Division of labour and innovation with indivisibilities: lessons from A. Smith

Bank i Kredyt 43 (6), 2012, 7–28

www.bankandcredit.nbp.pl www.bankikredyt.nbp.pl

Division of labour and innovation with indivisibilities: lessons from A. Smith

Krzysztof Makarski*

Submitted: 28 March 2012. Accepted: 22 May 2012.

Abstract

We study division of labour, innovation, and economic growth in a world with indivisibilities.

In order to increase the division of labour, more specialized, labour saving capital varieties must be invented, which can be done only subject to a minimum size requirement. Thus, the division of labour is limited by the size of the market. Furthermore, the division of labour has a major impact on labour productivity. Due to the minimum size requirement, producers want to charge nonlinear prices (two part tariffs). In the model we obtain interesting dynamics.

Depending on the parameters our economy can grow unboundedly, can grow up to a certain (even very high) level and then stagnate, or can be stuck in a poverty trap.

Keywords: innovation, indivisibility, division of labour, extent of the market JEL: J22, O30, O40, D90, D24, D40

* National Bank of Poland; Warsaw School of Economics; e-mail: kmakar@sgh.waw.pl.

K. Makarski

8

1. Introduction

Most of the economic literature on the division of labour, following A. Smith (1776) and A. Young (1928), emphasizes the importance of the extent of the market as a constraint on the division of labour. A good review of the literature is given by Lavezzi (2001) and Yang and Ng (1998). Since the classic paper of Dixit and Stiglitz (1977), the theory that formalizes economic of scale has been developed. This literature includes, among others, Ethier (1982), Krugman (1979), Judd (1985), Romer (1986; 1987), and Grossman and Helpman (1989). In these papers intermediate goods are aggregated into final goods with a Dixit-Stiglitz technology, thus it is optimal to have as many intermediate goods as possible. Since production of each intermediate good is subject to a fixed cost, the extent of the market limits the number of intermediate goods and therefore specialization.

Houthakker (1956) suggested another way to model the division of labour. In his view there is a trade-off between economies of specialization and the transaction costs associated with specialization. Papers that explore this line of reasoning include Baumgardner (1988), Kim (1989), Yang and Borland (1991), and Becker and Murphy (1992). This literature recognizes “coordination costs”, not the extent of the market, as the major constraint on the division of labour.

Literature on external economies of scale is also related to our topic. In a seminal paper Chipman (1970) argues that external economies of scale could be logically modelled as perfectly competitive. Markusen (1990) provides microfoundations for external economies of scale and shows, when ad hoc specifications used in trade and growth theories, are consistent with those microfoundations. In a more recent paper Grossman and Ross-Hansberg (2010) study external economies of scale at the industry level using the model of Bertrand competition. All those models are based on increasing returns of scale.

We follow the approach proposed by Edwards and Star (1987). They argue that “labour specialization results in scale economies only through indivisibility or other nonconvexity in the use of labour”. But, we introduce indivisibility on capital, rather than labour. In our model the division of labour is limited by the extent of the market without increasing returns on the firm level. Increasing returns arise endogenously and only in an aggregate production function. In our model, the division of labour and capital are related. In fact, the division of labour is embodied in the specialization of capital. The increase in capital diversity leads to more differentiated tasks being performed in the economy and increases the division of labour. As capital becomes more diversified, labour becomes more productive. Thus, technological progress is labour saving, as defined by Acemoglu (2003). The ideas that more advanced technology requires more advanced capital was explored by Boldrin and Levine (2002), but in their model there is no indivisibility and no division of labour.

In our model there are infinitely many technologies, which differ in the level of sophistication.

In order to use more sophisticated technology (with more diversified capital) the minimum size

requirement must be met. This indivisibility is not at the individual firm level, but rather at the

aggregate level. This captures the idea, that in order to use more advanced capital, firms need

access to engineers, educated labour force and so on. Thus, action of other firms matter. The key

mechanism in our model works as follows. As the extent of the market increases, there is more

innovation (i.e. more sophisticated technologies are adopted) and diversity of capital increases,

labour productivity increases, which in turn increases the extent of the market. This model is

Division of labour and innovation with indivisibilities... 9

consistent with A. Smith’s claim that the division of labour is limited by the extent of the market, but explores a different mechanism than increasing returns. Moreover, since trade increases the size of the market, it increases the division of labour and labour productivity. Thus it also increases GDP and welfare. It is also worth mentioning that this indivisibility is a source of nonconvexities in the aggregate production function. Perpetual growth requires the economy to adopt ever better technologies, but this process might stop if, at some point, the economy fails to fulfil the minimum size requirement.

