Table of Contents
Chapter 11
2. TC=800,000-5,000+100Q^2 and the price, P=$20,000
a) Marginal cost curve is given by the derivative of Total cost function.
MC=TC’=-5,000+200Q
=-5000+200Q
b) Marginal Revenue function is given by the derivative of the Revenue function.
(R) =Q*P=Q (20,000) =20,000Q
R’= 20,000
F (MR) =20,000
4. TC=$100,000+20Q
a) TC=$100,000+20Q
MC=TC’=$20
c) Let Q1 be the original Quantity demanded, Q2 be the new Quantity demanded, P1 be the original price and P2 be the new price.
Price elasticity of demand= %change of quantity/%change in price
Where, %change of quantity= ((Q2-Q1)/Q1)/100 and %change in price= ((P2-P1)/P1)/100
-1.5= (((Q2-Q1)/Q1)/100)/ (((P2-P1)/P1)/100)
P2= (P1 (Q2-Q1)-1.5Q2P1)/ -1.5Q2
New price= (P1 (Q2-Q1)-1.5Q2P1)/ -1.5Q2
d) Marginal revenue at the new price (MRP2) will be given by the derivative of the revenue function.
R =price*quantity=P2*Q2
P2= (P1 (Q2-Q1)-1.5Q2P1)/ -1.5Q2
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MR=R’=0.667P1-1.5P1^2
= 0.667P1-1.5P1^2
6. Q1=30,000 gallons @ Price, P=15 per year
Where; P1 is original price, P2 is the new price, Q1 is the original quantity and Q2 is the new quantity demanded.
Arc elasticity of demand= (P1+P2)/ (Q1+Q2)*((Q2-Q1)/ (P2-P1))
-2= (15+P2)/ (30,000+34,500)*((4,500)/ (P2-15)) and therefore P2=$13.9887
=15-13.9887
=$1.0113
Chapter 12
Determining equilibrium output and selling price of each firm.
P=600-2QC-QD, TCC=2,500+100QC, and TCD=20,000+125QD
a) i) Equilibrium output.
MCC=( TCC’) =100 and MCD= ( TCC’) =125
RC= QC*P=QC (600-QC-QD)
MRC=RC’=600-2QC-QD
RD=QD*p=QD (600-QC-QD)
MRD=RD’=600-QC-2QD
At equilibrium; MRC=MCC
600-2QC-QD=100…….. (1)
MRD= 600-QC-2QD=125……… (2)
Solving (1) and (2) simultaneously,
QC=175 and QD=150.
ii) Selling price, P=600-QC-QD=600-175-150=$ 275
b) Determining total profits for each firm.
Total profit=total revenue minus total cost,
TP=TR-TC.
TR=QC*P (firm C) =175*275=48,125
TR= QD*P (firm) 150*275=41,250
From TCC=2500+100QC
TCC=2500+175*100=42,500
TCD=20,000+125QD, TCD=20,000+125*150=38,750
Profit for firm C=TRC-TCC=$5,625
Profit for firm D=TRD-TCD=$2500
P=200-Qa-Qb where TCa=1,500+55Qa+Qa^2 and TCb=1200+20Qb+2Qb^2
a) Equilibrium output and the selling price
MCa=55+2Qa (From TCa’)
MCb=20+4Qb (From TCb’)
MRa=Qa (200-Qa-Qb) =200-2Qa-Qb (MRa’)
MRb=Qb (200-Qa-Qb) =200-Qa-2Qb (MRb’)
At long run equilibrium, MCa=MRa and MCb=MRb
55+2Qa=200-2Qa-Qb……… (1)
20+4Qb=200-Qa-2Qb…….. (2)
Outputs, Qa=30 and Qb=25
Selling price, P=200-30-25=$145
b) TPa = TRa -TCa
TRa=Qa*P=30*145=4,350t
TCa= TCa=1,500+55Qa+Qa^2=1500+55(30) +30^2=4,215
Total profit of A=4,350-4,215=$135
TPb= TRb-TCb
TRb=25*145=3,625
TCb=1200+20Qb+2Qb^2=1,200+20(25) +2(25^2) =2,950
Total profit B=3,625-2,950=$675
TP(a+b)= 675+135=$810
5. QT=QL+QF, P=20,000-4QT, MCL=5,000+5QL, MCF=2,000+4QF
a) At maximum profit, MC=MR but MCL=5,000+5QL
MRL=R’ but R=QL*P=QL (20,000-4QT) =20,000QL-4QLQT R’ =20,000-4QT
But MCL=MRL= 5,000+5QL=20,000-4QT………. (1)
MCF=2,000+4QF, MRF =R’ but R= QF*P
MRF=QF (20,000-4QT) =20,000QF-4QFQT f= 20,000-4QT
MRF=MCF = 20,000-4QT=2,000+4QF………. (2)
QL=857.14 and QF=1821.43
Max Profit=857.14 units.
ii) P=20,000-4QT But QT=QL+QF =857.14+1,821.43=2,678.57
P=20,000-4(2,678.57) = $9,285.72
b) Total market demand at the price established by the Alchem,
P= $9,285.72, P=20,000-4QT QT= (20,000-P)/4
Total demand=2,678.57
a) QT=QL+QF
QF=QT-QL =2678.57-857.14
= 1821.43 units.