H & Q PROBLEMSCH 7, 8, monopoly1 Problem 7 – 1 Determine the maximum profit and the corresponding...
-
Upload
lambert-parks -
Category
Documents
-
view
222 -
download
0
Transcript of H & Q PROBLEMSCH 7, 8, monopoly1 Problem 7 – 1 Determine the maximum profit and the corresponding...
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 .
H & Q PROBLEMS 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
H & Q PROBLEMS 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.
H & Q PROBLEMS 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
H & Q PROBLEMS 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.
H & Q PROBLEMS CH 7 , 8, monopoly 6
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
H & Q PROBLEMS 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+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.
H & Q PROBLEMS 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 - 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
H & Q PROBLEMS 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.
H & Q PROBLEMS 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
H & Q PROBLEMS 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.
H & Q PROBLEMS 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
H & Q PROBLEMS 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?
H & Q PROBLEMS 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 .
H & Q PROBLEMS 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 - 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.
H & Q PROBLEMS CH 7 , 8, monopoly 16
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.
H & Q PROBLEMS 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.
H & Q PROBLEMS 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/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
H & Q PROBLEMS 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.
H & Q PROBLEMS CH 7 , 8, monopoly 20
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
H & Q PROBLEMS 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=2.5q1
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).
H & Q PROBLEMS 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
H & Q PROBLEMS 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.
H & Q PROBLEMS 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
H & Q PROBLEMS CH 7 , 8, monopoly 25
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.
H & Q PROBLEMS CH 7 , 8, monopoly 26
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
H & Q PROBLEMS 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=0.25q2
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.
H & Q PROBLEMS 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 (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
H & Q PROBLEMS 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 (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
H & Q PROBLEMS 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 qq22=54 p=54 p22=162=162 SellerSeller’’s profit=4374s profit=4374 BuyerBuyer’’s profit=17496s profit=17496
H & Q PROBLEMS CH 7 , 8, monopoly 31
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.
H & Q PROBLEMS 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, qIf r=6, q22=54, =54, →→ p p2min2min=81=81 (P2 min) 81 <p 2
* < 486 (p2 max ) .