Y j R L divide goods into produced goods (outputs) > 0 output, call its price p < 0 input, call its price ω

Transcription

1 4 PARTIAL EQUILIBRIUM ANALYSIS 4.1 Perfectly Competitive Market Ref: MWG Chapter 10.C and 10.F (but also read 10.A &10.B) Recall: consumers described by preferences over consumption bundles represented by utility functions u i : X i R where X i R L + producers described by production sets Y j R L divide goods into produced goods (outputs) 1 and factors of production (inputs)j y j Y j y j > 0 output, call its price p y j < 0 input, call its price ω In P.E. analysis of perfectly competitive market:- focus on one produced good denoted by q 2

2 A. Consumers 1. happy to have good from any of the many producers/sellers good undifferentiated and a commodity 2. consumers have perfect knowledge about prices charged by various sellers 3. consumers are price-takers. From u i derive x i p, P, w i assume prices of all other outputs (supplied goods) constant w i wealth derived from ownership of firms and selling factors of production is fixed Summing horizontally, obtain market demand x (p) = IX x i p, P, w i i=1 3 B. Producers/Sellers 1. sellers happy to sell to any buyer. 2. perfect info. about competitors prices have ability to undercut competitors 3. resale cannot be controlled 4. producers also price-takers (relative to ruling market price) From SR prodn sets derive SR firm supply schedules max q j pq j c j (q j ) 4

3 Define q min j =argminc j (q j ) /q j. q>0 FONC p max c 0 ª j q j,c 0 j q min j, (with equality if q j > q min i.e. q j (p) = c 0 1 j (p) if p c 0 j q min j j ) Summing horizontally, obtain market supply JX q (p) = q j (p) j= Short-run Competitive Equilibrium J fixed (because of adjustment and sunk costs etc.) Short-run equil consists of (q 1,...,q J ) and x 1,...,x I and price p s.t. 1. preference maximization: x i = x i (p) 2. profit maximization: q j maximizes pq j c j (q j ), i.e.p max c 0 j 3. market clearing: (supply = demand): ª q j,c 0 j q min j, (with equality if q j >qj min ) JX q (p) = q j (p) = j=1 IX x i (p) =x (p) i=1 6

4 Special case: identical firms Symmetric SR equil consists of p, q, x 1,...,x I s.t. 1. preference max.: x i = x i (p) 2. profit max.:q argmax q pbq c (bq) 3. mkt clearing: Jq = x (p) Long-run Competitive Equilibrium Usualassumption nofactorsfixed for firm (recall 0 Y ) Free entry and exit. Firms described by LR cost function c j (q), wherec j (0) = 0. Again define q min j =argminc j (q j ) /q j q>0 8

5 LRCE consists of p, (J, (q 1,...,q J )) and x 1,...,x I s.t. 1. Preference max: x i = x i (p) 2. Profit max.:q j solves max q pq c j (q) 3. Market clearing 4. free entry: JX q (p) = q j (p) = j=1 IX x i (p) =x (p) i=1 pq 1 (p) c 1 (q 1 (p)) pq 2 (p) c 2 (q 2 (p))... pq J (p) c J (q J (p)) 0 pq J+1 (p) c J+1 (q J+1 (p)) 9 Special case: identical firms Symmetric LRCE consists of p, q, J, x 1,...,x I s.t. 1. preference max.: x i = x i (p) 2. profit max.:q solves max q pbq c (bq) 3. mkt clearing: Jq = x (p) 4. free entry pq c (q) =0 Short and Long Run Comparative Statics Suppose initially in LR equil with J 0 firms with c (q) =K + ψ (q), ψ (0) = 0, ψ 0 (0) > 0, ψ 00 (q) > 0 Now suppose demand drops (permanently), i.e. for demand x (p; θ), where θx (p, θ) < 0, consider increase in θ. 10

6 Short-run 1. π-max. condition dp 2. Market clearing condition J 0dq Substituting yields: dθ = ψ00 (q) dq dθ dθ = θ x (p, θ)+ p x (p, θ) dp dθ J 0dq dθ = θ x (p, θ)+ p x (p, θ) ψ00 (q) dq dθ dq dθ = θx (p, θ) J 0 p x (p, θ) ψ00 (q) < 0 Hence dp dθ = ψ00 (q) dq dθ < 0 In long-run price adjusts to c q 0 /q 0 as some firms exit industry and remaining firms expand output back to original optimal level (minimum efficient scale). 11 Two problems show up here: 1. J 0 and J 1 really need to be integers. 2. J needs to be big. When J is small, firms should recognise effect their output decisions have on prices and strategic interaction of their competitors. Resolution: model firms (or plant size) as literally infinitesimally small relative to market. A continuum of potential firms can be achieved as the limit of letting the number of firms but keeping the market size finite. 12

