Does competition increase pass-through?

How does market power affect the rate of pass-through from marginal cost to the market price? A standard intuition is that more competition makes prices more ”cost-reflective” and thus raises cost pass-through. This paper shows that this intuition is sensitive to the common assumption in the literature that firms’ marginal costs are constant. If firms have even modestly increasing marginal costs, more intense competition actually reduces pass-through. These results apply to the “normal” case where pass-through is less than 100%. They have implications for competition policy and environmental regulation.


Introduction
In recent years, there has been a resurgence of interest in cost pass-through as tool to understand market performance and the e¤ects of policy interventions across a wide range of …elds in economics including industrial organization, public economics, and international trade (Weyl & Fabinger 2013).
How does competition a¤ect pass-through?A common intuition is that …rms with market power have an incentive to "absorb"part of a cost change whereas, under perfect competition, price equals marginal cost so pass-through is 100%.This suggests that more intense competition leads to stronger pass-through.Perhaps most prominently, this intuition holds in a textbook linear Cournot model, with 50% pass-through under monopoly which rises up to 100% as the number of …rms grows large.
Yet this intuition and the existing literature on pass-through under imperfect competition (e.g., Bulow  ) maintain the assumption that …rms have constant marginal costs.On one hand, this is a substantive economic assumption which may be appropriate for some markets but less so for others.On the other hand, it obscures the comparison with the benchmark of perfect competition-precisely because it restricts competitive pass-through to a "knife-edge"rate of 100%.This paper revisits the basic question of how competition a¤ects cost pass-through.It generalizes earlier results from the pass-through literature and highlights their sensitivity to the assumption of constant marginal cost.The model has two key features.First, to facilitate the comparison with perfect competition, the industry sells a homogenous product and the setup nests monopoly, oligopoly and perfect competition as special cases.Second, …rms have convex cost functions, which can be justi…ed purely on technology grounds or by invoking the frictions that arise from principal-agent problems within the …rm (see especially Hart 1995).
The main point is that, if …rms have even modestly increasing marginal costs, the standard intuition is overturned-and more intense competition actually reduces passthrough.A less ‡exible production technology, with more steeply increasing marginal cost, always leads to lower pass-through.This holds in a textbook model of perfect competition and extends to imperfect competition.However, the e¤ect is stronger for a more competitive market because it has higher industry output.This helps explains why, in markets with a fairly in ‡exible production technology, more competition can be associated with less pass-through.Importantly, these results apply to the "normal" case where pass-through is less than 100%.
Consider comparing two markets with di¤erent intensities of competition.For a likefor-like comparison, suppose that any di¤erences in demand and cost conditions are controlled for.The analysis shows that, under plausible conditions, the more competitive market always has lower pass-through if cost convexity is su¢ ciently pronounced.For example, if demand is strictly convex and …rms' cost functions are at least quadratic, then the more competitive market passes on less of a (small) cost increase.
From a policy perspective, questions about pass-through and market power are especially salient across the energy industry.Pass-through of fuel costs to retail electricity prices has been an important concern of competition policy in the UK electricity sector (CMA 2015).Similarly, the extent to which a carbon price imposed on energy-intensive (and often signi…cantly concentrated) industries such as electricity, cement and steel is passed onto market prices is central to the e¤ectiveness of market-based regulation towards climate change (Fabra & Reguant 2014).
Section 2 sets up the model, and Section 3 presents a unifying equilibrium result on cost pass-through that holds under perfect and imperfect competition.Section 4 presents conditions under which more competition leads to lower cost pass-through, and Section 5 gives two illustrative examples.Section 6 concludes.

