Monopolistic Competition with Endogenous Specialization

The Review of Economic Studies, Ltd.

Monopolistic Competition with Endogenous SpecializationAuthor(s): Martin L. WeitzmanSource: The Review of Economic Studies, Vol. 61, No. 1 (Jan., 1994), pp. 45-56Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297876 .

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Review of Economic Studies (1994) 61, 45-56 0034-6527/94/00040045$02.00 ? 1994 The Review of Economic Studies Limited

Monopolistic Competition with

Endogenous Specialization MARTIN L. WEITZMAN

Harvard University

First version received September 1991 ;final version accepted July 1993 (Eds.)

In the familiar spatial model of monopolistic competition on the circle, a product is identified by a single locational characteristic representing its brand or variety. The ability of a variety to compete with other varieties a given distance away (its "specialization" as quantified by transportation losses or how rapidly the "melting iceberg" melts) is exogenously given in the standard model. Here, specialization is a choice variable selected by the firm. A monopolistically competitive general equilibrium is derived, where the degree of specialization is endogenously determined along with other variables. Implications are noted. The central contribution is to show that the effect of endogenizing specialization is to make the Hotelling-Lancaster-Chamberlin model of monopolistic competition isomorphic to the analytically much more tractable Dixit-Stiglitz-Ethier formulation, without sacrificing the appealing concept of product "distance".

INTRODUCTION

Adam Smith's famous dictum that the "division of labour is limited by the extent of the market" was intended as a generalization of his well-known allegory of the pin factory. As the size of the market for pins increases, each stage of 6peration of pin production becomes more specialized, resulting in greater overall output of pins per unit of input.

In some broad sense, Smith's basic message is that an economy develops by producing increasingly specialized goods or services at ever-lower cost. The aim of the present paper is to model this phenomenon by endogenizing an important parameter of monopolistic competition-the specialization of a product-that is almost always taken as given.

More specifically, I suppose the Hotelling-Lancaster-Chamberlin model of monopolistic competition on the circle. A consumer's utility for a good is assumed to decline exponentially with the distance in product space the good must travel to reach the consumer's ideal type. The principal innovation of this paper is to allow the firm to control the rate of exponential decay, at a cost, thus influencing demand for its product. With some specific but not unreasonable assumptions about functional forms, an explicit monopolistically competitive general equilibrium solution is derived. The reduced form of the solution is isomorphic to the Dixit-Stiglitz-Ethier model, thus demonstrating a strong, but hitherto unnoted, connection between the two seemingly different approaches to modelling monopolistic competition. Indeed, the primary technical contribution of the present paper is to show that the great analytic tractability of the "distanceless" Dixit-Stiglitz-Ethier model can be attained in the conceptually appealing Hotelling-Lancaster-Chamberlin model of monopolistic competition where there is a natural "distance" relation among goods, so long as specialization is endogenized in a certain way.

The model exhibits an externality-like increasing returns effect, where the degree of social increasing returns to scale can be derived explicitly as a function of the underlying

45



46 REVIEW OF ECONOMIC STUDIES

parameters. This Smithian externality occurs because a greater "extent of the market" results in firms offering a higher volume of more specialized products at cheaper prices, which yields social value to others greater than the private value to any one firm.

The externality-like social increasing returns feature of monopolistic competition has implications for economic growth, for trade, and for the existence of multiple equilibria, which are noted.

There seems not to be much history of this kind of formulation in the literature. A paper by von Ungern-Sternberg (1988) seeks to approximate the degree of "general purposeness" of a product by its per-unit-distance transportation cost, assumed linear. His paper has quite different aims, assumptions, conclusions and style from my own. It is a partial equilibrium treatment with an industrial organization flavour. His and mine are the only papers, to my knowledge, that attempt to endogenize "transportation cost competition" in the circular-road model of monopolistic competition.

THE MODEL

The paper presents a simple general equilibrium model of monopolistic competition having extreme symmetry properties. In order to obtain analytic solutions and sharp results, the model-perhaps it is more accurate to call it a generalizable example-postulates specific, but reasonable, functional forms.'

