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Chapter 1 Monopoly power

January 12, 2009

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1

Introduction

Standard monopoly results: • a monopoly firm prices above marginal cost; • monopoly pricing generates a deadweight loss because the quantity produced is too small (monopoly total surplus is less than the perfect competition total surplus).

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Harberger’s objection (Harberger, 1954) • Calculated that aggregate deadweight loss represented about 0.1% of US GNP. • Hence monopoly power is not empirically relevant..

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Objections to Harberger • Methodological criticism of his empirical approach (miss specification of demand, miss specification of the perfectly competitive profit). • His approach was partial equilibrium by taking the sum of deadweight losses over all the sectors in the economy: this may however lead to major mistakes (that may go both ways though). • Harberger’s low figure may reflect the effectiveness of US antitrust laws that go back to the late 19th century (Sherman Act and Claiton act).

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Demand elasticity and monopoly pricing. • By the inverse elasticity rule, monopoly markup is higher when demand is less elastic. • This is why demand elasticity is a key ingredient in empirical studies on market power. • No clear link between demand elasticity and the extent of the welfare loss due to monopoly. • Intuitively, a lower elasticity leads to a higher wedge between price and marg. cost but the quantity demanded is less affected by the increase in price: hence the impact on deadweight loss is amgibuous. • More formally we may consider how a change in elasticity affects the ratio of DWL to the 1st best social surplus (sum of DWL, CS and PS). 5

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• Consider constant elasticity demands, D(p) = p , p is price and  < 0 is price elasticity. • It can be shown that as  decreases from −1 to −∞, the ratio DW L/(DW L + P S + CS) increases (hence more elasticity leads to more inefficiency). • This is a special case of a more general result. • Consider D(p) such that D(p)ρ is linear in p for some real number ρ (ρ-linear demand). For instance constant elasticity demands are ρ-linear for ρ = 1/. • It can be shown that as ρ increases from −1 to +∞, the ratio first increases and than decreases to zero where the turning point is for some ρ > 0.

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• As ρ increases, the monopolist captures a larger share of the overall surplus and if that share is sufficiently high, the firm causes les inefficiency. • The limit corresponds to a rectangular demand where there is no deadweight loss and the firm captures the entire surplus.

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Outline 2 Some comparative statics 3 Some second order conditions 4 A durable good monopolist 5 Learning and experimentation 6 Cost distortions 7 Rent seeking behavior

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2

Some comparative statics

2.1

Change in marginal cost

• Consider two differentiable total cost functions C1 and C2 such that C20 > C10 for all positive quantities. • qim and pm i are monopoly quantity and price for cost function Ci . • Because qim and pm i maximize profit we have the two following inequalities: m m m m m pm 1 q1 − C1 (q1 ) ≥ p2 q2 − C1 (q2 )

(1)

m m m m m ) ≥ p q − C (q pm q − C (q 2 2 1 1 1 ). 2 2 2

(2)

and

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• Subtracting the two inequalities from each other and rearranging yields: [C2 (q1m ) − C1 (q1m )] − [C2 (q2m ) − C1 (q2m )] ≥ 0,

(3)

or equivalently Z

q1m

q2m

C20 (q) − C10 (q)dq ≥ 0.

(4)

• Since C20 − C10 > 0, we must have q1m > q2m and hence m (since demand is decreasing) pm 1 < p2 . • This shows that an increase in marginal cost leads to an increase in the monopoly price.

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Consider now the extent of the price increase caused by an increase in cost • Assume a constant marginal cost c > 0. From the inverse elasticity rule monopoly price pm satisfies D(pm ) p −c=− 0 m . D (p ) m

(5)

• Let g(pm ) = D(pm )/D0 (pm ). Standard comparative statics shows that 1 dpm = dc 1 + g 0 (pm )

(6)

• The impact of a cost increase is < 1 (resp. > 1) if and only if g 0 > 0 (rep. g 0 < 0). 11

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• Note that g 0 > 0 iff D is log-concave (i.e. lnD is concave). • This is not the case for constant elasticity demands. So for such demands a 1 Euro increase in marginal cost increases monopoly price by more than 1 Euro.

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2.2

Taxes

• Consider a unit tax t. Monopolist chooses price pm to solve: m m m max p D(p + t) − C(D(p + t)). (7) m p

• Nec. FOCs are: D(pm + t) + [pm − C 0 (D(pm + t))]D0 (pm + t) = 0.

(8)

• To restore efficiency, t must be set so that the price paid by consumers pm + t equals marg. cost C 0 (D(pm t )), so we have D(pm + t) t= 0 m < 0. (9) D (p + t) • Monopoly deadweight loss may be eliminated by using a unit subsidy. 13

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• In practice, this solution is not used much, in particular because it would require some tax revenue thus causing distortions elsewhere. • Typically, monopolies are regulated directly or government owned.

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Some second order conditions • For FOCs to be not only nec. but also suf. we need monopoly profit π m to be quasiconcave in price. • Formally, the set of prices p at which π m (p) ≥ K for some real number k should be convex. • So profit is quasiconcave iff it doe not have an interior local minimum. • At an interior local min., 1st derivative must be zero and 2nd derivative must be ≥ 0. • Hence profit is quaiconcave if whenever its 1st deriv. is 0 its second deriv. is < 0. • Assume constant marginal cost c. 15

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• Profit is (p − c)D(p). • 1st deriv. being 0 implies D(p) + (p − c)D0 (p) = 0. • 2nd deriv. is 2D0 (p) + (p − c)D00 (p) so that by substituting the zero 1st deriv. in the 2nd deriv. we have the following suf. condition for quasiconcavity: 2D0 (p)2 − D(p)D00 (p) > 0.

(10)

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ρ-concave demand functions • A function D > 0 with a convex domain is said to be ρ-concave for some real number ρ 6= 0 if Dρ is concave; D is zero-concave if it is logconcave. • If D is ρ-concave for some ρ, then it is ρ0 -concave for all ρ0 < ρ. • Assume D is strictly (-1)-concave. Then the 2nd deriv. of D−1 must be negative, which is equivalent to 2D0 (p)2 − D(p)D00 (p) > 0.

(11)

• So strict (-1)-concavity of demand is sufficient for quasiconcavity of profit (in fact, weak (-1)-concavity is sufficient as well). 17

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• Clearly this implies that logconcavity of demand is sufficient (this weaker assumption will be used in oligopolistic competition with product differentiation).

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A durable good monopolist • A monopolist sells over several periods a good for which each consumer needs only one unit (a durable good). • Then consumers may engage in temporal substitution and choose to wait for prices to go down before they buy. • Then the monopolist who may be tempted to lower its price in the future creates competition for its sales in the current period.

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• For instance if it charges the static monopoly price today and all those who are willing to pay that price buy it will be left with a residual demand in the future to which it can sell the good by lowering its price. • High valuation consumers may then choose to wait if the first-period price is too high. • The monopolist engages in dynamic price discrimination by selling first at a high price to consumers with high reservation prices and then lowering its price to sell to other consumers in future periods. • But such price discrimination is not profitable because high valuation consumers anticipate that prices will fall, which decreases their willingness to pay. 20

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• This may take an extreme form if price changes are very frequent. The Coase Conjecture As the frequency of price changes becomes increasingly high, the monopoly profit tends to zero and all consumers by the product at a price close to marginal cost. This result has been proved formally.

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Solutions to the Coase conjecture 1. Renting. 2. Most favored customer close whereby the firm commits to reimbursing a consumer if the price decreases.

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