=Paper= {{Paper |id=Vol-2795/short5 |storemode=property |title=Model of General Equilibrium in Multisector Economy with Monopolistic Competition and Hypergeometric Utilities |pdfUrl=https://ceur-ws.org/Vol-2795/short5.pdf |volume=Vol-2795 |authors=Vasily M. Goncharenko,Alexander B. Shapoval,Larisa V. Lipagina }} ==Model of General Equilibrium in Multisector Economy with Monopolistic Competition and Hypergeometric Utilities== https://ceur-ws.org/Vol-2795/short5.pdf 
Model of General Equilibrium in Multisector Economy
with Monopolistic Competition and Hypergeometric
Utilities
Vasily M. Goncharenkoa , Alexander B. Shapovala and Larisa V. Lipaginab
a
    National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia
b
    Financial University, Leningradsky pr., 49, 125167, Moscow, Russia


                                          Abstract
                                          We consider a general equilibrium model in a multisector economy with 𝑛 high-tech sectors where single-
                                          product firms compete monopolistically producing a differentiated good. Homogeneous sector is characterized
                                          by perfect competition. Workers attempt to find a job in high-tech sectors because of higher wages. However,
                                          it is possible for them to remain unemployed. Wages of employees in high-tech sectors are set by firms as a
                                          result of negotiations. Unemployment persists in equilibrium by labor market imperfections. All the consumers
                                          have identical preferences defined by a separable utility function of general form. We find the conditions such
                                          that the general equilibrium in the model exists and is unique. The conditions are formulated in terms of the
                                          elasticity of substitution between varieties of the differentiated good. We show that basic properties of the
                                          model can be described using families of hypergeometric functions.

                                          Keywords
                                          General equilibrium, monopolistic competition, variable elasticity of substitution, general utility function,
                                          hypergeometric functions




1. Introduction
In this paper we study a model of a multi-sector economy with monopolistic competition in 𝑛 high-
tech (industrial) sectors and perfect competition in the agricultural (traditional) sector. In our model
the consumer preferences are described by utility functions of a general type. If certain conditions are
met for elasticity of substitution between goods in industrial sectors, the existence of a symmetrical
general equilibrium can be proved in the model under the assumption that workers are mobile within
their sectors, but cannot move from sector to sector.
   Let us note that modeling of monopolistic competition is related to balancing between the need
to complicate the basic models to obtain adequate theoretical predictions and the possibility to get
analytical results. In particular, there is a natural question about the choice of the consumer utility
function. Going beyond the utility with constant substitution elasticity introduced in [1], which does
not allow describing the effects of market size, special utility functions were used in [2], [3]. As a
result, a deep theoretical analysis of the proposed structural models was obtained. In [4], [5] the
analytical approaches were developed to build a general equilibrium in economies where consumers
are empowered by separable preferences of a general type. However, the extension of their approaches
to the case of multi-sector economics, when the questions of trade or the heterogeneity of economic
agents are addressed, meets serious analytical difficulties. To identify the effects related to variable

MACSPro’2020: Modeling and Analysis of Complex Systems and Processes, April, 22–24, 2020, Venice, Italy
" vgoncharenko@hse.ru (V.M. Goncharenko); abshapoval@gmail.com (A.B. Shapoval); llipagina@fa.ru (L.V. Lipagina)

