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Problem 7 – 1

Problem 7 – 1. Determine the maximum profit and the corresponding price and quantity for a monopolist whose cost and demand functions are P= 20 – 0.5q C = 0.04q 3 – 1.94q 2 – 32.96q. 7 – 1 solution. TR = pq= 20q – 0.5q 2 MR= 20 – q MC = 0.12q 2 – 3.88q – 32.96

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Problem 7 – 1

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  1. Problem 7 –1 • Determine the maximum profit and the corresponding price and quantity for a monopolist whose cost and demand functions are • P= 20 – 0.5q • C = 0.04q3– 1.94q2 – 32.96q . CH 7 , 8, monopoly

  2. 7 – 1 solution • TR = pq= 20q – 0.5q2 • MR= 20 – q • MC = 0.12q2– 3.88q – 32.96 • dΠ/dq=0 MR=MC F.O.C. • MR = MC q= 6 , q=18 • d2Π/dq2 =2.88 - .24q <0 q=18 CH 7 , 8, monopoly

  3. Problem 7 – 2 • A monopolist uses on input x which she purchases at the fixed price r =5 to produce her output Q . Her demand and production functions are • P =85 – 3q • Q = 2(x)1/2 • Respectively.Determine the value of p , q , x, at which the monopolist will maximize her profit. CH 7 , 8, monopoly

  4. 7 – 2 , solution • Π=TR – TC = 85q – 3q2– 5x • Π=85(2(x)1/2) – 3(2(x)1/2)2– 5x • dΠ/dx =0 x=25 • Q = 2(25)1/2 = 10 • P= 85 – 3q = 55 • Π=425 CH 7 , 8, monopoly

  5. Problem 7-3 • Determine the maximum profit and the corresponding marginal price for a perfectly discriminating monopolist whose demand and cost functions are: • P = 2200 – 60q • C= 0.5q3– 61.5q2 +2740q respectively. CH 7 , 8, monopoly

  6. 7 – 3 solution • Π= TR – TC • TR = ∫0q P(q)dq • Π=∫0q (2200-60q)dq-(0.5q3-61.5q2+2740q) • dΠ/dq=0 ; q=12 q= 30 • If q=12 then d2Π/dq2>0 • If q=30 then d2Π/dq2<0 but ; • Π = - 1350 • Profit is negative , q=0 CH 7 , 8, monopoly

  7. Problem 7 – 4 • Let the demand and cost function of a multi-plant monopolist be ; • P=a – b(q1+q2) • C1=a1q1+b1q12 • C2=a2q2 +b2q22 where all the parameters are positive.Assume that an autonomous increase of demand increases the value of (a) , leaving the other parameters unchanged . Show that the output will increase in both plants with a greater increase for the plant in which marginal cost is increasing less fast. CH 7 , 8, monopoly

  8. Problem 7 – 4 , solution • Π=TR– TC1– TC2 • TR=pq where q=q1+q2 • TR=[a-b(q1+q2)](q1+q2) • Π=a(q1+q2) - b(q1+q2)2 - a1q1 - b1q12 – a2q2– b2q22 • dΠ/dq1=a – 2b(q1 + q2) –a1–2b1q1= 0 • dΠ/dq2=a – 2b(q1 + q2) –a2– 2b2q2=0 • 2(b+b1)q1+2bq2=a – a1 • 2(b+b2)q2+2bq1=a – a2 • 2(b+b1)dq1+2bdq2=da • 2(b+b2)dq2+2bdq1=da b1, b2, a1, a2 are parameters. • dq1=(2b2/ D)da ,dq2=(2b1/D)da , D=4[b(b1+b2)+b1b2]>0 • dq1/da=(2b2/D)>0 , dq2/da=(2b1/D)>0 • If b1>b2 then dMC1/dq1>dMC2/dq2 , then dq2>dq1 CH 7 , 8, monopoly

  9. Problem 7-5 • A revenue maximizing monopolist requires a profit of at least 1500.her demand and cost functions are • P= 304 – 2q • C = 500 + 4q + 8q2. • Determine her output level and price.Contrast these values with those that would be achieved under profit maximization. CH 7 , 8, monopoly

