Suppose that a state's tax code states that tax liability is % on the first of taxable earnings and % on the remainder. Find the constants a and b in the tax function T(x) such that and exists.

Question

Suppose that a state's tax code states that tax liability is  % on the first  of taxable earnings and  % on the remainder. Find the constants a and b in the tax function T(x)  such that  and  exists.

Quizplus · Accepted Answer

Let x be the taxable earnings. Then the tax function T(x) can be defined as:

T(x) = a*x + b       for  0 ≤ x ≤ 11eaa88b_945b_b565_a696_e3df34400ed5_TB5869_11
T(x) = a*x - c + b   for  x > 11eaa88b_945b_b565_a696_e3df34400ed5_TB5869_11

where c = 11eaa88b_945b_dc76_a696_e39829d2d4c8_TB5869_11 % is the rate applied to earnings above the threshold.

We need to find constants a, b, and c such that T(11eaa88b_945c_0388_a696_87550c0645e7_TB5869_11) and T(11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11) exist.

Since the maximum taxable earnings are 11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11, T(11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11) must exist. This means that we need to have:

a * 11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11 - c + b = T(11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11)

We can simplify this as:

a * 11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11 + (1 - c) * b = T(11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11)

Similarly, since the minimum taxable earnings are 0, T(0) must exist. This means that we need to have:

a * 0 + b = T(0)

We can choose any value for b because it only affects the vertical position of the tax function. So let's choose b = 0 for simplicity.

Now we have:

a * 11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11 + (1 - c) * 0 = T(11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11)

a * 0 + 0 = T(0)

Simplifying these equations, we get:

a * 11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11 = 11eaa88b_945c_0389_a696_f7d01dbd018b_TB5869_11 * 11eaa88b_945c_51ac_a696_9f6e4b188ea7_TB5869_11 / 100

a = 11eaa88b_945c_51ac_a696_9f6e4b188ea7_TB5869_11 / 100

a * 0 + 0 = 0

So we have:

a = 11eaa88b_945c_51ac_a696_9f6e4b188ea7_TB5869_11 / 100
b = 0
c = 11eaa88b_945b_dc76_a696_e39829d2d4c8_TB5869_11 / 100 = 11eaa88b_945c_78bd_a696_cdeb1f221899_TB5869_11 / 100

Therefore, the constants a and b are 11eaa88b_945c_51ac_a696_9f6e4b188ea7_TB5869_11/100 and 0, respectively, and the constant c is 11eaa88b_945c_78bd_a696_cdeb1f221899_TB5869_11/100. The answer is B.

Suppose That a State's Tax Code States That Tax Liability