Download H & Q , 6-1

Problem 7 –1
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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
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7 – 1 solution
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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
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Problem 7 – 2
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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
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7 – 2 , solution
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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
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Problem 7-3
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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
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7 – 3 solution
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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
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Problem 7 – 4
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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
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Problem 7 – 4 , solution
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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
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Problem 7-5
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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
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Problem 7-5 , solution
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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
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Problem 7-6
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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
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Problem 7-6 solution
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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
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Problem 7-7 H&Q
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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
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Problem 7-7 solution
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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
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Problem 7-8 , H & Q
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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
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Problem 7-8 , solution
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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
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Problem 7-9 H & Q
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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
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Problem 7-9 solution
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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
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Problem
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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
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Problem
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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
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Problem
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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
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Problem
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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
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Problem
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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
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Problem
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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
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Problem
8-4
H&Q
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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 .
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Determine the Cournot values for x1,x2,q1,,q2,Π1, Π2.
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H & Q PROBLEMS
CH 7 , 8, monopoly
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Problem
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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
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Problem
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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
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Problem
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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
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Problem
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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
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Problem
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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
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Problem
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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
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Problem
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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
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