CEP 4-18
Product Differentiation, Firm Size, and Entry
Richard L. Carson
January 2017
CARLETON ECONOMIC PAPERS
Department of Economics
1125 Colonel By Drive
Ottawa, Ontario, Canada
K1S 5B6
PRODUCT DIFFERENTIATION, FIRM SIZE, and ENTRY
Richard L. Carson
Carleton University
ABSTRACT
Using the ability of substitute products to constrain a firm’s pricing and output, this note gives a method of
determining the long-run equilibrium output of a firm operating under imperfect competition that differs from
standard methods of output determination and reveals properties of the equilibrium that standard methods
conceal. Under weak assumptions, a policy of free entry and exit is optimal even though all-around marginal-cost
pricing does not prevail. Under monopolistic competition, firm size will not be too small, nor will the supplier of
X have excess capacity.
JEL Classifications: D24, D43, D61.
Key Words: Imperfect Competition, Substitutability, Cost Effectiveness, Firm Size, Excess Capacity.
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PRODUCT DIFFERENTIATION, FIRM SIZE, and ENTRY
I.
Introduction
This note gives a method of finding the long-run equilibrium output of a firm operating under imperfect
competition—specifically under monopolistic competition or oligopoly—that differs from standard methods of
output determination and reveals properties of the equilibrium that standard methods conceal.1 Let X be a product
supplied by a firm under imperfect competition, with long-run general equilibrium output, xe, and suppose there is
a related good, Y, sold under perfect competition or at least under competitive pricing, with price equal to
minimum average cost in equilibrium.2 For example, X might be a differentiated version of a good or service and
Y a generic version. When two basic conditions are met, xe will be where the average cost of Y reaches its
minimum or where the production function for Y displays constant returns to scale. This is true as long as X and Y
are viable and the basic conditions hold. Differentiating a product does not change its long-run equilibrium
output at the level of the firm, which is entirely determined by technical properties of the production function for
the competitive good Y.
It will also be argued that the production function for X displays constant or decreasing returns to scale at
xe, even though the rent-inclusive average cost of X is falling as output expands. In this case, firm size is not too
small nor is there excess capacity. Moreover, a policy of free entry and exit will be optimal even though allaround marginal-cost pricing does not prevail. The reason is that marginal firms will price at or near marginal
cost, and the long-run equilibrium outputs of infra-marginal firms will be sticky in the face of shifts in product
demand. In what follows, we first set out basic assumptions and then use these to derive basic results. Next we
relax one of these assumptions to show how the outcome is changed. Finally, the question of entry is dealt with.
A concluding section summarizes results.
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II.
Assumptions
Let Px and Py be the demand prices of X and Y and x be the output of X by the supplier of this good. X
and Y are assumed to be measured in the same well-defined output units, and with x plotted on the horizontal
axis, ACy is the long-run average cost of Y that a supplier of Y would face. It is assumed to take the
conventional U shape, with a minimum at x = x*. Let ACRx be the “rent-inclusive” long-run average cost of x.
This is long-run average cost inclusive of all rents earned in the supply of X that persist over the long run.
These include the rent to the supplier’s market position, when this is protected by entry barriers, as well as
earnings of product-specialized inputs over and above the opportunity costs of these inputs in supplying other
goods. If all inputs are mobile between firms—including alternative possible suppliers of X—the latter rents are
part of the opportunity cost of a firm supplying X. Thus when inputs are mobile between firms and market
positions are freely transferable from one firm to another, rent-inclusive and opportunity costs will be the same
for each firm.
This is the assumption made here. A firm will then supply X over the long run if and only if it expects to
be able to produce where Px ≥ ACRx, but long-run competitive pressures will also force Px ≤ ACRx to hold at each
x. In long-run equilibrium, the firm supplying X therefore maximizes profit where Px = ACRx—and thus where
its demand is tangent to downward-sloping ACRx—and the firm supplying Y maximizes profit where Py = ACy =
MCy, where MCy is the marginal cost of Y. The outcomes, Px > ACRx or Py > ACy, can occur only in the short
run when sellers are able to earn disequilibrium or entrepreneurial profits (quasi-rents) that attract entry and are
competed away over the long run via expansion of supply by competitors.
