Ejercicios Cap 7 y 8

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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 = ∫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 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.
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 - 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 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 - 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 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.
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/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 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=88p1–4p12 +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 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).
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 infinitydoes 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=13x1-0.2x12
 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 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
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=0.25q22
 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 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 monopolysituation.
 Πbm=810(30)-6(30)2-450(30)=5400
 Πsm = 810q2 -13.5q22 = 12150
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 (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 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
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(270q2-2q22)-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 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 ) .