  10. Problem 7-5 , solution • Max TR = 304q – 2q2 • S.T. TR-TC=304q-2q2-500-4q-8q2 ≥ 1500 • dL/dq=304-4q+λ[300-20q] ≤0, q dL/dq=0. • dL/dλ=300q – 10q2– 2000 ≥0 , λ dL/dλ=0 • q>0 , 304 - 4q +λ[300-20q]=0 • λ #0 , 300q – 10q2– 2000 =0 , q=10,q= 20 • If q=10 , p=284, TR=2840 , Π=1500 • If q=20 , p=264, TR=5280 , Π=1580 , q=20 • Max TR-TC = 304q-2q2-500-4q-8q2, • q=15,p=274, Π=1750 CH 7 , 8, monopoly

  11. Problem 7-6 • Let the demand and cost functions of a monopolist be • P=100 – 3q+4(A)1/2 • C=4q2+10q+A • Where A is the level of her advertising expenditure.Find the values of A , q, and p, that maximize profit. CH 7 , 8, monopoly

  12. Problem 7-6 solution • Π=[100-3q+4(A)1/2]q-(4q2+10q+A) • dΠ/dA=2q(A)1/2 – 1=0, q=[(A)1/2]/2 • dΠ/dq =[100-6q-4(A)1/2] - (8q+10)=0 • Q=15 • A=900 • P=175 CH 7 , 8, monopoly

  13. Problem 7-7 H&Q • A monopolist uses only labor ,x, to produce her output,Q, which she sells in the competitive market at the fixed price p=2. Her production and labor supply functions are • Q=6x + 3 x2 - 0.02 x3 and r=60+3x . • Determine the values of x ,q, r at which she maximizes her profit. Is the monopolist’s production function strictly concave in the neighborhood of her equilibrium production point? CH 7 , 8, monopoly

  14. Problem 7-7 solution • Π=TR-TC • Π=2(6x+3x2 - 0.02x3) – (60+3x)x • dΠ/dx=0, 0.12x2– 6x +48=0 x=10,x=40 • If x=10, then ;dΠ2/dx2>0 • If x=40, then ;dΠ2/dx2<0 x=40 is maximizing the profit. • If x=40 , then dq/dx=6+6x - 0.06x2>0 • d2q/dx2=1/2>0 strictly convex . CH 7 , 8, monopoly

  15. Problem 7-8 , H & Q • Consider a market characterized by monopolistic competition .there are 101 firms with identical demand function and cost function; • Pk=150 – qk – 0.02Σ100qi • Ck=0.5qk3 - 20qk2 + 270qk • Determine the maximum profit and corresponding price and quantity for a representative firm. Assume that the number of firms in the industry does not change. CH 7 , 8, monopoly

  16. Problem 7-8 , solution • TR=pq=150qk- qk2– 0.02qkΣqi • dTR/dqk =150-2qk– 0.02 Σi100qi =MR qi=qk • d(TC)/dqk =1.5qk2– 40qk +270 =MC • MC=MR, qk=4 , qk=20 • qk=20 , pk=90 , Πk=400. CH 7 , 8, monopoly

  17. Problem 7-9 H & Q • A monopolist will construct a single plant to serve two spatially separated markets in which she can charge different prices without fear of competition or resale between markets. The market are 40 miles apart and are connected by a highway. The monopolist may locate her plant at either of the markets or at some point along the highway. Let z and (40 – z) be the distances of her plant from markets 1 and 2 respectively. the monopolist demand and production and cost function are affected by her location : • P1=100-2q1 , p2=120-3q2, , C=80(q1+q2) – (q1+q2)2 • Determine the optimal values for q1,q2,p1,p2, and z if the monopolist transport costs are T = 0.4zq1+0.5(40 – z) q2. CH 7 , 8, monopoly

