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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.

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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


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


= 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



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)


 RD=QD*p=QD (600-QC-QD)


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,


 TR=QC*P (firm C) =175*275=48,125

TR= QD*P (firm) 150*275=41,250

                     From TCC=2500+100QC


                      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


 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


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


QF=QT-QL =2678.57-857.14

= 1821.43 units.

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