Microeconomic Theory: Lecture 4...

Monopoly Price Discrimination Double Marginalization

Microeconomic Theory: Lecture 4Monopoly

Parikshit Ghosh

Delhi School of Economics

Summer Semester, 2014

Parikshit Ghosh Delhi School of Economics

Monopoly

Non-Discriminating Monopolist

The Monopolist�s ProblemI The monopolist realizes his quantity choice a¤ects marketprice through the demand function.

I Monopolist�s problem can be described either as one ofchoosing optimal price or optimal quantity.

I Inverse demand function p(q) and cost function c(q).I Maximizing pro�ts:

maxqp(q)q � c(q)

I First order condition:

p(qm) + qp0(qm)| {z } = c 0(qm)| {z }marginal revenue = marginal cost

Monopoly

Monopolist�s Optimum: Another LookI The FOC can be rewritten as

qmpm.dpdq

�= c 0(qm)

or, pm

�1� 1

�= c 0(qm)

I e(q) is the elasticity of demand at quantity q. The monopolistnever operates on the inelastic part of the demand curve.

I Yet another way to write it:

pm � c 0(qm)pm

I The mark-up is equal to the inverse of demand elasticity.

Monopoly

Comparison With Competitive Markets

I Competitive output level:

p(q�) = c 0(q�)

I The monopolist produces less than the competitive market.I Suppose qm � q� ) p(qm) � p(q�) and c 0(qm) � c 0(q�).I Subtracting the f.o.c�s yields a contradiction:

p(qm)� p(q�)| {z }+ qp0(qm)| {z } = c 0(qm)� c 0(q�)| {z }� � = +

I Unlike a competitive �rm, when a monopolist raises output,he earns lower revenue on previous units.

Monopoly

Monopoly in Pictures

Monopoly

Types of Price Discrimination

Price Discrimination: First DegreeI Each consumer buys 0 or 1 unit and is willing to pay up to v .I There is a continuum of consumers whose v follow adistribution with c.d.f F (v).

I The monopolist can charge each consumer his personal v .I Must choose a cuto¤ v � above which to sell:

maxv �

v �vf (v)dv � c (1� F (v �))

I First-order condition (using Leibnitz Rule) implies absence ofine¢ ciency:

�v �f (v �) + c 0(1� F (v �))f (v �) = 0

v �|{z} = c 0(1� F (v �))| {z }price to marginal customer = marginal cost

Monopoly

Price Discrimination: Second and Third Degree

I Second degree price discrimination arises when the monopolistcan charge di¤erent prices for di¤erent quantities.

I E.g., bulk discounts, multi-packs, frequent-�yer miles,buy-one-get-50%-o¤ on next purchase, etc.

I A way to extract consumer�s surplus from a single consumer.I Third degree price discrimination arises when observablydi¤erent groups are charged di¤erent prices.

I E.g., student/senior citizen discounts, country speci�c prices.I A cruder form of �rst degree price discrimination� usinggroup identity as a predictor of individual traits.

I A common instrument of price discrimination: screening.

Monopoly

Second Degree Price Discrimination

Monopoly

Two Part Tari¤

Monopoly

Two Part Tari¤

Monopoly

Two Part Tari¤

I Suppose a single consumer with income y has quasi-linearutility: u(q,m) = φ(q) +m.

I The monopolist can charge an entry fee (f ) and a price (p)per unit of consumption.

I Consumer�s optimum quantity choice (if she subscribes):

φ(q) + y � f � pq ) q(p) =�φ0��1

I Consumer subscribes if (participation constraint):

φ(q(p)) + y � f � pq(p) � y

Monopoly

Two Part Tari¤I The monopolist�s problem:

maxf ,p

pq(p) + f � c (q(p))

subject to φ(q(p)) + y � f � pq(p) � yI Participation constraint must be binding at optimum(otherwise simply increase f ).

φ(q(p))� c (q(p))) φ0(bq) = c 0(bq)I If the market were competitive (price taking behaviour):

φ0(q�) = c 0(q�)

I Again, two-part tari¤s remove the monopolistic distortion.

Monopoly

Monopolistic Screening

Airfares: More Than 2 Weeks

Monopoly

Airfares: Less Than 2 Weeks

Monopoly

Hardcover vs. Paperback: Prices

Monopoly

Hardcover vs. Paperback: Date and Sales

Monopoly

Cash Back Coupons

Monopoly

Rationing

Monopoly

Rationing

Monopoly

Price Discrimination by Screening

I Two types of consumers:I high value (value = vH , proportion = α).I low value (value = vL, proportion = 1� α).

I Consumer�s valuation is private information.I Cost of production is 0.I Uniform pricing strategy:

I charge p = vH if αvH � vL.I charge p = vL ifαvH < vL .

I Pro�t = maxfαvH , vLg. Assume αvH < vL.

