Time Lags And Indicative Planning In A Dynamic Model Of

Time Lags and Indicative Planning in a Dynamic Model of Industrialisation* by

JOSHUA S. GANS Melbourne Business School, University of Melbourne Carlton, Victoria, 3053, Australia E-Mail: [email protected]

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

This paper presents a model that combines the increasing specialisation and adoption of modern technologies views of industrialisation. Both of these views have been used in the recent literature to demonstrate the possibility of coordination failure. With simultaneous production across sectors, the model generates the indeterminancy of equilibria common to the recent literature. However, by positing quite natural time lags in production this indeterminancy is eliminated. It is shown that, in such a framework, even with optimistic expectations, firms prefer to delay industrialisation in the development trap. This suggests that policies aimed at transition by using indicative planning are unlikely to be successful. Journal of Economic Literature Classification Numbers: O14 & O20. Keywords: industrialisation, modernisation, specialisation, technology adoption, 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 helpful discussions and comments. I also thank the Fulbright Commission for financial support. All errors remain my responsibility.

2

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), RodriguezClaire (1996), and Ciccone and Matsuyama (1996) among others.

3 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 returns technology are no longer parameters that determine the range of equilibrium. They are replaced by a metaparameter describing the rate at which fixed costs are translated into lower marginal costs.

4 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 α L1Y−α , 1 > α ≥ 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 a general constant returns to scale production function with the restrictions discussed by Ciccone and Matsuyama (1996).

5 The intermediate input composite is assembled by final goods producers according to the following technology, σ

 k σ −1  σ −1 X =  ∫ xn σ dn , σ > 1 0  where xn denotes the amount of intermediate input of variety n that final good producers employ. The elasticity of substitution between different varieties, σ , 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 it optimal 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 −1 assumes that σ = (1 − α ) . 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 the model to be presented below could be generated under the labour units assumptions but only at the expense of additional restrictions of the kind explored by Ciccone and Matsuyama (1996).

6 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, xn =

ln . Ψ( Fn )

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, Ψ ′( Fn ) < 0 with a choice of zero fixed costs resulting in a constant returns to scale technology with Ψ(0) = 1 . 8 Upstream firms are able to choose their technology from a menu -- assumed here to be any positive real number.9 The actual technology adopted will depend upon demand conditions. As final goods producers earn zero profits, the inverse demand for a given intermediate input depends on the marginal cost of producing a unit of the composite, X. This is also the price of X and it is denoted by P. Thus, 1

k  k   k 1−σ  1−σ σ −1 σ P = min{x }k ∫ pn xn dn ∫ xn dn = 1 =  ∫ pn dn . 10 n n=0   0   0 0

Profit maximisation by final goods producers yields their demand for an individual variety, xn , σ

 P xn = X   ,  pn  where use is made of the assumption of constant returns to scale in final good production.11

8

Every result to come only requires that the marginal labour requirement when F n = 0 be some positive constant. 9 Dasgupta and Stiglitz (1980) and Vassilikas (1989) analyse continuous mechanisms for technological choice but in very different contexts to that here. 10 There is a formal difficulty here when k = 0. One could with additional notation assume that there is always an arbitrarily small subset of upstream that always chooses to produce. However, here it is more convenient to adopt the convention that when k = 0, P = ∞. 11 In deriving this demand function, the infinite product space is approximated as the limit of a sequence of finite economies. See Romer (1987, 1990) and Pascoa (1993) for a more complete discussion of this issue.

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

σ, their optimal pricing scheme if they undertake positive production in period t is, 1

k  1− σ 1− σ P =  ∫ ( σσ−1 wΨ( Fn )) dn = 0 

σ σ −1

1

wℑ1−σ ,

the standard constant mark-up over marginal costs. Using the optimal pricing rule, some simple substitutions show that, P =

σ σ −1

1

wℑ1−σ

σ

and xn = Ψ( Fn )−σ Xℑ1−σ , where,

[(

k

)]

ℑ = ∫ Ψ( Fn )1−σ dn = ℑ k, {Fn }n ≤ k . 0

The aggregate, ℑ, 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 σ

intermediate input producer is simply, ln = Ψ( Fn )1−σ Xℑ1−σ . As such, total labour demand in the upstream sectors is, k

