Time Lags and Indicative Planning in a Dynamic Model Of

8/14/2019 Time Lags and Indicative Planning in a Dynamic Model Of

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Time Lags and Indicative Planning in a Dynamic Model ofIndustrialisation*

by

JOSHUA S. GANSMelbourne Business School, University of Melbourne

Carlton, Victoria, 3053, Australia E-Mail: [email protected]

First Draft: January 25, 1994This Version: August 30, 1996

This paper presents a model that combines the increasing specialisationand adoption of modern technologies views of industrialisation. Both of theseviews have been used in the recent literature to demonstrate the possibility ofcoordination failure. With simultaneous production across sectors, the modelgenerates the indeterminancy of equilibria common to the recent literature.However, by positing quite natural time lags in production thisindeterminancy is eliminated. It is shown that, in such a framework, even withoptimistic expectations, firms prefer to delay industrialisation in thedevelopment trap. This suggests that policies aimed at transition by usingindicative planning are unlikely to be successful. Journal of Economic

Literature Classification Numbers: O14 & O20.

Keywords: industrialisation, modernisation, specialisation, technologyadoption, complementarities, irreversible investment, indicative planning.

* This paper draws on results from Chapter 4 of my Ph.D. dissertation from Stanford University (see Gans,

1994). I wish to thank Kenneth Arrow, Mark Crosby, Paul Milgrom and Scott Stern for helpfuldiscussions and comments. I also thank the Fulbright Commission for financial support. All errors remainmy responsibility.


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I Introduction

It is generally acknowledged that the process of industrialisation involves the

application of progressively modern technologies in production. The concept of

progressively modern is, however, imprecise. Nonetheless, a non-controversial position

would be to view industrialisation as involving the use of production techniques that are

more efficient at the margin. Indeed, one recent strand in the literature posits a direct

linkage between technology adoption and greater marginal efficiency. Production

processes are potentially carried out by two processes: one with no fixed costs and constant

marginal costs and the other with some fixed cost but lower marginal costs than the first

process. Thus, progressively modern technologies involve producers incurring greater

fixed costs in order to improve the marginal efficiency of variable inputs.1

Another strand of the recent literature has considered industrialisation as the use of

greater varieties of intermediate inputs in the production of final goods. Intermediate

inputs are imperfect substitutes for one another. Therefore, the employment of an

additional variety raises the efficiency of final good production at the margin. 2 Sometimes

this mechanism for industrialisation is interpreted as increasing returns due to

specialisation. The metaphor here is that additional varieties of intermediate inputs allow

each variety to perform a smaller range of tasks with greater efficiency.

This strand of the literature shares with the direct approach outlined above a

requirement that a fixed cost must be incurred before efficiency gains can be realised.

Here, the fixed costs are associated with the entry of new varieties into production. On the

other hand, in the direct strand, the fixed costs are associated with the modernisation of

production processes. Without the existence of such fixed costs, final goods producers

would demand a potentially infinite variety of intermediate inputs and would always adopt

the least cost technology. Industrialisation, whether it be from modernisation or entry,

1 This strand is best exemplified by the model of Murphy, Shleifer and Vishny (1989). An alternative

approach is developed by Baland and Francois (1995).2 The industrialisation as greater product variety view has been analysed by Romer (1987), Rodriguez-Claire (1996), and Ciccone and Matsuyama (1996) among others.


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becomes a problem because firms face a trade-off between the action generating greater

efficiency and the fixed costs of adoption or entry.

The purpose of this paper is to provide a unified model of these two concepts of

industrialisation. In so doing, the common element of both strands is maintained, that is,

that the size of the market is a critical ingredient in determining the possibility of

industrialisation. On the technological side, however, producers in the intermediate input

sector will face a multi-dimensional choice. They will face an entry decision of whether to

enter into production or not and a modernisation decision. The modernisation decision will

involve a choice from a menu of technologies rather than a simple binary choice between

some constant returns and increasing returns technology. Therefore, the level of fixed

costs becomes a choice variable of firms. By incurring greater levels of fixed costs, firms

obtain progressively higher levels of labour productivity. As such, an economy is

considered to be more industrialised the greater the level of fixed costs incurred by

intermediate input producers and the more varieties are available for final good production.3

Section II presents a static model. This model generates the multiplicity of

equilibria that is common to static models basing themselves on one aspect of

industrialisation. More significantly, the model demonstrates that the same assumptions on

parameters that allow the possibility of multiplicity for one aspect also generate the

possibility of multiplicity for the other. That is, the same conditions that generate strategic

complementarities between sectors in their entry decisions, ensure that complementarities

exist in their modernisation decisions. This is because each action affects others though the

same aggregate variables.

Section III then presents a dynamic version of the model in a discrete time setting.

In order to ensure that utility remains bounded, I assume some simple time lags in

production. These have the effect of limiting the period by period growth of the state

variable (i.e., the level of industrialisation). This also means that the model exhibits

multiple steady states as opposed to multiple equilibria, with states of persistent

3 In the model to presented here, therefore, the fixed and marginal cost components of the increasing returnstechnology are no longer parameters that determine the range of equilibrium. They are replaced by a meta-parameter describing the rate at which fixed costs are translated into lower marginal costs.


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industrialisation exhibiting increasing growth over time. No rational expectations path

exists from the development trap to a state of persistent industrialisation suggesting that

transition policies based on indicative planning are unlikely to be successful.

II Static Model

The model to be presented here is similar to the model of Gans (1995b) which itself

builds upon that of Ciccone (1993). The latter develops a model of industrialisation in

which the fixed costs of entry and modernisation are in final good units while the former

develops a continuous technological choice space.