The model
Consider a simple model of imperfect competition between n symmetric …rms that nests perfect competition and monopoly as special cases.
The inverse demand curve is p(X), where p is the market price, X is industry output and p 0 ( ) < 0. Let " D p(X)=Xp 0 (X) > 0 be the price elasticity of demand and let

D
Xp 00 (X)=p 0 (X) be a measure of demand curvature.Demand is concave if D 0 and convex otherwise; it is log-concave (i.e., the log of direct demand ln D(p) is concave in p) if D 1 and log-convex otherwise.
Demand curvature can also be expressed as is the superelasticity of demand, i.e., the elasticity of the elasticity (Kimball 1995).So demand is log-concave D 1 if and only if it is superelastic D 1.
where x i is its output (so X P i x i ), is a cost shifter, and which satis…es C 0 ( ) > 0; C 00 ( ) 0 (where b ) 0 be the elasticity of i's marginal cost which, given symmetry, will be identical across …rms with S i = S .This can be seen as a measure of the "in ‡exibility" of the production technology.
Remark 1.The model de…nes the elasticity of …rm i's marginal cost b C 0 (x i ) including the cost shifter .Many papers on pass-through focus on the case in which the initial value of the cost shifter is zero, = 0. Then marginal cost is (locally) identical including and excluding the cost shifter b C 0 (x i ) = C 0 (x i ), and so the cost elasticity S i = x i C 00 (x i )=C 0 (x i ) can equivalently be written without the cost shifter. 1This paper does not restrict atten-tion to = 0, though its …ndings also apply to this case.Firm i's pro…ts are given by i = p(X)x i C(x i ) x i .Each …rm chooses its quantity x i in a generalized version of Cournot competition.The industry's conduct parameter 2 [0; 1] serves as a summary statistic of the intensity of competition.Formally, …rms' equilibrium outputs (x i ) i=1:::n satisfy: Firm i, in deviating its output by (x i x i ), conjectures that industry output will change by (x i x i ) as a result.In this "conduct equilibrium", lower values of correspond to more intense competition.This setup can be viewed as a reduced-form representation of a dynamic game (Cabral 1995).The Cournot-Nash equilibrium, where each …rm takes its rivals'output as given, occurs where = 1, and perfect competition where = 0. Two regularity conditions will ensure a well-behaved interior equilibrium.First, a su¢ cient condition for an interior equilibrium is that p(0) > b C(0) = C 0 (0) + .Second, the condition D < 2, such that the industry's marginal revenue is downward-sloping, will ensure a well-behaved equilibrium, regardless of the intensity of competition.
The …rst-order condition for …rm i is: This says that a generalized version of …rm i's marginal revenue equals its marginal cost. 2n symmetric equilibrium, X = nx , and so the …rst-order condition becomes: Let b S ( =n) be an index of market power which is higher with a larger conduct parameter and/or fewer …rms.Writing p( b S ) for the equilibrium price, the role of this index is made precise as follows: , where the equilibrium market price p( b S ) rises with b S .
The setup facilitates comparative statics on competition via changes in b S (where both and n are exogenous).As expected, less intense competition leads to a higher market price (and lower industry output).Note also that, at equilibrium, the price elasticity of demand cannot be too low, with " D > b S (and so " D > 1 for monopoly).

Equilibrium cost pass-through
The analysis begins by deriving an expression for cost pass-through: the change in the equilibrium market price arising from a market-wide rise in marginal cost, dp=d .

