h & q problemsch 7, 8, monopoly1 problem 7 – 1 determine the maximum profit and the...

H & Q PROBLEMS CH 7 , 8, monopoly 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 .


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


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.


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


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.


7 – 3 solution Π= TR – TC TR = ∫0

q P(q)dq Π=∫0

q (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


Problem 7 – 4 Let the demand and cost function of a multi-plant

monopolist be ; P=a – b(q1+q2) C1=a1q1+b1q1

2

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.


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 - b1q1

2 – 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


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.


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


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.


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


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?


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 .


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 - 20qk

2 + 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.


Problem 7-8 , solution TR=pq=150qk- qk

2 – 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.


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.


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/dq2

2= -2 <0 d2Π/dq1

2= -4 <0 (d2Π/dq2

2) (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


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.


Problem 8-1 solution Π1=88p1–4p1

2 +2p1p2–10(88–4p1+2p2) Π2=56p2+2p1p2 – 4p2

2 – 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


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=2.5q1

2. 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).


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


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.


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


Problem 8-4 H & Q Let two duopolist have the production function as

follows ; q1=13x1-0.2x1

2

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.


Problem 8-4 solution Π1 =2(13x1-0.2x1

2)-x1[2+0.1(x1+x2)] Π2=3(12x2-0.1x2

2)-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


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=0.25q2

2

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.


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 (pp22 is set is set

by monopolistby monopolist ) = Πb=p1q1-p2q2 Πbm=3(270q2-2q2

2)-p2q2=810q2-6q22-p2q2

dΠbm/dq2=810 – 12q2 - p2 =0 Demand function of the buyer of qDemand function of the buyer of q22 ,,, , pp22=810-12q=810-12q22 Seller’s profit (of q2) in the case of monopoly situation =

Πs=p2q2-rx Πsm=q2(810-12q2)-6(0.25q2

2)=810q2 -13.5q22

dΠs/dq2=810-27q2=0 qq22=30=30 PP22= 810-12(30)=450450 p2 is determined by seller in the

monopoly situation. ΠΠbmbm=810(30)-6(30)2-450(30)=5400 5400 ΠΠsmsm = 810q2 -13.5q2

2 = 1215012150


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 (pp2 2 is is

set by the buyerset by the buyer) = Πsn= p2q2 - rx = p2q2 - 1.5q2

2

dΠsn/dq2= pp2 2 –– 3q 3q22=0 ; supply function for the seller of q=0 ; supply function for the seller of q2 2 .. Πbn =buyer’s profit in the case of monopsony situation = p1q1 – p2q2

Πbn = 3(270q2 – 2q22) – 3q2(q2)

d Πbn/dq2=810-18q2=0 q q22=45=45, pp22=3q2=135135 This price is set by the buyer of q2

ΠΠsnsn=3037.5 Π=3037.5 Πbnbn=18225=18225


Problem 8-8 solution c- quasi-competitive D=S , MC=P2

C=rx=1.5q22 MC=p = 3q2

P2=810 – 12q2 810 – 12q2= 3q2 qq22=54 p=54 p22=162=162 SellerSeller’’s profit=4374s profit=4374 BuyerBuyer’’s profit=17496s profit=17496


Problem 8-8 solution Collusion solution Πt= Πs+ Πb=[p2q2-rx]+[p1q1- p2q2] Πt =p1q1 – rx=3(270q2-2q2

2)-6(0.25q22)

Πt=810 – 7.5q22

d Πt/dq2=810 – 15q2=0 , qq22=54=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

ΠΠbmbm=p=p11qq11-p-p22qq22=p=p11(270q(270q22-2q-2q2222)-p)-p22qq22=0=0

If qIf q22=54 the p=54 the p2max2max=486=486.


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, qIf r=6, q22=54, =54, →→ p p2min2min=81=81 (P2 min) 81 <p 2

* < 486 (p2 max ) .

h & q problemsch 7, 8, monopoly1 problem 7 – 1 determine the maximum profit and the...

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