1. Suppose that the demand curve for a sub sandwich is q = 300 – 20p.
a. Calculate the revenue, marginal revenue
b. To determine the profit - maximizing, how many products are produced if subs can be
produced at a constant average and marginal cost of $7.
Bài làm
P = 15 – 0.05q
a. TR = Q*P = 15q- 0.05q^2
MR = TR’(Q) = 15 – 0.1q
b.
MC = 7 -> TC = 7Q
Tối đa hóa lợi nhuận : MR =MC
 15 – 0.1q = 7-> q= 80
π = TR –TC = - 0.05 q^2 + 8q
π’ = -0.1 + 8 <0
 Q = 80 sẽ tối đa lợi muận
2. Suppose at the firm have cost function follow C(q)= 300+20q
a. To calculate the Average Cost (AC), Marginal Cost (MC)
b. If the demand function is q= 350 – 10p, how many products are produced to maximize the
profit?
a.
MC = C(q)’ = 20
AC =
C( q) 300+20 Q 300
=
=
+ 20
Q
Q
Q
b.
Q = 350 – 10P -> P = 35 -
Q
10
TR = Q*P = 35q – q^2/10
 MR(Q) = 35 – q/5
To maximize the profit :
MC = MR - > 35 – q/5 = 20 -> q = 75
Π = TR – TC =
Π’ =
−q2
+15 q−300
10
−q
+15
5
Π’’ = -1/5 <0
 Q = 75 maximize the profit.
3. For the non-linear demand schedule p= 900-0.5q^2 what output will maximize the sales
revenue?
The non-linear demand : QD = a – bP -> QS
P = 900 – 0.5 Q2
TR = Pq =(900- 0.5 q^2)*q = 900q – 0.5q^3
Maximize the sales revenue : MR = 0
↔ 900 – 1.5q^2 = 0
 Q=
10 √ 6
-> aceppt Q =10 √ 6
−10 √ 6
4.
A firm faces the demand function p = 200-2q and total cost function TC= 2/3q^3 – 14q^2
+222q+60 . Derive expressions for the following functions and find out whether they have
maximum or minimum points. If they do, say what value of q this occurs at and calculate the
actual value of the function at this output.
a. Marginal cost (MC)
b. Average variable cost (AVC)
c. Average fixed cost (AFC)
d. Total revenue (TR)
p = 200-2q
TC= 2/3q^3 – 14q^2 +222q+60
a.
MC =TC’ = 2q^2 -28q +222
MC’ = 0 ↔ 4q -28 = 0 -> q =7
MC’’ = 4 >0 -> q =7 -> MC min
MCMIN = 124
b.
AVC =
VC 2 2
= q −14 q+ 222
Q 3
AVC’ =0 -> q = 10.5
AVC’’ =
4
>0
3
AVCMIN = 148.5
c.
AFC =
−60
FC 60
= -> AFC’ = 2 ≠0
Q Q
Q
 Không có max. min
d. TR = p*q = 200q – 2q^2
MR = 200 – 4q
TR ‘ = 0 -> q =50
TR’’ =-4 <0 -> TRMAX = 5000
5. Total cost function of a company A is: TC = q² + 10q + 100
a. Write functions for: FC, VC, AC, and MC
b. How many productions should the company produce to maximize the profit, if the price
on the market is P=27$. Calculate the profit at that price.
a.
TC = q² + 10q + 100
 AC =
TC q ²+10 q +100
100
=
=q+ 10+
Q
Q
Q
 MC = TC’ = 2q +10
TC = FC + VC = q² + 10q + 100
TC(0) = FC = 100 -> chi phí cố định ko đổi
FC = Q2 + 10Q
b. Maximize the profit ?
MR =Mc
Điều kiện cần Π max -> Π’ = 0
↔ 2Q +10 = 27 -> Q = 8.5
Điều kiện đủ Π max -> Π’ <0 , ∀ q € R
Π = TR –TC = 27q – (q² + 10q + 100)
Π = - 27.25
 Produce to maximize the profit : -27.75
6. Show that the function F(x) =3x3 + 2x +4 has the essential properties of a cost function.
Carefully graph corresponding average cost function and marginal cost function.
7. If the firm faces demand schedule p= 120-3q. And the total cost schedule TC= 120+36q +
1.2q2. What output levels, if any will
a) Maximize profit
b) Minimize cost.
p= 120-3q
TC= 120+36q + 1.2q2 - > MC = 36 + 2.4q
TR =(120 -3q)*q = 120q -3q2
a. Maximize profit -> Q?
MR = MC ; Π ‘’ <0
↔ 36 + 2.4q = 120 – 6q
↔ q =10
Π = TR –TC
Π = 120q -3q2 –(120+36q + 1.2q2)
Π = -4.2q2 + 84q -120
b. Minimize cost -> Q?
MC =0
↔ 36 +2.4Q = 0
 Q = -15
8.
a/ You have 600 USD, you sent it in your bank account. It the interest rate is 6% per year,
compound semiannually. How much your account will increase to after 6 years?
b/ You have started your first job and decide to put $400 a month into an annuity. The
annuity earns 6% interest per year, compounded monthly. How long (in months and years)
will it take for the account to be worth $550,000?
c/ You have started your first job and decide to put $400 a month into an annuity. The
annuity earns 6.5% interest per year, compounded monthly. How long (in months and years)
will it take for the account to be worth $600,000?