Sectoral Structure and Technology

The basic model to be considered here is of a closed economy consisting of two

production sectors -- an upstream and a downstream sector. The downstream (or final

output) sector consists of a continuum of firms producing a homogenous final good

denoted Y. Firms in this sector use a Cobb-Douglas technology, employing both labour,

LY, and a composite of intermediate inputs,X,

Y X LY= > 1 1 0, .

This production function exhibits constant returns to scale.4 In addition, it is assumed that

the downstream sector is competitive with all firms being price takers. I assume the good

they produce is the numeraire.

Households consume final goods not used in production and supply one unit of

labour inelastically for which they receive a competitive wage, w . The total labour

endowment is L .

4 The Cobb-Douglas assumption is not critical here. The results below could also be presented using ageneral constant returns to scale production function with the restrictions discussed by Ciccone andMatsuyama (1996).


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The intermediate input composite is assembled by final goods producers according

to the following technology,

X x dnn

k

=

1

1

0

, > 1

where xn denotes the amount of intermediate input of variety n that final good producers

employ. The elasticity of substitution between different varieties, , is constrained to be

greater than one implying that no single variety is necessary for production. It is assumed

that each variety, n, is produced by a single monopolist, regardless of their choice of

technology.5 Thus, there is potentially a continuum of such firms lying on the [0, k]

interval of the real line.6 Apart from the usual pricing decisions, potential producers in this

sector face two additional classes of decisions: (i) whether to enter production; and (ii) if

so, at what level of technology. The first class of decisions I term entry, while the second

is termed modernisation. Together these constitute industrialisation. I will deal with the

elements of each of these decisions in turn.

Entry Decisions

Entry into intermediate good production is costly. It is assumed here that a variety

cannot be produced without the firm incurring a unit charge in terms of the final good.7

The level of this charge is independent of both the technological choice and the actual level

of production. Thus, it is a pure sunk cost of entry. As will be apparent below, firms will

find it optimal to enter production if and only if they face non-negative profits upon entry

(given their optimal technological and pricing decisions).

5 The fixed costs associated with entry make this a reasonable assumption as potential entrants find itoptimal to produce a new variety rather than compete with incumbent firms. Strictly speaking, however,these firms are in monopolistic competition with each other as in Dixit and Stiglitz (1977).6 The model is similar to the set-up of Romer (1987), although it contains some additional generality for he

assumes that = ( )1 1 .7 Many models of industrialisation assume that the fixed costs of industrialisation are in labour units (e.g.,Murphy, Shleifer and Vishny, 1989; Rodriguez-Clare, 1996; and Ciccone and Matsuyama, 1996). This

assumption makes it more difficult to generate strategic complementarities. The substantive results of themodel to be presented below could be generated under the labour units assumptions but only at the expenseof additional restrictions of the kind explored by Ciccone and Matsuyama (1996).


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Technological Choice in the Upstream Sector

Upstream producers are able to choose, to some extent, their technology of

production. After entry, a typical technology has them using labour, ln, and producing

output, xn , according to,

x

l

Fn

n

n

=( )

.

The choice of Fn , itself, is assumed to be endogenous -- it represents a fixed cost (in final

good units) to the firm as well as a technological choice. Higher choices of Fn mean a

lower labour requirement, that is,


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Since intermediate input producers face demand curves with a constant elasticity, -

, their optimal pricing scheme if they undertake positive production in period tis,

P w F dn wn

k

= ( )

=

1

1

0

1

1

1

11

( ) ,

the standard constant mark-up over marginal costs.

Using the optimal pricing rule, some simple substitutions show that, P w=

1

11

and x F X n n= ( )

1 , where,

= = { }( )[ ] ( ) ,F dn k F nk

n n k

1

0

.

The aggregate, , is a measure of the overall level of industrialisation. It is increasing in

both the variety of intermediate inputs produced and the level of technology chosen by

upstream firms.

Now consider the labour market. To satisfy demand, the labour requirement for an

intermediate input producer is simply, l F Xn n= ( )1 1

. As such, total labour demand

in the upstream sectors is,

L l dn X X n

k

= = 0

11 .

For the final goods sector, note that the Cobb-Douglas production implies that,

LPX

wXY = ( ) = ( )( )

1 11

11

.

It is assumed that the labour market clears in every period. As such, L L LY X= + and,

therefore, X L= ( )

( )11

1 . Finally, it remains to find the wage level each period. This

can be found by looking at the marginal product of labour in the production of final goods

and using the solution forX: w = ( ) ( ) 1 11 1 1

( ) ( ) .

Substituting the relevant aggregate variables into this equation gives a convenient

reduced form for the payoffs of an intermediate input producer entering into production.

To examine the structure of these payoffs, consider upstream profits (of an entrant) when

wages are held constant,


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n n n n n n np w F x F F wL F = ( ) = ( )

( ) ( )1 11 1 .

The effect of rising industrialisation in this case is to depress profits and the marginal return

to modernisation. This is because increased entry and modernisation by others provides

more competition for any given upstream firm, reducing their total revenue. However,

when wages are varied profits become,

n n nL F F =

( )

( )1

1

1 1.

where = ( ) ( )

( ) ( ) ( )1 11

. The flows of income and goods leading to this

equation are depicted in Figure One. The wage effectexerts a positive feedback on both

entry and industrialisation decisions -- they reflect higher demand for final goods and

greater efficiency in its production, raising demand for intermediate inputs. If the so-called

increasing returns due to specialisation (( ) 1 1) outweigh the decreasing returns to

additional use of the intermediate input composite (), the game between intermediate input

producers exhibits strategic complementarities (with the wage effect outweighing the

competition effect). The greater the level of industrialisation, the greater the marginal return

to both entry and modernisation. Therefore, for the rest of this paper it is assumed that

1 .

