Imperfect Competition

Imperfect Competition 1-Pure Monopoly 2-Monopolistic competition 3-Oligopoly IMPERFECT COMPETITION

Pure Monopoly There is only on seller in the market Market demand curve is downward sloping She can either change price or quantity in order to maximize the profit In order to sell more , she should lower the price She is facing the market demand individually IMPERFECT COMPETITION

Monopoly demand Q = F(p) or P = F (q) Unique inverse P b P a P 1 Q1 Q Q IMPERFECT COMPETITION

Average and marginal revenue R=p(q)q total revenue MR=dR/dq=p+q(dp/dq)=p(1+(q/p)(dp/dq))= p(1-1/ e ) e=absolute value of elasticity dp/dq<0 MR<P p=a-bq then q=(p-a)/b TR = aq- bq2 , then MR = a – 2bq q = (a-MR)/2b p= a-bq q=(a-p)/b p D MR q P=MR ; q(Demand)= 2q(MR) IMPERFECT COMPETITION

MR=P(1- 1/ιeι ) If Q=Q* , ιeι=1 , MR=0 , R(Q)=MAX If Q<Q* , ιeι>1 , MR>0 , If Q>Q* , ιeι<1 MR<0 , P Monopolist will always produce in the elastic portion of the demand curve MR P * D = AR Q* Q IMPERFECT COMPETITION

e>1 e=1 Demand, Total Revenue and Elasticity Demand, Total Revenue and Elasticity Demand, Total Revenue and Elasticity e<1 demand Max TR TR elasticity IMPERFECT COMPETITION

Profit maximizationcost function • П=p(Q)Q – C(Q) = TR(Q) – TC(Q) • dП/dQ = dTR(Q)/d(Q) – dTC(Q)/Q = 0 • MR(Q) = MC(Q) F.O.C. • MR>0 , Monopolist always choose a point on the elastic portion of the demnad. • dMR(Q)/dQ < dMC(Q)/dQ S.O.C. • MC must cut MR from below • If first and second order condition satisfies for More than a point , the one which yield greater profit will be chosen. IMPERFECT COMPETITION

Figure 1 and 2 satisfies the S.O.C. but 3 does not MC MC p p p MR D D MR D MR MC q q q IMPERFECT COMPETITION 2 1 3

Profit maximization :production function • П=TR(q) - r1x1 – r2x2 • Q= h(x1,x2) • dП/dxi = MR(q)hi – ri = 0 • MR(q)hi = ri • MRPxi= ri IMPERFECT COMPETITION

VMPxi = ri F.O.C. • S.O.C. П11<0 , П22<0 , • П11П22 – П212>0 • Пii= MR(q)hii+ dMR(q)/dq hi2<0 • MR’(q)<-MR(q)hii/hi2=-rihii/hi3 MC=ri/hi MC=MR • MR‘ (q ) is negative for monopolist . So hii could be positive ( MPxi is increasing) and monopolist may produce where production function is not concave .(the condition for concavity requires hii to be negative ). IMPERFECT COMPETITION

Price discrimination • Selling at more than one price to increase profit • Buyers should be unable to buy from one market and sell it in other one • Personal services ; electricity , gas, water • Saptially seperated markets, domestic and export markets IMPERFECT COMPETITION

IMPERFECT COMPETITION

Price discrimination • П=R1(q1)+R2(q2)-C(q1+q2) • qi= Sale in the ith market • Ri(qi) = piqi revenue in the ith market • dП/dqi=MR(qi)-MC(q1+q2) = 0 i=1,2 • MR(q1)=MR(q2)=MC(q1+q2) • P1(1-1/e1)=P2(1-1/e2) • Greater elasticity lower price • S.O.C. dП/dqi <0 dMRi(qi)/dq<dMC(q)dqi=1,2 IMPERFECT COMPETITION

Perfectly discriminating monopolist The monopolist is able to subdivide her market to such a degree that she could sell each successive unit of her commodities for the maximum amount that consumers are willing to pay. The consumers should have different elasticity's of demand for the monopolist output. IMPERFECT COMPETITION

