=Paper=
{{Paper
|id=Vol-1987/paper17
|storemode=property
|title=Monopolistic Competition Model with Different Technological Innovation and Consumer Utility Levels
|pdfUrl=https://ceur-ws.org/Vol-1987/paper17.pdf
|volume=Vol-1987
|authors=Igor A. Bykadorov
}}
==Monopolistic Competition Model with Different Technological Innovation and Consumer Utility Levels==
https://ceur-ws.org/Vol-1987/paper17.pdf
Monopolistic Competition Model with Different
Technological Innovation and Consumer Utility Levels
Igor A. Bykadorov
Sobolev Institute of Mathematics SB RAS
Acad. Koptyug avenue 4, 630090 Novosibirsk, Russia
Novosibirsk State University
Pirogova street 2, 630090 Novosibirsk, Russia
Novosibirsk State University of Economics and Management
Kamenskaja street 56, 630099 Novosibirsk, Russia
bykadorov.igor@mail.ru
Abstract
We consider a monopolistic competition model with the endogenous
choice of technology. We study the impact of technological innovation
on the equilibrium and socially optimal variables. We obtained the
comparative statics of the equilibrium and socially optimal solutions
with respect to the technological innovation parameter and utility level
parameter.
1 Introduction
We study a monopolistic competition model with endogenous choice of technology in the closed economy case.
We consider “technological innovation” parameter α that influences on costs. Moreover, we consider “consumer
utility level” parameter β that influences on utility. The aim is to make comparative statistics of equilibrium
and social optimal solutions with respect to parameters α and β.
Our key findings are:
• When parameter α increases,
– consumption and investments in R&D both increase;
– the behavior of the equilibrium and socially optimal variables does not depend on the properties of the
costs as a function of investments in R&D;
– the behavior of the equilibrium variables depends on the elasticity of demand only;
– the behavior of the socially optimal variables depends on the elasticity of utility only;
– the equilibrium variables depend on the elasticity of demand and the socially optimal 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, individual consumption, and mass of
firms depend on the behavior of the demand elasticity;
Copyright ⃝
c by the paper’s authors. Copying permitted for private and academic purposes.
In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of
the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org
108
– the behavior of the social optimal individual investments in R&D, individual consumption, and mass
of firms depend on the behavior of the utility elasticity;
– the behavior of the equilibrium total investments in R&D depends on the behavior of the elasticities
of both demand and marginal costs;
– the behavior of the social optimal total investments in R&D depends on the behavior of the elasticities
of both utility and marginal costs.
We discuss the generalization the results to another monopolistic competition models.
The paper concerns with [Antoshchenkova & Bykadorov, 2017]. Our research technique uses
[Zhelobodko et al., 2012].
2 The Basic Model of Closed Economy
In this section we set the basic monopolistic competition model for closed economy (one country case). We will
use the ArrowPratt measure of concavity defined for any function g (z) as
g ′′ (z) z
rg (z) = − .
g ′ (z)
Note that for sub-utility function u(·), ArrowPratt measure ru means the “relative love for variety.”
Denote by L the number of consumers and let [0, N ] be the endogenous interval of the firms.
2.1 Main Assumptions of Monopolistic Competition
Due to [Chamberlin, 1933] and [Dixit & Stiglitz, 1977], 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 measured in 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 by other 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 industry/economy;
• free entry drives all profits to zero;
• labor supply/demand in each country is balanced.
2.2 Consumer
Each consumer maximizes the total utility function under budget constraint by choosing an infinite-dimensional
consumption vector X = (xi )i∈[0,N ] with coordinates xi : [0, N ] → R+ . Since consumers are identical, we omit
the index of a consumer: ∫N
u (xi ) di → max
0 X
∫ ∫
N p x di ≤ w + 0N πi di = 1.
0 i i L
Here N is number (mass) of firms determined endogenously. Scalar xi is consumption of variety i by each
consumer. We assume that sub-utility function u (·) satisfies the conditions
u(0) = 0, u′ (xi ) > 0, u′′ (xi ) < 0,
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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 we consider the general equilibrium model, wage can be
normalized to w ≡ 1.
The First Order Condition (F OC) for the consumer’s problem entails the inverse demand for variety i:
u′ (xi )
p (xi , λ) = , (1)
λ
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 he spends f units of labor as fixed costs, then the
total costs of producing y units of output are c(f )y + f units of labor. It is natural to suppose that the function
c(f ) satisfies the condition c′ (f ) < 0.
Using (1) the profit maximization problem of the producer i with respect to xi and fi can be formulated as
( ′ )
u (xi )
πi (xi , fi , λ) = (p (xi , λ) − c (fi )) Lxi − fi = − c (fi ) Lxi − fi → max .
λ xi ≥0,fi ≥0
2.4 Equilibrium
The producers are assumed identical, and hence the producer’s problem acquires the same form for each producer.
Accordingly, further analysis focuses on the symmetric equilibria xi = x, fi = f for any i.
The F OC for the producer’s problem are
u′′ (x)x + u′ (x)
− c(f ) = 0, c′ (f )Lx + 1 = 0. (2)
λ
while the Second Order Conditions (SOC) are
(u′′′ (x) + 2u′′ (x)) c′′ (f )x
− (c′ (f )) > 0.