To emphasise, the pathways through which entry and modernisation decisions

affect industrialisation and, in turn, how industrialisation effects those decisions are

essentially the same. Both entry and modernisation decisions reduce the revenues of others

through the competition effect and raise them through the wage effect. Moreover,

industrialisation impacts upon these decisions through a single variable in the profit

function -- raising the gross profits (and marginal profits) net of modernisation and entry

costs. Thus, it impacts upon these decisions in a very similar manner.12 Both aspects of

industrialisation, therefore, have the same economic interpretation.

12 Slight differences do occur because the entry decision depends on gross profits, while the modernisationdecision depends on marginal profits. Their qualitative aspects are, however, the same.


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Equilibria

In order to simplify the exposition of what follows, I will adopt the following

functional form for ( )Fn ,

( ) ( ) ,F Fn n= + >1 0 .

This functional form captures the notion that greater sunk costs reduce the marginal labour

requirement and also imposes diminishing returns to this process. To ensure is concave

in Fn , it is assumed that < 1 1 .

Since the reduced form profit function already takes into account labour and good

market clearing, only equilibria in the game between intermediate input producers need be

considered. Suppose that m k upstream firms are active. Let

B Fn F n nn( ) argmax ( ; ) be the best response set for their modernisation decisions.13

The pair ( ,{ } )m Fn n m constitutes apure strategy Nash equilibrium if:

(i) ( )F Bn n for all n m ;

(ii) ( ) ( ) = + F dnnm

1 10

;

(iii) max ( , ) ,F n nn

F n m < >0 .

Thus, in equilibrium, all firms choose the technology that maximises profits and these

decisions generate a consistent level of industrialisation. In addition, in equilibrium, if they

choose to enter, non-active firms earn negative profits.

The following proposition summarises the possible equilibria arising in this model.

Proposition 1 (Static Equilibria). Suppose that the initial level ofindustrialisation is 0. The following characterise, completely, the set of pure

strategy Nash equilibria:(i) If L <

1

0

1

1

( )

, then for all parameters, there exists a development trap

with no further entry (or modernisation) by intermediate input producers;

(ii) If 111

11

11

( )

( ) ( )

> k L k and L <

11 0

11

( )

( )

, there exists an entry

equilibrium with no modernisation (i.e., k firms enter into production butFn = 0 for all n);

(iii) If L k > 1

1

1

1

( )

( )

and1 > , there exists an industrialisation equilibrium

(i.e., k firms enter into production and Fn > 0 for all n).

13 This involves an implicit assumption that firms producing low ordered varieties will enter first in anyequilibrium. This is a reasonable assumption given the symmetry among upstream producers producingmodern varieties and the fact that basic input producers do not face an entry charge.


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PROOF: Note first that the strategic complementarities and symmetry in payofffunctions ensure that any equilibrium is symmetric. Second, observe that anyupstream firm who enters into production chooses their technology according to thefollowing best response function,

F Ln*

max , ( )

( ) ( )

= ( )

0 1 1

1

1

11 1

.

Note that this is not positive under the condition for case (i). Moreover, under thatcondition, n( ; )0 00 < for all n. This remains true so long as no upstream firmchooses to enter or modernise further. Thus, = 0 is an equilibrium.

Now suppose that all k upstream firms entered but none invests in a moremodern technology than their initial level. In this case, = kand individual profitsare:

n k Lk( ; )( )

0 11

1=

This is positive so long as,

L k>

11

1

( )

.Observe, however, that

F L kn*

( )

( )

= 0 11

1

1

.

For there to exist a range ofL such that these two inequalities hold requires that:

11

1 11

1

1

1

1

( )

( ) ( )

> > k L k .

Finally, suppose all producers of modern varieties enter and adopt some

positive level of modernisation, i.e., F> 0. Then = + k F( ) ( )1 1 where,

( )( )

F Lk= ( )

1 11

1

11

.This is positive by the conditions of the proposition.

Several remarks on this proposition are in order. First, the presence of the entry cost

makes a development trap generic to the model. If the labour endowment is large enough

or ifk is large enough, there exist multiple equilibria in this model. Both the entry and

industrialisation equilibria Pareto dominate the development trap since positive output (and

hence, consumption) occurs in these cases. The additional condition for the existence of an

industrialisation equilibrium (that 1 > ) is a sufficient condition for global concavity of

aggregate consumption in the level of modernisation. As such, it does not appear to be

excessively restrictive here. Note too that (along with the condition for strategic

complementarity) this condition implies that firm profit functions are concave in technology

choice.


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III A Dynamic Model of Industrialisation

The above model shares with other static models of the big push the idea that

temporary government intervention can potentially facilitate a change from the development

trap to persistent industrialisation. It also shares with those models the possibility that

generating optimistic expectations or some form of indicative planning could achieve this

task without the need for direct government intervention.

As has been noted elsewhere (e.g., Krugman, 1991), in order to properly analyse

this latter possibility one needs to move from a static model to consider dynamics

explicitly.14

Taken literally, the economy could easily move back and forth between the two[equilibria]. The problem is that, in a completely static framework, one cannot capturethe difficulty of the transition in the process of industrialization, which may beresponsible for stagnation. In order to understand the self-perpetuating nature ofunderdevelopment and the inability of the private enterprise system to break away fromthe circularity, it is necessary to model explicitly the difficulty of coordination.(Matsuyama, 1992a, p.348)

Matsuyama extends the Murphy, Shleifer and Vishny (1989) model to a dynamic setting.