( 1+r )n −1
 FV =A[
]
r
( 1+3 % )12−1
 FV = 600[
]= 8515.3
3%
a.
( 1+r )n −1
]
r
( 1+ 0.5 % )n −1
FV = 400(
-> n = 35 years
0.5 %
( 1+r )n −1
b. FV =A[
]
r
FV =A[
600000=400(
( 1+ 0.54 % )n−1
¿
0.54 %
-> n = 410.04 months
≈ 34 years
9. At 9 percent annual interest rate, which of the following has the largest present value. a)
$420 two years from now
b) $220 after each of the next two years, or
c) $220 now and $180 two years from now
d) $120 now, $100 two years from now and $80 after each of the next three years to
forever?
a)
FV = PV*(1+r%)n
420 = PV*( 1 + 9%)2 – > PV = 354
PV =A[
1− (1+ r )−n
]
r
b)
PV =220*[
1− (1+ 9 % )−2
]
9%
-> PV = 387
c)
PV = 220
FV = PV1*(1+r%)n
180 = PV1*(1+9%)2 -> PV 1= 152
PV= 220 + 152 = 372
d)
PV= 120
FV = PV1*(1+r%)n
100 = PV1 *(1+9%)2 -> PV1 = 84
PV2=A[
1− (1+ r )−n
]
r
PV2 = 80(
1−( 1+9 % )−3
= 202
9%
PV tổng = 202 + 84 + 120 = 406
10. The supply and demand functions for a commodity are q=12p-4; q=8-4p.If an excise tax of T
is imposed, what are the selling price and quantity sold, in equilibrium? Comments?
Q = f(P)
The supply : QS= 12p -4
The demand: Qd=8-4p
Equilibrium :QS = QD
↔ 12p – 4 = 8 – 4p - > p = 0.75 ; Q = 5
Ruvenue : P*Q = 5*0.75 = 3.75
If an excise tax of T is imposed :
P = f(Q)
1
1
Q+ (1)
12
3
−1
Q+2 (2)
PD =
4
PS =
PS - P D = T
1
1 −1
Q+ –(
Q+2 ¿ –T =0
12
3
4
1
5
↔ Q− – T =0
3
3
↔
↔ Q = 3T + 5
3
3
T+
12
4
−3
3
T+
(2) -> PD =
4
4
(1) -> PS =
11. Assume a market has the demand function q = 40-0.5p and the supply function q = 2p - 4.
The government currently imposes a per-unit tax of £3. If this tax is slightly increased will the
tax yield rise or fall?
Q = f(P)
The supply : QS= 2p - 4
The demand: Qd=40-0.5p
Equilibrium :QS = QD
↔ 12p – 4 = 8 – 4p - > p = 17.6 ; Q = 31.2
Doanh thu : P*Q = 549.12
If an excise tax of T is imposed :
P = f(Q)
PS = −2 Q+ 80(1)
PD =
1
Q+2 (2)
2
PD – P S = 3
1
Q+2–¿+ 80¿ –3 =0
2
5
↔ Q−81 =0
2
↔ Q = 32.4
↔
(1) -> PS =15.2
(2) -> PD = 18.2
12. A multi-plant monopoly operates two plants whose total cost schedules are TC1 = 8.5 +
0.03q12 TC2 = 5.2 + 0.04q22 . If it faces the demand schedule p = 60 - 0.04q where q = q1 +
q2, how much should it produce in each plant in order to maximize profits?
q = q1+q2
TR = p*q = [60- 0.04*(q1+q2)]*( q1+q2)
TR = 60*( q1+q2) – 0.04 *( q1+q2)2
Π = 60*( q1+q2) – 0.04 *( q1+q2)2 –(8.5 + 0.03q12 ) –(5.2 + 0.04q22)
Π = -13.7 +60q1 + 60q2 – 0.07q12 – 0.08q2 – 0.08q2q1
(1) Điều kiện cần để Π max :
dπ
dπ
' = 0 và
' =0
dq 1
dq 2
dπ
' = 60 – 0.14q1 - 0.08q2 = 0
dq 1
dπ '
=60−0.16 q 2−0.08 q 1=0
dq 2
↔ q1 = 300 ; q2 = 225
(2) Điều kiện đủ để Π max :
''
dπ
' ' < 0 và dπ <0
dq 1
dq 2
↔
''
dπ
' ' = -0.14 <0 và dπ =−0.16<0
dq 1
dq 2
Vậy tại mức sản lượng q1= 300 và q2 = 225 thì sẽ tối đa hóa lợi nhuận.
13. A firm produces two products which are sold in two separate markets with the demand
schedules p1 = 600 - 0.3q1; p2 = 500 - 0.2q2 .Production costs are related and the firm faces
the total cost function TC = 16 + 1.2q1 + 1.5q2 + 0.2q1*q2. If the firm wishes to maximize
total profits, how much of each product should it sell? What will the maximum profit level
be?
14. Derive reduced form equations for equilibrium price and output in terms of (a) a per-unit
sales tax t (b) a lump sum tax T(c) a percentage profits tax c in each of the cases below:
(Please select a case) (i) p = 450 - 2q and TC = 20 + 0.5q 2 (ii) p = 200 - 0.3q and TC = 10 +
0.1q 2 (iii) p = 260 - 4q and TC = 8 + 1.2q 2 In each case assume that the demand schedule
and total cost function apply to a single firm in an industry.