We also perform numerical simulations to show dynamic behaviour of the model. We find that the model − depending on parameters or initial conditions − can deliver quite rich dynamics.

First, in the model there can be both unbounded growth and no growth. Second, poverty traps are possible. Third, it is possible to grow for some time (even very long) and develop quite sophisticated technologies and then stagnate.

This environment creates problems for the standard notion of equilibrium, so we have to make modifications. The main modification is that the producers are allowed to choose nonlinear prices, in the form of two part tariff pricing structures. With introduction of the two part tariff pricing structures equilibrium exists and is efficient. The two part tariff pricing structures were studied by Brown (1991) in the context of natural monopoly.

The paper is organized as follows. In section 2 we analyse the efficient division of labour.

In section 3 we show problems with the standard notion of competitive equilibrium in this environment. In section 4 we propose the notion of equilibrium and establish its properties. Then we conclude the paper.

2. Environment and effi cient division of labour

We consider a finite or infinite period economy, ¹ with a measure one of homogeneous agents. Each consumer has the constant relative risk aversion (CRRA) utility function = ( 1)/(1 )

= ≤

≥

∈

) ( c _t c ¹ _t u

)

0 (

= T t t

t u c

, , : ) ,

{( j n j n N

J j n }

jnt 0,

k k _jnt

1 for = ,)

, (

1 for > ,} ), , ( {

= min

+

1

j l

k nf

j l y

k y nf

n jnt jnt

nt j n jnt jnt jnt

jnt jnt

jnt k l

y , , 0

k >

f , f _kk < 0 , f _l > 0 , f _ll < 0 ) 1

( ) = ,

( k _jnt ⁿ l _jnt k _jnt ⁿ l _jnt f

} { N T

1 +

+

+

+

+

+

+

+

+ +

+ _ +

+ +

+

+ k jnt

n b _n /

jnt

otherwise ,

0 , / : =

1

>

/1) 0( = for

for

= , 1 :

=

1

1 1

n k b

n

b k

n

n jnt jnt

jnt

jnt jnt

jnt

n

n b

b ₁ >

= 1

) ,

( jnt

n J

j l

J nnt n jnt n J n

t j y

c

) , 1 ( ) , (

=

T t n j jnt jnt jnt

t y k l

c , ( , , ) _{( , )} } _{= 0} {

) max (

0 0 =

} = ) , ) ( , , (

{ , ^t

T t T t t J n j l jnt k jnt y jnt c t

c u

J nnt n jnt n J n

t j k y

c

) , 1 ( ) , (

≠

n b k _{jnt 1 n} /

t, if k _{jnt 1} > 0 n N , n

n k _{j n t 1} = 0 N

n

n , j n k _{jnt 1} > 0 j n , k _{j n t 1} > 0 n

n >

n j ,

, 0

1 =

t n

k j

0

t > ( j , n ) J , ,

,

> 1

j , ₁ = _{j 1 nt 1}

+

+ +

_ 1nt 1 + 1

jnt + 1

jnt + j

+ +

_ 1nt 1 j

jnt k

k

= l

l y ₌ y

jnt n j

nt k

k =

nnt

nt y

y =

) , (

= _nt ⁿ _nt

nt f k l

y

N nt nt n N

t n k y

c ₁ =

0

=

or ₁

1 n nt

nt b k

k 1

nt =

N

n l

T t N n nt nt nt

t y k l

c , ( , , ) } _{= 0} {

) max (

0 0 =

} = ) , , (

{ , ^t

T t T t t l nt k nt y nt

c t u c

θ

Σ δ

θ – –

–

γ

γ

κ

κ κ

κ

κ κ

∞

γ ^α

α

γ

–

–

∈ ∪

≥

≥

Σ

Σ Σ

Σ

Σ Σ

∈

∈ ∈

J

∈

∈

∈ ∈

∈

∈

∈

∈ ∈

∈

N n ∈

∈

δ

≥

≤

≥

≤

≤

≤

Σ ≤ Σ

Σ Σ

Σ Σ

γ

δ

n jnt

nt j l

l = , where c ^t and 1/θ denote, respectively, consumption and the elasticity of intertemporal substitution consumption. Agents discount future with the discount factor δ < 1 and the total lifetime utility is given by

(1)

There is an infinite number of technologies in the consumption good sector indexed by n ^∈ N, where N denotes the set of natural numbers (to avoid confusion we want to clarify that it does not include zero). The production process of technology n ^∈ N is divided into n steps. Let

) 1)/(1 (

=

= ≤

≥

∈

) ( c _t c ¹ _t u

)