                                       Β© 2020 Copyright for this paper by its authors.
                                       Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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                  ISSN 1613-0073       CEUR Workshop Proceedings (CEUR-WS.org)
elasticity of substitution (VES) between products, it is sufficient to introduce a non-specific separable
utility function at the lower level and consider the Cobb-Douglas utility at the upper level. This is
what we do in this work.
   We are going to show that the basic properties of monopolistic competition models are described
using utility functions, which are a family of hypergeometric functions. We will demonstrate our idea
using the general equilibrium model which shows the dependence of the employment structure on
consumer demand and easily extends to other models.
   Hypergeometric functions and series, as well as their generalizations, play an important role in
many fields of modern applied mathematics. Although they first appeared in the middle of the sev-
enteenth century as elementary generalizations of converging series for the sum of a decreasing geo-
metric progression, and then were used by L. Euler in the study of solutions of differential equations,
now hypergeometric functions can be found in a broad range of mathematical knowledge varying
from algebraic topology to theoretical economics. A modern overview of applied research where
these functions and their properties play a key role can be found, for example, in [6].
   A power series of the form
                      π‘Žπ‘ 𝑧 π‘Ž(π‘Ž + 1)𝑏(𝑏 + 1) 𝑧 2 π‘Ž(π‘Ž + 1)(π‘Ž + 2)𝑏(𝑏 + 1)(𝑏 + 2) 𝑧 3
                 1+        +                   +                                   + ….                    (1)
                       𝑐 1     𝑐(𝑐 + 1)     2!          𝑐(𝑐 + 1)(𝑐 + 2)        3!
is called the hypergeometric series of the complex variable 𝑧 ∈ β„‚, depending on the parameters π‘Ž, 𝑏 ∈ β„‚
and 𝑐 ∈ β„‚ ⧡ {0, βˆ’1, βˆ’2, …} (see [7, 8]).
   If the complex plane is cut along the ray [1; +∞), then the series (1) defines an analytical function in
the complex plane, which is denoted by 2 𝐹1 (π‘Ž, 𝑏; 𝑐; 𝑧) and is called hypergeometric function. The indexes
in the entry 2 𝐹1 (π‘Ž, 𝑏; 𝑐; 𝑧) denote the number of parameters in the numerator and denominator of the
power series coefficients (1), respectively. Let us remark that generalized hypergeometric functions
𝑝 πΉπ‘ž (π‘Ž1 , … , π‘Žπ‘ ; 𝑏1 , … , π‘π‘ž ; 𝑧) are also used in applied economics problems (see, for example, [8])

                                                                                  𝑝
                                                  ∞                           ∞   ∏ (π‘Žπ‘˜ )𝑛
                                                            (π‘Ž1 )𝑛 … (π‘Žπ‘ )𝑛 𝑧 𝑛                𝑧𝑛
                 𝑝 πΉπ‘ž (π‘Ž1 , … , π‘Žπ‘ ; 𝑏1 , … , π‘π‘ž ; 𝑧) = βˆ‘                  β‹…    = βˆ‘ π‘˜=1π‘ž      β‹… ,
                                                        𝑛=0 (𝑏1 )𝑛 … (π‘π‘ž )𝑛 𝑛!    𝑛=0 ∏ (𝑏 )   𝑛!
                                                                                          π‘˜ 𝑛
                                                                                      π‘˜=1

where (𝑐)𝑛 – the Pohhammer symbol that is calculated according to the rule
                                         π‘›βˆ’1
                                 (𝑐)𝑛 = ∏(𝑐 + π‘˜) = 𝑐(𝑐 + 1) … (𝑐 + 𝑛 βˆ’ 1).
                                         π‘˜=0

   It is easy to see that the row for 0 𝐹0 (𝑧) is an exponent, and 1 𝐹0 (π‘Ž; 𝑧) is a row for (1 βˆ’ 𝑧)βˆ’π‘Ž . There-
fore, the hypergeometric functions are often considereded as generalizations of the exponential ones.
Further, for simplicity of notation when working with hypergeometric functions, indexes will be
omitted: 𝐹 (π‘Ž, 𝑏; 𝑐; 𝑧) = 2 𝐹1 (π‘Ž, 𝑏; 𝑐; 𝑧).
   Applications of hypergeometric functions in economics are also diverse. Thus, solutions for gener-
alizations of the Sollow model of economic growth can be written in terms of hypergeometric func-
tions (see [9]). Hypergeometric functions play a key role in describing solutions to the Uzava-Lucas
model of endogenous growth in a two-sector economy, [10]. In finance, the variety of solutions of
the Black-Scholes model containing all previously known analytically solvable cases were obtained
(see [11]) as well as new families of solutions.
   Note that we are not the first to use hypergeometric functions in models of monopolistic competi-
tion. For example, in a recent paper [12], hypergeometric functions set examples of demand functions
in which the distribution of firm productivity coincides with the distribution of sales. But it is we who
propose to use a family of such functions in order to exhaust all the studied values of the elasticity of
substitution between products.
   The use of a family of hypergeometric functions allows, on the one hand, to use the accumulated
methods of their research in analytics, and, on the other hand, allows us to draw conclusions that are
stable with respect to the variation of the utility function.


