a Quantity demanded depends on the price of tea, average exp., etc.. It is not monotonic, e.g.. Goes through origin only 7.. Objective function parallel to first constraint 11... No stati
Trang 1Chapter 2
2.4 1. 73
101
4
19 30
5 213
37
1
12 13
9 1839
1 2
2.5 1 (a) 1
5
2
2 1 3 (a) 1 (b) 5 4 15, 41
3,2
1
5,1
2
7,
7
9,
5
11,
3
13,
1 15 2.6 1 36.914 2 751.4 3 435.1096 4 36,082
8 (a) 0.1 (b) 0.001 (c) 0.000001
9 (a) 0.452 (b) 2.431 (c) 0.075 (d) 0.002
9 −157
17 16
5 11.641754 6 531,441 7 0.0015328 8 36
9.−618.47021 10 25.000655
Trang 22.9 1.±25 2 2 3 0.2 4 7 5 2.4494897
8 5.2780316 9 0.03423 10 87.977857
2.10 1 270,818.98 2 220.9478 3 2.8563× 109 4 1.5728× 108
5 1.2683 6 16,552,877 7 93.696376 8 4.38228
9 5.1331868
Chapter 3
3.1 1 (a) 0.01x (b) 0.5x (c) 0.5x 2 0.01rx + 0.5wx + 0.5mx
3 (a) x
12(b)
xp
12 4 (a) 0.1x kg (b) 0.3x kg (c) x(0.1m + 0.3p)
5 0.5w + 0.25 6 Own example 7 10.5x + 6y 8 3q− 6000
3.2 1 456 2 77.312 3 r + z, 9% 4 Own example
5 1.094 6 £465.58 7 £2,100 8 (a) 99+ 0.78M (b) £2,166
3.3 1 30x + 4 2 24x − 18y + 7xy − 12 3 6x + 5y − 650
4 9H− 120
3.4 1 6x2− 24x 2 x2+ 4x + 9 3 2x2+ 6x + xy + 3y
4 42x2− 16y2− 34xy + 6y 5 33x + 2y − 20y2+ 62xy − 21
6 120+ 2x + 54y + 40z − x2+ 6y2+ xy + 4xz + 8yz
7 200q − 2q2 8 13x + 11y 9 8x2+ 60x + 76
10 4,000 + 150x
3.5 1 (x + 4)2 2 (x − 3y)2 3 Does not factorize
4 Does not factorize 5 Own example
3.6 1 3x + 7 − 20x−1 2 x + 9 3 4y + x + 12 4 200x−1+ 21
5 179x 6 2(x + 3) + 4 − x − 3 − x + 2 = 9 7 Own example
3.7 1 4 2.111 3 7 4 14 5 82 6 20 p 7 33%
8 40p 9 £3,062.50 10 4 m 11 26
3.8 1.1n
n
i=1H i, 173.7 cm 2 35 3 60
4
n
i=16,000(0.9)
i−1,16,260 tonnes 5 (a) 1
n n
i=1R i , £4,425
(b)1
3
n−1
n−3R i ,
£4,933 6 13.25%, 8.2
3.9 1 (a)≤ (b) < (c) ≥ (d) > 2 (a) > (b) ≥ (c) > (d) >
3 (a) Q1< Q2 (b) Q1= Q2 (c) Q1> Q2 4 P2> P1
Trang 3Chapter 4
4.1 1 (a) Quantity demanded depends on the price of tea, average exp., etc
(b) Q tdependent, all others independent
(c) Qt = 99 − 6P t − 0.5Y t + 0.8A + 1.2N + 1.4P c
(suggested number assumes tea is an inferior good)
2 (a) 202 (b) 7 (c) 6, x≥ 0 3 Yes; no
4.2 1.◦F= 32 + 1.8◦C 2 P = 2,400 − 2Q
3 It is not monotonic, e.g TR= 200 when q = 5 or 10
4 T= (0.0625X − 25)2; no 5 Own example
4.3 (Answers to 1 to 5 give intercepts on axes)
1 x = −12, y = 6 2 x = 31
3, y = −40 3 P = 60, Q = 300
4 P = 150, Q = 750 5 K = 24, L = 40 6 Goes through origin only
7 Goes through (Q = 0, TC = 200) and (Q = 10, TC = 250)
8 Horizontal line at TFC= 75 9 Own example
10 (a) and (d); both slope upwards and have positive intercepts on P axis
4.4 1 Q = 90 − 5P ; 50; Q ≥ 0, P ≥ 0 2 C = 30 + 0.75Y
3 By £20 to £100 4 P = 12 − 0.015Q 5 £6,440