Production of X is assumed to use product-specialized inputs, which add value to X, but are not used to
produce Y. Let Fx be a fixed cost of X in the form of rent to the supplier’s market position and to its productspecialized inputs. Then ACRx = ACx + Fx/x, where ACx is the average cost of X, exclusive of this rent. (In
general, ACx and ACy are not the same.) A differentiated product is assumed to possess unique attributes,
features, or properties whose cost-effective supply requires inputs specialized to these unique elements and thus
to the product. For example, suppose a restaurant supplies cuisine with a unique flavor and ambience. Its
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product-specialized inputs are its recipes, together with the tacit knowledge and the talent—embodied in a
master chef and his/her team—used to create the cuisine from the recipes. These inputs could be bought by
another owner which would then acquire the unique cuisine in question. Thus the rent earned by these productspecialized inputs is part of the opportunity cost of the restaurant supplying the cuisine that they produce. The
recipes, tacit knowledge, and talent in question are fixed stocks, and their remuneration is therefore a cost that is
fixed or “quasi-fixed”, in the sense that it does not vary with output once a decision has been taken to produce a
product.
Because production of X uses product-specialized inputs that are not used to produce Y and because the
supplier of X may be able to exercise market power, Px > Py will hold. In long-run general equilibrium, (Px 
Py) ≥ (ACRx  ACy) must also hold for the firm supplying X if ACRx and ACy are evaluated at x = xe. Since Px =
ACRx in equilibrium, (Px  Py) < (ACRx  ACy) would imply that Py > ACy is possible. In general, either (Px  Py)
> (ACRx  ACy) or (Px  Py) = (ACRx  ACy) can hold, equality implying that xe = x*. Figures (a) and (b)
illustrate these possibilities. In Figure (a), the equilibrium price and output of X are assumed to be at E1 where
xe < x* and (Px  Py) > (ACRx  ACy). In Figure (b), equilibrium price and output are at E2, where xe = x*, and
thus (Px  Py) = (ACRx  ACy). For example, in traditional monopolistic competition [Chamberlin 1933], E1 and
E2 are two possible points of tangency between demand and ACRx.
We shall argue below that in an economy with free entry and exit, E2 is the more likely outcome. In the
context above, one could say that the supplier of X adds value to Y internally, within its production process, by
adding quality. However, a firm could also add value to Y externally by supplying a third product, Z, that is
used with Y. For example, let Y be a generic bread with a bland taste and X be a differentiated bread with a
unique taste. Suppose an entrepreneur markets a spread, Z, that improves the taste of the generic bread, but
clashes with the taste of the differentiated bread. Then the combination of generic bread with spread will
compete with the differentiated bread. If both X and Y survive in long-run equilibrium, the basic assumption of
this paper says that at least one product, Z, is also viable over the long-run that meets two conditions—a
substitutability condition and a cost-effectiveness condition.
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The substitutability condition requires Z and X to be substitutes in demand, given the presence of Y
(although not necessarily otherwise). Let the Z industry and the X industry denote the industries in which the
suppliers of these two products operate. Then an expansion of supply in the Z industry that reduces Pz, the
demand price of Z at any given output, z, is also assumed to reduce Px at any given x. Likewise, an expansion
of supply in the X industry that reduces Px at any given x is assumed to lower Pz as well at any given z. Most
products face competition from outside the immediate industry in which they operate. By itself, therefore, the
substitutability condition is not an overly strong requirement.
The cost-effectiveness condition says that a supplier of Z adds value to Y at least as cost effectively as
the supplier of X. Let Z have rent-inclusive average cost, ACRz, defined as for X above. At any given x, the
quality increase from Y raises price by (Px  Py) and rent-inclusive average cost by (ACRx  ACy). To be at least
as cost effective in adding value to Y as is the supplier of X, a supplier of Z must be able to produce an output
scaled to have a price equal to (Px  Py) at a rent-inclusive average cost no greater than (ACRx  ACy).
More precisely, suppose the supplier of X is producing an output, x0, which can be any output. Then a
supplier of Z will be said to be at least as cost effective in adding value to Y as the supplier of X, provided at
least one output, z0, exists such that:
ACRz ≤ (ACRx  ACy)
(1).
whenever Px ≤ ACRx if z is scaled in such a way that Pz = (Px  Py). Here Px, ACRx, ACy, Pz, and ACRz are all
evaluated at x = x0 and z = z0. For every x0, there is assumed to be at least one such z0.