2. Model
Supply. Let us consider an economy consisting of 𝑛 hi-tech sectors where 𝑁𝑖 (𝑖 = 1, … , 𝑛) single-
product firms monopolistically compete in each sector, and a homogeneous sector with the perfect
competition. Due to the perfect competition firms price their products at the marginal cost in the
homogeneous sector. Prescribing the index 0 to this sector, we assume that prices 𝑝0 are equal to
wages 𝑀0 as productivity can be assigned to 1.
  In the 𝑖-th hi-tech sector a firm produces a good πœ‰π‘– with sector specific fixed costs 𝑐𝑖 and variable
                                                                                         πœ‘

costs associated with wages 𝑀(πœ‰π‘– ) of employees working with an inverse productivity 𝑐𝑖𝑣 . The price
𝑝(πœ‰π‘– ) can be found from the optimization problem to maximize the profit πœ‹(πœ‰π‘– )

                                                                                                      (2)
                                                                        πœ‘
                         πœ‹(πœ‰π‘– ) = 𝑝(πœ‰π‘– )𝑄(πœ‰π‘– ) βˆ’ 𝑐𝑖𝑣 𝑄(πœ‰π‘– )𝑀(πœ‰π‘– ) βˆ’ 𝑐𝑖 𝑀(πœ‰π‘– ) ⟢ max,

where 𝑄(πœ‰π‘– ) is the aggregate demand for the good πœ‰π‘– by all the consumers in the economy. Labor is
supposed to be the only production factor and a firm hires

                                                                                                      (3)
                                                                    πœ‘
                                           𝑙(πœ‰π‘– ) = 𝑐𝑖𝑣 𝑄(πœ‰π‘– ) + 𝑐𝑖

workers to produce 𝑄(πœ‰π‘– ) goods.
   The mass 𝑁𝑖 of firms in the 𝑖-th sector is defined by the free entry condition πœ‹(πœ‰π‘– ) = 0. In each
hi-tech sector we have 𝐿𝑖 employed and 𝐿𝑒𝑖 unemployed workers. If 𝐿0 is a number of workers in the
homogeneous sector and 𝐿𝑛+1 is the total number of unemployed, that is,
                                                         𝑛
                                                 𝐿𝑛+1 = βˆ‘ 𝐿𝑒𝑖 ,
                                                        𝑖=1

then the total number  of workers in the economy
                                                               𝑛
                                           = 𝐿0 + 𝐿𝑛+1 + βˆ‘ 𝐿𝑖 .
                                                              𝑖=1


Demand. The aggregate demand 𝑄(πœ‰π‘– ) for a good πœ‰π‘– is made of individual preferences of consumers.
Their incomes depend on the sector where they work. So, there are 𝑛 + 2 types of the incomes 𝑦𝑖 , 𝑖 =
0, 1, … , (𝑛 + 1) in the economy. We suppose that a consumer with an income 𝑦𝑗 decides upon her
demand in two steps.
   First, she differentiates between production sectors represented by consumption indices 𝐻𝑖𝑗 , 𝑖 =
0, … , 𝑛, maximizing the upper-tier utility

                                                                                                      (4)
                                             𝛽     𝛽      𝛽
                                     π‘ˆ = 𝐻0𝑗0 𝐻1𝑗1 … 𝐻𝑛𝑗𝑛 . ⟢ max,
                                                                        𝑛
Here the exponents 𝛽𝑖 ∈ (0, 1), 𝑖 = 0, 1, … , 𝑛 are such that βˆ‘ 𝛽𝑖 = 1. Because of Cobb-Douglas form of
                                                                       𝑖=0
utility the consumers spend their income proportionally to the exponents 𝛽𝑖 that is, a consumer with
income 𝑦𝑗 spends 𝛽𝑖 𝑦𝑗 for the good of the 𝑖-th sector.
   Second, each consumer from the 𝑗-th sector where 𝑗 = 0, … , 𝑛 + 1 chooses the demand π‘žπ‘— (πœ‰π‘– ) for each
𝑖 = 1, … , 𝑛 maximizing the consumption index

                                       𝐻𝑖𝑗 = ∫ 𝑒𝑖 (π‘žπ‘— (πœ‰π‘– )) π‘‘πœ‰π‘– ⟢ max,                                         (5)
                                             𝑁𝑖

with a low-tier four times differentiable increasing concave utility function 𝑒𝑖 (𝜘), Optimization prob-
lem (5) subject to budget constraint

                                             ∫ 𝑝(πœ‰π‘– )π‘žπ‘— (πœ‰π‘– )π‘‘πœ‰π‘– ≀ 𝛽𝑖 𝑦𝑗                                        (6)
                                            𝑁𝑖

reflects preferences for the 𝑖-th sector differentiated good.

