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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 . H & Q PROBLEMS CH 7 , 8, monopoly 1 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 H & Q PROBLEMS CH 7 , 8, monopoly 2 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. H & Q PROBLEMS CH 7 , 8, monopoly 3 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 H & Q PROBLEMS CH 7 , 8, monopoly 4 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. H & Q PROBLEMS CH 7 , 8, monopoly 5 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 H & Q PROBLEMS CH 7 , 8, monopoly 6 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. H & Q PROBLEMS CH 7 , 8, monopoly 7 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 H & Q PROBLEMS CH 7 , 8, monopoly 8 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. H & Q PROBLEMS CH 7 , 8, monopoly 9 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 H & Q PROBLEMS CH 7 , 8, monopoly 10 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. H & Q PROBLEMS CH 7 , 8, monopoly 11 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 H & Q PROBLEMS CH 7 , 8, monopoly 12 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? H & Q PROBLEMS CH 7 , 8, monopoly 13 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 . H & Q PROBLEMS CH 7 , 8, monopoly 14 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. H & Q PROBLEMS CH 7 , 8, monopoly 15 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. H & Q PROBLEMS CH 7 , 8, monopoly 16 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. H & Q PROBLEMS CH 7 , 8, monopoly 17 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 H & Q PROBLEMS CH 7 , 8, monopoly 18 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. H & Q PROBLEMS CH 7 , 8, monopoly 19 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 H & Q PROBLEMS CH 7 , 8, monopoly 20 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). H & Q PROBLEMS CH 7 , 8, monopoly 21 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 H & Q PROBLEMS CH 7 , 8, monopoly 22 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 quasicompetitive solution. H & Q PROBLEMS CH 7 , 8, monopoly 23 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 H & Q PROBLEMS CH 7 , 8, monopoly 24 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. H & Q PROBLEMS CH 7 , 8, monopoly 25 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 H & Q PROBLEMS CH 7 , 8, monopoly 26 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. H & Q PROBLEMS CH 7 , 8, monopoly 27 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 H & Q PROBLEMS CH 7 , 8, monopoly 28 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 H & Q PROBLEMS CH 7 , 8, monopoly 29 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 H & Q PROBLEMS CH 7 , 8, monopoly 30 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. H & Q PROBLEMS CH 7 , 8, monopoly 31 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 ) . H & Q PROBLEMS CH 7 , 8, monopoly 32