2
ru′ (x) < 2, − (3)
λ
Like in the standard monopolistic competition framework, the firms enter into the market until their profit
remains positive. Therefore, free entry implies the zero-profit condition
u′ (x) f
− c(f ) = . (4)
λ Lx
The labor balance condition can be written as
∫N
(c (fi ) xi L + fi ) di = N (c(f )xL + f ) = L. (5)
0
Summarizing, we define the symmetric equilibrium as a bundle (x∗ , p∗ , λ∗ , f ∗ , N ∗ ) satisfying the following:
• the rational consumption condition (1);
• the rational production conditions (2) and (3);
• the free entry condition (4) and the labor balance condition (5).
Proposition 1. [Antoshchenkova & Bykadorov, 2017] The equilibrium consumption/investment couple
(x∗ , f ∗ ) is the solution of the system
ru (x)x f
= , (1 − rln c (f ) + rc (f )) (1 − ru (x)) = 1,
1 − ru (x) Lc(f )
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under the conditions
ru (x) < 1, (2 − ru′ (x)) rc (f ) > 1.
The equilibrium mass of firms N ∗ , price p∗ and markup are
L c (f ∗ ) p∗ − c (f ∗ ) N ∗f ∗
N∗ = , p∗ = , = r u (x∗
) = .
c (f ∗ ) x∗ L + f ∗ 1 − ru (x∗ ) p∗ L
2.5 Social Optimality
Let us consider another optimization problem:maximize the total (social) utility subject to the labor balance
condition. Within the framework of our model where xi = x, fi = f for any i, the problem is
N u(x) → max
N,x,f
N (c(f )xL + f ) = L,
i.e.,
Lu (x)
→ max.
c (f ) xL + f x,f
A bundle (xopt , f opt , N opt ) that solves this problem will be called socially optimal.
In what follows we use the concept of elasticity. The elasticity of a one-variable function g(x) is
g ′ (z)z
εg (z) = .
g(z)
Note that rg (z) = −εg′ (z) and rg (z) + εg (z) = rln g (z).
Proposition 2. [Antoshchenkova & Bykadorov, 2017] In the social optimality setting, the F OC is
r (x) − r (x) = c(f )xL ,
ln u u
c(f )xL + f
c′ (f )xL = −1,
while the SOC is
εc + ru (x)rc (f ) ≡ rln c (f ) − (1 − ru (x)) rc (f ) > 0.
3 Generalization 1: the Case c = c(f, α)
Let us study a more complicated case where the cost function depends on investments and also on the parameter
α showing how technological innovation affects the costs. Let c = c(f, α) with
∂c ∂2c ∂c ∂2c
< 0, > 0, < 0, < 0.
∂f ∂f 2 ∂α ∂f ∂α
The solution is the same as in the case c = c(f ), Proposition 1 and Proposition 2 remain valid under the
notation
∂2c f ∂ 2 ln c f
rc := rc (f, α) := − · > 0, rln c := rln c (f, α) := · .
∂f 2 ∂c ∂f 2 ∂ ln c
∂f ∂f
· x , Ef /α , EN/α , EN f /α , Ep/α with respect to the parameter α. Note that
dx α
We study the elasticities Ex/α = dα
( )
du x ∂c α ∂c f ∂ ∂c α ∂2c α
εu = · > 0, εc/α := · < 0, εc/f := · < 0, εc′f /α := · = · > 0.
dx u ∂α c ∂f c ∂α ∂f ∂c ∂f ∂α ∂c
∂f ∂f
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3.1 Comparative Statics w.r.t. α
Here we study the behavior of the equilibrium and socially optimal solutions when the technological innovation
parameter α increase. More precisely, we study the signs of the derivatives w.r.t. α. We present the results
in terms of the elasticities w.r.t. α. By total differentiation w.r.t. α the equations for the equilibrium (see
Proposition 1) and social optimality (see Proposition 2)
Proposition 3. The elasticities of the equilibrium variables x∗ , f ∗ , N ∗ , N ∗ f ∗ and p∗ w.r.t. α are
εc′f /α − (1 − ru ) rc εc/α (2 − ru′ ) εc′f /α − (1 − ru ) εc/α
Ex∗ /α = > 0, Ef ∗ /α = > 0, EN ∗ /α = εru ·Ex∗ /α −Ef ∗ /α ,
(2 − ru′ ) rc − 1 (2 − ru′ ) rc − 1
−ru εc′f /α + (1 − ru ) ((2 − ru′ ) rc − 1 + ru rc ) εc/α
EN ∗ f ∗ /α = E p∗ −c(f ∗) = εru · Ex∗ /α , Ep∗ /α = < 0.
p∗ /α (2 − ru′ ) rc − 1
The elasticities of the socially optimal variables xopt , f opt , N opt and N opt f opt w.r.t. α are
( )
εc/f · εu · εc/α · rc/f − εc′f /α εc′ /α + Exopt /α
Exopt /α = > 0, Ef opt /α = f > 0,
ru · rc/f + εc rc/f
( εc′ /α )
( ) εεu · rc/f
EN opt /α = −εu · εc/α + Exopt /α , EN opt f opt /α = · εu · εc/α − f .
ru · rc/f + εc/f rc/f
Let us compare the signs of the resulting elasticities of the equilibrium and socially optimal variables (Propo-
sition 3). We summarize the results in Table 1 and Table 2. Note that the symbol “?” in the tables means that
the sign of corresponding elasticity is not uniquely determined.
