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You are required to write a research essay addressing all of following points:
“Australia has too many taxes and too many complicated ways of delivering multiple policy objectives through the tax and transfer systems. The capacity of the legislative and operating platforms of these systems, and their human users, to deal with the resulting complexity has been overreached. The tax and transfer architecture is overburdened and beginning to fail in dealing efficiently and effectively with multiplying policy goals and demands. Rationalisation of the tax and transfer architecture should now be a strategic priority.”Discuss. What are some of these multiple policy objects the Australian tax and transfer system tries to achieve? How does the system overreach and is hence overburdened currently? How do you propose to rationalise the tax and transfer system of Australia to overcome these shortcomings?
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If two points are considered such as (x1, f1) and (x2, f2), from the Lagrange interpolation the following can be written ("Newton-Cotes Formulas -- From Wolfram Math world"):
P2(x) = [(x-x2)/(x1-x2)]*f1 + [(x-x1)/(x2-x1)]*f2
= (x – x1 – h)/(-h) *f1 + [(x-x1)/h]*f2
= (x/h) *(f2-f1) + (f1+(x1/h)*f1 – (x1/h)*f2)
The Lagrange interpolating polynomial can be written as
P3(x) = [(x-x2)(x-x3)/(x1-x2)(x1-x3)]*f1 + [(x-x1)(x-x3)/(x2-x1)(x2-x3)]*f2 + [(x-x1)(x-x2)/(x3-x1)(x3-x2)]*f3
= 1/h^2{x^2(1/2 * f1 – f2+ ½ *f3) + x[- ½ (2x1 + 3h)f1 + (2x1+2h)f2 – ½ (2x1+h)f3] + [ ½ (x1+h)(x1+2h)f1-x1(x1+2h)f2+1/2 x1(x1+h)f3]} ("Newton-Cotes Formulas -- From Wolfram Math world")
After integrating the above equation the following can be written:
x1x5fxdx=245h7f1+3f2+3f3+f4-380h5()
This is known as the Boole’s rule.
Here, h = 0.5
x0 = 0
x1 = x0 + h = 0 + 0.5 = 0.5
x2 = 0.5 + 0.5 = 1
x3 = 1 + 0.5 = 1.5
Now from Simpson’s 3/8 rule, the following can be written:
01.5(x3+t)vdt = (3*0.5)/8 [(Xn^3+0) v0+3(Xn^3 + 0.5) v1+3(Xn^3+1) v2 + (Xn^3+1.5) v3]
Therefore, from the main integral equation, it can be written that
V (Xn) = Xn^3 + 0.25[(3*0.5)/8 {(Xn^3+0) v0+3(Xn^3 + 0.5) v1+3(Xn^3+1) v2 + (Xn^3+1.5) v3}]
Now, substituting
X0 = 0, x1 = 0.5, x2 = 1 and x3 = 1.5, the followings can be written:
V0 = 0.0703125v1 + 0.140625v2+0.1953125v3
V1 = 0.125 + 0.005859v0 + 0.08789v1+ 0.158203v2+0.0761718v3
V2 = 1+0.046875v0 + 0.2109375v1+0.28125v2+0.1171875v3
V3 = 3.375 + 0.1582v0+6.921875v1+0.61523v2+0.22851v3
By solving the above four equations, the followings can be obtained
V0 = -6.63980987677947
V1 = -3.1896396387399073
V2 = -4.7730785407988545
V3 = -29.41093974978935
Hence the integral equation can be written as
V(x) = -1.8096351x^3 + 0.28125
3.