4.5 1 3.75, 0.75, 0.375,−0.75 2 P = 12, Q = 40; £4.50; 10 3 (a) 2/3 (b) 3
4 (c) (i) (a) (ii) (b) 5 (a), (d) 6 APC= 400Y−1+ 0.5 > 0.5 = MPC
7 (a) 0.263 (b) 0.714 (c)1.667 8 Own example
4.6 1.−1.5 (a) becomes −1 (b) becomes −1 (c) no change (d) no change
2 K = 100, L = 160, PK= £8, PL= £5
3 Cost £520 > budget; PLreduced by £10 to £30
4 (a)−10 (b) −1 (c) −0.1 (d) −0.025 (e) 0 5 No change
6 Height £120, base 12, slope−10 = −(wage) 7 Own example
4.7 1 Sketch graphs 2 Sketch graphs 3 Steeper
4 Like y = x−1; £260 5 Own example
4.8 1 Sketch graphs 2 Own example 3 π = 50x − 100 − 0.4x3; inverted U
4 (a) 40= 3250q−1
(b) Original firms’ π per unit = £27.50 but new firms’ AC = £170 > price
4.9 Plot Excel graphs
4.10 1 (a) 16L−1 (b) 0.16 (c) constant
2 (a) 57,243.34L−15 (b) 57.243 (c) constant
3 (a) 322.54L−1 (b) 3.2254 (c) increasing
4 (a) 3,125L −1.25 (b) 9.882 (c) increasing
Trang 45 (a) 23,415,916L −1.6667 (b) 10,868.71 (c) decreasing
6 (a) 4,093.062L −1.7714 (b) 1.173 (c) decreasing
4.11 1 MR= 33.33 − 0.00667Q for Q ≥ 500
2 MR= 76 − 0.222Q for Q ≥ 22.5
3 MR= 80 − 0.555Q for Q ≥ 562.5
4 MC= 30 + 0.0714Q for Q ≥ 56
5 MC= 56 + 0.1333Q for Q ≥ 30
6 MC= 3 + 0.0714Q for Q ≥ 59
Chapter 5
5.1 1 q = 40, p = 6 2 x = 67, y = 17 (approximately) 3 No solution exists
5.2 1 q = 118, p = 256 2 (a) q = 80, p = 370 (b) q falls to 78, p rises to 376
3 Own example 4 (a) 40 (b) rises to 50 5 x = 2.102, y = 62.25
5.3 1 A = 24, B = 12 2 200 3 x = 190, y = 60
5.4 1 x = 30, y = 60 2 A = 6, B = 36 3 x = 25, y = 20
5.5 1 x = 24, y = 14.4, z = 19.2 2 x = 4, y = 6, z = 4
3 A = 6, B = 22, C = 2 4 x = 17, y = 4, z = 8
5 A = 82.5, B = 35, C = 6, D = 9
5.6 1 q = 500, p = 275 2 K = 17.5, L = 16, R = 10
3 (a) p rises from £8 to £10 (b) p rises to £9
4 Y = £3,750 m; government deficit £150 m
5 Y = £1,625 m; balance of payments deficit £15 m
6 L = 80, w = 52
5.7 1 p = 184 + 0.2a, q = 43.2 + 0.06a, p = 216, q = 52.8
2 p = 84 + 0.2t, q = 32 − 0.4t, p = 85, q = 30
3 p = 122.4 + 0.2t, q = 13.8 − 0.1t, p = 123.4, q = 13.3
4 (a) Y= 100/(0.25 + 0.75t), Y = 250 (b) Y = 110/(0.25 + 0.75t), Y = 275
5 p = (4200 + 3800v)/(9 + 5v),
q = (750 − 50v)/(9 + 5v)
p = 494.30, q = 76.94
5.8 1 q1= 60, q2= 80, p1= £10, p2= £8
2 q1= 40, q2= 50, p1= £6, p2= £4
3 p1= £8.75, q1= 60, p2= £6.10, q2= 550
Trang 54 £81 for extra 65 units
5 £7.50 for extra 25 units
6 q1= 48, q2= 39, p1= £12, p2= £8.87
7 (a) 190 units (b) £175 for extra 75 units
5.9 1 q1= 180, q2= 200, p = £39 2 q1= 1,728, q2= 780, p = £190.70
3 q1= 1,510, q2= 1,540, qA= 800, qB= 2,250, PA= £500, PB= £625
4 q1= 160, q2= 600, qA= 2931
3, qB= 2662
3, q C = 200, PA= £95,
PB= £80, P c= £60