Elasticity of substitution between hi-tech varieties. The function

                                                             𝑒 β€² (𝜘)
                                                  πœŽπ‘– (𝜘) = βˆ’ ′′𝑖
                                                            𝑒𝑖 (𝜘)𝜘

to be interpreted the elasticity of the substitution between hi-tech goods turns out to be key part to
write down the solution of the consumer’s problem (5), (6). Namely, its first order conditions with
respect to the price 𝑝(πœ‰π‘– ) for a particular good πœ‰π‘– are

                                            E𝑝(πœ‰π‘– ) π‘žπ‘— (πœ‰π‘– ) = βˆ’πœŽπ‘– (π‘žπ‘— (πœ‰π‘– )),                                  (7)

so that the elastisity E𝑝(πœ‰π‘– ) π‘žπ‘— (πœ‰π‘– ) of the demand π‘žπ‘— (πœ‰π‘– ) with respect to its price 𝑝(πœ‰π‘– ) is opposite to the
elasticity of substitution.
   Exploring a multi-sector economy, we introduce the mean value of individuals’ elasticity of substi-
tution
                                                      𝑛+1
                                                          π‘žπ‘— (πœ‰π‘– )𝐿𝑗
                                             S(πœ‰π‘– ) = βˆ‘              πœŽπ‘– (π‘žπ‘— (πœ‰π‘– ))                              (8)
                                                      𝑗=0 𝑄(πœ‰π‘– )
                                                                                    𝑛+1
between varieties (MES) where 𝑄(πœ‰π‘– ) is the aggregated demand 𝑄(πœ‰π‘– ) = βˆ‘ π‘žπ‘— (πœ‰π‘– )𝐿𝑗 . One can easily
                                                                                    𝑗=0
check that property (7) is extended to S(πœ‰π‘– ):

                                              E𝑝(πœ‰π‘– ) 𝑄(πœ‰π‘– ) = βˆ’S(πœ‰π‘– ),                                         (9)

Labor Market. Workers are motivated to find a job in the hi-tech sectors as wages there are larger.
Nevertherless, to be employed in each sector including homogeneous one, they have to get some
sector specific skills. So, workers should choose a sector they would like to get a job, get required
skills, and then enter the sector job market. As soon as the sector is chosen they can’t change their
choice. Let us remark that an arbitrary number of workers can be employed in the homogeneous
sector. In any hi-tech sector firms hire the required number of employees while the other become to
be unemployed. We assume that the labor market exhibits some frictions, so that rejected candidates
cannot find a job in another sectors as they are not qualified for it. Following Stole and Zwiebel [13],
one can write out the wages 𝑀(πœ‰π‘– ) = 𝑀𝑖 obtained as a result of bargaining:
                                                                                     πœ‘
                                                    𝑝(πœ‰π‘– )        𝑙(πœ‰π‘– )2 βˆ’ (𝑐𝑖 )2
                                          𝑀(πœ‰π‘– ) =    𝑣    + 𝑀0                    .                                       (10)
                                                   ( 𝑐𝑖         )     2𝑙(πœ‰π‘– )2
   We assume that workers’ choice of the labor market is balanced on average. As the probabilities
to be employed and unemployed respectively in sector 𝑖, 𝑖 = 1, … , 𝑛, are 𝐿𝑖 /(𝐿𝑖 + 𝐿𝑒𝑖 ) and 𝐿𝑒𝑖 /(𝐿𝑖 + 𝐿𝑒𝑖 )
they face identical expected incomes that coincide with the incomes got by workers employed in the
homogeneous sector:
                                         𝑦𝑖 𝐿𝑖     𝑦𝑛+1 𝐿𝑒𝑖
                                                 +          = 𝑦0 .                                      (11)
                                       𝐿𝑖 + 𝐿𝑖 𝐿𝑖 + 𝐿𝑒𝑖
                                               𝑒