П = F(Q) - MC(Q) dП/dQ=0 F(Q) – MC(Q) = 0 F . O . C . ; Marginal price =Marginal cost S .O . C . ; Slope of demand <Slope of marginal cost P IMPERFECT COMPETITION

Multi plant monopolist • Output (q) will be produced in two separate plant (i) • qi = production in plant i • Output of Plants will be sold in a single market. • П = R(q1+q2) – C1(q1) – C2(q2) • Ci(qi) = cost of production in plant i • dП/dq1= MR(q1+q2) – MC(q1) = 0 • dΠ/dq2 = MR(q1+q2) – MC(q2) = 0 • MC(q1) = MC(q2) = MR(q1+q2) F.O.C. • dMC(qi)/dqi>dMR(q1+q2)/dqi S.O.C. IMPERFECT COMPETITION

Multi product monopolist • Two distinct product • Q1=F1(p1,p2) Q2=F2(p1,p2) • P1=f1(q1,q2) p2=f2(q1,q2) • R1(q1,q2)=p1q1 R2(q1,q2)=p2q2 • Π=R1(q1,q2) + R2(q1,q2) - C1(q1) - C2(q2) • dΠ/dq1=dR1/dq1+dR2/dq1 – MC1(q1)=0 • dΠ/dq2=dR1/dq2 +dR2/dq2 – MC2(q2)=0 IMPERFECT COMPETITION

Multi product monopolist • dR1/dq1+dR2/dq1=MC1(q1) • dR1/dq2+dR2/dq2=MC2(q2) • If q1 increase by one unit and q1 is a substitute for q2 (dR2/dq1<0) , then Revenue increase by (dR1/dq1+dR2/dq1) • Cost increase by MC(q1) • For profit maximization these two should be the same for one unit increase in q1 IMPERFECT COMPETITION

Monopoly taxation • 1- Lump-sum tax • Π=R(q)-C(q)-T • dΠ/dq=MR(q)-MC(q)=0 MR=MC • Same output as before the tax • Only monopoly profit will decrease • 2-Profit tax 0<t<1 • Π=R(q)-C(q)-t{R(q)-C(q)}=(1-t){R(q)-C(q)} • dΠ/dq=(1-t){MR(q)-MC(q)}=0 MR=MC • Same output as before the tax • Only monopoly profit will decrease IMPERFECT COMPETITION

Monopoly taxation continued • 3- specific sale tax T=αq • Π=R(q)-C(q)-αq • dΠ/dq=MR(q)-MC(q) -α =0 • Profit maximization condition will change • dq/dα=1/(dMR(q)/dq-dMC(q)/dq) • (dMR(q)-dMC(q))/dq<0 S.O.C. • dq/dα<0 • Increase in tax rate(α) will lead to decrease in quantity produced and Increase in price IMPERFECT COMPETITION

Monopoly taxation cont. • T = sR(q) 0<s<1 • Π=R(q)-C(q)-sR(q)=(1-s)R(q)-C(q) • dΠ/dq=(1-s)MR(q)-MC(q)=0 • (1-s)MR(q)=MC(q) • Taking total differential • dq/ds=MR(q)/{(1-s)(MR’(q)-MC’(q)} <0 By S.O.C. < 0 IMPERFECT COMPETITION

Revenue maximizing monopoly Max R(q) s.t. Π=R(q)-C(q) ≥Π0 L= R(q) +λ{R(q) - C(q) – Π0} dL/dq = MR(q) + λ {MR(q)-MC(q)} ≤0, q dL/dq=0 dL/dλ=R(q) - C(q) – Π0 ≥0 , , λdL/dλ= 0 If Π0 =Π* =maxΠ then MR(q) – MC(q) = 0 q=q* If Π0> Π* no solution IMPERFECT COMPETITION

TC Profit when revenue is maximized TR is max C , TR TR MR=MC max Π = Π* q=q* MR>0 MC>0 q q* qm(TR is max) IMPERFECT COMPETITION