Under free entry and exit, a product is likely to exist that meets both conditions. Suppose that Z meets
the substitutability condition, but not necessarily the cost-effectiveness condition. Suppose that another
product, T, meets the cost-effectiveness condition, but not necessarily the substitutability condition. If output
units of T are scaled in such a way that its price, Pt, equals (Px  Py), which in turn equals Pz, suppose that Z
does not meet the cost-effectiveness condition. But then Pz = Pt = ACRt ≤ (ACRx  ACy) < ACRz, and Z cannot
survive in long-run equilibrium. Thus if any product, T, that is viable in long-run equilibrium meets the costeffectiveness condition, all viable products other than X and Y must meet this condition, including products that
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meet the substitutability condition. Given the lack of restrictions on T, a product, Z, that meets both conditions
is likely to be viable in long-run equilibrium under conditions of all-around free entry and exit. Then fewer
resources will be crowded into the sector of non-differentiated products that are priced competitively and more
resources will be in the sector of differentiated products than when the differentiated sector is protected by entry
barriers.
III.
Basic Results
Assume that a product, Z, is available that meets both the substitutability and the cost-effectiveness
conditions. Then the supplier of X must produce on a scale that is efficient for Y in order to cost compete with
the composite good whose units consist of a unit of Z plus a unit of Y and whose average cost equals (ACRz +
Py). To see this, suppose that x is set where Px = ACRx and where (Px  Py) > (ACRx  ACy) or, equivalently, ACy
> Py as in Figure (a). But then long-run general equilibrium cannot prevail, since (1) implies that a supplier of
Z can produce where Pz > ACRz. As a result, the prospect of quasi-rents will attract entry of new competitors
into the Z industry, and this additional competition will generate downward pressure on Pz and Px. When Pz =
ACRz is reached, Px < ACRx must also hold if the supplier of X is still producing where ACy > Py, since Pz = (Px 
Py) = ACRz ≤ (ACRx  ACy) then holds as well. The supplier of X is failing to cover its opportunity cost and must
therefore change its output or exit the market.
If this supplier survives over the long run, it must produce where (Px  Py) = (ACRx  ACy) = Pz, because
it is otherwise impossible for Px = ACRx to hold, together with (1) and Pz = ACRz. At any other output, the
demand for X lies below ACRx, owing to competition from the Z industry and from Y. This gives xe = x*, the
output at which ACy reaches its minimum value and at which:
Py = MCy = ACy,
(2).
where MCy is the marginal cost of Y. x* is also the output at which the production function for Y exhibits
constant returns to scale. Its value depends only on technical properties of that function, and the equilibrium
outputs of X and Y are the same at the firm level. As a result, xe will be invariant to shifts in demand as long as
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the supplier of X can cover its costs and at least one product is available that meets the substitutability and costeffectiveness conditions. Demand shifts will affect Px and the rent earned from supplying X, as well as the
number of firms in the X industry, but not xe.
Although long-run equilibrium is where ACRx is downward-sloping, ACx could still be constant or
upward-sloping where x = x* = xe. Let Lex = (Px – MCx)/Px be the Lerner index of market power for X. If we
measure returns to scale using ACx rather than ACRx, we remove the effect of Fx on economies of scale. When
the supplier of X faces fixed input prices and uses no indivisible inputs other than those that are specialized to X,
measuring returns to scale from ACx is the same as measuring them from the production function for X. If Rx
denotes returns to scale in the production of X at any given output—or the percentage increase in x that results
from increasing each input by one percent—Rx = ACx/MCx. At any given x, it would be reasonable to expect
that production returns to scale would be at least as great for a more homogeneous version of a product as for a
more differentiated version. Returns to scale increase as a product becomes more standardized and as its
production process becomes more routinized. Thus ACx/MCx ≤ ACy/MCy at any given x.
Suppose then that differentiating the firm’s product (supplying X rather than Y) does not raise returns to
scale at any output, when these returns are measured as ACx/MCx and ACy/MCy. As a result, MCx ≥ ACx at x =
xe = x*. There is no excess capacity in production of X nor is firm size too small. If fx = Fx/Pxx is the share of
product-specialized rent in product value, we have:
Lex ≤ fx,
(3).
when both Lex and fx are evaluated at xe = x*. Thus if free entry and exit leads to competitive pressures on X that
are “strong”, in the sense of keeping fx low, these same pressures will ensure approximate marginal-cost pricing.
The reason is that the supplier of X produces where ACx is constant or upward-sloping, reflecting constant or
decreasing returns to scale in production. ACRx then slopes downward at x* solely because Fx is positive. In the
simplest case, Lex equals one over the elasticity of demand for X at the profit maximum. Thus if fx is no more
than 10 percent (more generally  percent), the elasticity of demand will be at least 10 (more generally 1/),
owing to competition from the Z industry and from Y.
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In general, Z can be supplied under either perfect or imperfect competition, and in the latter case, a
symmetrical relationship may hold between X and Z. Suppose that Z is a differentiated product and that V is a
generic version of Z, which is sold under competitive pricing. Let z* be the output of Z at which ACv reaches its
minimum. In the bread example, V could be a bland spread that complements the differentiated bread (X).