5 q1= 15.47, q2= 27.34, q3= 26.17, p = £14.20
5A.1 1 8.4 of A, 4.64 of B (tonnes); (a) no change (b) no B, 12.16 of A
2 A = 13, B = 27 3 12 of A, 5 of B 4 22.5 of A, 7.5 of B
5 6 of A, 32 of B 6 Own example
7 13.64 of A, 21.82 of B; £7092; surplus 2.72 of R, 22.72 of mix additive
8 Produce 15 of A, 21 of B 9 30 of A, B= 0
10 Objective function parallel to first constraint
11 24,000 shares in X, 18,000 shares in Y, return £8,640
12 Own example
5A.2 1 C = 70 when A = 1, B = 1.5, slack in x = 30 2 A = 3, B = 0
3 Q = 2.5, R = 1.5; excesses 62.5 mg of B, 27.5 mg of C
4 10 of A, 5 of B; space for 50 extra loads of X
5 Zero R, 15 tonnes of T; G exceeds by 45 kg
6 100 of A, 40 of B 7 Own example
5A.3 1 2 of A, 1 of B 2 7.5 of X and 15 of Y (tonnes) 3 8
4 Own example
Chapter 6
6.1 1 2 or 3 2 10 or 60 3 When x= 2 4 0.5 5 9
6.2 1 10 or−12.5
2 £16.353 (a) 1.01 or 98.99 (b) 11.27 or 88.73 (c) no solution exists
4 Own example
6.3 1 x = 15, y = 15 or x = −3, y = 249
2 x = 1.75, y = 3.15 or x = −1.53, y = 20.97 3 16.4
4 q1= 3.2, q2= 4.8, p1= £136, p2= £96
5 p1= £15, q1= 80, p2= £8.50, q2= 70
6.4 1 52 2 1069 3 10
Trang 6Chapter 7
7.1 1 £4,630.50 2 £314.70 3 £17,623.16
4 £744.71 5 £40,441.40 6 £5,030.03
7.2 1 £43,747.41; 12.68% 2 £501,159.74; 7.44% 3 (a) APR 11.35%
4 £2,083.61; 19.25% 5 £625; 5.09% 6 19.28%
7 0.01467% 8 £494,531.25; 4.5%
7.3 1 £6,301.69 2 £355.89 3 No, A = £9,106.27
4 £6,851.65 5 (a) £9,638.58 (b) £11,579.83 (c) £13,318.15
6 5 7 5.27 years 8 12.1 years 9 5.45 years 10 3.42 years
11 10.7% 12 9.5% 13 7.5% 14 0.8% 15 10.3% 16 8.4%
17 (b) as PV= £5,269.85
7.4 1 (a) £90.75 (b)−£100.07 (c) −£474.01 (d) £622.86 (e) £1,936.87
(f) £877.33 (g) £791.25 (h) £992.16
2 B, PV= £6,569.10 3 (a) All viable (b) A best, NPV= £6,824.68
4 Yes, NPV= £7,433.56 5 Yes, NPV= £4,363.45
6 (a) Yes, NPV= £610.02 (b) no, NPV = −£522.30
7 B, NPV= £856.48
7.5 1 rA= 20%, rB= 41.6%, rC= 20%, rD= 20%;
B consistently best, but others have same IRR with different NPV ranking
2 (a) A, rA= 21.25%, rB= 20.42% (b) B, NPVB= £2,698.94,
NPVA= £2,291.34 3 IRR= 16.93%
7.6 1 (a) 2.5, 781.25, 50,857.3 (b) 3, 121.5, 14,762 (c) 1.4, 10.756, 139.6
(d) 0.8, 19.66, 267.8 (e) 0.75, 0.57, 9.06
2 5,741 (to nearest whole unit)
3 A, £1,149.32; B, £2,980.91; C, £45,216.47
4 Yes, NPV= £3,774.71 5 £4,149.20
7.7 1 (a) k = 1.5, not convergent (b) k = 0.8, converges on 600
(c) k = −1.5, not convergent (d) k =1
3converges on 54 (e) converges on 961.54 (f) not convergent
2 £3,076.92 3 Yes, NPV= £50,000
4 (a) £240,000 (b) £120,000 (c) £80,000 (d) £60,000 5 £3,500
7.8 1 £152.59 2 £197.38 3 £191.46
4 £794.66 5 (a) 14.02% (b) 26.08% (c) 23.86% (d) 14.71%