LX = ∫ ln dn = Xℑ1−σ . 1

0

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

1 PX 1−α σ 1− α 1−σ ( α ) = X ( α )( σ −1 )ℑ . w

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

therefore, X = L ( ασ(σ−−α1) )ℑσ −1 . Finally, it remains to find the wage level each period. This 1

can be found by looking at the marginal product of labour in the production of final goods and using the solution for X: w = σ1 (α (σ − 1)) ((1 − α )σ ) α

1− α

α

ℑσ −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,

8

π n = ( pn − wΨ( Fn )) xn − Fn − 1 = Ψ( Fn )1−σ wL ( σ α−α )ℑ−1 − Fn − 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, α − ( σ −1 ) σ −1

π n = ΛL Ψ( Fn )1−σ ℑ where Λ = (α (σ − 1)) ((1 − α )σ ) α

1− α

α σ (σ − α )

− Fn − 1.

. The flows of income and goods leading to this

equation are depicted in Figure One. The wage effect exerts 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 ( (σ − 1)−1 ) outweigh the decreasing returns to additional use of the intermediate input composite (α), 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

σ −1≤ α . 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 modernisation decision depends on marginal profits. Their qualitative aspects are, however, the same.

9 Equilibria In order to simplify the exposition of what follows, I will adopt the following functional form for Ψ( Fn ) , Ψ( Fn ) = ( Fn + 1)−θ , θ > 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 π is concave in Fn , it is assumed that θ <

1 σ −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

m≤k

that

upstream

firms

are

active.

Let

Bn ( ℑ) ≡ arg max Fn π n ( Fn ; ℑ) be the best response set for their modernisation decisions.13 The pair ( m,{Fˆn}n ≤ m ) constitutes a pure strategy Nash equilibrium if: ˆ ) for all n ≤ m ; (i) Fˆn ∈ Bn ( ℑ ˆ = m ( Fˆ + 1)θ (σ −1) dn ; (ii) ℑ

∫

0

n

ˆ ) < 0, ∀n > m . (iii) max Fn π n ( Fn , ℑ 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 of industrialisation is ℑ0 . The following characterise, completely, the set of pure strategy Nash equilibria: α − ( σ −1 ) (i) If L < Λ1 ℑ0− σ −1 , then for all parameters, there exists a “development trap” with no further entry (or modernisation) by intermediate input producers; − α −( σ −1 ) − α −( σ −1 ) − α −( σ −1 ) (ii) If θ (σ 1−1) Λ k σ −1 ≥ L > Λ1 k σ −1 and L < Λθ (1σ −1) ℑ0 σ −1 , there exists an “entry equilibrium” with no modernisation (i.e., k firms enter into production but Fn = 0 for all n); − α −( σ −1 )

(iii) If L > θ (σ 1−1) Λ k σ −1 and α1 > θ , 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 any equilibrium. This is a reasonable assumption given the symmetry among upstream producers producing modern varieties and the fact that basic input producers do not face an entry charge.

10

PROOF: Note first that the strategic complementarities and symmetry in payoff functions ensure that any equilibrium is symmetric. Second, observe that any upstream firm who enters into production chooses their technology according to the following best response function, 1 α − ( σ −1 ) 1−θ ( σ −1 )   Fn* = max 0, Λθ (σ − 1) L ℑ σ −1 − 1 .   Note that this is not positive under the condition for case (i). Moreover, under that condition, π n (0; ℑ0 ) < 0 for all n. This remains true so long as no upstream firm chooses to enter or modernise further. Thus, ℑ = ℑ0 is an equilibrium. Now suppose that all k upstream firms entered but none invests in a more modern technology than their initial level. In this case, ℑ = k and individual profits are: α − ( σ −1 ) π n (0; k ) = ΛLk σ −1 − 1 This is positive so long as, − α −( σ −1 ) L > Λ1 k σ −1 . Observe, however, that − α −( σ −1 ) Fn* = 0 ⇒ L ≤ θ (σ 1−1) Λ k σ −1 . For there to exist a range of L such that these two inequalities hold requires that: − α −σ( σ−1−1 ) − α −( σ −1 ) 1 ≥ L > Λ1 k σ −1 ⇔ σ 1−1 > θ . θ ( σ − 1) Λ k Finally, suppose all producers of modern varieties enter and adopt some positive level of modernisation, i.e., Fˆ > 0. Then ℑ = k ( Fˆ + 1)θ (σ −1) where,

(

(

)

α − ( σ −1 )

)

1

1−θα Fˆ = θ (σ − 1)ΛLk σ −1 − 1. 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 if k 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 > θα ) 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.

