Non‐Homothetic Multisector Growth Models

• Date: 01 May 2015
• DOI: http://doi.org/10.1111/rode.12139
• Author: Ulla Lehmijoki,Bjarne S. Jensen,Elena Rovenskaya
• Published date: 01 May 2015

Non-Homothetic Multisector Growth Models

Bjarne S. Jensen, Ulla Lehmijoki, and Elena Rovenskaya*

Abstract

Multisector growth (MSG) models are dynamic versions of computable general equilibrium (CGE) models.

Non-homothetic preference (utility) functions are required for the evolution of factor allocations and

industrial structures in accordance with consumption expenditure patterns implied by the non-unitary

income elasticities observed in all budget data since Engel in the 1850s. But comparative static general

equilibrium solutions and particularly solving the dynamics of MSG models require explicit specifications

of all demand and cost (price) functions. On the demand side, the constant differences of elasticity of sub-

stitution (CDES) non-homothetic indirect utility functions and Roy’s identity provide the explicit

Marshallian demand functions and budget shares. Sectorial constant elasticity of substitution (CES) cost

functions and Shephard’s lemma provide the explicit relative commodity price functions and the sectorial

cost shares and capital–labor ratios. Walrasian equilibria are given by one equation and the multisector

dynamics by three differential equations. Benchmark solutions are given for three cost regimes of a

10-sector MSG model. History patterns of industrial/allocational evolutions are recognized.

1. Introduction

“Early on one would have expected much more work on multisector growth models

than there has been” (Solow, 2005, p. 4). Long ago Kuznets (1966, p. 1) wrote: “We

identify the economic growth of a nation as a sustained increase in per capita or per

worker product, most often accompanied by an increase in population and usually by

sweeping structural changes.” A standard textbook such as Barro and Sala-i-Martin

(2004) does not go beyond two final goods.

The economic literature has paid some attention to dynamic general equilibrium

models with the industrial patterns of structural change, e.g. Echevarria (1997) and

Kongsamut et al. (2001) that study three-sector growth models, but the sectorial produc-

tion functions in Kongsamut et al. (2001) are identical up to proportionality constant and

hence the relative prices of three different goods are not affected by factor prices, factor

allocations or change over time. The factor reallocations and sectorial changes in output

composition are entirely driven by differences in the income elasticities of demand (here

the linear expenditure system (LES) and its non-homothetic preferences are used in an

intertemporal optimization). Echevarria (1997) uses three Cobb–Douglas (CD) technol-

ogies and non-homothetic preferences that converge to CD preferences. The time paths

in this three-sector model are asymptotically equivalent to the steady-state values in the

CD growth model with unitary demand elasticities. Echevarria (2010) extends this three-

sector model in modified forms to include internatonal trade. Buera and Kaboski (2009)

analyses further the balanced growth paths of Kongsamut et al. (2001) and the biased

* Jensen: Department of Environmental and Business Economics, University of Southern Denmark, Niels

Bohrs, Vej 9, 6700 Esbjerg, Denmark. E-mail: bsj@sam.sdu.dk. Lehmijoki: Helsinki Center of Economic

Research (HECER), University of Helsinki, Finland. Rovenskaya: Faculty of Compuational Mathematics

and Cybernetices, Lomonosov Moscow State University, Moscow, Russia. International Institute for

Applied Systems Analysis (IIASA), Laxenburg, Austria. For many discussions and comments, we are

indebted to the late Arkady V. Kryazhimskiy, Steklov Institute of Mathematics, Russian Academy of Sci-

ences, Moscow, Russia, and IIASA, Laxenburg, Austria.

Review of Development Economics, 19(2), 221–243, 2015

DOI:10.1111/rode.12139

© 2015 John Wiley & Sons Ltd

productivity growth story of Ngai and Pissarides (2007). Bonatti and Felice (2008) study

the transformation of a two-sector economy with a technologically progressive and stag-

nant industry. Buera and Kaboski (2012) examines the role of high-skilled labor behind

the growth of the service sector.

Jensen and Lehmijoki (2011) studied N-sector growth models with homothetic con-

sumer preferences and simulated a growing 10-sector constant elasticity of substitution

(CES) economy. Our objective is to equip multisector growth models with non-

homothetic preferences such that its consumer budget shares can actually be sensibly

related to observed Engel curve patterns. Our outlook is much in accordance with the

perspectives in Herrendorf and Valentinyi (2012) and Herrendorf et al. (2013), combin-

ing both the evolution of price and income elasticities as determinants of structural trans-

formations. Instead of their three-sector model and non-homothetic LES demand, we

apply the non-homothetic constant differences of elasticity of substitution (CDES)

demand system of Houthakker (1960), Jensen et al. (2011) to a MSG (10-sector) model.

A richer system than LES is needed, as LES only allows for price inelastic demands.

As to strategy and methodology, we use Shephard’s lemma and Roy’s identity to

combine the supply and demand sides of general equilibrium models (see Roy, 1947).

In this way the Walras–Pareto general equilibrium is parametrically specified. Hence

the multisector equilibrium dynamics can be explicitly derived and solved (Pareto,

1971; Walras, 1954).

The paper is organized as follows. Section 2 starts with the central theoretical core of

the growth theory consisting of cost/production and utility functions. As is well-known

from duality theory, there are two approaches to the deriving systems of optimal con-

sumer and factor demand functions. One procedure starts from functional forms of

direct utility and production functions (satisfying proper regularity properties) and then

uses Lagrangian techniques to solve the optimization problem. The second procedure

begins with differentiable functional forms of indirect utility and cost functions to

obtain consumer and factor demands by differentiation. The difficulty with the first

methodology is that it is usually impossible to obtain the consumer and factor demands

as explicit functions in the parameters of direct utility/production functions. Hence we

proceed with the second method. National income accounting of a multisector economy

is integrated with Walras law and general equilibrium conditions.

In section 3, the general equilibrium of the non-homothetic multisector economy

with two primary factors (labor and capital) is expressed compactly by just one equa-

tion. Section 4 introduces the equations of factor accumulation, which give the laws of

motion that must be solved. The dynamics of our non-homothetic multisector

economy can be expressed by three differential equations. Section 5 presents in detail

the benchmark solutions of the general equilibrium dynamics for the CDES 10-sector

economy with three different parameter regimes. Section 6 concludes.

2. Structure of Non-Homothetic N-Sector General Equilibrium Model

CES Sectoral (Industry) Cost Functions and Relative Price Functions

Consider an economy consisting of Nindustries (sectors) and two primary factors,

labor Land capital Kwith (w,r) as the factor prices of labor and capital services. Yiis

the total output of sector (commodity), i=1,...,N.

Our industry (sector) cost function, Ci(w,r,Yi), must be a smooth regular cost func-

tion, cf. Diewert (1974, p. 111), McFadden (1978, p. 13), Lehmijoki (1984, p. 386), Varian

(1992, p. 72), Silberberg and Suen (2001, p. 205), i.e. (i) homogeneous of degree one in

222 Bjarne S. Jensen et al.

© 2015 John Wiley & Sons Ltd