6 Loan is marginally better deal (PV of payments= £6,348.33 + £1,734
deposit= £8,082.33, less than cash price by £12.67)
Trang 77.9 1 6.82 years 2 After 15.21 years 3 4%
4 Yes, sum of infinite GP= 1,300 million tonnes 5 4.85%
Chapter 8
8.1 1 36x2 2 192 3 21.6 4 260x4 5 Own example
8.2 1 3x2+ 60 2 250 3 −4x−2− 4 4 1
5 0.2x−3+ 0.6x −0.7 6 Own example
8.3 1 120− 6q, 20 2 25 3 14,400 4 £200 5 Own example
8.4 1 7.5 2 12q2− 40q + 60
3 (a) 1.5q2− 6q + 25 (b) 0.5q2− 3q + 25 + 20q−1 (c) q − 3 − 20q−2
4 MC constant at 0.8 5 Own example
8.5 1 4 2 (a) 80 (b) 158.33 (c) 40 or 120 3 6
8.6 1 50−2
3q 2 900 3 24− 1.2q2
8.7 1 0.8 2 Proof 3 0.16667 4 1
8.8 1 £77.50 2 Own example 3 Rise, maximum TY when t= £39 8.9 1 (a) 0.8 (b) 4,400 (c) 5 (d) 120 (e) Yes, both 940
Chapter 9
9.1 1 62.5 2 150 3 (a) 500 (b) 600 (c) 300 4 50
9.2 1 1,200, max 2 25, max 3 4,096, max 4 4, not max
9.3 1 6, min 2 14.4956 min 3 0, min 4 3, not min
5 No stationary point exists
9.4 1 (a) MC= 2q2− 28q + 222, min when q = 7, MC = 124
(b) AVC= 2
3q2− 14q + 222, min when q = 10.5, AVC = 148.5
(c) AFC= 50q−1, min when q → ∞ =, AFC → 0
(d) TR= 200q − 2q2, max when q = 50, TR = 5,000
(e) MR= 200 − 4q, no turning point, end-point max when q = 0
(f) π = −2
3q3+ 12q2− 22q − 50, max when q = 11, π = 272.67
π min when q = 1, π = −602
3
2 Own example 3 (a) 16 (b) 8 (c) 12
4 No turning point but end–point min when q= 0
5 No turning point but end–point min when q= 0
6 Max when x = 63.33, no minimum
Trang 89.5 1 π max when q = 4 (theoretical min when q = −1.67 not realistic)
2 (a) Max when q= 10 (b) no min exists
3 π max when q = 12.67, gives π = −48.8 4 5,075 when q = 10 5 27.6
when q= 37
9.6 1 15 orders of 400 2 560 3 480 every 4.5 months 4 140
9.7 1 (i) (a) q = 90 − 0.2t, p = 270 + 0.4t (b)&(c) q = 90, p = 270
(ii) (a) q = 250 − 1.25t, p = 125 + 0.375t (b)&(c) q = 250, p = 125
(iii) (a) q = 25 − 0.9615t, p = 160 + 0.385t (b)&(c) q = 25, p = 160
(there is no tax impact for (b) and (c) in all cases)
2 q = 100, p = 380 (no tax impact)
Chapter 10
10.1 1 (a) 3+ 8x, 16 + 4z (b) 42x2z2, 28x3z (c) 4z + 6x−3z3, 4x − 9x−2z2
2 MPL= 4.8K 0.4 L −0.6 , falls as L increases
3 MPK= 12K −0.7 L 0.3 R 0.4 ,MPL= 12K 0.3 L −0.7 R 0.4,
MPR= 16K 0.3 L 0.3 R −0.6
4 MPL= 0.7, does not decline as L increases 5 No 6 1.2x −0.7
j
10.2 1 (a) 0.228 (b) falls to 0.224 (c) inferior as ∂q/∂m < 0 (d) elasticity with respect
to ps = 0.379 and so a 1% increase in both prices would cause a percentage rise in q of 0.379 − 0.228 = 0.151%
2 (a) Yes, MUAand MUBwill rise at first but then fall;
(b) no, MUAfalls but MUBcontinually rises, therefore law not obeyed;