11 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 capture the difficulty of the transition in the process of industrialization, which may be responsible for stagnation. In order to understand the self-perpetuating nature of underdevelopment and the inability of the private enterprise system to break away from the 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: σ

∞  σ −1 σ −1 σ X (t ) =  ∫ xn (t ) dn , σ > 1. 0 

14 15

The former possibility is discussed in Gans (1995a). 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.

12 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 of X(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 t allows 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, t

Fn (t ) = ∑ fn (s), s=0

where fn (s) is the amount of the final good purchased in period s. Then in t, and in subsequent periods, the firm is able to produce xn (t ) without additional investment according to: ln (t ) = Ψ( Fn (t )) xn (t ) .

Thus, by incurring sunk costs, intermediate input

producers require only Ψ( 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 of fn (t ) 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 a continuous time setting.

13 As before, higher accumulations of Fn (t ) mean a lower labour requirement, that is, Ψ ′( Fn (t )) < 0. 17 Households now solve an intertemporal maximisation problem. In each period, they choose their level of consumption of the final good to solve: ∞

max{C ( t )} ∞

t =0

∞

subject to

∑( t =1

1 1+ r ( t )

∑ ( ) U (C(t )) t =1

1 t 1+ δ

∞

) C (t ) ≤ L ∑ ( t

t =1

1 1+ r ( t )

) w (t ) + v( 0 ) t

where v(0) is the value of share holdings in upstream firms, U(.) is a continuously differentiable, non-decreasing, strictly concave function, δ > 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 )) 1 + r (t + 1) for all t, = U ′(C(t + 1)) 1+δ ∞

∑( t =1

1 1+ r ( t )

) (C(t ) − Lw(t )) = v(0). t

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, ℑ(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 + 1) = X (t )α LY (t )1−α .

17

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

14 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, α

π n (t ) = ΛL Ψ( Fn (t ))1−σ ℑ(t − 1) σ −1 ℑ(t )−1 − fn (t ) . where Λ is as before. Observe that if σ − 1 ≤ α , then, from a system-wide point of view, there exists a positive feedback between the past technological choices of intermediate input producers and the firm’s 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 fn (t ) and ℑ(t − 1) is nonnegative if and only α

if ℑ(t − 1) σ −1 ℑ(t )−1 is nondecreasing in ℑ(t − 1) for all fn (t ) . Observe that holding the current increment to industrialisation, ∆ℑ(t ) ≡ ℑ(t ) − ℑ(t − 1) , constant, this is equivalent to, α

∂ ∂ℑ( t −1)

ℑ(t − 1) σ −1 ≥ 0, ℑ(t − 1) + ∆ℑ(t )

which is true if and only if σ − 1 ≤ α . 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 firm’s 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

15 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

(

)

Bn, t {ℑ(τ ), ℑ(τ − 1)}τ ≥ t ≡ arg max{ fn (τ )}

τ ≥t

∑ (

1 τ ≥ t 1+ r (τ )

)

t −τ

π n ( fn (τ ); ℑ(τ ), ℑ(τ − 1))

{

}

be the best response set for an active firm n ≤ k (t ). A strategy pair, ( k (t ),{ fˆn (τ )}n ≤ k ( t ) )

τ ≥t

constitutes a pure strategy Nash equilibrium if, for all t: ˆ (τ ), ℑ ˆ (τ − 1)} (i) { fˆn (τ )}τ ≥ t ∈ Bn, t {ℑ τ ≥ t for all active n; ˆ (t ) = (ii) ℑ ∫

k (t )

0

(∑

(iii) max{ fn (τ )} τ ≥t

(

t

)

fˆ (τ ) + 1

s=0 n

∑ (

1 τ ≥ t 1+ r (τ )

)

t −τ

θ (σ −1)

)

dn ;

π n (τ ) < 1, ∀n > k (t ) ;

(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, δ. 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.