The systems of coupled equations that are given are written below:
xt= ?xt-ty(t)
yt= ?xtyt- ?y(t)
Here, =4
=2
=3
And =3
h = 0.01 and interval [0, 10]
Here, x0 = 2
y0 = 1
The solution is given in the below table:
Table 1
x | y=f(x) |
2 | 1 |
3.96 | 12.6816 |
5.92 | 32.0464 |
7.88 | 59.0944 |
9.84 | 93.8256 |
11.8 | 136.24 |
13.76 | 186.3376 |
15.72 | 244.1184 |
17.68 | 309.5824 |
19.64 | 382.7296 |
21.6 | 463.56 |
23.56 | 552.0736 |
25.52 | 648.2704 |
27.48 | 752.1504 |
29.44 | 863.7136 |
31.4 | 982.96 |
33.36 | 1,109.8896 |
35.32 | 1,244.5024 |
37.28 | 1,386.7984 |
39.24 | 1,536.7776 |
41.2 | 1,694.44 |
43.16 | 1,859.7856 |
45.12 | 2,032.8144 |
47.08 | 2,213.5264 |
49.04 | 2,401.9216 |
51 | 2,598 |
52.96 | 2,801.7616 |
54.92 | 3,013.2064 |
56.88 | 3,232.3344 |
58.84 | 3,459.1456 |
60.8 | 3,693.64 |
62.76 | 3,935.8176 |
64.72 | 4,185.6784 |
66.68 | 4,443.2224 |
68.64 | 4,708.4496 |
70.6 | 4,981.36 |
72.56 | 5,261.9536 |
74.52 | 5,550.2304 |
76.48 | 5,846.1904 |
78.44 | 6,149.8336 |
80.4 | 6,461.16 |
82.36 | 6,780.1696 |
84.32 | 7,106.8624 |
86.28 | 7,441.2384 |
88.24 | 7,783.2976 |
90.2 | 8,133.04 |
92.16 | 8,490.4656 |
94.12 | 8,855.5744 |
96.08 | 9,228.3664 |
98.04 | 9,608.8416 |
100 | 9,997 |
Figure 1
y0 = 20
The solution is given in the below table:
Table 2
x | y=f(x) |
2 | 20 |
3.96 | 31.6816 |
5.92 | 51.0464 |
7.88 | 78.0944 |
9.84 | 112.8256 |
11.8 | 155.24 |
13.76 | 205.3376 |
15.72 | 263.1184 |
17.68 | 328.5824 |
19.64 | 401.7296 |
21.6 | 482.56 |
23.56 | 571.0736 |
25.52 | 667.2704 |
27.48 | 771.1504 |
29.44 | 882.7136 |
31.4 | 1,001.96 |
33.36 | 1,128.8896 |
35.32 | 1,263.5024 |
37.28 | 1,405.7984 |
39.24 | 1,555.7776 |
41.2 | 1,713.44 |
43.16 | 1,878.7856 |
45.12 | 2,051.8144 |
47.08 | 2,232.5264 |
49.04 | 2,420.9216 |
51 | 2,617 |
52.96 | 2,820.7616 |
54.92 | 3,032.2064 |
56.88 | 3,251.3344 |
58.84 | 3,478.1456 |
60.8 | 3,712.64 |
62.76 | 3,954.8176 |
64.72 | 4,204.6784 |
66.68 | 4,462.2224 |
68.64 | 4,727.4496 |
70.6 | 5,000.36 |
72.56 | 5,280.9536 |
74.52 | 5,569.2304 |
76.48 | 5,865.1904 |
78.44 | 6,168.8336 |
80.4 | 6,480.16 |
82.36 | 6,799.1696 |
84.32 | 7,125.8624 |
86.28 | 7,460.2384 |
88.24 | 7,802.2976 |
90.2 | 8,152.04 |
92.16 | 8,509.4656 |
94.12 | 8,874.5744 |
96.08 | 9,247.3664 |
98.04 | 9,627.8416 |
100 | 10,016 |
Figure 2
Figure 3
Figure 4
d3y/dx3 + 3d2y/dx2 + x dy/dx= x2y
It is considered that
x1 = y
x2 = dy/dx
And x3 = d2y/dx2
Differentiating the above equations, the followings can be written:
x1’ = dy/dx = x2
x2’ = d2y/dx2 = x3
x3’ = d3y/dx3 = x3 = x2x1 – 3x2-xx2
The first order differential equations along with their initial values are written below:
x1’ = x2 x1(0)=2
x2’ = x3 x2(0)=1
x3’ = x2x1 – 3x2-xx2 x3(0) = -1
Let us again consider the first order initial value problem to be
dydx=fx,y, yx0=y0
Range-Kutta method is another numerical method of solving differential equations. This method is also known as predictor-corrector method. The Range-Kutta method can be explained with the help of following expressions:
xn+1 =xn+h