   A flat tax with some rate 𝛼 ∈ (0, 1) is applied to the wages of all employed workers (including
workers employed in the homogeneous sector) and distributed equally between unemployed agents
as an unemployment benefit. This sets the (netto) income of employed in the hi-tech sectors, employed
in the homogeneous sector, and unemployed workers to, respectively,

                                     𝑦𝑖 = (1 βˆ’ 𝛼)𝑀𝑖 , 𝑖 = 1 … , 𝑛, 𝑦0 = (1 βˆ’ 𝛼)𝑀0 ,
                                                 𝛼(𝐿0 𝑀0 + 𝐿1 𝑀1 + … + 𝐿𝑛 𝑀𝑛 )
                                         𝑦𝑛+1 =                                .                                           (12)
                                                              𝐿𝑛+1

3. Definition of Equilibrium and Assumptions
The set of prices {𝑝̂ (πœ‰π‘– )}, individual demands {π‘žΜ‚ 𝑗 (πœ‰π‘– )}, firms’ outputs {𝑠̂ (πœ‰π‘– )}, the number 𝑁̂ 𝑖 of firms,
                                                                                                 𝑒
wages 𝑀̂ 𝑖 , incomes 𝑦̂ 𝑗 , the numbers 𝐿̂𝑖 of workers in each sector, and the number 𝐿̂𝑖 of rejected job
market candidates in hi-tech sectors (𝑖 = 1, … , 𝑛, 𝑗 = 0, 1, … , 𝑛 + 1, πœ‰π‘– ∈ [0, 𝑁̂ 𝑖 ]) constitute a general
equilibrium, if the following conditions are satisfied:
    β€’ For any fixed 𝑗 = 0, 1, … , 𝑛 + 1, individual demands {π‘žΜ‚ 𝑗 (πœ‰π‘– )}πœ‰π‘– ∈[0,𝑁𝑖 ],𝑖=1,…,𝑛 solve a consumer’s
      problem (4)–(6) with 𝑝(πœ‰π‘– ) = 𝑝̂ (πœ‰π‘– ), 𝑁𝑖 = 𝑁̂ 𝑖 , 𝑦𝑗 = 𝑦̂ 𝑗 , 𝑖 = 1, … , 𝑛, πœ‰π‘– ∈ [0, 𝑁̂ 𝑖 ];

    β€’ For any fixed 𝑖 = 1, … , 𝑛, and πœ‰π‘– ∈ [0, 𝑁̂ 𝑖 ] a firm’s optimization problem with respect to a
      single price 𝑝(πœ‰π‘– ) is defined by Equation (2), where 𝑀(πœ‰π‘– ) = const = 𝑀̂ 𝑖 and 𝑄(πœ‰π‘– ) = βˆ‘π‘—=0
                                                                                                𝑛+1
                                                                                                    π‘žπ‘— (πœ‰π‘– )𝐿̂𝑗
      depends implicitly on 𝑝(πœ‰π‘– ). Namely, for any fixed 𝑗 = 0, 1, … , 𝑛+1, we define individual demands
      {π‘žπ‘—βˆ— (πœ‚π‘˜ )}πœ‚π‘˜ ∈[0,𝑁̂ π‘˜ ],π‘˜=1,…,𝑛 as the solutions of a consumer’s problem (4)–(6) with
          – the price for the specific variety πœ‰π‘– equalled to 𝑝(πœ‰π‘– ),
          – the equilibrium prices for the other varieties: 𝑝(πœ‚π‘˜ ) = 𝑝̂ (πœ‚π‘˜ ) if π‘˜ β‰  𝑖 or π‘˜ = 𝑖 and πœ‚π‘– β‰  πœ‰π‘– ,
          – and π‘π‘˜ = 𝑁̂ π‘˜ , π‘˜ = 1, … , 𝑛, 𝑦𝑗 = 𝑦̂ 𝑗 .
       Then these π‘žπ‘—βˆ— (πœ‰π‘– ) are substituted for π‘ž(πœ‰ ) into the Equation for 𝑄(πœ‰π‘– ). Finally, we require that
       𝑝̂ (πœ‰π‘– ) solves a firm’s optimization problem formulated above.