16

Proposition 2 (Persistent Industrialisation). Let utility be linear. Suppose that σ −1 at some time t, ℑ(t ) > ℑ* , where ℑ* = ( 1+δ δ ΛL ) σ −1−α . Then ℑ(τ ) − ℑ(t ) ≤ ℑ(τ + 1) − ℑ(τ ) ≤ ... for τ > t. Suppose that in period t, ℑ(t ) > ℑ* . Entry and technological choices are considered in turn. First, given the shock in period t, new varieties enter in period t+1 until the difference in discounted cash flows from entering in t+1 as opposed to t+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 ℑ(t ) close to ℑ* . Let ∆k (t + 1) = k (t + 1) − k (t ) . For an upstream firm, the difference in discounted sum of cash flows between entering t+1 as opposed to t+2 is, PROOF:

α

ΛL ℑ(t ) σ −1 ( ℑ(t ) + ∆k (t + 1))−1 − ( 1+δ δ ) . Setting this equal to zero gives a unique solution: α

∆k (t + 1) = ℑ(t ) σ −1 δ (1σ+−δ 1) ΛL − ℑ(t ) . ∆k (t + 1) is positive since ℑ(t ) > ℑ* . This, in turn, implies that ℑ(t + 1) > ℑ(t ) , meaning that ∆k (t + 2) > 0 since the right hand side of the equation is increasing in ℑ(t ) . Note too that the finiteness of ∆k (t + 1) puts a bound on period by period utility. A similar reasoning applies to the technological decisions. The proof then follows 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, α

ℑ(t ) = ΛL (1+δ δ ) ℑ(t − 1) σ −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, ℑ(t ) > ℑ* , 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 ∆ℑ(t + 1) ≡ ℑ(t + 1) − ℑ(t ) ),

17 α

g( ∆ℑ(t + 1), ℑ(t )) ≡ ΛL 1+r (rt(+t +1)1) ( f + 1)θ (σ −1) −1 ℑ(t ) σ −1 − ℑ(t ) − ∆ℑ(t + 1) = 0 . When utility is linear, the g(.) (i) is positive at ∆ℑ(t + 1) = 0 since ℑ(t ) > ℑ* ; (ii) becomes negative as ∆ℑ(t + 1) grows large; (iii) is strictly decreasing in ∆ℑ(t + 1) ; and (iv) is strictly increasing in ℑ(t ) , once again, since ℑ(t ) > ℑ* . The first three properties guarantee that ∆ℑ(t + 1) is positive and finite (as depicted in Figure 2(a)), while the last guarantees that ∆ℑ(t + 2) > ∆ℑ(t + 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 ∆ℑ(t + 1) and ℑ(t ) . From the household Euler condition, 1 + r (t + 1) = (1 + δ )

(

α

U ′ Λ σα L ℑ(t − 1) σ −1 − ∆k (t )

(

U ′ Λ σα L ℑ(t )

α σ −1

)

− ∆k (t + 1) − F

)

.

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

This means that any of the above properties could be violated.

Therefore, we need additional conditions to assure that any solution, ∆ℑ(t + 1) , to the general firm Euler condition is positive, finite and increasing in ℑ(t ) .

Let

ℑ* ≡ {ℑ(t ) g(0, ℑ(t )) = 0} . The sufficient conditions are: (i) Marginal utility is bounded from below, lim U ′(C(t )) = µ < ∞ ; C ( t )→ 0

(ii) There exists no ∆ℑ(t + 1) > 0 with the property that g( ∆ℑ(t + 1), ℑ(t )) > 0, ∀ℑ(t ) < ℑ* ; (iii) g( ∆ℑ(t + 1), ℑ(t )) ( g(0, ℑ(t )) ) is non-decreasing (increasing) in ℑ(t ) , for all ∆ℑ(t + 1) and ℑ(t ) > ℑ* . 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 ∆ℑ(t + 1) , although this is sufficient for (ii) to hold. All that is required is that the highest value of g occurs at ∆ℑ(t + 1) = 0 when ℑ(t ) < ℑ* . 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 ℑ(t ) = ℑ . *