Lemma 2
The equilibrium rate of cost pass-through equals: The expression for pass-through from Lemma 2 nests various results from prior literature. 3First, under perfect competition ( b S = 0), the …rst-order condition (2) de…nes …rm i's supply curve; letting " S i px 0 i (p)=x i (p) > 0 be …rm i's price elasticity of supply, at symmetric equilibrium, " S i = " S and S = 1=" S .This leads to the textbook result that competitive pass-through = " S =(" S + " D ) is driven by the ratio of demand and supply elasticities-and is never greater than 100%.(1 D )] is additionally determined by market structure-as then given by the competition index b S (1=n).Lemma 2 shows that, more generally, pass-through is determined by four factors: the price elasticity of demand " D , demand curvature D , the elasticity of marginal cost S , and the intensity of competition b S .The role of the demand elasticity " D is predicated on the presence of the cost elasticity, S > 0, which is often assumed away in prior literature.
It is easy to see that pass-through, in general, is always lower for a less ‡exible production technology, that is, @ =@ S < 0, all else equal.In this sense, a basic insight from perfect competition extends to settings with imperfect competition.In the limit, pass-through tends to zero, !0, as technology becomes entirely in ‡exible, S ! 1, for example, because …rms face binding capacity constraints.In such a situation, the change in marginal cost induces no change in output-and hence also no price change.
It is well-known that, under imperfect competition, it is possible for pass-through to exceed 100%.Proposition 1 makes precise that this occurs whenever b S ( D 1) S (" D b S ).Several things are needed: (i) there is market power b S > 0; (ii) demand is log-convex D > 1 (equivalently, superinelastic D < 1); and (iii) the elasticity of marginal cost S cannot be too large (for example, if S > maxf0; (" D 1) 1 g S then < 1 for any b S 2 [0; 1] and D < 2).
4 Does competition increase pass-through?
What is the equilibrium impact of more competition on cost pass-through?Answering this question requires some care because varying the intensity of competition via b S can, in general, also a¤ect the (equilibrium) values of the demand and cost parameters (" D ; D ; S ) as none of these are necessarily constants.
Two approaches are presented.First, the "cross section" approach compares passthrough in two di¤erent markets on a like-for-like basis, where one market is more competitive than the other but identical in terms of (" D ; D ; S ).Second, the "time series" approach compares pass-through in the same market following an exogenous increase in its intensity of competition, taking into account any knock-on e¤ects on (" D ; D ; S ).
Under both approaches, it will turn out that cost convexity makes the standard intuition-more competition raises pass-through-quite fragile.Firm conduct is more competitive in market 1 because there are more …rms or because rivalry is more intense for the same number of …rms.

Varying competition between markets
The markets may di¤er in terms of their demand and cost functions.Lemma 2 makes clear that the relevant demand and cost conditions for pass-through are given by (" D ; D ; S ).The idea here is that an econometric analysis will control for any di¤erences between the markets in terms of the values of (" D ; D ; S ).Proposition 1 Consider two markets 1 and 2 with identical demand conditions (as given by " D ; D ) and cost conditions (as given by S ) where market 1 is more competitive than market 2 with b S 1 < b S 2 .Equilibrium cost pass-through is lower in the more competitive market 1, ( b S 1 ) < ( b S 2 ), if and only if demand and cost conditions satisfy: which always holds for a su¢ ciently large elasticity of marginal cost S .
Proposition 1 yields the opposite of the standard intuition that more competition leads to higher pass-through.All else equal, whenever costs are su¢ ciently convex in that S > 1 D , pass-through is lower in the market with more intense competition.For example, the condition always holds for pass-through j !0 of a small new tax if demand is strictly convex D > 0 and costs are at least as convex as a quadratic cost function, C(x i ) = kx 2 i (as then S 1).More generally, the condition always holds for a su¢ ciently large S , regardless of demand conditions and competitive intensity.
In the special case with constant marginal cost, S = 0, the condition from Proposition 1 boils down to demand being log-convex D > 1 (equivalently, superinelastic D < 1).In such circumstances, therefore, both markets feature pass-through in excess of 100% but it is closer to 100% in the more competitive market, ( b S 2 ) > ( b S 1 ) > 1.By contrast, with non-constant marginal cost, S > 0, more competition can yield lower pass-through even in the "normal"case in which it lies below 100%.
Figure 1 illustrates Proposition 1 by plotting pass-through against the cost elasticity S for the two polar cases: perfect competition in market 1 ( b S 1 = 0) and monopoly in market 2 ( b S 2 = 1).Demand is taken to be linear D = 0 with a price elasticity of demand (at equilibrium) set at " D = 2.For "small"cost elasticities, S 1, pass-through rates are higher in the competitive market ( b S 1 ) ( b S 2 ).This is in line with the standard intuition.However, for "large" cost elasticities, S > 1, this relationship ‡ips to ( b S 1 ) < ( b S 2 ) and it is the monopoly market that features stronger pass-through.This is the opposite of the standard intuition.(In the limiting case as S ! 1, pass-through converges to zero for both market structures.) What is driving this result?Recall that a less ‡exible production technology always means lower pass-through, @ =@ S < 0. A key observation is that this e¤ect is mitigated by market power in the following sense: Lemma 3. Equilibrium cost pass-through satis…es @ @ b S @ @ S (" D ; D ; S ; b S ) 0 if and only if the cost elasticity satis…es S D is a su¢ cient condition.
Lemma 3 shows that, for modest values of S , the pass-through function is supermodular in the cost elasticity and market power.A less ‡exible production technology means lower pass-through-and more strongly so for a more competitive market.This helps explains why, in markets with a fairly in ‡exible production technology, more competition can be associated with less pass-through.

