3. Some economists have proposed to increase the tax on gasoline, then rebate the tax revenue to consumers through a payroll tax reduction or some other means. In this exercise you explore whether such a policy is effective in reducing the consumption of gasoline and whether it increases or decreases consumers' utility. Let an agent's utility function be U (x1, x2) = ln x1 + ax2. Interpret good 1 as gallons of gasoline and good 2 as consumption on other goods. Let P2 = 1.

3.1 Form the Lagrangian function, find the first-order necessary conditions for utility maximization, and solve these first-order conditions to find the demand functions for goods 1 and 2, as a function of the price p and the income m.

3.2 Now suppose the government introduces a $1-a-gallon gasoline tax. Hence, the consumer now pays P + 1 for each gallon of gasoline, and $1 is collected by the government in taxes for each gallon the consumer purchases. The consumer's budget constraint is now (p1 + 1)x1 + x2 = m. Given this budget constraint, find the consumer's demand functions, and explain how the imposition of the tax affects the consumption of good 1.

3.3 Assume p = 1. Compute the total revenue that the tax raises. Denote the tax revenue by R.

3.4 Suppose now that the tax revenue is given back to consumers: the consumer income is now m + R. Find the consumer's demand functions after this raise in income, maintaining the assumption that a gallon of gasoline costs P1 +1 to the consumer.

3.5 Assume a = 1/4, m = 10. Does the consumption of gasoline decrease with the introduction of the policy? What is the effect on the consumer's utility?

Critics of tax-cum-rebate argue that paying the revenue raised by the tax back to the consumers would have no effect on demand since they could just use the rebated money to purchase more gasoline. You showed that this argument is actually incorrect. In general, the consumer is made worse off. However, we are ignoring important considerations such as the fact that other consumers could benefit from better quality of the air they breath.