(c) yes, both MUAand MUBcontinually fall
3 No, MU will never reach zero for finite values of A or B.
4 3,738.46; balance of payments changes from 4.23 deficit to 68.85 surplus
5 25+ 0.6q2+ 2.4q1q2 6 0.45; 1.81818; 55
10.3 1.−2K 0.6 L −1.5 , 2.4K −0.4 L −0.5 2 QLL= 6.4, MPLfunction has constant slope; QLK=
35+2.8K, position of MPLwill rise as K rises; QKK= 2.8L, MPKhas constant slope, actual
value varies with L; QKL= 35 + 2.8K, increase in L will increase MPK, effect depends on
level of K.
3 TC11= 0.008q2,TC22= 0, TC33= 0.008q2
TC12= 1.2q3= TC21,TC23= 9 + 1.2q1= TC32
TC31= 0.016q1q3+ 1.2q2= TC13
10.4 1 q1= 12.46; q2= 36.55 2 p1= 97.60, p2= 101.81
3 q1= 0, q2= 501.55 (mathematical answer gives q1= −1,292.24,
q2= 1,701.77 so rework without market 1)
4 £575.81 when q1= 47.86 and q2= 39.01 5 q1= 266.67, q2= 333.33
6 q1= 1,580.2, q2= 1,791.8 7 K = 2,644.2, L = 3,718.5
Trang 98 £29,869.47 when K = 1,493.47 and L = 2,489.12
9 Because max π = £18,137.95 when K = 2,176.5 and L = 2,015.22
10 K = 10,149.1, L = 9,743.1
10.5 1 (a) 12K −0.4 L 0.4 dK + 8K 0.6 L −0.6 dL
(b) 14.4K −0.7 L 0.2 R 0.4 dK + 9.6 0.3 L −0.8 R 0.4 dL + 19.2K 0.3 L 0.2 R −0.6 dR
(c) (4.8K −0.2 + 1.6KL2) dK + (3.5L −0.3 + 1.6K2L) dL
2 (a) Yes (b) no, surplus (c) no, surplus
3 40x −0.6 z −0.45 + 12x 0.4 z −0.7
4.∂Q A
∂P A +∂Q A
∂M
dM
dP A
Chapter 11
11.1 1 K = 12.6, L = 21 2 K = 500, L = 2,500 3 A = 6, B = 4
4 141.42 when K = 25, L = 50 5 Own example
6 (a) K = 1,000, L = 50 (b) K = 400, L = 20
7 1,950 when K = 60, L = 120 8 L = 241, K = 201, TC = £3,617
11.2 See answers to 11.1
11.3 1 See answers to 11.1 2 L = 38.8, K = 20.7, TC = £3,104.50
3 C1= £480,621, C2= £213,609
4 L = 19.04, K = 8.18, TC = £1,145.30
11.4 1 x = 30, y = 30, z = 90 2 877.8 when K = 15, L = 45, R = 13
3 x = 50, y = 100, z = 150 4 79,602.1 when x = 300, y = 300, z = 1,875
5 K = 26.7, L = 33.3, R = 8.9, M = 55.6 6 Own example
7 L = 60, K = 45, R = 40
Chapter 12
12.1 1 9 2 Answer given 3 3M(1 + i)2 4 0.6x(3 + 0.6x2) −0.5
5 0.5(6 + x) −0.5 6 MRP
L= 60L −0.5 − 8, L = 16 7 169 units
8 £8 9 0.000868
12.2 1 (6x + 7) −0.5 ( 39x2+ 36.4x − 5.7) 2 12
3 76.5L −0.5 ( 0.5K 0.8+ 3L0.5 ) −0.4 4 312.5 5 £190
6 Own example 7 (a)−0.05(60 − 0.1q) −0.5
(b) rate of change of slope= −0.0025(60 − 0.1q) −0.5 < 0 when q < 600
(c) 400