    β€’ The market clearance condition is hold: 𝑠̂ (πœ‰π‘– ) = βˆ‘π‘›+1           Μ‚
                                                          𝑗=0 π‘žΜ‚ 𝑗 (πœ‰π‘– )𝐿𝑗 , 𝑖 = 1, … , 𝑛.
                                                                   𝑒
    β€’ Equations (3)–(10) hold with 𝐿𝑗 = 𝐿̂𝑗 , 𝐿𝑒𝑖 = 𝐿̂𝑖 , 𝑦𝑗 = 𝑦̂ 𝑗 , 𝑀𝑖 = 𝑀̂ 𝑖 , 𝑝(πœ‰π‘– ) = 𝑝̂ (πœ‰π‘– ), and 𝑄(πœ‰π‘– ) = 𝑠̂ (πœ‰π‘– ),
      where 𝑖 = 1, … , 𝑛, 𝑗 = 0, 1, … , 𝑛 + 1, πœ‰π‘– ∈ [0, 𝑁̂ 𝑖 ]. Additionally, βˆ«π‘Μ‚ 𝑖 𝑙(πœ‰π‘– ) π‘‘πœ‰π‘– = 𝐿̂𝑖 .

    β€’ Firms are free to enter: 𝑝̂ (πœ‰π‘– )𝑠̂ (πœ‰π‘– ) βˆ’ 𝑐𝑖𝑣 𝑠̂ (πœ‰π‘– )𝑀̂ (πœ‰π‘– ) βˆ’ 𝑐𝑖 𝑀̂ (πœ‰π‘– ) = 0, 𝑖 = 1, … , 𝑛, πœ‰π‘– ∈ [0, 𝑁̂ 𝑖 ].
                                                                          πœ‘
  We discuss the existence and uniqueness of the equilibrium in the model under the following As-
sumptions:
Assumption 1. Differentiable functions πœŽπ‘– (𝜘) are supposed to be monotonous and satisfy the condi-
tion
                            πœŽπ‘– (𝜘) > 1, for all 𝑖 = 1, … , 𝑛 and 𝜘 β‰₯ 0.                       (13)
Assumption 2. If 𝐿𝑗 is the equilibrium number of employed workers in sector 𝑗 then we assume that
𝐿𝑗 β‰₯ 1 and
                                𝐿𝑗
                   |πœŽπ‘–β€² (𝜘)| <       for all 𝑖 = 1, … , 𝑛, 𝑗 = 0, … , 𝑛 + 1, and 𝜘 β‰₯ 0,      (14)
                               2𝑛𝐢𝑖
where 𝐢𝑖 = 𝑐𝑖 /𝑐𝑖𝑣 is the ratio of the fixed to variable costs.
              πœ‘


Assumption 3. The diversity of the equilibrium individual demands is supposed to be limited:
                      πœŽπ‘– (π‘žπ‘— (πœ‰π‘– ))
                                      <2    for all 𝑖 = 1, … , 𝑛, 𝑗, 𝑗 β€² = 0, … , 𝑛 + 1, and 𝜘 β‰₯ 0.         (15)
                      πœŽπ‘– (π‘žπ‘— β€² (πœ‰π‘– ))

If πœŽπ‘– increases, we assume additionally that there exists some 𝛿𝑖 > 0 such that inequalities

                               πœŽπ‘–β€² (π‘žπ‘— (πœ‰π‘– ))π‘žπ‘— (πœ‰π‘– ) < 𝛿𝑖 < πœŽπ‘– (π‘žπ‘— (πœ‰π‘– )) βˆ’ 1,   𝑖 = 1, … , 𝑛,             (16)

holds for all equilibrium individual demands π‘žπ‘— (πœ‰π‘– ), 𝑖 = 1, … , 𝑛, 𝑗 = 0, … , 𝑛 + 1.
   Zhelobodko et al. [4] used an analogue of Assumptions 1 and 2 to prove the existence and unique-
ness of the equilibrium in a single-sector economy. Assumption 2 links together the equilibrium
number of employed workers in each sector, variability of the elasticity of substitution, and the ratio
of fixed to variable costs. It is easy tio see that small values of 𝐢𝑖 correspond to low fixed or/and high
variable costs. Both effects can be attributed to less efficient economies.
   Assumption 3 is most restrictive. It appears because the economy is multi-sector. Nevertheless, one
can check that if the economy (number of individuals ) is large enough then Assumption 3 follows
from Assumption 2.