It is supposed that consumers derive their ultimate satisfaction from characteristics which are embodied in specific commodities. People differ in their preferences for the underlying characteristics. A buyer of a given attribute preference type wants commodities to be as close as possible to his or her type. Suppose there is a natural one dimensional preference ordering for attributes which allows them to be meaningfully represented as points on a circle of circumference H.

The population of buyers is assumed to be uniformly distributed in attribute preference types around the circle. To say that a buyer is "located" at a particular point on the circle means that the buyer most prefers the variety represented by that point. A buyer consuming c units of the good at distance h from his own location is postulated to achieve a utility level

U(c, h) = cesh. (1)

The expression (1) seems to be the most reasonable simple formula that gives concrete results. In effect, there is an iceberg technology with a constant melting rate. So far as the consumer is concerned, the tradeoff between output and specialization is as if the product decays exponentially at rate s per unit distance that the product must move from production to consumption.

The parameter s is a measure of the specialization of the product or the inverse of its range of competitiveness. The utility isoquant (1) can be rewritten to express the attribute distance over which one unit of the good is competitive with another product delivering

1. The model of this paper is patterned closely on an earlier general equilibrium model of Weitzman (1982), as modified by Meade (1986) and Solow (1986). In its turn, this earlier model had its origins in the pioneering work of Hotelling (1929), Chamberlin (1933), Lancaster (1979), Salop (1979), and others who developed the basic apparatus of monopolistic competition on the circle. The underlying norm is the 'most perfect' spatial competitive equilibrium that could possibly exist under increasing returns.



WEITZMAN ENDOGENOUS SPECIALIZATION 47

total net utility value u < 1 as:

h =- (log (1 /u)). (2) s

A consequence of the formulation put forth is that each buyer specializes in the consumption of just one variety. An alternative interpretation avoids this extreme feature while yielding the same aggregate behaviour. Think of each buyer as a composite of random preference atoms distributed uniformly around the attribute circle. Every atom is itself like a hypothetical modular consumer with the extreme preferences given by (1). The demand attributed to each buyer is the expected demand over uniformly distributed preference atoms. This interpretation, which makes everyone a generalist in consumption, is ultimately equivalent to the conceptually easier-to-envision world where buyers are complete specialists. Actually, other, even more complicated spending patterns could also be accommodated. The crucial assumption is that whenever there is aggregate income I to be spent, in the final reduced form it should be spread out evenly among all preference types.

While there is a continuous spread of expenditures around the circle, due to increasing returns to scale there will only be a finite number of varieties produced. In equilibrium, the firms are assumed to be located at equal distance from each other on the circle, because each firm attempts to avoid the competition of neighbouring firms by getting as far away from them as possible.

Suppose, then, there are n firms. In equilibrium, each firm will cover a market span equal to H/n. Every firm will compete with neighbouring firms in two parameters. The first is price, p. The second is product specialization s. The analysis of this second, non- price, competition constitutes a central focus of the present paper.

At this stage I postulate a single homogeneous "generalized primary factor", which alone is used in production and whose total aggregate availability to the economy is simply given as X. For simplicity, X might be envisioned as inelastically supplied labour, although a much more general interpretation as total aggregate input is possible.

Consider the general form of the problem facing a monopolistically competitive firm. Think of s as a shift parameter of non-price competition that can be lowered to increase demand-at some cost.2 Let the demand function be written in the form

D(p, s). (3)

The cost function is represented as

C(y, s), (4)

where y (=D) is the firm's output or volume of production. The first-order condition for maximizing profits with respect to price p yields the

familiar condition:

p=PC. (5)

2. A story could be told in the context of the present model about wrapping ice blocks in varying amounts of insulation. The general point is that the firm has an option of increasing its degree of non-price ability-to- compete at some cost.