18 two examples of g satisfying these conditions. Note that in each ∆ℑ(t + 1) > 0 and ∆ℑ(t + 2) ≥ ∆ℑ(t + 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, ℑ(t ) > ℑ* , where ℑ* ≡ {ℑ(t ) g(0, ℑ(t )) = 0} . Then ℑ(τ ) − ℑ(t ) ≤ ℑ(τ + 1) − ℑ(τ ) ≤ ... for τ > t. First, observe that (i) guarantees that as ∆ℑ(t + 1) → ∞ , g( ∆ℑ(t + 1), ℑ(t )) → ∞ . This along with (iv), the continuity of U(.) and the fact that ℑ(t ) > ℑ* ensures there exists at least one solution to g( ∆ℑ(t + 1), ℑ(t )) = 0, with ∆ℑ(t + 1) > 0 , by Theorem 1 of Milgrom and Roberts (1994).20 This, in turn, implies that ℑ(t + 1) > ℑ(t ) , meaning that ∆ℑ(t + 2) ≥ ∆ℑ(t + 1) since the right hand side of the equation is non-decreasing in ℑ(t ) . Note too that (i) guarantees the finiteness of ∆ℑ(t + 1) and hence, puts a bound on period by period utility. The proof then follows by induction. PROOF:

It is worth emphasising here that these propositions guarantee that only ℑ(t + 1) − ℑ(t ) and hence, C(t + 1) − C(t ) 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(0, ℑ(t )) , the growth rate in consumption is bounded away from zero, for all time after ℑ(t ) > ℑ* . 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 and concavity assumptions on U(.), so long as the solution to the household’s problem was an interior one. 21 The easiest way to see this is to examine utility of the form, U ( C ( t )) = 1 C ( t )1 − γ , 0 ≤ γ < 1 . All of 1−γ the conditions (i) to (iv) hold for γ close enough to 0.

19 The Impossibility of Indicative Planning The model under conditions (i) to (iii) has a very interesting implication. As a model of dynamic 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 < ℑ* .

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 − k0 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 + 1)θ (σ −1) > ℑ* ). 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, ΛL ( fn + 1)θ (σ −1) k0 σ −1 (( k − k0 )( f + 1)θ (σ −1) + k0 ) α

−1

+( 1+1δ )ΛL ( fn + 1)θ (σ −1) (( k − k0 )( f + 1)θ (σ −1) + k0 )

α +1−σ σ −1

− fn − 1

.

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

( 1+1δ )ΛL ( fn + 1)θ (σ −1) ((k − k0 )( f + 1)θ (σ −1) + k0 )

α +1−σ σ −1

− ( 1+1δ )( fn + 1) .

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,

20

( 1+δδ )( fn + 1) ≥ ΛL ( fn + 1)θ (σ −1) k0 ((k − k0 )( f + 1)θ (σ −1) + k0 ) α σ −1

−1

.

When fn = 0, this inequality holds, strictly, by the condition for the development trap (i.e., that k0 < ℑ* ). 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, ℑ(0) , if ℑ(0) < ℑ* then the economy is in a development trap for all t. 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 to the one presented here. 23 Matsuyama (1991) states that one state is accessible from another if there exists a rational expectations/perfect foresight equilibrium path from one that state that reaches or converges to the other. A state is absorbing if, within a neighbourhood of it, no other state is accessible.

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 25 26

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

22 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

23 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.

24

Appendix A In this appendix, I derive πn(t). Under these assumptions of Section III,  P(t )  xn (t ) = X (t )   pn (t ) 

σ

1

k (t ) k ( t )   k ( t )  1− σ σ −1 σ P(t ) = min{xn ( t )}nk=( t0)  ∫ pn (t ) xn (t )dn ∫ xn (t ) dn = 1 =  ∫ pn (t )1−σ dn ,   0   0 0

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 ) =

σ σ −1

w(t )ℑ(t ) 1−σ 1

σ

−σ 1−σ and xn (t ) = Ψ( Fn (t )) X (t )ℑ(t ) , where now, k (t )

ℑ(t ) =

∫ Ψ( F (t ))

1− σ

n

[(

)]

dn = ℑ k (t ), {Fn (t )}n ≤ k ( t ) .