  18. Problem 7-9 solution • Π=(100-2q1)q1+(120-3q2)q2-[80(q1+q2) –(q1+q2)]-[0.4zq1+0.5(40-z)q2] • dΠ/dq1=(100-4q1)-[80-2(q1+q2)]-0.4z=0 • dΠ/dq2=(120-6q2)-[80-2(q1+q2)]-0.5(40-z)=0 • d2Π/dq22= -2 <0 • d2Π/dq12= -4 <0 • (d2Π/dq22) (d2Π/dq12) – (d2Π/dq1dq2)2=4>0 • q1=30 - 0.15z • q2=20+ 0.05z , substitute q1, q2 in the profit function; • Π=500 - 2 z +0.0425 z2 • d Π/dz=-2+0.085z=0 , z=23.53 , d2 Π/d z2 <0 • So when z=23.53, profit (Π=476.47) ,is not maximum. • If z=40 , Π=488 If z=0 , then Π=500 and maximum , q1=30 , p1=40 , q2=20 ,p2=60 CH 7 , 8, monopoly

  19. Problem 8-1 H&Q • Consider a duopoly with product differentiation in which the demand and cost functions are: • q1=88 – 4p1 + 2p2 , C1=10q1 • q2=56+2p1– 4p2 , C2=8q2 • For firms 1 and 2 respectively. Derive a price reaction function for each firm on the assumption that each maximizes its profit with respect to its own price. Determine the equilibrium values of price quantity and profit for each firm. CH 7 , 8, monopoly

  20. Problem 8-1 solution • Π1=88p1–4p12 +2p1p2–10(88–4p1+2p2) • Π2=56p2+2p1p2– 4p22– 8(56 +2p1-4p2) • d Π1/dp1=128 – 8p1+2p2=0 • d Π2/dp2=88 + 2p1 - 8p2=0 • P1=16+(1/4)p2 p1=20 , q1=38, Π1=400 • P2=11+(1/4)p1 p2=20 , q2=32 Π2 =400 CH 7 , 8, monopoly

  21. Problem 8-2 H&Q • Let duopolist ,1, producing a differentiated product ,face an inverse demand function given by • P1=100 – 2q1– q2 and having a cost function C1=2.5q12. Assume that duopolist , 2, wishes to maintain a market share of 1/3. Find the optimal price , output, and profit for duopolist one . Find the output of duopolist (2). CH 7 , 8, monopoly

  22. Problem 8-2 solution • K=1/3=q2/(q1+q2) q2=0.5q1 • Π1=p1q1-C1=(100-2q1-q2)q1-2.5q12 • Π1=100q1-5q12 • d Π1/dq1=0 q1=10 q2=5 • P1=100-2(10)-5=75 • Π1=500 • Q=q1+q2=10+5=15 CH 7 , 8, monopoly

  23. Problem 8-3 H&Q • Let n duopolist face the inverse demand function p=a – b(q1+….qn) and let each have the identical cost function Ci=cqi. • Determine the cournot solution. Determine the quasi-competitive solution . As n tends to infinity does the Cournot solution converge to the quasi-competitive solution. CH 7 , 8, monopoly

  24. Problem 8-3 solution • Cournot solution; Πi=pqi-Ci=aqi – bqi(q1+q2+….qn) -cqi • dΠ1/dq1=a - 2bq1- b(q2+q3+….qn) - c=0 • ….. • dΠn/dqn=a - 2bqn-b(q1+q2+...qn-1)–c=0 • ,n, equations and ,n, unknowns , q1=…….qn qi=(a-c)/(b + bn), i=1,2,….n • Quasi-competitive solution; • p=MCi , i=1,2,…n • a-b(q1+q2+q3+…qn)=c, n,identical equations • qi=(a-c)/nb i=1,2,…n CH 7 , 8, monopoly

  25. Problem 8-4 H & Q • Let two duopolist have the production function as follows ; • q1=13x1-0.2x12 • q2=12x2-0.1x22 , where xi is the input • Assume that the input supply function is r=2+0.1(x1+x2) where r is the supply price of input , and q1 , and q2 , are sold in the competitive markets for price p1=2 ,p2=3 • Find the input reaction function . • Determine the Cournot values for x1,x2,q1,,q2,Π1,Π2. CH 7 , 8, monopoly