In his model, firms face adjustment costs in adopting the modern technology or switching

back to the traditional one. As such, they need to anticipate not only the current movements

of others but their future movements as well. In this set-up, Matsuyama finds that

indicative planning will not be sufficient to generate an escape from a development trap if

adjustment costs are large or the discount rate is high.

Ciccone and Matsuyama (1996) also offer an explicitly dynamic model of the big

push.15 Their model has the same structure as the static model above although they do not

consider a modernisation choice and entry costs are in labour (rather than final good) units.

The only other significant difference between their model and the one in the previous

section is that:

X t x t dnn( ) ( )=

1

1

0

, >1.

14 The former possibility is discussed in Gans (1995a).15 Murphy, Shleifer and Vishny (1989) also offer a simple dynamic model of industrialisation. It is ,however, only a two period model whereas the alternatives here and elsewhere have a long time horizon.


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There is potentially an infinite number of entrants in any period t. Using this framework

they analyse several models in a continuous time setting. They provide several examples of

models that exhibit multiple dynamic equilibria and thus, allow the possibility of indicative

planning. In those models, however, growth in the industrialisation equilibrium involves

constant per capita consumption. They do present one model with rising per capita

consumption and multiple steady states without the possibility of indicative planning.

However, that involves constant growth in the industrialisation steady state. Allowing the

possibility of increasing growth may be more consistent with the empirical reality of

industrialisation (see Romer, 1986).

In this section, I wish to consider an alternative approach to dynamics using the

model of section II. In so doing, I will use the form ofX(t) above but, for reasons that will

soon become apparent, use a discrete rather than continuous time setting.16 Households

and firms in this model solve intertemporal maximisation problems. For upstream firms,

incurring entry costs in period tallows them to start production in period t and successive

periods. Their technological choices involve sunk costs as well, although these can be

spread over time. By accumulating quantities of the final good over time, upstream

producers can increase their labour productivity. Thus, suppose that, at time t, the

cumulative amount of the final good purchased by firm n is,

F t f sn ns

t

( ) ( ),==

0

where f sn( ) is the amount of the final good purchased in period s. Then in t, and in

subsequent periods, the firm is able to produce x tn( ) without additional investment

according to: l t F t x t n n n( ) ( ( )) ( )= . Thus, by incurring sunk costs, intermediate input

producers require only ( )Fn units of labour to produce a unit of intermediate input in

subsequent periods. To make the choice space of upstream firms continuous, I suppose

that their choice off tn( ) is endogenous in each period and can take any positive real value.

16 All dynamic recent models of industrialisation and endogenous growth that I am aware of use acontinuous time setting.


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As before, higher accumulations of F tn( ) mean a lower labour requirement, that is,

0 is the subjective discount

rate and r(t) is the interest rate. The solution to this optimisation problem is characterised

by the familiar Euler condition and the binding budget constraint:

+

=+ +

+U C t

U C t

r t( ( ))

( ( ))

( )

1

1 1

1 for all t,

11

1

0+=

( ) ( ) = r tt

t

C t Lw t v( ) ( ) ( ) ( ).

That is, to justify any rising growth in consumption, the interest rate must rise over time.

As discussed in depth by Romer (1986), a problem arises in contexts such as these:

with net profits increasing in the level of industrialisation, ( )t , utility could become

unbounded. Indeed, in this framework, from any positive level of industrialisation, all

intermediate input producers choose to enter and modernise in a single period, leading to

nonsensical infinite production. To avoid this difficulty, here I exploit the structure of the

positive and negative feedbacks in the model in section II by introducing time lags into

production.

For final good production, it is now assumed that:

Y t X t L t Y( ) ( ) ( )+ =1 1 .

17 Some depreciation could be included in this specification, although it would not alter the results to comein any substantive manner.


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That is, production of final goods takes one period. This is the reason why I have used a

discrete time setting. Allowing for this possibility means that the positive feedback (i.e.,

wage effect) from industrialisation will be delayed one period. As will be shown, this leads

to a mixture of substitution and complementarity in cash flows that results in smoothed

industrialisation across time.

Appendix A derives the relevant aggregate variables as a period-bu-period general

equilibrium of the model. Substituting these into the cash flow equation gives a convenient

reduced form for the cash flow of an intermediate input producer producing a positive

output in period t,

n n nt L F t t t f t ( ) ( ( )) ( ) ( ) ( )= 1 11 1 .

where is as before. Observe that if 1 , then, from a system-wide point of view,

there exists a positive feedbackbetween the past technological choices of intermediate

input producers and the firms current choice. To see this more clearly, suppose that there

is no further increase in overall industrialisation in period t. Then the mixed partial

derivative of the profit function with respect to f tn ( ) and ( )t 1 is nonnegative if and only

if ( ) ( )t t1 1 1

is nondecreasing in ( )t 1 for all f tn( ) . Observe that holding the

current increment to industrialisation, ( ) ( ) ( )t t t 1 , constant, this is equivalent

to,

+

( )

( )

( ) ( )t

t

t t1

1

10

1

,

which is true if and only if 1 . Then, ceteris paribus, the greater the past level of

industrialisation, the greater is the marginal return to both entry and modernisation.