0 (

= T t t

t u c

, , : ) ,

{( j n j n N

J j n }

jnt 0,

k k _jnt

1 for = ,)

, (

> 1 ,} for ),

, ( {

= min

+

1

j l

k nf

j y

l k y nf

n jnt jnt

nt j jnt n jnt jnt

jnt jnt

jnt k l

y , ,

> 0

f k , f _kk < 0 , f _l > 0 , f _ll < 0 ) 1

( ) = ,

( k _jnt ⁿ l _jnt k _jnt ⁿ l _jnt f

} N {

T

1 +

+

+

+

+

+

+

+

+ +

+ _ +

+ +

+

+ k jnt

n b _n /

jnt

otherwise ,

0 , / : =

1

>

/1) 0( = for

for

= , 1 :

=

1

1 1

n k b

n

b k

n

n jnt jnt

jnt

jnt jnt

jnt

n

n b

b ₁ >

= 1

) ,

( jnt

J n

j l

J nnt n n jnt J n j

t y

c

) , ( ) 1 , (

=

T t n j jnt jnt jnt

t y k l

c , ( , , ) ( , ) } = ₀

{

) max (

0 0 =

} = ) , ) ( , , (

{ , ^t

T t t T t J n j l jnt k jnt y jnt

c t u c

J nnt n jnt n J n

t j k y

c

) , 1 ( ) , (

≠

n b k _{jnt 1 n} /

t, ^if k _{jnt 1} > 0 n N , n

n k _{j n t 1} = 0 N

n

n , j n k _{jnt 1} > 0 j n , k _{j n t 1} > 0 n

n >

n j ,

, 0

1 =

t n

k j

0

t > ( j ) , n J , ,

,

1

j > , ₁ = _{j 1 nt 1}

+

+ +

_ 1nt 1 + 1

jnt + 1

jnt + j

+ +

_ 1nt 1 j

jnt k

k

= l

l y ₌ y

jnt n j

nt k

k =

nnt

nt y

y =

) , (

= _nt ⁿ _nt

nt f k l

y

N nt nt n N

t n k y

c ₁ =

0

=

or ₁

1 n nt

nt b k

k

= 1

N nt

n l

T t N n nt nt nt

t y k l

c , ( , , ) } = 0

{

) max (

0 0 =

} = ) , , (

{ , ^t

T t T t t l nt k nt y nt

c t u c

θ

Σ δ

θ – –

–

γ

γ

κ

κ κ

κ

κ κ

∞

γ ^α

α

γ

–

–

∈ ∪

≥

≥

Σ

Σ Σ

Σ

Σ Σ

∈

∈ ∈

J

∈

∈

∈ ∈

∈

∈

∈

∈ ∈

∈

N n ∈

∈

δ

≥

≤

≥

≤

≤

≤

Σ ≤ Σ

Σ Σ

Σ Σ

γ

δ

n jnt

nt j l

l =

, be the set of all possible steps of all possible technologies, the pair (j, n) denotes step j of technology n. The stock of capital in period t is denoted as k k _{jnt jn jn} ≥ 0, where k jnt _jn

k _jn is interpreted as the capital specific to step j of technology n. This capital can be used in the production of the consumption good. The production process that uses technology n is represented by the following production function

1 We denote the time horizon of the economy as

) 1)/(1 (

=

= ≤

≥

∈

) ( c _t c ¹ _t u

)

0 (

= T t t

t u c

, , : ) ,

{( j n j n N

J j n }

jnt 0,

k k _jnt

1 for = ,)

, (

1 for >

,}

), , ( {

= min

+

1

j l

k nf

j y

l k y nf

n jnt jnt

nt j n jnt jnt jnt

jnt jnt

jnt k l

y , , 0

k >

f , f _kk < 0 , f _l > 0 , f _ll < 0 ) 1

( ) = ,

( k _jnt ⁿ l _jnt k _jnt ⁿ l _jnt f

} N { T

1 +

+

+

+

+

+

+

+

+ +

+ _ +

+ +

+

+ k jnt

n b _n /

jnt

otherwise ,

0 , / : =

1

>

/1) 0( = for

for

= , 1 :

