A consumer has preferences u(x) = 2x 1 2 1 + x2. The price of good 1 is p1 > 0 and the price of good 2 is 1. You may restrict your attention to interior solutions throughout.

(a) Explain whether these preferences are i) monotonic, ii) quasilinear, iii) essential and/or iv) convex.

(b) Find the consumer’s (Marshallian) demand functions and indirect utility function. For full credit, use a Lagrangian and show your work.

(c) Suppose that there were a a tax of t per unit on good 1. How much revenue would be raised (in terms of the exogenous variables)? How much worse off does the consumer become, in dollars, from such a tax (in terms of exogenous variables)?

(d) Suppose that p1 = 1 2 and the tax is also t = 1 2 . What was the original amount of good 1 purchased? What would be the values of revenue and consumer welfare loss from the formulas that you calculated in part (c). Which is larger? Briefly explain why.

(e) Suppose that later, with the tax still in place, you give back to the consumer the amount of money that was raised by the tax. With the rebate and the tax in place, what would be the new optimal consumption bundle? What would be the consumer’s utility? By how much does the consumer’s utility change relative to the pre-rebate level? Explain the economic intuition for the change in welfare you found from the rebate relative to the original (pre-tax) level of utility.