Varying competition within a market
Now consider the second approach: the same market, with the same demand and cost functions, is observed "over time"and competition (exogenously) intensi…es, as measured by a lower b S . Write the price in terms of the conduct parameter p( b S ), and think of the (equilibrium) values of the demand and cost parameters as )).How does more competition a¤ect pass-through?
Let S i x i C 000 (x i )=C 00 (x i ) be the elasticity of the slope of i's marginal cost which, given symmetry, will again be identical across …rms with S i = S (also recalling that b C 00 ( ) = C 00 ( ) and b C 000 ( ) = C 000 ( )).
Proposition 2. (a) Equilibrium cost pass-through is lower with more competition, d ( b S )=d b S > 0, if and only if demand and cost conditions and …rm conduct satisfy: ) for which S > 0 and S > (1 2 D ) are then necessary.
Proposition 2 delivers a similar conclusion to Proposition 1: Under plausible conditions, more competition reduces pass-through-and the standard intuition is overturned.
There is a simple set of su¢ cient conditions.First, demand is log-concave, which is a common assumption in economic theory (e.g., Bagnoli & Bergstrom 2005), and is more convex at a higher price d D (p)=dp 0, which applies, for example, for any demand curve of the family p(X) = X , which has constant curvature D = 1 .Second, …rms'costs and marginal costs are su¢ ciently convex, that is, S > 0 , C 00 ( ) > 0 and S > 0 , C 000 ( ) > 0 are both positive and su¢ ciently large.
To see the role of su¢ cient cost convexity, consider a market with a single …rm and linear demand (n = 1, D = 0).Initially the …rm is a price-taker ( b S = 0) and then it becomes a monopolist ( b S = 1).Let x c x(0) denote the competitive output and x m x(1) the monopoly output, where x m < x c .Cost pass-through under monopoly m is higher than with perfect competition c whenever: which holds if and only if R x c x m C 000 (y)dy = [C 00 (x c ) C 00 (x m )] > .So competition reduces pass-through if C 00 ( ) > 0 and C 000 ( ) is large enough.The condition from Proposition 2 provides a general result for the case of a small change in competitive intensity.

Illustrative examples
A couple of examples are useful to illustrate the issues that arise and the di¤erences between the "cross section"and "time series"approaches.For simplicity, these examples consider pass-through j =0 where the initial value of the cost shifter is zero.
Example 1. Demand is linear p(X) = X while …rms'cost functions are quadratic C(x i ) = kx 2 i .This corresponds to D = 0 and S = 1.Using Lemma 2, pass-through j =0 = (1 + " D ) 1 < 1 then is always incomplete-and depends directly only on the demand elasticity " D .
Between markets: Conditional on the demand elasticity " D , competition has zero impact on pass-through in the cross section, i.e., ( b S 1 ) = ( b S 2 ).This is a knife-edge case of the condition of Proposition 1.This is at least partly inconsistent with the standard intuition.
Within market: With linear demand, the elasticity " D (p( b S ) itself varies along the demand curve-with a positive superelasticity D = 1 + " D > 0. So a higher b S implies a higher price p( b S ), a higher elasticity " D (p( b S )), and hence lower pass-through with d ( b S )=d b S < 0, thus violating the condition of Proposition 2. So here competition increases pass-through via a demand-superelasticity channel.This is in line with the standard intuition.
Example 2. Demand is exponential p(X) = log X and …rms'cost functions have constant elasticity C(x i ) = kx i , with > 1.This corresponds to D = 1 (equivalently, unit superelasticity D = 1) and S = 1 > 0. Using Lemma 2, pass-through )( 1)] 1 < 1 is again always incomplete but now also more involved.Between markets: Conditional on the demand elasticity " D and the cost elasticity , the more competitive market 1-by inspection or by Proposition 1-always has lower pass-through in the cross section, i.e., ( b S 1 ) < ( b S 2 ) This is the opposite of the standard intuition.
Within market: Pass-through declines with more competition, ))=d b S = , so the condition of Proposition 2 always holds.So here competition reduces pass-through via a competition-plus-demand channel.This is again the opposite of the standard intuition.