Suppose further that when we interchange the roles of Y and V and of X and Z above, X meets the
substitutability and cost-effectiveness conditions with respect to Z. Then ze = z*.
IV.
Further Results
Suppose xe ╪ x* and let S be a product that meets the substitutability condition. Then S cannot meet the
cost-effectiveness condition in long-run general equilibrium and, indeed, no product can meet it. If one product
did meet this condition, suppliers of S would have to meet it as well by accepting a lower rent or else exit the
market. Yet, even if no product meets the cost-effectiveness condition, the presence of products that meet the
substitutability condition alone will place limits on Lex and on how far xe can depart from x*.
To see this, suppose that for any output, x = x0, produced by a supplier of X, a supplier of S is able to
produce an output, s = s0, at which:
ACRs ≤ Kx(ACRx  ACy),
(1a).
for some constant, Kx > 1, which does not depend on x0. Here s is scaled in such a way that Ps = (Px  Py), and
ACRx, ACy, and ACRs are all evaluated at x = x0 and s = s0. For every such x0, there is assumed to be at least one
s0. If (ACx  ACy) has a positive lower bound over all values of x at which ACRx is downward-sloping, a finite
Kx for which (1a) holds will exist. In fact, there will be a range of parameter values for which (1a) holds, and Kx
is chosen as the minimum value in this range. Inequality (1a) generalizes (1) and becomes (1) as Kx tends to
one. From (1a), and arguing as above, xe must be where:
(ACy  Py) ≤ (Kx – 1)(ACRx  ACy)
(2a).
in long-run general equilibrium. Inequality (1a) can also be derived from (2a), which generalizes (2) and
becomes (2) as Kx tends to one. If two products, S1 and S2, meet the substitutability condition, with associated
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values Kx1 and Kx2, where Kx1 < Kx2, S2 cannot survive in long-run general equilibrium. There can be only one
value of Kx ≥ 1 which holds for every product meeting the substitutability condition that survives in long-run
general equilibrium.
Again assuming that Rx = ACx/MCx ≤ Ry = ACy/MCy at any given output, we have:
Lex ≤ [fx + (Ry – 1)]/Ry = fx + (1 – fx)[(Ry – 1)/Ry]
(3a).
in long-run general equilibrium. Equation (3a) generalizes (3) and becomes (3) as Kx tends to one, since from
(2a), Ry then tends to one as well. The right-hand side of (3a) is increasing in both fx and Ry. When xe is where
Ry > 1, (3a) gives a looser constraint on Lex for any given fx than does (3). If the supplier of X maximizes profit
where Px > MCx, it will produce where Ry ≥ Rx > (1 – fx). Therefore Lex can be no greater than fx plus a fraction
of (Ry – 1). When xe is where Ry < 1, (3a) gives a tighter constraint on Lex than does (3). Approximate
marginal-cost pricing prevails when Kx is close to one and fx is small.
V. Entry
Suppose that, for every product, X, supplied under oligopoly or monopolistic competition, products with
the properties of Y(X) and Z(X) are supplied as well, and also that all-around free entry and exit prevail. Let
MCx and ACx be valued at x = x*. Then marginal firms earn rents equal to or just above x*(MCx – ACx) and
price at or just above marginal cost; this is also the outcome when there is a viable product, S(X), that meets the
substitutability condition and almost meets the cost-effectiveness condition. Infra-marginal firms earn rent
significantly above x*(MCx – ACx) as a share of product value and price significantly above marginal cost.
However, they maintain constant or nearly constant output in the face of demand shifts that do not drive them
from the market, unless these shifts drive out all products that meet both the substitutability and costeffectiveness conditions.
When firms set prices above corresponding marginal costs, a policy of free entry and exit is in general
not optimal [Mankiw and Whinston 1986]. The reason is that entry or exit potentially creates external effects
on welfare that do not become part of the profits or losses of the entering or exiting firms. One such externality
is the “business stealing” effect. Suppose that a new entrant reduces the demand for X. If this causes an output
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decrease of x, the loss to consumers would be roughly Pxx, whereas the cost saving is MCxx, leaving a net
welfare loss of (Px  MCx)x that the entrant does not take into account in making its entry decision. However,
there is no business-stealing effect for goods priced at marginal cost (Px  MCx) nor is there any for products
whose equilibrium output remains constant (x = 0) in the face of entry.
Thus under the assumptions above, the business-stealing effect will be small under free entry and exit.