4. Equilibrium and its Properties
Proposition 1. Let Assumptions 1–3 be satisfied. Then a general equilibrium exists, and it is unique.
This equilibrium is symmetrical with respect to varieties of the 𝑖-th differentiated product: 𝑝(πœ‰π‘– ) and π‘žπ‘— (πœ‰π‘– )
depends on 𝑖 but not on specific varieties πœ‰π‘– . We denote

                             𝑝𝑖 = 𝑝(πœ‰π‘– ),     π‘žπ‘–π‘— = π‘žπ‘— (πœ‰π‘– ),     𝑄𝑖 = 𝑄(πœ‰π‘– ),     S𝑖 = S(πœ‰π‘– )              (17)

the symmetrical equilibrium variables. Then they are given by the following expressions:

                                         𝑄𝑖 = 𝐢𝑖 (S𝑖 βˆ’ 1) , 𝑖 = 1, … , 𝑛,                                   (18)
                                                S𝑖 𝑐𝑖𝑣 𝑀𝑖
                                           𝑝𝑖 =           , 𝑖 = 1, … , 𝑛,                                   (19)
                                                S𝑖 βˆ’ 1
                                        (1 βˆ’ 𝛼)𝛽𝑗 𝑄𝑖 (S𝑗 + 1)𝑄𝑖
                                  π‘žπ‘–π‘— =               =            , 𝑗 = 1, … , 𝑛,                          (20)
                                             𝐿𝑗            S𝑗
                                                            𝑄𝑖
                                                     π‘žπ‘–0 = ,                                                (21)
                                                            
                                                                𝛼𝑄𝑖
                                                   π‘žπ‘–,𝑛+1 =          .                                     (22)
                                                                𝐿𝑛+1
The following Equations determine the equilibrium wages, number of employed and unemployed workers,
and number of firms:
                                             S𝑖 + 1
                                         𝑀𝑖 =       𝑀0 , 𝑖 = 1, … , 𝑛                                      (23)
                                               S𝑖
                                           S𝑖
                         𝐿𝑖 = (1 βˆ’ 𝛼)𝛽𝑖         , 𝑖 = 1, … , 𝑛, 𝐿0 = (1 βˆ’ 𝛼)𝛽0 ,                         (24)
                                          S𝑖 + 1
                                                         𝑛
                                                             𝛽𝑗
                                   𝐿𝑛+1 =  𝛼 + (1 βˆ’ 𝛼) βˆ‘            ,                                     (25)
                                           (            𝑗=1 S𝑗 + 1 )
                                           𝛽𝑖 /(S𝑖 + 1)
                                 𝐿𝑒𝑖 =    𝑛                𝐿𝑛+1 ,        𝑖 = 1, … , 𝑛,                     (26)
                                         βˆ‘π‘—=1 𝛽𝑗 /(S𝑗 + 1)
                                                (1 βˆ’ 𝛼)𝛽𝑖 
                                         𝑁𝑖 =    πœ‘          ,      𝑖 = 1, … , 𝑛.                           (27)
                                                𝑐𝑖 (S𝑖 + 1)
   In the frame of this paper it is impossible to give the detailed economic interpretation of the Propo-
sition 1. Let us do several remarks. Equilibrium variables depend on the MES S𝑖 . If S𝑖 is large and the
demand is elastic, consumers are almost indifferent between varieties. On the contrary, small values
of S𝑖 enlarge the diversity of the differentiated good. Therefore, the quantity 1/S𝑖 is interpreted as
the love for variety in various papers (see, f.e., Zhelobodko et al. [4])
   Equation (18) also relates the output of varieties to the efficiency of the economy: more efficient
economies produce more amount of specific varieties. The number of unemployed workers expect-
edly increases with the taxation rate 𝛼: a growth of the unemployment benefit subdues the risk of
unemployment, Equation (25). The size of each sector can be measured in terms of the number of
firms operated in the sector or the number of workers employed by those firms. As in other similar
models, these two quantities are proportional, Equations (24) and (27).