Revenue maximizing cont. • If Π0<Π* then Π should be greater than or equal to Π0 , • 1-When Π0 is less than profit at q=qm(when TR is maximized) , the solution is where TR is maximized [ MR =0 (q=qm)] • dL/dq=MR(q)+λ{MR(q)-MC(q)} ≤ 0 • If MR =0 so maximized TR is the solution, so there is no constraint., so λ=0 IMPERFECT COMPETITION

Revenue maximizing cont. 2- if Π is greater than the profit where q=qm(TR is max) and is less than the maximum profit ,the solution for q is when q*<q<qm . Profit tax will alter the output of revenue maximizing monopoly ; Max R(q) s.t. (1-t){R(q)-C(q)}=Π0 Taking total differential ; dq/dt= {R(q)-C(q)}/(1-t){MR(q)-MC(q)}<0 By S.O.C MR<M.C IMPERFECT COMPETITION

Inefficiency of Monopoly: MC(y*+1) < p(y*+1), so both seller and buyer could gain if (y*+1) level of output is produced. Market is Pareto inefficient $/output unit p(y) CS p(y*) MC(y) PS y* y IMPERFECT COMPETITION MR(y)

Inefficiency of Monopoly: DWL = gains from trade not achieved $/output unit p(y) p(y*) MC(y) DWL y* MR(y) y IMPERFECT COMPETITION

Inefficiency of Monopoly Inefficiently low quantity, inefficiently high price $/output unit p(y) p(y*) MC(y) DWL p(ye) y y* ye IMPERFECT COMPETITION MR(y)

Inefficiency resulting from two-price monopoly is lower than one-price monopoly Z<W The Efficiency Losses from Single-Price and Two-Price Monopoly Efficiency loss Z < W IMPERFECT COMPETITION

Welfare loss from monopoly pricing; Comparing to perfect competition wThe Welfare Loss from a Single-Price Monopoly Loss = (Π+s1+s2)–s2 Monopoly profit IMPERFECT COMPETITION

Monopsony • The sole purchaser in the market • Producer of q is the sole purchaser in the labor market and sell her output in the competitive market. • q=h(x) q=output x=input • r=price of x , r=g(x) , g’>0 R(q)=pq • TC= rx = x g(x) • Marginal cost of labor = d(TC)/dx= g(x) +xg’(x) IMPERFECT COMPETITION

monopsony • Π=TR – TC= ph(x) – g(x)x • dΠ/dx=ph’(x) – g(x) – xg’(x)=0 • Ph’(x)= g(x) +xg’(x) F.O.C. • VMPx=MCx(marginal factor cost) • d2Π/dx2=ph’’(x) – 2g’(x) – xg’’(x)<0 S.O.C. IMPERFECT COMPETITION

MCx=dc/dx C g(x)=supply x r1 r0 VMPx x x0 x1 IMPERFECT COMPETITION

Monopsony If monopsony is a monopolist in the output market , p=F(q) q=h(x) r=g(x) Π=pq – g(x)x =F(q) h(x) – xg(x) dΠ/dx = [ dF(q)/dq ] [ dq/dx ] h(x) + [ dh(x)/dx ] F(q) – [ dg(x)/dx ] x – g(x) =0 [dh(x)/dx ] { [dF(q)/dq] (h(x)] + F(q) } = g(x) + x { dg(x)/dx } VMPx MCx IMPERFECT COMPETITION

Monopolistic competition Number of sellers is sufficiently large that the actions of an individual seller have no perceptible influence upon her competitors. Each seller has a negatively sloped demand curve for her distinct product . Pk=Ak – akqk – Σi bkiqi i≠k dpk/dqi= - bki <0 i= 1,,,,n IMPERFECT COMPETITION

Monopolistic competition • If bki=b , ak=a , Ak=A , Ck(qk)=C(qk) • Pk =A – aqk – bΣqi i=1…..n • Πk=qk(A – aqk – bΣqi) – C(qk)i≠k • Representative firm assumes that when she maximizes profit ,the other fellows do not change their output level ,so she can move along her individual demand curve . IMPERFECT COMPETITION

Imperfect Competition