  26. Problem 8-4 solution • Π1 =2(13x1-0.2x12)-x1[2+0.1(x1+x2)] • Π2=3(12x2-0.1x22)-x2[2+0.1(x1+x2)] • dΠ1/dx1=24-x1-0.1x2=0 • dΠ2/dx2=34-0.8x2-0.1x1=0 • X1=24 – 0.1x2 • X2=42.5 – 0.125x1 reaction functions. • x1 =19.5 x2=40 • q1=177.45 q2=320 , Π1 =200 , Π2=640 CH 7 , 8, monopoly

  27. Problem 8-8 H & Q • Let the buyer and seller of q2 in a bilateral monopoly situation have the following production functions; • q1=270q2-2q22 , x=0.25q22 • Assume that the price of q1 is 3 and the price of x is 6. • Determine the values of p2 ,q2, and the profit of buyer and seller for the monopoly ,monopsony, and quasi-competitive solution. • Determine the bargaining limits for p2 under the assumption that the buyer can do no worse that monopoly situation and the seller can do no worse than monopsony situation . • Compare the results. CH 7 , 8, monopoly

  28. Problem 8-8 solution • a – monopoly situation (seller of q2 is dominating the market) • Buyer’s profit (of q2) in the case of monopoly situation (p2 is set by monopolist ) = Πb=p1q1-p2q2 • Πbm=3(270q2-2q22)-p2q2=810q2-6q22-p2q2 • dΠbm/dq2=810 – 12q2 - p2 =0 • Demand function of the buyer of q2,, p2=810-12q2 • Seller’s profit (of q2) in the case of monopoly situation = Πs=p2q2-rx • Πsm=q2(810-12q2)-6(0.25q22)=810q2 -13.5q22 • dΠs/dq2=810-27q2=0 q2=30 • P2= 810-12(30)=450 p2 is determined by seller in the monopoly situation. • Πbm=810(30)-6(30)2-450(30)=5400 • Πsm= 810q2 -13.5q22 = 12150 CH 7 , 8, monopoly

  29. Problem 8-8 solution • b- monpsony solution (buyer of the q2 is dominating the market) • Πsn=seller’s profit in the case of monopsony situation (p2 is set by the buyer) = • Πsn= p2q2 - rx = p2q2 - 1.5q22 • dΠsn/dq2= p2 – 3q2=0 ; supply function for the seller of q2 . • Πbn =buyer’s profit in the case of monopsony situation = • p1q1 –p2q2 • Πbn =3(270q2– 2q22) –3q2(q2) • d Πbn/dq2=810-18q2=0 q2=45, p2=3q2=135 • This price is set by the buyer of q2 • Πsn=3037.5 Πbn=18225 CH 7 , 8, monopoly

  30. Problem 8-8 solution • c- quasi-competitive • D=S , MC=P2 • C=rx=1.5q22 MC=p = 3q2 • P2=810 – 12q2 • 810 – 12q2=3q2 q2=54 p2=162 • Seller’s profit=4374 • Buyer’s profit=17496 CH 7 , 8, monopoly

  31. Problem 8-8 solution • Collusion solution • Πt= Πs+Πb=[p2q2-rx]+[p1q1- p2q2] • Πt =p1q1 – rx=3(270q2-2q22)-6(0.25q22) • Πt=810 – 7.5q22 • d Πt/dq2=810 – 15q2=0 , q2=54 • The maximum price that the seller of q2 could charge is P2max which makes the buyer’s profit equal to zero when seller of q2 is dominating the market ,or when the seller has monopoly power. P2=P2max,if Πbm=0 • Πbm=p1q1-p2q2=p1(270q2-2q22)-p2q2=0 • If q2=54 the p2max=486. CH 7 , 8, monopoly

  32. Problem 8-8 solution • The minimum price that the seller of q2 • Will accept (p2min) is that price which makes the seller’s profit equal to zero, when buyer is dominating the market . • If Πsn =0, p2=p2min • Πsn=p2q2-rx= p2q2-r(0.25q22)=0 • If r=6, q2=54, → p2min=81 • (P2 min) 81 <p 2* < 486 (p2 max ) . CH 7 , 8, monopoly

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