In (5), the markup coefficient p is related to the price elasticity E of the underlying demand function (3) by the well known formula

E

E-1 (6)

The first-order condition for maximizing profits with respect to the shift parameter s is:

=C C2 (7)

Condition (7) can be interpreted as saying that in profit maximizing equilibrium the difference between price and marginal cost, or extra net income obtained from selling an extra unit of the good at the same price, should be equal to the marginal effort cost of making consumers want to buy an extra unit. Although it is typically neglected in standard monopolistic competition theory, condition (7) is, I feel, every bit as important as (5) in understanding the behaviour of the firm.

Returning to the specific model of this paper, suppose that n - 1 of the n symmetric, equally-spaced firms each set their price at value - and their specialization parameter at value s-.

Allow any one firm to vary its price p and specialization parameter s. Suppose the firm is able to attact customers with attribute types in a range of -h/2 to +h/2 centred on the particular variety the firm is producing. This is the firm's 'market area' in attribute space. The marginal buyer must be located right on the boundary at an attribute distance h/2 from the firm and H/n - h/2 from the firm's nearest neighbour. By (1), the buyer who is just indifferent between purchasing from the firm or its nearest neighbour must satisfy the condition

- h/2 e-5fn-2)

(8) p P

Expression (8) can be solved for h(p, s) to yield

sH - -logp + logfi

h(P s) n(s + s)/2 (9)

In what follows here, I assume that all goods are produced for final demand. (An extension to the case of intermediate goods is discussed later.) Suppose that the total income to be spent on all goods is I, distributed uniformly around the circle. Then the demand for this firm's product is

D(p, s) Ih(p, s) (10) pH

where the market area h(p, s) is defined by (9).




Calculating the elasticity of the demand function (10) and evaluating it at the symmetric Nash equilibrium point p =p, s = s yields3

E=1+ n

11 Hs

Combining (6) with (1 1), the corresponding markup formula is

I=1+Hs (12) n

Differentiating (10) with respect to s and evaluating at p=p, s = s yields

is D2 =- ~~ '(13) 2pn

In the present model a firm's costs are expressed in terms of "generalized input", assigned a numeraire price of unity. In order to obtain sharp analytic solutions, the following assumption is made about the form of the cost tradeoff between output and specialization.

Assumption: the cost function can be expressed in the form

(y, s) = (ysa). (14) The function T(g) is intended to be well-behaved and have the traditional character-

istics of a cost function. From (14), there is a constant elasticity tradeoff between y and s. If specialization is

decreased by one percent, the cost increase is the same as if production output y had been increased by q percent. An equivalent way of stating the tradeoff, which will be more relevant to the present paper, is that when specialization is raised by one percent, output can be increased by q percent without altering costs. By varying the elasticity parameter a, situations can be modelled where output trades off more or less readily with specialization.

Let

i?'(>-5v(() G(15)

stand for the elasticity of cost with respect to output, or the ratio of marginal to average cost. It is assumed that

4G'()>0; 4 (0)=0; 4(co)> 1. (16)

The reader should readily become convinced that (16) accommodates a large family of reasonable cost functions, including the classical U-shaped average cost curves that reflect the presence of a fixed cost element.

Using (12) and (14), the price markup condition (5) becomee

P=(1 ?+ )'s-a. (17)

Free entry and exit under monopolistic competition, along with the usual assumption that firms are able to relocate costlessly around the circle, results in the zero-profit

3. Tedious calculations show that the symmetric Nash equilibrium of this model is locally stable. 4. Tedious calculation shows that the second-order conditions for a maximum are reached under not

unreasonable sufficient conditions. (For example, if I" = 0 and E> 3/2.)




condition

py=(ys-a). (18)

When there are no pure profits, total income X of "generalized primary factor input" must equal national product npy. (Remember that all prices are expressed in terms of generalized factor input as numeraire.) Thus, in effect (18) can be written as

pyX= $ (19) n

Also, since all income is assumed to be spent in equilibrium, aggregate spending I of formula (13) must be equal to aggregate income X, or

I=X. (20)

Using (13), (14), the profit maximizing condition (7) becomes

p-_T5-a = yas (21) Is

2pn

Using (19), (20), condition (21) can be simplified to

p = T's-a(1 + 2a). (22)

