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### Tax Deduction

German income tax laws provide a tax deduction of 18% of gross income. This deduction, however, must never exceed € 6,000 minus 16% of gross income.

a) Determine the interval in which the tax deduction is exactly 18% of gross income.

b) Determine the highest possible amount of tax deduction.

c) Describe the tax deduction by a deduction function D(I), with D tax deduction, and I income (both in €).
(Hint: you need 3 intervals with three different assignment equations.)

Draw the graph of D(I) for $$0 ≤ I ≤ 50'000$$.

### Part a)

As defined in the exercise, we have
D = deduction (€), and
I = income (€).

To start with, sketch the graph of D for "small" incomes". This is how the graph should look like. Whenever I increases by € 10,000, D increases by € 1,800, because this is 18% of € 10,000.

Thus, our function reads   $$D(I)=0.18I$$ , provided I is "small".

For "big" incomes, 18% of I would be more that € 6,000 minus 16% of I. In that case, the tax deduction must be

$$D(I)= 6,000–0.16I$$.

The straight line showing this second assignment must have a negative slope.

To tell between "small" and "big" incomes, determine the income I for which 18% of that income is just the same as € 6,000 minus 16% of that income. Graphically, this is the income for which both lines intersect.

Here is the calculation: 0.18I = 6,000 – 0.16I

Now add 0.16I to both sides. We obtain 0.18I+0.16I = 0.34I = 6,000.

To isolate I, divide by 0.34. The result is I = € 17,647.05.

This means that up to a gross income of € 17,647.05, tax deduction is 18% of that gross income.

### Part b)

In part a) we have seen that tax deduction increases with gross income up to an income level of € 17,647.05. For bigger incomes, tax deduction decreases. Thus, tax deduction reaches its maximum for I = € 17,647.05. At that level, tax deduction is

Dmax $$= 0.18 \cdot 17,647.05=€\;3,176.45$$

### Part c)

We have done most of the work above:

(i) For gross income not exceeding € 17,647.05, we have $$D(I) = 0.18I$$. Thus, D is proportional to I.

(ii) For gross incomes exceeding € 17,647.05, tax deduction is € 6,000 – 16% of gross income. The assignment equation reads

$$D(I) = 6,000−0.16I.$$

Here again, D depends on I, and the graph is a straight line pointing downwards. This assignment is valid till the graph hits the income line: it wouldn't make sense to have a negative tax deduction (lawmakers forgot to mention this, but here we take account of it).

But where does this happen ? The condition is that tax deduction according to our second assignment is zero. This means that

$$D(I)=6,000−0.16I=0$$.

Add 0.16I on both sides : 6,000 = 0.16I. Then, divide by 0.16 obtain I=37,500.

(iii) So our last finding is that from I = 37,500 on, there is no tax deduction at all, which means D(I)=0 .

To put it all together, we have the following three assignment equations for D: ### Graph and modified conditions

Try to figure out how the graph would change if the initial percentage of 18% was modified. Do likewise with the amount of € 6,000 or with the second percentage of 16%.

To check your findings, have a look at the simulation.

### Part a)

$$0.18I=6000-0.16I \quad \Rightarrow \quad I=17,647.05$$

### Part b)

Dmax $$= 0.18 \cdot 17,647.05=3,176.45$$

### Part c) The diagram to this exercise can be seen here.