yn+1 =yn+k1+2k2+2k3+k46
Where, k1 = hf(xn,yn)
k2 = hf(xn+h2,yn+k12)
k3 = hf(xn+h2,yn+k22)
k4 = hf(xn+h,yn+k3)
For the first equation, the calculation is done in the table:
x1’ = x2 x1(0)=2
Table 3
x | y=f(x) |
0 | 2 |
0.06 | 2.0018 |
0.12 | 2.0072 |
0.18 | 2.0162 |
0.24 | 2.0288 |
0.3 | 2.045 |
0.36 | 2.0648 |
0.42 | 2.0882 |
0.48 | 2.1152 |
0.54 | 2.1458 |
0.6 | 2.18 |
0.66 | 2.2178 |
0.72 | 2.2592 |
0.78 | 2.3042 |
0.84 | 2.3528 |
0.9 | 2.405 |
0.96 | 2.4608 |
1.02 | 2.5202 |
1.08 | 2.5832 |
1.14 | 2.6498 |
1.2 | 2.72 |
1.26 | 2.7938 |
1.32 | 2.8712 |
1.38 | 2.9522 |
1.44 | 3.0368 |
1.5 | 3.125 |
1.56 | 3.2168 |
1.62 | 3.3122 |
1.68 | 3.4112 |
1.74 | 3.5138 |
1.8 | 3.62 |
1.86 | 3.7298 |
1.92 | 3.8432 |
1.98 | 3.9602 |
2.04 | 4.0808 |
2.1 | 4.205 |
2.16 | 4.3328 |
2.22 | 4.4642 |
2.28 | 4.5992 |
2.34 | 4.7378 |
2.4 | 4.88 |
2.46 | 5.0258 |
2.52 | 5.1752 |
2.58 | 5.3282 |
2.64 | 5.4848 |
2.7 | 5.645 |
2.76 | 5.8088 |
2.82 | 5.9762 |
2.88 | 6.1472 |
2.94 | 6.3218 |
3 | 6.5 |
Figure 5
For the second equation, the calculation is done in the table:
x2’ = x3 x2(0)=1
Table 4
x | y=f(x) |
0 | 1 |
0.06 | 1.0018 |
0.12 | 1.0072 |
0.18 | 1.0162 |
0.24 | 1.0288 |
0.3 | 1.045 |
0.36 | 1.0648 |
0.42 | 1.0882 |
0.48 | 1.1152 |
0.54 | 1.1458 |
0.6 | 1.18 |
0.66 | 1.2178 |
0.72 | 1.2592 |
0.78 | 1.3042 |
0.84 | 1.3528 |
0.9 | 1.405 |
0.96 | 1.4608 |
1.02 | 1.5202 |
1.08 | 1.5832 |
1.14 | 1.6498 |
1.2 | 1.72 |
1.26 | 1.7938 |
1.32 | 1.8712 |
1.38 | 1.9522 |
1.44 | 2.0368 |
1.5 | 2.125 |
1.56 | 2.2168 |
1.62 | 2.3122 |
1.68 | 2.4112 |
1.74 | 2.5138 |
1.8 | 2.62 |
1.86 | 2.7298 |
1.92 | 2.8432 |
1.98 | 2.9602 |
2.04 | 3.0808 |
2.1 | 3.205 |
2.16 | 3.3328 |
2.22 | 3.4642 |
2.28 | 3.5992 |
2.34 | 3.7378 |
2.4 | 3.88 |
2.46 | 4.0258 |
2.52 | 4.1752 |
2.58 | 4.3282 |
2.64 | 4.4848 |
2.7 | 4.645 |
2.76 | 4.8088 |
2.82 | 4.9762 |
2.88 | 5.1472 |
2.94 | 5.3218 |
3 | 5.5 |
Figure 6
For the second equation, the calculation is done in the table:
x3’ = x2x1 – 3x2-xx2 x3(0) = -1
Table 5
x | y=f(x) |
0 | -1 |
0.06 | -1.996858 |
0.12 | -2.995264 |
0.18 | -3.976966 |
0.24 | -4.923712 |
0.3 | -5.81725 |
0.36 | -6.639328 |
0.42 | -7.371694 |
0.48 | -7.996096 |
0.54 | -8.494282 |
0.6 | -8.848 |
0.66 | -9.038998 |
0.72 | -9.049024 |
0.78 | -8.859826 |
0.84 | -8.453152 |
0.9 | -7.81075 |
0.96 | -6.914368 |
1.02 | -5.745754 |
1.08 | -4.286656 |
1.14 | -2.518822 |
1.2 | -0.424 |
1.26 | 2.016062 |
1.32 | 4.819616 |
1.38 | 8.004914 |
1.44 | 11.590208 |
1.5 | 15.59375 |
1.56 | 20.033792 |
1.62 | 24.928586 |
1.68 | 30.296384 |
1.74 | 36.155438 |
1.8 | 42.524 |
1.86 | 49.420322 |
1.92 | 56.862656 |
1.98 | 64.869254 |
2.04 | 73.458368 |
2.1 | 82.64825 |
2.16 | 92.457152 |
2.22 | 102.903326 |
2.28 | 114.005024 |
2.34 | 125.780498 |
2.4 | 138.248 |
2.46 | 151.425782 |
2.52 | 165.332096 |
2.58 | 179.985194 |
2.64 | 195.403328 |
2.7 | 211.60475 |
2.76 | 228.607712 |
2.82 | 246.430466 |
2.88 | 265.091264 |
2.94 | 284.608358 |
3 | 305 |
Figure 7
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