It is worth noting, however, that the firms current technological choice is a

strategic substitute with the current choices of other intermediate input producers. So while

a greater level of past industrialisation raises the marginal returns to entry and technological

decisions today, greater current industrialisation dampens those incentives. The former

(complementary) effect emerges because greater past industrialisation pushes up current

wages which in turn raises demand for intermediate inputs through higher aggregate


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demand. On the other hand, the latter (substitution) effect occurs because of the reduction

in current intermediate input prices caused by lower marginal costs of production and the

competition of entrants.

Equilibrium Defined

Given the dynamic context, the definition of what constitutes an equilibrium in the

game between intermediate input producers needs to be restated. Let

B fn t t f rt

n ntn t, ( ) ( )

( ), ( ) argmax ( ); ( ), ( ) { }( ) ( ) ( ) { } +

1 111

be the best response set for an active firm n k t

( ). A strategy pair, ( ( ),{ ( )} )

( )k t f

n n k t t

{ }

constitutes apure strategy Nash equilibrium if, for all t:

(i) { ( )} { ( ), ( )},f Bn t n t t ( )1 for all active n;

(ii) ( ) ( )( )( )

= +( )=

t f dnnstk t

10

1

0;

(iii) max ( ) , ( )( ) ( )f r

t

ntn tn k t

{ } +

( ) < > 11 1 ;

(iv) r(t) satisfies the household Euler condition.

Thus, in equilibrium, all firms choose the technology that maximises discounted cash flows

and these decisions generate a consistent level of industrialisation. In addition, in

equilibrium, if they chose to enter, non-active firms would earn negative profits. Finally,

the rate of interest satisfies the intertemporal optimisation condition for households.

Linear Utility

As will be discussed further below, the time structure of production makes the

specification of industrialising equilibria very difficult. However, one can show that

persistent industrialisation is possible.18 In order to make clear the forces driving this

result, I will start with the case of linear utility (i.e., U(C(t)) = C(t) for all t) and generalise

this in Proposition 2 below. In this simple case, the interest rate, r, is constant and equal

to the subjective discount rate, .

18 This result is related to the Momentum Theorem, initially stated in Milgrom, Qian and Roberts (1991)for contracting problems, and extended in Gans (1994, Chapter 3) to game theoretic contexts.


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Proposition 2 (Persistent Industrialisation). Let utility be linear. Suppose that

at some time t, > ( ) *t , where = ( )+

* 11

1

L . Then

+ ( ) ( ) ( ) ( ) ... t 1 for> t.

PROOF: Suppose that in period t, > ( ) *t . Entry and technological choices areconsidered in turn. First, given the shock in period t, new varieties enter in periodt+1 until the difference in discounted cash flows from entering in t+1 as opposed tot+2 fall to zero for all firms. Without loss of generality, assume that entering firms

do not adopt more modern technologies, as would be case for ( )t close to * . Letk t k t k t ( ) ( ) ( )+ = + 1 1 . For an upstream firm, the difference in discounted sumof cash flows between entering t+1 as opposed to t+2 is,

L t t k t + + +( ) ( ( ) ( )) ( )

1 1 11

.

Setting this equal to zero gives a unique solution:

k t t L t ( ) ( ) ( )( )+ = +

11 1

1

.

k t( )+1 is positive since > ( ) *t . This, in turn, implies that + > ( ) ( )t t1 ,meaning that k t( )+ >2 0 since the right hand side of the equation is increasing in( )t . Note too that the finiteness of k t( )+ 1 puts a bound on period by periodutility. A similar reasoning applies to the technological decisions. The proof thenfollows by induction.

This proposition says that once industrialisation reaches a critical level, the process will

persist and continue of its own accord. Note too that, under persistent industrialisation, the

state variable of industrialisation evolves according to,

= + ( ) ( )( )t L t 1 1 1

,

a unique path. Thus, in the spirit of big push theories of industrialisation, the economy

can be stuck in a development trap from which an escape could be made provided sufficient

coordination of the decisions of intermediate input producers is achieved.

General Utility Functions

With more general utility functions, the result here becomes more complicated as the

interest rate, r(t), changes over time. Suppose that in period t, > ( ) *t , and k t( )+ 1

firms choose to enter in t+1 with firms modernising to a level, f. In this case, the relevant

Euler condition for intermediate input producers becomes (with + + ( ) ( ) ( )t t t1 1 ),


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g t t L f t t t r t

r t( ( ), ( )) ( ) ( ) ( ) ( )( )

( )

( ) + + + =+ ++ 1 1 1 0

1 1

1

1 1 1

.

When utility is linear, the g(.) (i) is positive at + =( )t 1 0 since > ( ) *t ; (ii) becomes

negative as +( )t 1 grows large; (iii) is strictly decreasing in +( )t 1 ; and (iv) is strictly

increasing in ( )t , once again, since > ( ) *t . The first three properties guarantee that

+( )t 1 is positive and finite (as depicted in Figure 2(a)), while the last guarantees that

+ > +( ) ( )t t2 1 and that industrialisation is increasing over time.

These four properties are potentially violated when utility takes a more general form

and the interest rate varies over time. Observe that the interest rate depends both on

+( )t 1 and ( )t . From the household Euler condition,

1 1 11

1

1

1

+ + = + ( )

+ ( )

r t

U L t k t

U L t k t F ( ) ( )

( ) ( )

( ) ( )

.

With strictly concave utility, one can see that r(t+1) is decreasing in +( )t 1 and

increasing in ( )t . This means that any of the above properties could be violated.

