The Laffer curve y = R(t) proposes a simple theoretical relationship between a government's tax revenue R(t) and tax rate t: R(t) = tb(t), where b(t) is the tax base for tax rate t that is, the total dollar amount of economic activity taxable at tax rate t. The dependence of b(t) on t comes from things like changes in people's spending habits or companies' investment plans based on the tax rate. Because t represents a tax rate, it is restricted to be in the interval [0, 1]. (For example, if you have a tax base of $20 billion and a tax rate of 1 4 , $5 billion goes to the government through taxes, and $15 billion remains with the economic actors.) In this assignment, you may use without proof the assumption that b(t) is continuous on the interval [0, 1] and differentiable on the interval (0, 1). You may also look up terms and arguments online, though it is certainly possible to get full marks on the assignment without doing so.

In the remainder of this assignment, you may assume that b (t) < 0 and b (t) < 0 on the interval (0, 1).

(a) Find R(0).

(b) In a paragraph, explain why it is plausible that R(1) = 0.

(c) Determine the concavity of R(t).