Monopoly

Price Discrimination by ScreeningI Monopolist can impose a burden B on consumers (delay,coupons, uncertainty, etc.).

I Cost of the burden is cH for high value types and cL for lowvalue types (cH > cL).

I The burden reduces the willingness to pay of all customers.Its direct impact on pro�ts is negative. Why will themonopolist hurt his own interests?

I By imposing the burden, the monopolist can gather valuablemarket information which allows him to price discriminate.This indirect bene�t may compensate for the direct loss oflower reservation prices.

I The screening technique is useful in other contexts (employersseeking dedicated workers, governments targeting anti-povertyprogrammes at the poor).

Monopoly

I Monopolist o¤ers a menu (pH , 0) and (pL,B), satisfying:I Self Selection Constraint: H-type chooses (pH , 0), L-typechooses (pL,B).

pH � pL + cH (IC-H)

pL + cL � pH (IC-L)

I Participation Constraint: Both types want to buy.

vH � pH � 0 (PC-H)

vL � pL � cL � 0 (PC-L)

I Monopolist solves: maxpH ,pL αpH + (1� α)pL subject to theseconstraints.

Monopoly

I Assumption A: Relative to the L-type, the H-type gainsmore from the good than she su¤ers from the burden:

vH � vL � cH � cL

I Step 1: PC-H is implied by the other constraints and can bedropped.

vL + (vH � vL) � (pL + cL) + (cH � cL) (combining PC-L and Assumption A)

or, vH � pL + cHvH � cH (using IC-H)

Monopoly

I Step 2: PC-L must be binding at the optimum.I If not, increase both pH and pL by ε > 0.I IC�s continue to hold, pro�ts have increased.

I Step 3: IC-H must be binding at the optimum.I Otherwise, increase pH by ε > 0.I The other remaining constraints continue to be satis�ed.

I Step 4: Binding IC-H implies IC-L, so it can be dropped.

Monopoly

I Optimal solution:

p�L = vL � cLp�H = vL + cH � cL

I Assume vH > vL + cH � cL.I Pro�t from price discrimination:

π� = α(vL + cH � cL) + (1� α)(vL � cL)= vL + αcH � cL

I Price discrimination is better than uniform pricing ifαcH � cL > 0.

Monopoly

Vertical Mergers

Horizontal vs. Vertical MergersI Downstream product X (furniture), upstream product Y(wood). Fixed coe¢ cient technology (1:1).

I Each supplied by a separate monopolist, with marginal costscx and cy .

I Inverse demand function for the �nal good: p = p(x). LetR(x) = p(x)x be downward sloping.

I Let the price charged by the upstream monopolist be q.I Downstream problem:

maxxR(x)� (cx + q)x

I First order condition:

R 0(x) = cx + q

Monopoly

Vertical Mergers

Upstream ProblemI The upstream monopolist takes into account the downstreamdemand function:

maxyyq � cy y � max

yy�R 0(y)� cx

�� cy y

I First order condition (replacing y by x��):

R 0(x��) + x��R 00(x��) = cx + cy

I Suppose the two monopolies merged. Then the �rm will solve:

maxxR(x)� (cx + cy )x

I First order condition shows vertical merger increases e¢ ciency:

R 0(x�) = cx + cy ) x� > x��

Monopoly

Tax Harmonization

Centre and State Taxes

I Let q = a� bp be the demand function.I Supply is horizontal at some price bp (net of taxes).I The Centre �rst chooses a tax tc . Then the State chooses itsown tax ts .

I Both governments aim to maximize tax revenue.I Centre and State do not coordinate when choosing their taxpolicies. They do not maximize total government revenue.

I This can be ine¢ cient. Cumulative tax rates are too high.

Monopoly

Tax Harmonization

State�s ProblemI The State solves:

maxtsts [a� b(bp + tc + ts )]

I State�s FOC de�nes its reaction function ts (tc ):

a� b(bp + tc ) = 2btsI Di¤erentiating w.r.t tc :

t 0s (tc ) = �12

I If Central taxes are higher, State will lower its own taxes tosome degree but not completely.

Monopoly

Tax Harmonization

Centre�s Problem

I Centre chooses tc to solve:

maxtctc [a� b(bp + tc + ts (tc ))]

I Replacing State�s reaction function:

t�c = argmaxtc

12tc (a� bbp � btc )

=a� bbp2b

I Plugging back:

t�s =a� bbp4b

Monopoly

Tax Harmonization

I Consider a harmonized single tax rate to maximize taxcollection for Centre and State. It solves

maxtt [a� b (bp + t)]

I FOC is gives the optimal tax rate

t� =a� bbp2b

I Since t�s = t�, we have t�s + t

�c > t

�.I A harmonized single tax will

I Decrease tax burden and increase consumer�s surplusI Increase tax collection which can be suitably shared.

Monopoly

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