x∗, f ∗, N∗, N∗ f ∗ and p∗w.r.t. β are
Ex∗/β = − rc · εru/β
(2 − ru′)rc − 1, (38)
E f ∗/β = 1
rc· Ex∗/β, (39)
EN∗/β = −cx∗N∗ · Ex∗/β, (40)
EN∗ f ∗/β = (1 − ru) · εεc
rc· Ex∗/β, (41)
Ep∗/β = rc · ru(2 − ru′) · rc − 1
· εru/β . (42)
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Monopolistic competition with investments in productivity 1815
Table 3 Equilibrium: comparative statics w.r.t. β
∂ru∂β
< 0∂ru∂β
> 0
ε′c > 0 ε′
c = 0 ε′c < 0 ε′
c > 0 ε′c = 0 ε′
c < 0
Ex∗/β > 0 > 0 > 0 < 0 < 0 < 0
E f ∗/β > 0 > 0 > 0 < 0 < 0 < 0
EN∗/β < 0 < 0 < 0 > 0 > 0 > 0
EN∗ f ∗/β < 0 = 0 > 0 > 0 = 0 < 0
Ep∗/β < 0 < 0 < 0 > 0 > 0 > 0
Table 4 Social optimality: comparative statics w.r.t. β
∂εu
∂β> 0
∂εu
∂β< 0
ε′c > 0 ε′
c = 0 ε′c < 0 ε′
c > 0 ε′c = 0 ε′
c < 0
Exopt /β > 0 > 0 > 0 < 0 < 0 < 0
E f opt /β > 0 > 0 > 0 < 0 < 0 < 0
ENopt /β < 0 < 0 < 0 > 0 > 0 > 0
ENopt f opt /β < 0 = 0 > 0 > 0 = 0 < 0
The signs of the elasticities can be found in Table 3.The elasticities of the socially optimal variables xopt , f opt , Nopt and Nopt f opt
w.r.t. β are
Exopt/β = rc · εεu/β
εc + rurc, (43)
E f opt/β = 1
rc· Exopt/β, (44)
ENopt/β = − εu · Exopt/β, (45)
ENopt f opt/β = εu · εεc
rc· Exopt/β . (46)
The signs of the elasticities can be found in Table 4.
Proof of Proposition 5 (sketch) Let us totally differentiate (12) and (13) w.r.t. β. Using(12) and (13), the identity for elasticity (37) and the identity for Arrow–Pratt measure(30), we get (38) and (39).
Further, due to (9) and (5),
EN∗/β = − L
N∗ · Lc( f ∗)x∗
(Lc( f ∗)x∗ + f ∗)2· Ex∗/β,
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1816 I. Bykadorov
i.e., (40) holds due to (9). Moreover, due to (40) and (39),
EN∗ f ∗/β = EN∗/β + E f ∗/β = −N∗ · rc · c ( f ∗) · x∗ + 1
rc· Ex∗/β,
so (41) holds due to (37), (9), (12) and (13).Further, due to (17), (39), (12), (38) and (30), we get
Ep∗/β = c ( f ∗)p∗ ·
(εcrc
+ x∗1−ru(x∗,β)
· ∂ru(x∗,β)∂x
)· Ex∗/β − εc · εru/β
1 − ru (x∗, β)
= −c ( f ∗)p∗ · (εc − rc · ru (x∗, β)) · (1 − ru (x∗, β)) + ru (x∗, β)
(1 − ru (x∗, β))2 · ((2 − ru′) · rc − 1)· εru/β
= c ( f ∗)p∗ · rc ( f ∗) · ru (x∗, β)
(1 − ru (x∗, β)) · ((2 − ru′ (x∗, β)) · rc ( f ∗) − 1)· εru/β,
i.e., (42) holds.As to social optimality, let us totally differentiate (21) and (22) w.r.t. β. Using (21),
we directly get (43) and (44). Further, due to (24), (43) and (44), we directly get (45).Finally, due to (45), (44) and (21), we get
ENopt f opt/β = ENopt/β + E f opt/β = 1 − rc(f opt
)εu
(xopt , β
)
rc ( f opt )· Exopt/β
=(1 − εc
(f opt
) − rc(f opt
)) · εu
rc ( f opt )· Exopt/β .
Therefore, due to (37), we get (46).
Remark 4 Tables 3 and 4 show that he equilibrium variables depend on the behavior ofelasticity of demand w.r.t. β in a similar way as the socially optimal variables dependon the behavior of elasticity of utility w.r.t. β.
5 Conclusions
Weconsider amonopolistic competitionmodelwith the endogenous choice of technol-ogy. We study the impact of technological innovation on the equilibrium and sociallyoptimal variables, namely, consumption, costs, the mass of firms and prices (in theequilibrium case). We obtain the comparative statics of the equilibrium and sociallyoptimal solutions with respect to the technological innovation parameter and utilitylevel parameter. The analysis shows, see Remarks 3 and 4, that the equilibrium vari-ables depend on the elasticity of demand in a similar way as the socially optimalvariables depend on the elasticity of utility.
The results can be generalized to another monopolistic competition models: retail-ing [10], market distortion [4], international trade [8], and to the marketing models:
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Monopolistic competition with investments in productivity 1817
optimization of communication expenditure [6] and the effectiveness of advertising[5], pricing [7].
Acknowledgements The author would like to thank Jacques-Francois Thisse for guiding and advising,Kristian Behrens, Richard Ericson, Federico Etro, Sergey Kokovin, Peter Neary, Alexander Shepotylo,Philip Ushchev and Natalya Volchkova for valuable comments, Evgeny Zhelobodko (1973–2013), IrinaAntoshchenkova and Ekaterina Shelkova for participation in the early stages of this research, the anonymousreferees for valuable comments.
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