Innovation, specialization and growth in a model of structural change

Rainer Andergassen; Franco Nardini; Massimo Ricottilli

doi:10.1515/bejm-2017-0095

Article

Innovation, specialization and growth in a model of structural change

Rainer Andergassen , Franco Nardini and Massimo Ricottilli

Published/Copyright: March 21, 2018

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From the journal The B.E. Journal of Macroeconomics Volume 18 Issue 2

Abstract

In this paper we investigate the process of creation and destruction of industries as it stems from productivity increasing innovations and from the induced changes of consumption patterns. In our model industries whose demand increases experience an expansion of the number of intermediate goods and hence of their research effort, while those whose demand declines undergo a cost-cutting restructuring with a corresponding reduction of the number of intermediates. We show that if aggregate consumption is concentrated on high (low) priority goods in the early (later) stages of the economy’s development and spread out more evenly in an intermediate stage, then the diversification of the economy over the development path is inversely U-shaped: a result that is consistent with the empirical evidence in Imbs and Wacziarg 2003 “Stages of Diversification.” American Economic Review 93: 63–86.

Keywords: development; innovation; specialization; structural change

JEL Classification: O10; O30; O40

Acknowledgement

The authors are indebted to Joseph Zeira, Josef Zweimüller and two anonymous referees for helpful comments and suggestions. The usual disclaimer applies.

Appendix

The consumption function. The solution to the consumer’s problem is as follows. Let jt∗ be the optimal variety at time t, that is U(jt∗,t)>U(j,t) for each j other than jt∗, then (9) follows and from which the indirect utility can be obtained. Assumption 1 states sufficient conditions for the existence of jt∗. More particularly, it guarantees that the utility is monotonically increasing in j for the consumption bundles with variety lower than jt∗ and monotonically decreasing for baskets with variety larger than jt∗, that is,

(18)U(j,t)<U(j+1,t) for each j=1,...,jt∗−1

and

(19)U(j+1,t)<U(j,t) for each j=jt∗,...,J−1

To see this, taking into account (9), write U(j,t)≶U(j+1,t), i.e. the inequality of indirect utilities, as Ω(j,t)≶χ(j,t), where Ω(j,t)≡(∑h=1jdhdjpj+1,t)αj+1Πh=1j(∑h=1jdhdjph,t)αj+1−αj. Assumptions made on the production and demand side of the economy guarantee that if at time t = 0 prices of higher priority goods (in reference to jt∗) are lower than those of lower priority goods then this is true also for t > 0. As a consequence, Assumption Assumption 1 (1) and (3) guarantee that Ω(j,t) is increasing in j. Conditions (18) and (19) can be written as follows

Ω(j,t)<χ(j,t) for each j=1,...,jt∗−1

and

Ω(j,t)>χ(j,t) for each j=jt∗,...,J−1

Since at each time period t, Ω(j,t) and χ(j,t) cross at most once, an unique jt∗ exists.

Changes in prices or income lead to a revision of the individual’s quantity choice and eventually also of the variety one. Suppose that at time t price p_j,t, j⩽jt∗, declines. This event decreases Ω(jt∗,t) and may trigger an increase in the variety of the consumption bundle. In this case, a re-balancing of the individual’s consumption pattern occurs, decreasing the consumption of some or all high priority goods to accommodate the consumption of an additional variety of lower priority. If the variety consumed remains unchanged, the individual’s consumption of good j simply increases. ■

Derivation of equilibrium income. Given the assumption r = 0 and since in equilibrium h_f = h, (12) becomes V=πh; consequently the first order conditions for problem (13) yield

(20)hj,tk=hj,t=Vj,ta,πj,t=F,and hj,t=Fa.

Employment in both final and intermediate good producer can be characterized in terms of aggregate output. Since lj,ty=βj,t1bj,tYtpj,t and xj,t(k)=βj,t1aj,t(k)Ytpj,t, the number of workers engaged in producing a final good j is

(21)lj,ty=βj,tYteλ(1+δkj,t)

From (21) and (5) it follows that employment by intermediate good producers in each industry is

(22)lj,tx=∫0kj,tlj,t(k)dk=βj,t∫0kj,tbj,tηaj,t(k)Yteλ(1+δkj,t)dk=βj,tYtδkj,teλ(1+δkj,t)

Summing over j in the above formula and in (21), total manufacturing employment turns out to be:

(23)Lt=∑j=1J(lj,ty+lj,tx)=Yteλ.

The third of (20) and (15) allow us to calculate the employment H_t that followers hire to conjure up the next round of innovations.

Considering (23) and having in mind that the assumption that the overall employment is constant L=Lty+Ltx+Ht, H_t is, then, equal to

(24)L−Yteλv=Ht

Finally from (24), (15) and the third of (20),

(25)L=Y{1eλ+1Fa(1−e−λ)}.

aggregate final good output (16) follows. ■

Proof of Proposition 1.

Using the accounting definition of aggregate income and taking the time derivative, aggregate growth,^[25] is:

Y⋅tYt=∑j=1Jβj,t(p⋅j,tpj,t+yj,t⋅yj,t).

In this model, however, the nominal wage rate is kept constant and productivity gains translate into proportionally lower prices such that the real wage rate and likewise real monopolists’ profits grow in step with productivity. The long-run real growth Y_R,t turns out to be:

YR,t⋅YR,t=∑j=1Jβjyj,t⋅yj,t,

from which we can characterize the long-run growth rate, recalling that the Poisson arrival rate is hj,t=Fa , as:

gYR=Fa∑j=1Jβj(ekjλ−1).

Since k_j = β_jf(L), the expression in (17) follows. ■

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Published Online: 2018-03-21

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Keywords for this article

development; innovation; specialization; structural change