7 4.2 Welfare in Partial Equilibrium (Ref: 10.E) Inverse demand of consumers p i x i,aggregateinversedemandp (x). Consider differential change P dx 1,...,dx I,dq 1,...dq J satisfying i dxi = P j dq j and let dx denote P i dxi and dq denote P j dq j. Change in aggregate Marshallian surplus is then IX ds = p i x i JX dx i c 0 j (q j ) dq j i=1 j=1 Since p i x i = P (x) for all i, c 0 j (q j)=c 0 (q) for all j, anddx = dq, weget ds =(P (x) c 0 (x)) dx, sos (x) =S 0 + Z x Alternatively, measure of welfare is CS + tax revenue + PS. Z p c x (p) dp +(p c p F ) x (p)+ 0 Z pf 0 (P (s) c 0 (s)) ds q (p) dp 13 Consumers: For each i, CS i = R p c x i (p) dp, socs= P i CSi assuming either income effects negligible or no income effects (e.g. quasi-linear utility). assuming all other markets operate efficiently, or no (compensated) cross-price effects Producers: PS is EV for producers. I.e. What is equivalent transfer to selling q at price p F. Hotelling s Lemma π (p) =q (p) p Implicit welfare function is sum of corresponding money metric utility functions dollar is a dollar for given base prices 14

8 4.3 Monopoly (Ref: 12.B) Single producer with cost function c (q) who knows market demand x (p). or equivalently FONC max hqi Pblm: max hpi px (p) c (x (p)) p (q) q c (q) where p (.) is inverse demand q : q m p 0 (q m )+p(q m ) c 0 (q m )=0 p (q m ) c 0 (q m )= q m p 0 (q m ) p c0 p = 1 ε p (where ε p q 0 (p) p/q) 15 Welfare Z q c DWL = (p (s) c 0 (s)) ds q m = [4CS + 4PS] [p m c 0 (q m )] q m Z p m = x (p) dp + p c Z p c c 0 (q m ) q (p) dp [p m c 0 (q m )] q m 16

9 Sufficient conditions for non-discriminating monopoly pricing. 1. Unit-demand by consumers & monopolist unable to discern any particular consumer s preferences 2. Multi-unit demand by consumers monopolist unable to discern any particular consumer s preferences resale of good is costless & monopolist cannot control (competitive) resale market. Denote price for q units by R (q). Since resale costless & resale mkt comp., unit price of good in resale mkt equals min q {R (q) /q} consumers will only buy qty q =argmin q {R (q) /q} and then resell each unit for R (q ) /q Hence monopolist just as well off charging price per unit of p (q )= R (q ) /q Price Discrimination (a) First-Degree (or Perfect) Price Discrimination Suppose consumer i has quasi-linear utility u i (q i )+m i [with u i (0) = 0] Monopolist makes take-it-or-leave-it offer (q i,t i ) to each consumer. Given offer (q i,t i ),consumeri willing to accept offer iff u i (q i ) T i 0. Consumer willing to pay T i = u i (q i ), leaving her with zero surplus. Monopolist s pblm becomes X I ³ max u PI i (q i ) c hq 1,...,q I i i=1 i=1 q i Solution: FONC ³ qi : u 0 (qi ) c 0 P I j=1 q j 0 (with = if qi > 0) Solution maximizes aggregate surplus. 18

10 (b) Second-Degree Price Discrimination (Monopolistic Screening) (ref: 14.C pp , particularly p ) Suppose two types of consumers λ : High marginal value u H (q) T, u H (0) = 0; 1 λ : Low marginal value u L (q) T, u L (0) = 0. u 0 H (q) u 0 L (q),forallq Monopolist s pblm is to offer menu of (qty, price) pairs that maximizes profits. Assume monopolist has constant MC of production = c. 19 I. Type Observable: back to 1 st -degree price discrimination. Offer: 1. (qh,t H ) to High MV customers; 2. (ql,t L ) to Low MV customers; where, u H (q H) T H = 0 and u 0 H (q H)=c & u L (q L) T L = 0 and u 0 L (q L)=c. 20