12.3 1 (24 + 6.4x − 4.5x 1.5 − 3x 2.5 )(8− 6x1.5 ) −1.5
2 (18,000 + 360q)(25 + q) −1.5 3.−0.113
Trang 104 q = 1,3333,d TR/dq = −0.00367 5 L = 4.8, H = 7.2
6 Adapt proof in text for MC and AC to AVC= TVC(q)−1
12.4 1 (a) 12.5x2+ C (b) 5x + 0.6x2+ 0.05x3+ C (c) 24x5− 15x4+ C
(d) 42x + 18x−1+ C (e) 60x 1.5 + 220x −0.2+ C
2 (a) 4q + 0.05q2 (b) 42q − 9q2+ 2q3 (c) 35q + 0.3q3
(d) 62q − 8q2+ 0.5q3 (e) 185q − 12q2+ 0.3q4
12.5 1 (a) £750,000 (b) £81,750 (c) £250,000 (d) £67,750
2 £49,600 3 Own example
Chapter 13
13.1 1 20 2 No production in period 4 3 (a) Unstable (b) stable
13.2 1 P t = 4 + 0.25(−2) t 2 Stable, 118.54 3 404.64
13.3 1 2,790.625; yes 2 39,946.789 3 492.57 4 1,848.259
13.4 1 2,460.79 2 No, 1,976.67 < 1,980 3 P tx= 562 − 63(0.83) t , 555.27
Chapter 14
14.1 1 64.44 million 2 61,062 units 3 16.8 million tonnes
4 Usage in million units: (a) 94.6, yes (b) 137.6, yes (c) 200.2, no
(d) 291.31, no 5 56,609 units 6.e31,308.07
14.2 1 2%; 9.84 million; no 2 9%, 401,767,300 barrels
3 £122,197.54 4 587
14.3 1 0.48% 2 2.05%; 3.49% 3 0.83%, 621.43 million tones
4.e6,446.39 million 5 8.8% 6 5.83% 7 6.18%
8 9% discrete (equivalent to 8.62% continuous)
14.4 1 (a) 200e0.2t, 1477.81 (b) 45e1.2t, 7323965.61 (c) 14e−0.4t,0.26
(d) 40e1.32t, 21614597.49 (e) 128e−0.03t, 99.69 2 10 %, 6.77
14.5 1.−20e0.4t+ 200, 52.22, unstable 2.−19.2e −1.5t+ 32, 31.99, stable
3.−20e−0.75t+ 120, 119.53, stable 4 75e0.08t − 300, −188.11, unstable
14.6 1 7e−0.325t+ 30, stable 2.−7.25e −0.96t + 26.25, difference 0.01
3 Yes, as predicted spot price is $27.56
4 32.54e −0.347t + 17.46, £23.20, 3 periods 5 $44.01
Trang 1114.7 1 25e−0.2t + 180, 183.38 2.−63.33e −0.195t + 1583.33, 1574.32
3 12.22e−0.036t + 2027.78, 2036.3 4.−9.49e −0.226t + 141.49, 140.495
5.−18.154e −0.176t + 346.15, 343.015
Chapter 15
2 Yes
3 a yes, 6 70
27 23
b no c yes,
14 3 14
−3 2
4
1.4 1.2 0.6 0.6 0.2 1.6 0.8 0.5
0.8 0.24 0.4 0
15.2 1.
27 39
2 a 98 84
163 134
b 71 24 13
c not possible 3 PR= 5262 31.530 5671 4355
15.3 1 a 17.5 45 19
b
136130 67.5 47.5 9051 4044
c
907 4505 400 1030 9932 7386
15.4 1 a
25 41 94
x y z
=
9532 61
63 42 84
x y z
=
5628 34
c
59 44 20
x y z
=
9532 61
d not possible
2 (b) and (c) 3 (b) not square, (c) rows linearly dependent
15.5 1 A 2 B.0 C.56 D.137 E.119
15.6 1 a.−39 b.15 c −4 d 28 e 50 f 4
2 A.636 B.−101 C −4462