0

The aggregate, ℑ(t ) , is therefore a measure of the overall level of industrialisation in period t. Now consider the labour market. As before, 1− σ

ln (t ) = Ψ( Fn (t ))

X (t )ℑ(t )

σ 1−σ

k (t )

∫ l (t )dn = X (t )ℑ(t )

and LX (t ) =

n

1 1−σ

.

0

For the final goods sector, since production is lagged one period, producers choose intermediate inputs and labour to maximise:

(

1 1+ r ( t +1)

)Y (t + 1) − w(t ) L (t ) − P(t ) X (t ). Y

The Cobb-Douglas assumption means that the interest rate drops out with, LY (t ) =

1 P(t ) X (t ) 1−α ( α ) = X (t )( 1−αα )( σσ−1 )ℑ(t )1−σ . w(t )

with period by period labour market clearing implying, X (t ) = L ( ασ(σ−−α1) )ℑ(t ) σ −1 . 1

25 Finally, it remains to find the wage level each period. Observe, first, that in each period the cash flow of an upstream firm is,

π n (t ) = w(t )Ψ( Fn )1−σ L ( σ α−α )ℑ(t )−1 − fn (t ) . Inserting this into the national income identity, Y (t ) − F(t ) − ( k (t ) − k (t − 1)) = w(t ) L + Π(t ) − ( k (t ) − k (t − 1)) where Π(t ) = w(t ) L ℑ(t )

−1

(

k (t )

α σ −α

) ∫ Ψ( Fn )

1− σ

0

and Y (t ) = ((1 − α )σ )

1− α

k (t )

dn −

∫ f (t )dn n

0

(α (σ − 1))1−α ( σ −1 α ) L ℑ(t − 1) . α σ −1

Therefore, w(t ) = ( σ α−α )

α Y (t ) 1 1− α α = α (α (σ − 1)) ((1 − α )σ ) ℑ(t − 1) σ −1 . L

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.

26

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 ℑ(t ) > ℑ* ,     U ′  Λ ασ L ℑ( t ) σ −1  α   1 σ −1 Λ + ℑ − ℑ(t ) > 0 ; g(0, ℑ(t )) = L  (1+ δ )  ( t ) 1  α    U ′  Λ ασ LL ℑ( t −1) σ −1 − ∆k ( t )     α

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) σ −1 hold and suppose that at some time t, ℑ(t ) > ℑ* , where ℑ* = ( ΛL 1+δ δ ) σ −1−α . Then ℑ(τ ) < ℑ(τ + 1) < ... for τ > t. (i) and (iv) ensure that ∆ℑ(t + 1) > 0 for all t, by the intermediate value theorem. Hence ℑ(t + 1) > ℑ(t ) for all t. Note too that the finiteness of ∆ℑ(t + 1) puts a bound on period by period utility. The proof then follows by induction.

PROOF:

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, ℑ(t ) is such that g( ∆ℑ(t + 1), ℑ(t )) > 0 for some ∆ℑ(t + 1) > 0 . Then ℑ(τ ) − ℑ(t ) ≤ ℑ(τ + 1) − ℑ(τ ) ≤ ... for τ > t. PROOF: (i) and the condition of the corollary ensure that ∆ℑ(t + 1) is positive and finite by the intermediate value theorem. Hence ℑ(t + 1) > ℑ(t ) . By (iii) and Theorem 1 of Milgrom and Roberts (1994), this implies that ∆ℑ(t + 2) > ∆ℑ(t + 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

27 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 ℑ* such that g(0, ℑ* ) > 0 , then g(0, ℑ(t )) > 0 , for all ℑ(t ) > ℑ* . Corollary 3 (No Guaranteed Development Trap/Persistent Positive Growth). Assume that only (ii) and (iv)’ hold and suppose that at some time t, ℑ(t ) is such that g(0, ℑ(t )) > 0 .. Then, ℑ(τ ) < ℑ(τ + 1) < ... for τ > t. PROOF: (ii) and (iv)’ ensure that ∆ℑ(t + 1) > 0 for all t, by the intermediate value theorem. Hence ℑ(t + 1) > ℑ(t ) for all t. Note too that the finiteness of ∆ℑ(t + 1) puts a bound on period by period utility. The proof then follows by induction.