Assume that in a given country, tax revenues, \(T\), depend on income, \(I\), according to the formula

\[ T=-4,000+0.2 I \]

Thus, for example, when a household has an income of \(\$ 50,000\), its tax burden is \(-4,000+\) \(0.2 \times 50,000\), or \(\$ 6,000\). Is this a progressive tax schedule? [Hint: Compute average tax rates at several different levels of income.]

Now let's generalize the tax schedule in this problem to:

\[ T=a+t I \]

where \(a\) and \(s\) are numbers. (For example, in the tax schedule above, \(a=-4,000\) and \(t=0.2\).) Write down a formula for the average tax rate as a function of the level of income. Show that the tax system is progressive if \(a\) is negative, and regressive if \(a\) is positive. [Hint: The average tax rate is \(T / I\).]