=

1

1 1

n k b

n

b k

n

n jnt jnt

jnt

jnt jnt jnt

n

n b

b ₁ >

= 1

) ,

( jnt

J n

j l

J nnt n jnt n J n

t j y

c

) , 1 ( ) , (

=

T t n j jnt jnt jnt

t y k l

c , ( , , ) ( , ) } = ₀

{

) max (

0 0 =

} = ) , ) ( , , (

{ , ^t

T t T t t J n j l jnt k jnt y jnt

c t u c

J nnt n jnt n J n

t j k y

c

) , 1 ( ) , (

≠

n b k _{jnt 1 n} /

t, if k _{jnt 1} > 0 n N , n

n k _{j n t 1} = 0 N

n

n , j n k _{jnt 1} > 0 j n , k _{j n t 1} > 0 n

n >

n j ,

,

= 0

1 t n

k j

0

t > ( j , n ) J , , ,

1

j > , ₁ = _{j 1 nt 1}

+

+ +

_ 1nt 1 + 1 jnt + 1

jnt + j

+ + _ 1nt 1 j

jnt k

k

= l

l y ₌ y

jnt n j

nt k

k =

nnt

nt y

y =

) , (

= _nt ⁿ _nt

nt f k l

y

N nt nt n N

t n k y

c ₁ =

0

= or ₁

1 n nt

nt b k

k

= 1

N nt

n l

T t N n nt nt nt

t y k l

c , ( , , ) } = 0

{

) max (

0 0 =

} = ) , , (

{ , ^t

T t T t t l nt k nt y nt

c t u c

θ

Σ δ

θ – –

–

γ

γ

κ

κ κ

κ

κ κ

∞

γ ^α

α

γ

–

–

∈ ∪

≥

≥

Σ

Σ Σ

Σ

Σ Σ

∈

∈ ∈

J

∈

∈

∈ ∈

∈

∈

∈

∈ ∈

∈

N n ∈

∈

δ

≥

≤

≥

≤

≤

≤

Σ ≤ Σ

Σ Σ

Σ Σ

γ

δ

jnt n j

nt l

l =

. All the results in the paper hold for both fi nite and infi nite version of the model.

) 1)/(1 (

=

= ≤

≥

∈

) ( c _t c ¹ _t u

)

0 (

= T t t

t u c

, , : ) ,

{( j n j n N

J j n }

jnt 0,

k k _jnt

1 for = ,)

, (

1 for >

,}

), ,

( {

= min

+

1

j l

k nf

j l y

k y nf

n jnt jnt

nt j n jnt jnt jnt

jnt jnt

jnt k l

y , , 0

k >

f , f _kk < 0 , f _l > 0 , f _ll < 0 ) 1

( ) =

,

( k _jnt ⁿ l _jnt k _jnt ⁿ l _jnt f

} N {

T

1 +

+

+

+

+

+

+

+

+ +

+ _ +

+ +

+

+ k jnt

n b _n /

jnt

otherwise ,

0 , / : =

> 1

/1) 0( = for

for

= , 1 :

=

1

1 1

n k b

n

b k

n

n jnt jnt

jnt

jnt jnt

jnt

n

n b

b 1 >

= 1

) ,

( jnt

n J

j l

J nnt n jnt n J n

t j y

c

) , ( 1 ) , (

=

T t n j jnt jnt jnt

t y k l

c , ( , , ) _{( , )} } _{= 0} {

) max (

= 0

= 0 ) }

, ) ( , , (

{ , ^t

T t T t t J n j l jnt k jnt y jnt c t

c u

J nnt n jnt n J n

t j k y

c

) , ( 1 ) , (

≠

n b k _{jnt 1 n} /

t, if k jnt 1 > 0 n N , n

n k j n t 1 = 0 N

n

n , j n k _{jnt 1} > 0 j n , k _{j n t 1} > 0 n

n >

n j ,

, 0

1 =

t n

k j

0

t > ( j , n ) J , ,

,

1

j > , ₁ = _{j 1 nt 1}

+

+ +

_ 1nt 1 + 1

jnt + 1

jnt + j

+ +

_ 1nt 1 j

jnt k

k

= l

l y ₌ y

jnt n j

nt k

k =

nnt

nt y

y =

) , (

= _nt ⁿ _nt

nt f k l

y

N nt nt n N

t n k y

c ₁ =

= 0

or ₁

1 n nt

nt b k

k 1

nt =

n N l

T t N n nt nt nt

t y k l

c , ( , , ) } = 0

{

) max (

0 0 =

} = ) , , (

{ , ^t

T t T t t l nt k nt y nt

c t u c

θ

Σ δ

θ – –

–

γ

γ

κ

κ κ

κ

κ κ

∞

γ ^α

α

γ

–

–

∈ ∪

≥

≥

Σ

Σ Σ

Σ

Σ Σ

∈

∈ ∈

J

∈

∈

∈ ∈

∈

∈

∈

∈ ∈

∈

N n ^∈

∈

δ

≥

≤

≥

≤

≤

≤

Σ ≤ Σ

Σ Σ

Σ Σ

γ

δ

n jnt

nt j l

l =