Dividing the left- (right-) hand side of (22) into the left- (right-) hand side of (18) yields, after re-arranging,

ySa (ySa) 1 (23)

TI(yS-a) 1+2a

By definition (15), equation (23) is equivalent to the condition

4 (ys-a) 1 (24)

Condition (24) can be rewritten in the more amenable form

yS-a = A, (25)

where A is defined to be the unique, by (16), solution to the equation

1 + 2a * (26)

Combining (18), (19), (25) yields, after re-arrangement

n - X. (27)

In equation (27) and what follows, the expression within the square brackets is a constant. This mode of formulation is intended to highlight the basic relation between the equilibrium values of the fundamental variables of this model to the level of aggregate generalized primary factor X.




Generalized input employed per firm is, from (27),

x x= ---= [T(A)]* (28)

n

The extremely simple forms of (27), (28) foreshadow the analytical tractability of the entire general equilibrium system. The result of a complicated balance of forces is that each firm has the same input level independent of X, so that the total number of firms is linearly proportional to the total availability of factor input.

Comparing (22) with (17), we have

n H (29) s 2a

It follows immediately from (29) and (11), (12) that

E=[1 + 1 (30)

p=[1+2a]. (31)

Combining (27) with (29) yields

F2a1

S[I j-X. (32) Equations (32) and (25) imply

\a-j

HT (A))

Combining (19) and (27) with (33) determines

P= [2- T(])' xa (34) Except with probability measure zero, the economic system is not delivering to

any customer the characteristic type most preferred by that customer. Let 0 represent the average utility delivered per unit of good produced. From (1),

T H/2n

eh/vdh

0 0 (35) H/2n

Substituting from (29) into (35) yields

[ =

(36)

The "real price" of goods relative to generalized input is then

p*=P (37) 0




"Real" national product, appropriately measured by (1), is

Y= nyO. (38)

Substituting from (27), (33), (36) determines "real" national product to be

[ _A (2a) o XI+a (39)

(Note that what I am calling "real" national product differs from "traditionally measured" national product only by the multiplicative constant 0, making no substan- tive analytical difference between the two concepts.)

Equations (27), (28), (30), (31), (33), (34), (39) relate all equilibrium values of the fundamental variables of this model to some constant [within square brackets] times some simple function of the level of aggregate "generalized primary factor" X.

ANALYSING THE MODEL

In Adam Smith's famous paradigm, "the division of labour is limited by the extent of the market". Smith's "extent of the market" might be identified in this model with X. If the example of the pin factory is interpreted ultimately as a parable about the development of an entire economy, as presumably was intended, then the present model would appear to capture fairly the essence of Smith's famous dictum. As the amount of generalized factor input X increases, so too does the variety of different products n, the degree of specificity of each product s, and the economy's overall productivity Y/X.

As the economy develops, here identified with larger values of X, by (27) the variety of products n increases linearly with X. The resultant crowding of firms around the product attribute circle might be thought to intensify competitiveness. But there is another side of the issue. With more crowding, a firm does not need to produce such a versatile product. Due to the increased availability of similar products, the firm cannot now aspire anyway to capture as wide a market area as before. Thus, firms decide it is more profitable to compete by offering a greater volume of more specialized products at cheaper prices, and the entire economy develops along these lines.5 Actually, in the particular parameterization of this model, the effects of increased numbers of firms and increased specialization of products exactly offset each other, so that each firm's elasticity of demand remains constant, by (30).

From the fundamental reduced form expression (39), the system as a whole displays increasing returns to scale, where a > 0 is a measure of the degree of increasing returns to scale. The parameter a measures the elasticity of productivity Y/X with respect to primary factor X.

In this model a is a positive constant. The percentage by which division-of-labour productivity is increased by a one percent increase in the size of the market never dimin- ishes. (Presumably Smith intended something like this because he viewed division of labour as a major ongoing engine of growth.) In the traditional formulation of the circle model, where s is a given constant, total factor productivity reaches an asymptotic upper limit as the amount of primary factor increases to infinity.