11 II. Type Unobservable: Menu must satisfy self-selection as well as participation constraints. Formally pblm becomes: max λt H +(1 λ) T L (λq H +[1 λ] q L ) c h(q H,T H ),(q L,T L )i s.t. self-selection constraints 1. u H (q H ) T H u H (q L ) T L 2. u L (q L ) T L u L (q H ) T H and participation constraints 3. u H (q H ) T H 0 4. u L (q L ) T L Lemma 1. We can ignore (3) Proof Whenever constraints (1) & (4) are satisfied, we have u H (q H ) T H u H (q L ) T L u L (q L ) T L 0 [Since u H (0) = 0 = u L (0) and u 0 H (ˆq) u0 L (ˆq), ˆq u H (q) u L (q)] Lemma 2. In an opitmal contract u L (q L ) T L =0. Proof Suppose not, that is, there is an optimal solution ³ˆq H, H ˆT, ³ˆq L, L ˆT with u L (q L ) T L >ε>0 Now consider an alternative menu in which T H and T L are both increased by ε. New contract still satisfies (4). (1) and (2) are still satisfied since ε subtracted from boht sides. But profit has been increased by ε, so original menu not optimal. A contradiction. 22

12 Lemma 3. In any optimal solution (i) q L q L and (ii) q H = q H For proof refer to discussion and diagram in lecture. Lemma 4. In any optimal solution q L <ql. For proof refer to discussion and diagram in lecture. Solution: ³ˆq H, H ˆT, ³ˆq L, L ˆT where ˆq H = qh (i.e. u 0 H (ˆq H )=c) ˆT H = u H (qh) R ˆq L q=0 [u0 H (q) u 0 L (q)] dq ˆT L = u L (ˆq L ) and where ˆq L satisfies λ [u 0 H (ˆq L ) u 0 L (ˆq L )] = (1 λ)[u 0 L (ˆq L ) c] λu 0 H (ˆq L )+(1 λ) c = u 0 L (ˆq L ) 23 inefficiency of package designed for Low MV consumers (i.e. ˆq L <ql ) designed to make ³ˆq L, L ˆT less attractive for High MV consumers. In practice, monopolist often encourages self-selection not by adjusting quantities offered to consumers but by adjusting quality of good offered to different groups. e.g. airlines unrestricted fare designed for business traveler v. restricted fare (incl. Saturday night stay) designed for non-business traveler. first-classtravelv.coach. 24

13 (c) Third-Degree Price Discrimination Monopolist charges different price per unit to (identifiably) different groups of consumers. e.g. child & adult prices for movie tickets, senior citizens discounts. Compare π from serving only High MV consumers with π from serving both types. Mathematically, max hq H,q L i p H (q H ) q H + p L (q L ) q L c (q H + q L ) Optimal solution: MR H (q H ) = c 0 (q H + q L ) MR L (q L ) c 0 (q H + q L ), with equality if q L > Static Models of Oligopoly (Ref. 12.C) Game Theory & Strategic interactions Industrial Organisation Basic problem profit offirm not simply based on that firm s own actions but also depends on actions of competitors. Postulate firms have beliefs aboutwhatotherfirms are doing. Optimization now involves: choosing best action given belief about what other firms are doing. Equil concept is Nash equilibrium (consistent and self-fulfilling beliefs) Formally Nash equilibrium is profile of actions (strategies) (s 1,...,s J ) such that for each firm j, s j is best/optimal response for firm j given its belief that other firms are undertaking s j =(s 1,...,s j 1,s j+1,...,s J ). 26

14 4.4.1 Cournot (Quantity) Competition Model Each firm simultaneously chooses quantity q j Price is determined ³ to equate S = D. P That is, p = p j q j where p (q) is inverse demand. Define b j (q j ) as firm j s optimal quantity decision given belief other firms are choosing q j =(q 1,...,q j 1,q j+1,...,q J ) That is, b j (q j ) = argmax hq ji ³ p q j + X q k q j c j (q j ) k6=j FONC ³ p 0 q j + X ³ q k q j + p q j + X q k c 0 k6=j k6=j j (q j )=0 (that is, view firm j acting as monopolist on residual demand curve.) 27 Hence a Nash equil strategy profile (q1,...,q J ) satisfies for each j : Ã JX! Ã JX! p 0 qk qj + p qk c 0 j q j =0 k=1 k=1 Example: Say p (q) =a bq, c j (q j )=cq j. Then there is exists a unique symmetric Nash equilibrium (q,...,q ) where p 0 (Jq ) q + p (Jq ) c 0 (q )=0 That is, And Jq = q = a c (J +1)b J (a c) (J +1)b > a c = q m 2b 28

