Distortionary Taxes with Redistribution: Consider a 2-person exchange economy in which I own 200 units of x1 and 100 units of x2 while you own 100 units of x1 and 200 units of x2.
A: Suppose you and I have tastes that are quasilinear in x1, and suppose that I sell x1 to you in the competitive equilibrium without taxes.
(a) Illustrate the no-tax competitive equilibriumin an Edgeworth Box.
(b) Suppose the government imposes a per-unit tax t (paid in terms of x2) on all units of x1 that are traded. This introduces a difference of t between the price received by sellers and the price paid by buyers. How does the tax result in kinked budget constraints for us?
(c) True or False: The tax can never be so high that I will turn from being a seller to being a buyer.
(d) Illustrating the tax in the Edgeworth box will imply we face different budget lines in the box —but demand and supply of x1 still has to equalize. Illustrate this and show how a difference between economy’s endowment of x2 and the amounts consumed by us emerges. What’s that difference?
(e) Suppose the government simply takes the x2 revenue it collects, divides it into two equal piles and gives it back to us. In a new Edgeworth box, illustrate our indifference curves through the final allocation that we will consume. How can you tell that the combination of the tax and transfer of x2 is inefficient?
B: Suppose that our endowments are as specified at the beginning. My tastes can be represented by the utility function uM(x1,x2) = x2 + 50lnx1 and yours by the utility function uY (x1,x2) = x2 + 150lnx1.
(a) Derive our demand functions and use them to calculate the equilibrium price p defined as the price of x1 given that the price of x2 is normalized to 1.
(b) How much of x1 do we trade among each other?
(c) Now suppose that a per-unit tax t (payable in terms of x2) is introduced. Let p be the price buyers will end up paying, and let (p −t) be the price sellers receive. Derive the equilibrium levels of p and (p −t) as a function of t.
(d) Consider the case of t = 0.25. Illustrate that the post-tax allocation is inefficient.
(e) Suppose the government distributes the x2 revenue back to us—giving me half of it and you the other half. Does your previous answer change?
(f) Construct a table relating t to tax revenues, buyer price p, seller price (p−t), my consumption level of x1 and your consumption level of x1 in 0.25 increments from 0 to 1.25. (This is most easily done by putting the relevant equations into an excel spreadsheet and changing t)
(g)Would anything in the table change if the government takes the x2 revenue it collects and distributes it between us in some way?