Conclusions and policy implications
Existing literature on imperfect competition typically assumes that …rms have constant marginal costs.As a result, pass-through analysis has focused on demand-side properties.More competition then raises pass-through as long as it lies below 100%.
This paper has shown that this result is perhaps surprisingly fragile.If …rms have increasing marginal costs, then more competition may reduce pass-through.A rough intuition is that a more competitive industry has higher output, and with convex costs is therefore more exposed to a cost increase.
These results may have implications for competition policy, for example, for understanding how cost savings from horizontal mergers are passed on to consumers and for evaluating the "passing-on defense"(Verboven & Van Dijk 2009) whereby cartel damages are limited because a¤ected …rms pass the overcharge onto their own customers.
In a more competitive market, the market price will be more re ‡ective of marginal cost.It does not follow that price changes will necessarily be more re ‡ective of cost changes.

Appendix
Proof of Lemma 1.The expression for L " D [p(X) b C 0 (x)]=p(X) follows by rearranging (3) and using the de…nitions of " D and b S .Di¤erentiating (3) shows that: where the denominator of this expression is positive because (n + ) > D given that n 1, 2 [0; 1] and D < 2 as well as C 00 (x) 0 and p 0 (X) < 0.
Proof of Lemma 2. By construction, cost pass-through satis…es using the de…nition D Xp 00 (X)=p 0 (X) and where the denominator is again positive.Now rewrite the last term as follows: which uses Lemma 1 and the de…nitions " D p(X)=Xp 0 (X), S i , and b S ( =n).Combining ( 5) and ( 6) and some rearranging yields the expression for (" D ; D ; S ; b S ).
Proof of Proposition 1.Given the assumptions, follows by inspection of Lemma 2.
Proof of Lemma 3. Di¤erentiating the expression for equilibrium cost pass-through from Lemma 2 gives: Di¤erentiating again for the cross-partial e¤ect gives: @ @ b S @ @ S (" D ; D ; S ; b S ) = It is immediate that @ @ b S @ @ S > 0 if S 1 D and some further rearranging shows that @ @ b S @ @ S > 0 > 0 if and only if where the last step uses the de…nitions D Xp 00 (X)=p 0 (X) and S i x i C 000 (x i )=C 00 (x i ) (at symmetric equilibrium) and combines the result for dp=d b S from Lemma 1, see (4), with the result for from Lemma 2, see (5).Now using which can be rearranged as claimed.

Consider two markets, 1 and 2 ,
with di¤erent values of the intensity of competition, b S 1 and b S 2 , where b S 1 < b S 2 .

Figure 1 :
Figure 1: Equilibrium cost pass-through for di¤erent elasticities of marginal cost S under perfect competition ( b S 1 = 0) and monopoly ( b S 2 = 1)

1 ,
which always holds for su¢ ciently large elasticities of marginal cost S and its slope S ; (b) Equilibrium cost pass-through lies below 100%, ( b S ) and is lower with more competition d ( b S )=d b S > 0 if: Demand is log-concave D 1 (equivalently, superelastic D 1) and demand curvature is non-decreasing d D (p)=dp 0; Costs are su¢ ciently convex in that ( S ; S ) satisfy (" D b S

( 1 D
) + (" D b S ) S i ( S + D ), and the condition follows immediately as claimed.For part (b), under the assumption d D =dp 0, it follows that d d b S [ b S (1 D )] (1 D ) since dp=d b S > 0 by Lemma 1. Therefore a su¢ cient condition for the condition from part (a) is: & P ‡eiderer 1983; Kimmel 1992; Anderson & Renault 2003; Weyl & Fabinger 2013; Mrázová & Neary 2017