If X survives the entry of a new competitor, xe will remain unchanged if Z(X) is present or will change by a
relatively small amount if S(X) is present and Kx is close to one. If X is forced to exit, it is likely to have been
earning a small amount of rent before entry occurred and therefore to have been pricing nearly at marginal cost.
The second type of externality created by entry or exit is the “product diversity” effect. When entry
adds a new product—which is not a perfect substitute for any product or combination of products already
supplied—product variety increases. However, entry may also cause previously-existing products to disappear,
which reduces product variety. If customers value variety, an increase in product diversity increases welfare,
and this increase accrues as consumer surplus, but not as profit [Mankiw and Whinston, 1986, p. 54]. Entry or
exit of firms facing horizontal demand does not alter product variety and therefore creates no product-diversity
effects. Thus when (3) holds, firms forced out of an industry by entry are likely to cause small productdiversity effects since such firms are initially earning small amounts of rent and are therefore facing highly
elastic demand. This is also true of firms on the margin of entering or staying out of an industry. Plausibly, the
product-diversity effect is also subject to diminishing returns. As the diversity of products grows, the additional
welfare gain from further diversity is falling. In short, free entry and exit helps to keep business-stealing and
product-diversity effects small when (3) holds, and it may well be impossible to know whether the net bias is in
favor of too many competitors or too few, relative to the socially optimal number.3
VI. Conclusion
Given products X and Y plus a third product, Z, that meets the substitutability condition and a fourth
product, T, that meets the cost-effectiveness condition, the long-run equilibrium output of X at the firm level will
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be where the average cost of Y reaches its minimum and therefore the same as that of Y. The differentiated
version of a product (X) has the same equilibrium output, xe = x*, as the non-differentiated version (Y).
Equilibrium output, xe, will also be where rent-inclusive average cost (ACRx) is downward-sloping, but where
rent-exclusive average cost (ACx) is constant or upward-sloping and where the production function for X displays
constant or decreasing returns to scale. Firm size will not be too small, nor will the supplier of X have excess
capacity.
Even if a product such as T is unavailable, the existence of a product, S, that meets the substitutability
condition, but not the cost-effectiveness condition, puts a limit on how far xe can depart from x* and on how far
price can diverge from marginal cost. The value of x* is wholly determined by technical properties of the
production function for Y. Thus when Z is available or when only S is available, but Kx is close to one, products
supplied under imperfect competition will be priced approximately at marginal cost when they are marginal, in
the sense that the share of product-specialized rent in product value is low. Infra-marginal firms that earn
significant rent will also price significantly above marginal cost. Nevertheless, because the equilibrium outputs of
such firms are sticky in the face of entry or exit, a policy of free entry and exit is efficient.
NOTES
1.
For surveys of models of imperfect competition, see Mansfield and Yohe [2004], Nicholson and Snyder [2012], and Shapiro
[1989]. See Chamberlin [1933] for the basic model of monopolistic competition.
2.
The market for Y may be more vulnerable to “hit-and-run” entry than is the market for X, which causes Y to be priced
competitively, even though Y is not supplied under perfect competition. See Baumol [1982]. Also perfect competition is
compatible with some product differentiation. See Rosen [1974].
3. For further discussion of this question, including what is meant by the “socially optimal” number of firms, see Mankiw and
Whinston [1986].
REFERENCES
1. Baumol, W.J. Contestable Markets: An Uprising in the Theory of Industry Structure. American Economic
Review, March 1982, 72 (1), pp. 1-15.
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2. Chamberlin, E. The Theory of Monopolistic Competition. Cambridge, MA: Harvard
University Press, 1933.
3. Mankiw, N.G. and Whinston, M.D. Free Entry and Social Inefficiency. Rand Journal of Economics,
Spring 1986, 17 (1), pp. 48-58.
4. Mansfield, E. and Yohe, G. Microeconomics. New York: Norton, 11th ed., Chs. 10-13, pp. 356-493.
5. Nicholson, W. and Snyder, C. Microeconomic Theory: Basic Principles and Extensions. Mason, OH:
South-Western, a part of Cengage Learning, 2012, 11th ed., Chs. 14-15, pp. 501-569.
6. Rosen, S. Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition. Journal of
Political Economy, January/February 1974, 82 (1), pp. 34-55.
7. Shapiro, C. “Theories of Oligopoly Behavior.” Ch. 6 of Schmalensee, R. and Willig, R.D. Handbook of
Industrial Organization. Amsterdam: North-Holland, 1989, vol. I, pp. 329-414.
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