Therefore, we need additional conditions to assure that any solution, +( )t 1 , to the

general firm Euler condition is positive, finite and increasing in ( )t . Let ={ }* ( ) ( , ( ))t g t0 0 . The sufficient conditions are:

(i) Marginal utility is bounded from below, lim ( ( ))( )C t

U C t

= < 0

;

(ii) There exists no + >( )t 1 0 with the property that g t t t ( ( ), ( )) , ( ) * + > < 1 0 ;

(iii) g t t( ( ), ( )) + 1 ( g t( , ( ))0 ) is non-decreasing (increasing) in ( )t , for all +( )t 1

and > ( ) *t .

Of these conditions, only (ii) appears to differ significantly from the properties listed for the

linear case. It does not require that g be nonincreasing in +( )t 1 , although this is

sufficient for (ii) to hold. All that is required is that the highest value of g occurs at

+ =( )t 1 0 when < ( ) *t .19 This guarantees that entry and modernisation can only

possibly occur if past industrialisation reaches a critical value. Figures 2(b) and 2(c), give

19 It is not sufficient for this condition to hold only for = ( ) *t .


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18

two examples of g satisfying these conditions. Note that in each + >( )t 1 0 and

+ +( ) ( )t t2 1 guaranteeing the conclusion of Proposition 2.

Assuming conditions (i) to (iii) it becomes possible to generalise Proposition 2 to

more general utility functions.

Proposition 2 (Persistent Industrialisation). Assume the conditions (i) to (iii)

hold and suppose that at some time t, > ( ) *t , where ={ }* ( ) ( , ( ))t g t0 0 . Then + ( ) ( ) ( ) ( ) ... t 1 for> t.

PROOF: First, observe that (i) guarantees that as + ( )t 1 ,g t t( ( ), ( )) + 1 . This along with (iv), the continuity of U(.) and the fact that > ( ) *t ensures there exists at least one solution to g t t( ( ), ( )) + =1 0, with + >( )t 1 0 , by Theorem 1 of Milgrom and Roberts (1994).20 This, in turn,

implies that + > ( ) ( )t t1 , meaning that + +( ) ( )t t2 1 since the right handside of the equation is non-decreasing in ( )t . Note too that (i) guarantees thefiniteness of +( )t 1 and hence, puts a bound on period by period utility. Theproof then follows by induction.

It is worth emphasising here that these propositions guarantee that only + ( ) ( )t t1 and

hence, C t C t ( ) ( )+ 1 is increasing over time. They do not guarantee that the growth rate

in consumption is rising (in contrast to endogenous growth theory -- Romer, 1986),

although that is possible. In Appendix B it is shown that, by dropping condition (iii) and

replacing it with an alternative bound on g t( , ( ))0 , the growth rate in consumption is

bounded away from zero, for all time after > ( ) *t . Thus, in contrast to neoclassical

growth theory, positive per capita growth persists over time. Proposition 2 also ensures

that industrialisation ensues so long as industrialisation exceeds a critical value. This

property has an interesting implication (as will be shown below). It also holds for all utility

functions with a sufficiently high intertemporal elasticity of substitution.21 Nonetheless, it

is shown in the appendix that without (ii), if it is ever the case that growth become positive

(not just at a critical level of industrialisation), then positive growth would persist

thereafter.

20 That theorem shows that the result here would also hold for some relaxation of the continuity andconcavity assumptions on U(.), so long as the solution to the households problem was an interior one.

21 The easiest way to see this is to examine utility of the form, U C t C t ( ( )) ( ) ,=


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The Impossibility of Indicative Planning

The model under conditions (i) to (iii) has a very interesting implication. As a

model ofdynamic coordination failure this one differs from analogous static models (like

that of section II) in that optimistic expectations would not generate an escape from the

development trap. In many models of coordination failure, there exist rational expectations

paths from the development trap to industrialisation. Here, however, there exists no

rational expectations or perfect foresight paths from non-industrialisation to

industrialisation.

To see this, suppose that, the economy is at some low level of economic activity,

k0 < *. Also, for this demonstration, suppose that utility is linear (this will not be

necessary for Proposition 3 below). Now suppose that, beginning in the development trap,

all potential intermediate input producers expect k k 0 others to enter and adopt some

modern technology in the current period. Let the expected level of technology be some

constant,f> 0, and the new number of intermediate input producers be high enough such

that the resulting expected level of industrialisation would make these decisions profitable

when considered overtime (i.e., k f( )( ) *

+ >

11

). The question must be asked: is it

profitable for a given modern input producer to enter and modernise their technology this

period? A producer could, after all, wait one period before taking either of these actions.

To consider the optimal decision, all that is relevant are the cash flows of firms in the

current and next period. The two period cash flow from entering and modernising today is,

L f k k k f k

L f k k f k f

n

n n

( ) ( )( )

( ) ( )( )

( ) ( )

( ) ( )

+ + +( )

+( ) + + +( )

+

+

1 1

1 1 1

1

0 0

1

0

1

11

1 0 1 0

1

1

1

.

And the two period cash flow from waiting until tomorrow to enter and modernise is,

11

1

0

1

01

11 1 1

11

+

+( ) + + +( ) ( ) ++

L f k k f k f n n( ) ( )( ) ( )

( ) ( ) .

Thus, there is a trade-off between the earnings from production and higher productivity

today and deferring the sunk costs of entry and modernisation. An intermediate input

producer will choose to wait rather than produce if the following inequality is satisfied,


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1

1

0 0

1

0

1

1 1 11+ ( ) + + + +( )( ) ( ) ( )( )( ) ( )f L f k k k f k n n .