5. Two Families of Utilities
Let us consider two families of utilities. The first of them is defined by
                         ⎧    𝐴                             2       1 𝜘
                                    (𝜘(𝜘 + 2)) 𝐴 𝐹 (1, 2 βˆ’ ; 2 βˆ’ ; βˆ’ ) , if 𝐴 > 1,
                                                π΄βˆ’1
                         βŽͺ
                         βŽͺ
                  𝑒(𝜘) = ⎨ 2(𝐴 βˆ’ 1) √                       𝐴       𝐴 2
                         βŽͺ
                         βŽͺln
                         ⎩ (  𝜘  + 1 +    𝜘 2 + 2𝜘 ,
                                                   )                       if 𝐴 = 1,
where 𝐴 β‰₯ 1 is a parameter. Then the elasticity of substitution is
                                    𝜎 (𝜘) = 𝐴(1 + 1/(𝜘 + 1),               𝐴 β‰₯ 1.
  The second family of utility functions is given by equation
                        {
                            𝐴 1βˆ’ 𝐴1
                                                               ; 2 βˆ’ 𝐴1 ; βˆ’2𝜘 ) , if 𝐴 > 1,
                                                1            1
                           π΄βˆ’1 𝜘    (2𝜘 + 1)1+ 2𝐴 𝐹 (1, 2 βˆ’ 2𝐴
                𝑒(𝜘) = √                  √
                          2 2𝜘 + 1 + ln √2𝜘+1βˆ’1
                                            2𝜘+1+1
                                                                                  if 𝐴 = 1,
In this case the elasticity of substitution is 𝜎(𝜘) = 𝐴(2 βˆ’ 1/(𝜘 + 1). The function 𝑒(𝜘) defined above are
represented respectively by the sum of two power functions up to terms of higher order:
                         2βˆ’1/𝐴 𝐴      1     π΄βˆ’1           1          1
                𝑒(𝜘) =           𝜘 1βˆ’ 𝐴 βˆ’            𝜘 2βˆ’ 𝐴 + 𝑂(𝜘 3βˆ’ 𝐴 ),                𝐴 > 1,   𝜘 β‰ͺ 1,
                          π΄βˆ’1 (           2𝐴(2𝐴 βˆ’ 1)        )
                          𝐴        1     π΄βˆ’1          1          1
                𝑒(𝜘) =        𝜘 1βˆ’ 𝐴 +           𝜘 2βˆ’ 𝐴 βˆ’ 𝑂(𝜘 3βˆ’ 𝐴 ),   𝐴 > 1,   𝜘 β‰ͺ 1,
                         π΄βˆ’1(          𝐴(2𝐴 βˆ’ 1)        )
where minus is used ahead of big-O just to stress that the next term of the series is negative.
   If an arbitrary function from any of these families represent the low-tier utility of consumers then
for any choice of the economy primitives, that is, variable {𝑐𝑖𝑣 }𝑛𝑖=1 and fixed {𝑐𝑖 }𝑛𝑖=1 costs and the
                                                                                     πœ‘

exponents 𝛽𝑖 of the Cobb–Douglas upper tier utility, there exists such a value of ξˆΈΜ‚ that for any  > ξˆΈΜ‚
Assumption 2 holds. So, all the Assumptions 1–3 are satisfied.


6. Conclusion
In the paper we introduced the model of a multi-sector economy with high-tech sectors producing
differentiated goods and firms competing monopolistically. A homogeneous sector is characterized by
perfect competition. The conditions are found of symmetric equilibrium to exist and unique which
allow to find individual consumer demands, product prices, wages, the number of employed and
unemployed, and the number of firms in each sector. Two one-parameter families of hypergeometric
functions are defined as examples of utility functions of a general type that satisfy the conditions of
existence and uniqueness. The examples found correspond to the decreasing and increasing elasticity
of substitution between products. When changing parameters, all possible values of the equilibrium
elasticity of substitution are exhausted. So, the complete analysis of the equilibrium in the model is
possible.


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