5. It is perhaps easier to think of high-volume specialized products occurring in intermediate goods than in consumption goods. An equivalent version of the model could emphasize this aspect by interpreting all of the n goods as intermediate inputs to a single final good Y. In this interpretation, Y might stand for pins while the n goods represent various sub-operations performed in making pins.




The ultimate source of increasing returns throughout the system is the ability to substitute volume for versatility as embodied in (14). Essentially, the economy develops by producing increasingly specialized goods at lower cost.

When a-+O+, it is extremely cheap to produce high general-purpose products and the economy in effect approaches the perfectly competitive outcome. As a becomes larger, versatility is relatively more expensive. With greater a, each fractional increase in specialization allows a greater fractional increase in output without altering costs. Thus, as firms substitute toward higher values of s when X (and n) increases, other things being equal that permits higher levels of y the larger is a.

It is not difficult to extend the model to include intermediate goods.6 The presence of intermediate production amplifies the degree of overall increasing returns, in effect making the coefficient a larger. As the economy develops, cheaper intermediate inputs increasingly substitute for primary factor in the production process.

Note that the economy of scale the system as a whole displays is strictly an external economy so far as any single agent is concerned. If an individual in equilibrium could obtain one percent more X, he could increase his private value by just one percent. But for the economy as a whole, the elasticity of social output with respect to social input is 1 +a. The discrepancy between social and private returns is due to an externality-the benefits of division of labour accrue throughout the economy and cannot be fully captured by the particular agent that increases the size of the market.

There is an interesting isomorphism between the reduced form of the present model of competition on the circle and the reduced form of the Dixit-Stiglitz model of monopolistic competition as modified for production by Ethier, Romer, and others.7 The latter approach begins with a utility function typically of the CES family

U({c}) ) = (Yc')I/P (40)

which generates iso-elastic demand functions. Not all possible goods can be produced, due to some fixed cost. Costs are given by some function T(ci) having similar properties to (15), (16).

When amount X of the primary factor is available, it is not difficult to show that the relation between aggregate utility and aggregate primary factor for the Dixit-Stiglitz model is of the general form (39). Previously it had been thought that the Dixit-Stiglitz model of a representative consumer and the Hotelling-Lancaster model of attributes on the circle were two different approaches to monopolistic competition that happened to give similar results in practice. This paper shows there is a deeper connection.

The Dixit-Stiglitz model has the disturbing property that there is no sense in which different goods are nearer or farther from each other. All goods are equally distant from each other. The model has been widely used primarily because of its great analytical tractability. Perhaps not surprisingly, I find the generalized Hotelling-Lancaster approach of the present paper more appealing because it seems richer and more convincing as a description of how goods (and firms) are related to each other. However, since the reduced forms (39) are isomorphic, the aggregate behaviour of both models is identical. In other words, the Hotelling-Lancaster spatial model of monopolistic competition can also have great analytical tractability, without sacrificing its appealing spatial structure, when the degree of specialization is endogenized.

6. This extension is carried out in Weitzman (1991). 7. See Dixit and Stiglitz (1977), Ethier (1982), Romer (1987, 1989), and the references cited therein.




IMPLICATIONS FOR TRADE, MULTIPLE EQUILIBRIA, AND GROWTH

Suppose there are J "countries", indexed by j= 1, 2, .. ., J. The word "country" is being used here in a generic sense to denote any economic sub-group of a larger population. Suppose countryj commits Xj units of effective8 generalized primary factor to participation in trade.

Let

X= Xj (41)

be aggregate world primary factor committed to trade. Then, from (39), total world product or income derived from trade is

Y(X) = [ Y(1)I X I". (42)

The national income or product of country j deriving from trade would then be

yi =(J) *Y (43)

Combining (42) with (43), we obtain

Y = [ Y(1 )]XjXa. (44)

Equation (44) shows clearly the externality inherent in the present model. Let XJ be interpreted generally as the degree of economic "activity" or "participation" of sub-group j. Then the income Yj of any sub-group j is directly proportional to its participation Xj, but is "enhanced" by total group participation X raised to the power a. If Xj is small relative to X, then agent j can effectively take X in (44) as given when calculating its privately optimal contribution of Xj. When every sub-group is so calculating, however, the result will be socially sub-optimal levels of participation. In a private equilibrium each agent will under-invest in trade and division of labour relative to what is socially optimal. Thus, in the world of this model there is a theoretical case for co-ordinating economic activity to achieve greater input and output levels.