When fn = 0, this inequality holds, strictly, by the condition for the development trap ( i.e.,

that k0 < *). Moreover, it is easy to show that, from low levels of industrialisation, the

left hand side increases with fn faster than the right hand side. This means that it is always

optimal to wait.

This argument leads to the following proposition for general utility functions.

Proposition 3. Assume conditions (i) to (iii) hold. Given any initial level of

industrialisation, ( )0 , if < ( ) *0 then the economy is in a development trap for allt. Otherwise, it is in a state of persistent industrialisation.

The optimality of waiting means that no rational expectations/perfect foresight path exists

from the development trap to persistent industrialisation. The reason for this is that if it is

always optimal for one intermediate input producer to wait, by symmetry, it is optimal for

all firms to do so.22 As a consequence no industrialisation occurs and hence, any

expectations to the contrary would not be fulfilled. Observe that this result holds for any

positive discount rate. Thus, the non-industrialisation equilibrium is absorbing in the sense

of Matsuyama (1991, 1992a).23 Note, however, this fact is a direct result of the assumed

time lag in production of the final good. This assumption makes modernisation and entry

today strategic substitutes with similar decisions on the part of other producers. It is also

important to note that there does not exist a rational expectations path from industrialisation

to the development trap. This latter feature is a direct consequences of the irreversibility of

entry and technology adoption.

When a development trap is purely the result of coordination failure, it is often

argued that the role for the government is to coordinate the expectations of individual

agents, making them consistent with those for persistent industrialisation. This is also the

stated goal of indicative planning. If possible, such a policy would be costless (save,

22 This result is similar in flavour to the example of Rauch (1993) although in a very different context tothe one presented here.

23 Matsuyama (1991) states that one state is accessible from another if there exists a rationalexpectations/perfect foresight equilibrium path from one that state that reaches or converges to the other. Astate is absorbing if, within a neighbourhood of it, no other state is accessible.


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21

perhaps, the costs of communication), and firms would modernise on the basis of

optimistic expectations.

The above proposition shows that this solution will not work. This is essentially

because the problem, while one of a failure to coordinate investment, is not one of a failure

to coordinate expectations. If a government were to announce that firms should modernise

to a certain degree, even if this were believed perfectly by firms, each individual firm

would still have an incentive to wait one period before modernising. And, in that case, the

optimistic expectations created by the government would not be realised and the policy

would be ineffective.

Irreversibility and the time lag of production mean that history rather than

expectations matter for the selection of persistent industrialisation as opposed to a

development trap.24 The previous level of industrialisation determines whether the

economy will continue to industrialise in the future. However, it does not specify the

precise path this could take and there could be a multiplicity of steady states involving

persistent industrialisation. The selection of these could depend on expectations. This is

why it is difficult to characterise the industrialising paths of the economy. It is also difficult

to characterise the optimality, or otherwise, of industrialisation. Industrialisation clearly

involves foregone consumption in its initial periods. Therefore, to examine welfare issues

would involve some specification of household preferences. This issue is beyond the

scope of the current paper.

In summary, the above model exhibits, in a certain sense, both the development

traps and persistent industrialisation that are the hallmark of the big push theories of

industrialisation. It is important to note, however, that the distinction between this model

and other models of coordination failure lies solely in the assumption of a time lag to

production.25 With linear utility, this makes the steady state completely determinant. It is

worth noting therefore, that for a small open economy with perfect international capital

mobility and non-tradable intermediate inputs,26 that even with general utility functions the

24 See Krugman (1991a) for an extensive discussion of this point.25 It also relies to some extent on condition (ii) as is demonstrated in the appendix.26 As in Rodriguez-Claire (1996).


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interest rate will not depend on the state of industrialisation. In this case, the uniqueness

results of the linear utility case will hold.

VII Conclusion

This paper has done two things. First, a model that combines both the

modernisation and specialisation views of industrialisation has been constructed. In so

doing, it was shown that the qualitative characteristics and hence, conclusions of the both

views were essentially the same. Thus, both viewpoints are complementary.

Second, this model was put into an explicit dynamic framework. In order to prove

the existence of a dynamic equilibrium, time lags into final good production were

introduced. This change meant that the wage effect from industrialisation was delayed

relative to competition effect. Firms would then have an incentive to industrialise over time

rather than in a single period. This eliminated the possibility of unbounded utility as

discussed in Romer (1986). This change also implied that policies for industrialisation

based on indicative planning or optimistic expectations were unlikely to be successful.

Even if firms were optimistic about future industrialisation they would have an incentive to

delay their own decisions. Since this applied to all firms, optimistic expectations would not

be realised.

It is worth emphasising here that the proof of existence of a dynamic equilibrium

and its characterisation is distinct from those usually undertaken in the growth literature. In

the recent literature on industrialisation or new growth theory, persistent growth

conclusions are found by assuming a specific functional form for utility functions, and

solving for balanced growth paths of interest rates and other state variables. Then it is

shown how these imply that positive growth will persist over time. In contrast, here I used

the monotone methods of Milgrom and Roberts (1994), to show that momentum, once

begun, will persist over time. This allowed a characterisation of dynamic paths as

involving persistent growth without looking for balanced growth paths or imposing specific


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functional form assumptions on utility. This approach allowed a clearer understanding of

the assumptions that allowed for persistent growth over time. A direction for future

research would be to use this approach directly on endogenous growth models (e.g.,

Romer, 1990) and examine the criticality of function form assumptions.