15 4.4.2 Betrand (Price) Competition Model Take J =2.Eachfirm simultaneously picks price p j x (p j ) if p j <p k x j (p j,p k )= x (p j ) /2 if p j = p k 0 if p j >p k If c 1 (q) =c 2 (q) =cq then the unique Nash equil is p 1 = p 2 = c Capacity Constraints Often natural to suppose firms operate under conditions of eventual DRS (especially in SR when some factors, e.g. capital stock, are fixed). With capital constraints, no longer sensible to assume price announcement is commitment to provide any demanded quantity since costs of order larger than capacity are infinite (or at least prohibitively large). 29 So assume price announcements are commitments to supply up to capacity. Assume capacities commonly known among firms. example with ½ K + ψq q q c (q) = q> q where q =3x (ψ) /4. Consider following Notice p 1 = p 2 = ψ no longer an equilibrium. More generally, whenever q <x(ψ), each firm can assure itself strictly positive level of sales and strictly positive operating profits by setting its price below p ( q) but above ψ. 30

16 Consider following two-stage game 1. Each firm simultaneously sets its capacity. 2. Given capacity choices, firms simultaneously set prices. Solving game backwards:- i.e. requiring price choice to be N.E. given capacity choices, the unique subgame perfect Nash equilibrium involves Cournot outcome in capacity choices (Kreps-Scheinkman, 1983). That is, may view Cournot quantity competition as capturing LR competition through capacity choice, with price competition occurring in SR given chosen levels of capacity Product Differentiation Firm j faces continuous demand function x j (p j,p j ). Solve: max (p j c) x j (p j, p j ) p j where p j is firm j s expectation about rivals price choices. p is a Nash equilibrium, if for all j p argmax p j (p j c) x j (p j, p j ) 32

17 Spatial Models of Production Differentiation. Example: Linear City Model of Product Differentiation. Mass 1 of consumers uniformly distributed along unit interval. Consumer s location indexed by z [0, 1] c is constant MC of production for both firms. Totalcostofbuyingfromfirm j for consumer located at distance d from j is p j + td, which yields net utility v p j td. Given p 1 and p 2, ẑ is consumer who is indifferent between buying from either firm, i.e., p 1 + tẑ = p 2 + t (1 ẑ) ẑ = t + p 2 p 1 2t 33 Lemma. If v>c+2t then firms never want to set prices at level that causes some consumers not to purchase from either firm. Proof. Suppose to contrary, in situation where for each z (ẑ 1, ẑ 2 ), min (p 1 + tz, p 2 + t (1 z)) >v And v p 1 tz 0 for all z [0, ẑ 1 ] v p 2 t (1 z) 0 for all z [ẑ 2, 1]. Consider a small reduction in firm 1 s price p 1.Aslongasfirm 1 does not start to compete with firm 2, its sales are given by consumer at location x 1 who is indifferent between buying from firm 1 and not buying at all: 34

18 So v p 1 tx 1 =0 Demand x 1 (p 1 )= (v p 1) t π 1 (p 1 )=(p 1 c) x 1 (p 1 )= (p 1 c)(v p 1 ). t Differentiating wrt p 1, π 0 1 (p 1 )= (v + c 2p 1). t By assumption, some consumers strictly prefer not to buy rather than buy from firm 1. This is true for consumers at location z =1,whichmeans p 1 >v t. Hence π 0 (c v +2t) 1 (p 1 ) < < 0, by assumption. t Therefore, by lowering p 1 slightly, firm 1 could raise its profit, which contradicts assumption firm 1 is playing its best response. 35 So Similarly, x 1 (p 1,p 2 )= 0 if p 1 >p 2 + t (t + p 2 p 1 ) /2t if p 1 [p 2 t, p 2 + t] 1 if p 1 <p 2 t x 2 (p 1,p 2 )= 0 if p 2 >p 1 + t (t + p 1 p 2 ) /2t if p 2 [p 1 t, p 1 + t] 1 if p 2 <p 1 t So firm j s pblm, given firm k is setting price p k :- (p j c)(t + p k p j ) max p j [ p k t, p k +t] 2t 36

19 FONC t + p k + c 2p j Hence, the best response function is given by 0 if p j = p k =0 if p j ( p k t, p k + t) 0 if p j = p k + t b j ( p k )= Symmetric solution p 1 = p 1 = p,i.e., p k + t if p k c t (t + p k + c) /2 if p k (c t, c +3t) p k t if p k c +3t p = b (p ) p =(t + p + c) /2 p = c + t. 37