The same point can be made somewhat differently as follows. Suppose that each country's supply of primary factor devoted to international trade is a function of the income it yields, reflecting the alternative domestic uses to which primary factor can be put. People can specialize into a trade niche or produce more general products for home use. Supply of XJ available for trade can then be written as the upward sloping supply function

Xi= S1( Yj), (45)

where the derivative of (45), SJ, is positive. It is not difficult to see that the system of equations (41), (44), (45) has the potential

for generating a large number of multiple equilibria, which can, in general, be Pareto ordered. There may be low-level equilibria where every agent participates minimally because every other agent is participating minimally and therefore private output per unit of private input, which from (44) is proportional to X'1, is low. Or, there can be high-level equilibria where agents participate relatively fully because other agents are participating relatively fully and the return to private participation from (44) is relatively high. The

8. By definition, each unit of effective generalized primailrlly l.-Actor lhas (lie sa;.me productivity in each country.




present model thus exhibits the "strategic complementarities" which may lie behind some macroeconomic failures of co-ordination.9

Thus far, the endowment of total generalized primary factor X has been taken as exogenously given while the major variables of interest have been derived as functions of X. How the economy develops thus depends, in a well-defined way, on how X grows. At this point the model is waiting to be closed by specifying a reasonable mechanism for X to grow over time. Virtually any structure of economic growth could be grafted onto the mechanical apparatus of the present model, with results that are not difficult to predict. (Indeed, the externality-like increasing returns structure is particularly compatible with models of the "new" growth theory.'0)

As just one example, take the standard Solow growth model." Suppose that aggregate primary factor input X is itself produced by an aggregate production function homogeneous of degree one in capital and labour with Hicks-neutral technical change. In the model here, the effect of endogenous specialization is to scale down the contribution of the Solow residual, other things being equal. The fraction a of output growth due to economies of scale must in effect be subtracted from raw output growth before calculating the true residual. The increasing returns of the present model essentially amplifies by (1 +a) the growth contributions of labour, capital, and techno- logical change, thus making product specialization, too, an engine of economic growth.

CONCLUSION

This paper attempts to model Adam Smith's insight that specialization from division of labour plays an important role in economic development. The model is at a very high level of abstraction, offering insights into very long-term trends or very general tendencies, at best. The model suggests a way in which the idea that "the division of labour is limited by the extent of the market" can be translated into a workable parable about how economic development proceeds.

Technically, the basic contribution is to show an isomorphism between the conceptually more appealing Hotelling-Lancaster-Chamberlin model of spatial monopolistic competition and the analytically more tractable Dixit-Stiglitz-Ethier formulation-when the degree of specialization in the former is modelled appropriately as a choice variable. Thus, any "endogenous growth" or other applications of the latter formulation can be given a much more natural and appealing (in my opinion) interpretation in terms of an economy developing by providing increasingly specialized products at lower cost.

While it would be dangerous to attempt to draw hard conclusions from such an abstract formulation, the model suggests, in the spirit of Adam Smith, that increased specialization from division of labour may be an important engine of economic expansion.

Acknowledgements. This paper is a revised and abridged version of my earlier paper "Volume, Variety, and Versatility in Growth and Trade." For their helpful comments on the original version, I am grateful to K. J. Arrow, A. Mas-Colell, P. Romer and E. Maskin.

9. For more on "strategic complementaries" see Cooper and John (1988) and the references contained therein.

10. For an extremely lucid survey, see Romer (1989) and the references cited therein. 11. See, for example, Solow (1970).




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Monopolistic Competition with Endogenous Specialization