Policy Application: Distortionary Taxes in General Equilibrium: Consider, as in exercise 16.10, 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 identical homothetic tastes.
(a) Draw the Edgeworth Box for this economy and indicate the endowment allocation E.
(b) Normalize the price of good x2 to 1. Illustrate the equilibrium price p∗ for x1 and the equilibrium allocation of goods in the absence of any taxes. Who buys and who sells x1?
(c) Suppose the government introduces a tax t levied on all transactions of x1 (and paid in terms of x2). For instance, if one unit of x1 is sold from me to you at price p, I will only get to keep (p −t ). Explain how this creates a kink in our budget constraints.
(d) Suppose a post-tax equilibrium exists and that price increases for buyers and falls for sellers. In such an equilibrium, I will still be selling some quantity of x1 to you. (Can you explain why?) How do the relevant portions of the budget constraints you and I face look in this new equilibrium, and where will we optimize?
(e) When we discussed price changes with homothetic tastes in our development of consumer theory, we noted that there are often competing income (or wealth) and substitution effects. Are there such competing effects here relative to our consumption of x1? If so, can we be sure that the quantity we trade in equilibrium will be less when t is introduced?
(f) You should see that, in the new equilibrium, a portion of x2 remains not allocated to anyone. This is the amount that is paid in taxes to the government. Draw a new Edgeworth box that is adjusted on the x2 axes to reflect the fact that some portion of x2 is no longer allocated between the two of us. Then locate the equilibrium allocation point that you derived in your previous graph. Why is this point not efficient?
(g) True or False: The deadweight loss from the distortionary tax on trades in x1 results from the fact that our marginal rates of substitution are no longer equal to one another after the tax is imposed and not because the government raised revenues and thus lowered the amounts of x2 consumed by us.
(h) True or False: While the post-tax equilibrium is not efficient, it does lie in the region of mutually beneficial trades.
(i) How would taxes that redistribute endowments (as envisioned by the Second Welfare Theorem) be different than the price distorting tax analyzed in this problem?
B: Suppose our tastes can be represented by the utility function u(x1, x2) = x1x2. Let our endowments be specified as at the beginning of the problem.
(a) Derive our demand functions for x1 and x2 (as functions of p —the price of x1 when the price of x2 is normalized to 1).
(b) Derive the equilibrium price p∗ and the equilibrium allocation of goods.
(c) Now suppose the government introduces a tax t as specified in A (c). Given that I am the one that sells and you are the one that buys x1, how can you now re-write our demand functions to account for t?
(d) Derive the new equilibrium pre- and post-tax prices in terms of t.
(e) How much of each good do you and I consume if t = 1?
(f) How much revenue does the government raise if t = 1?
(g) Show that the equilibrium allocation under the tax is inefficient.