Table 1: Equilibrium: Comparative statics w.r.t. α
ru′ < 0 ru′ = 0 ru′ > 0
Ex∗ /α >0 >0 >0
Ef ∗ /α >0 >0 >0
EN ∗ /α <0 <0 ?
EN ∗ f ∗ /α <0 =0 >0
Table 2: Social Optimality: Comparative statics w.r.t. α
ε′u > 0 ε′u = 0 ε′u < 0
Exopt /α >0 >0 >0
Ef opt /α >0 >0 >0
EN opt /α <0 <0 ?
EN opt f opt /α <0 =0 >0
Therefore, the equilibrium variables depend on the elasticity of demand in a similar way as the socially optimal
variables depend on the elasticity of utility.
4 Generalization 2: the Case u = u(x, β)
Now let us consider the situation when sub-utility function u depends not only on consumption x, but also on
parameter β. We can interpret this parameter as the level of consumption utility (consumption quality). Thus,
∂u(x, β)
u = u(x, β). Of course, it is natural to assume that > 0. But under comparative statics with respect
∂β
to β, as we will see, the signs of equilibrium variables depends essentially on the partial elasticity w.r.t. β of the
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relative love for variety ru ,
2
∂ β ∂ ∂ u(x, β) 1 ∂u(x, β) 1
εru /β := (ru (x, β)) · ≡ · · · 2 · β,
∂β ru (x, β) ∂β ∂x2 ∂u(x, β) ∂x ∂ u(x, β)
∂x ∂2x
while the signs of socially optimal variables depends essentially on the partial elasticity w.r.t. β of the elasticity
of sub-utility u,
( )
∂ β ∂ ∂u(x, β) 1 1
εεu /β := (εu (x, β)) · ≡ · · u(x, β) · · β.
∂β εu (x, β) ∂β ∂x u(x, β) ∂u(x, β)
∂x
Proposition 4. The elasticities of the equilibrium variables x∗ , f ∗ , N ∗ , N ∗ f ∗ and p∗ w.r.t. β are
rc · εru /β 1
Ex∗ /β = − , Ef ∗ /β = · Ex∗ /β , EN ∗ /β = −cx∗ N ∗ · Ex∗ /β ,
(2 − ru′ )rc − 1 rc
(1 − ru ) · εεc rc (ru − 1)
EN ∗ f ∗ /β = · Ex∗ /β , Ep∗ /β = εru /β · εc/f · .
rc (2 − ru′ )rc − 1
The elasticities of the socially optimal variables xopt , f opt , N opt and N opt f opt w.r.t. β are
rc · εεu /β 1 εu · εεc
Exopt /β = , Ef opt /β = · Exopt /β , EN opt /β = −εu · Exopt /β , EN opt f opt /β = · Exopt /β .
εc + ru rc + rc rc rc
Let us summarize the results of Proposition 4 in Table 3 and Table 4.
Table 3: Equilibrium: Comparative statics w.r.t. β
∂ru ∂ru
<0 >0
∂β ∂β
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
Ef ∗ /β >0 >0 >0 <0 <0 <0
EN ∗ /β <0 <0 <0 >0 >0 >0
EN ∗ f ∗ /β <0 =0 >0 >0 =0 <0
Table 4: Social Optimality: Comparative statics w.r.t. β
∂εu ∂εu
>0 <0
∂β ∂β
ε′c > 0 ′
εc = 0 ε′c < 0 ε′c > 0 ′
εc = 0 ε′c < 0
Exopt /β >0 >0 >0 <0 <0 <0
Ef opt /β >0 >0 >0 <0 <0 <0
EN opt /β <0 <0 <0 >0 >0 >0
EN opt f opt /β <0 =0 >0 >0 =0 <0
Therefore, the equilibrium variables depend on the behavior of elasticity of demand w.r.t. β in a similar way
as the socially optimal variables depend on the behavior of elasticity of utility w.r.t. β.
5 Conclusions
We consider a monopolistic competition model with the endogenous choice of technology. We study the impact
of technological innovation on the equilibrium and socially optimal variables, namely, consumption, costs, the
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mass of firms and prices (in the equilibrium case). We obtained the comparative statics of the equilibrium and
socially optimal solutions with respect to the technological innovation parameter and utility level parameter.
The results can generalize to another monopolistic competition models: retailing [Bykadorov et al., 2014],
market distortion [Bykadorov et al., 2016], international trade [Bykadorov et al., 2015], and to the marketing
models: optimization of communication expenditure [Bykadorov et al., 2002] and the effectiveness of advertising
[Bykadorov et al., 2009a], pricing [Bykadorov et al., 2009b].
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