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Appendix A

In this appendix, I derive n(t). Under these assumptions of Section III,

x t X t P t

p tn

n

( ) ( )( )

( )=

P t p t x t dn x t dn p t dnx t n n

k t

n

k t

n

k t

n nk t( ) min ( ) ( ) ( ) ( ){ ( )}

( ) ( ) ( )

( )= =

=

=

0

1

0 0

1

0

1

1

1

,

where here it is supposed that only a subset [0,k(t)] of firms choose to produce in period t.

Using the optimal pricing rule, some simple substitutions show that, P t w t t ( ) ( ) ( )

=

1

11

and x t F t X t t n n( ) ( ( )) ( ) ( )=

1

, where now,

= = { }( )[ ] ( ) ( ( )) ( ), ( )( )

( )t F t dn k t F t n

k t

n n k t 1

0

.

The aggregate, ( )t , is therefore a measure of the overall level of industrialisation in period

t.

Now consider the labour market. As before,

l t F t X t t n n( ) ( ( )) ( ) ( )= 1 1

and L t l t dn X t t X n

k t

( ) ( ) ( ) ( )

( )

= = 0

11 .

For the final goods sector, since production is lagged one period, producers choose

intermediate inputs and labour to maximise:

11 1 1+ +( ) + r t YY t w t L t P t X t ( ) ( ) ( ) ( ) ( ) ( ).

The Cobb-Douglas assumption means that the interest rate drops out with,

L tP t X t

w tX t t Y( )

( ) ( )

( )( ) ( )= ( ) = ( )( ) 1 1 1

11

.

with period by period labour market clearing implying,

X t L t ( ) ( )( )= ( ) 11

1 .


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Finally, it remains to find the wage level each period. Observe, first, that in each

period the cash flow of an upstream firm is,

n n nt w t F L t f t ( ) ( ) ( ) ( ) ( )= ( )

1 1 .

Inserting this into the national income identity,

Y t F t k t k t w t L t k t k t ( ) ( ) ( ( ) ( )) ( ) ( ) ( ( ) ( )) = + 1 1

where

( ) ( ) ( ) ( ) ( )( ) ( )

t w t L t F dn f t dnn

k t

n

k t

= ( ) 1 1

0 0

and Y t L t ( ) ( ) ( ) ( )= ( ) ( ) ( ) 1 1 11 1 1 1

.

Therefore,

w tY t

Lt( )

( )( ) ( ) ( )= ( ) = ( ) ( )

1 11 1 1 1 .

Wages reflect the previous technological choices of intermediate input producers only

because of the time lag in final good production. Substituting w(t) into the above yields the

relevant equation.


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Appendix B

The first result here will show what conditions on g(.) guarantee persistent positive

growth as opposed to rising increments to consumption over time (as proved in Proposition

2). For this purpose, condition (iii) can be dropped as g need not increase over time, but it

needs to be replaced with (iv) below to ensure that it remains positive as the level of

industrialisation rises.

(iv) For all > ( ) *t ,

g t L t t U L t

U LL t k t

( , ( )) ( ) ( )( )

( )

( ) ( )

0 011

1

1

1

11 =

>+

+

;

Figure 3(a) provides an example of what happens under these new conditions

demonstrating graphically the following result.

Corollary 1 (Persistent Positive Growth). Assume conditions (i), (ii) and (iv)

hold and suppose that at some time t, > ( ) *t , where = ( )+

* L 11

1

. Then

< + t.

PROOF: (i) and (iv) ensure that + >( )t 1 0 for all t, by the intermediate valuetheorem. Hence + > ( ) ( )t t1 for all t. Note too that the finiteness of +( )t 1puts a bound on period by period utility. The proof then follows by induction.

A version of Proposition 3 can be proved for this case.

Turning to examine the role of condition (ii), if is removed one can prove the

following corollary.

Corollary 2 (No Guaranteed Development Trap). Assume that only (i) and (iii)hold and suppose that at some time t, ( )t is such that g t t ( ( ), ( )) + >1 0 for some

+ >( )t 1 0 . Then + ( ) ( ) ( ) ( ) ... t 1 for> t.PROOF: (i) and the condition of the corollary ensure that +( )t 1 is positive andfinite by the intermediate value theorem. Hence + > ( ) ( )t t1 . By (iii) andTheorem 1 of Milgrom and Roberts (1994), this implies that + > +( ) ( )t t2 1 .The proof then follows by induction.

Figure 3(b) demonstrates this possibility. What Corollary 2 says is that if it is ever the case

that industrialisation rose to a high enough level (perhaps due to a temporary shock), then

persistent industrialisation will persist thereafter. It differs from Proposition 2, in that it

does not rule out the possibility that a path to persistent industrialisation could exist from


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the development trap equilibrium. In this case, Proposition 3 would not hold and indicative

planning could succeed.

Finally, a similar version of Corollary 1 holding for persistent positive growth can

be proved using the following condition.

(iv) If there exists some * such that g( , )*0 0 > , then g t( , ( ))0 0 > , for all > ( ) *t .

Corollary 3 (No Guaranteed Development Trap/Persistent Positive Growth).Assume that only (ii) and (iv) hold and suppose that at some time t, ( )t is such thatg t( , ( ))0 0 > .. Then, < + t.

PROOF: (ii) and (iv) ensure that + >( )t 1 0 for all t, by the intermediate valuetheorem. Hence + > ( ) ( )t t1 for all t. Note too that the finiteness of +( )t 1puts a bound on period by period utility. The proof then follows by induction.

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Time Lags and Indicative Planning in a Dynamic Model Of