Problem Set 4
Due: Friday, December 6, 5:00 p.m. Eastern Time
Submission Instructions: Upload a Single PDF File to Canvas, under “Assignments”
Applied Econ 440.602: Macroeconomic Theory
Fall, 2024
1. Trade Barriers and Exchange Rates. This question is meant to illustrate how tariffs and trade costs can influence exchange rates. Suppose there is a single good that’s traded in both the United States and Europe. Let Pt denote the price, in dollars, of the good in the United States; let Pt* denote the price, in euros, of the good in Europe. Let Et denote the exchange rate, in euros per dollar. American consumers can purchase goods from either a foreign supplier or a domestic supplier. There are two wrinkles, one related to policy, and one related to technology.
The U.S. government levies a tax on goods imported from Europe at rate τ . More precisely, if you paid your European supplier X euros in Rotterdam, then the U.S. government taxes you τ × X euros, or the dollar equivalent of τ × X euros. This arrangement is like U.S. Customs looking at your receipt and charging the tax based on the amount you paid, taking into account the value of the currency that you used to make the purchase. Assume that American consumers do not have to pay taxes on goods purchased from American suppliers.
In class, we assumed that trade was totally frictionless. Now, we’ll consider the cost of moving goods across borders. Assume that an American importer loses a fraction ι of goods in transit. That is, if you load X units of goodsonto your boat in Rotterdam, then ι × X units falloff the boat, leaving you with (1 − ι) × X units of goods by the time the boat arrives in New York. (That’sastylized story about transit costs, but it’s equivalent to assuming that the cost of shipping is proportional to the quantity of goods being shipped.)
(a) How much does it cost, in euros, for an American to buy one unit of goods from a European supplier? This is the total cost, including taxes and transport costs, so your answer should include τ and ι .
(b) How much does it cost,in dollars, for an American to acquire the euros necessary to purchase one unit of goods from a European supplier?
(c) What does the principle of no arbitrage imply about the relationship between the cost of goods from European suppliers and the cost of goods from American suppliers? Provide a brief (one sentence) answer in words, and provide an equation.
(d) Does the dollar appreciate or depreciate if the U.S. government increases the tariff τ? Provide a brief (one or two sentence) explanation.
(e) Does the dollar appreciate or depreciate if an improvement in shipping technology causes ι to fall? Provide a brief (one or two sentence) explanation.
2. Monetary Policy with Pegged Exchange Rates. When we studied closed economies, we combined an IS curve with an LM curve that treated the domestic interest as an autonomous choice of the central bank. This question will have you analyze how monetary policy changes with a pegged exchange rate, and the relationship between macroeconomic outcomes across countries. In the following, the “domestic country” will be the one with a fixed exchange rate; the “foreign country” will be the one that issues the anchor currency.
In the domestic country, assume that the IS curve takes the form.
The notation above is the same as what we used in our discussion of the IS curve in the closed economy. Implicitly, equation (1) assumes that the domestic country’s net exports are zero. That’s arather strong assumption, but it will highlight the role of financial linkages between countries, even when there isn’t necessarily trade of goods and services between countries. Assume that interest rate parity applies:
where Et is the exchange rate (in units of foreign currency per unit of domestic currency), Et(e)+1 is the expected future exchange rate, and i* is the foreign nominal interest rate.
Assume that, in the domestic country, the price of goods is perfectly sticky, so the real interest rate equals the nominal interest rate: r = i. Assume that, in the foreign country, the expected inflation rate is equal to the current inflation rate, so Fisher equation becomes i* = r* + π * , where r* is the foreign real rate, and π *e is the expected inflation rate in the foreign country.
(a) Assume that the foreign central bank sets interest rates according to the following rule:
where Y* is foreign output, and λπ and λy are parameters that describe how the foreign central bank responds to macroeconomic conditions, and i0(*) is an intercept term. Do you think it’s more reasonable to assume that λπ is positive or negative? What about λy? Provide a brief economic explanation of your reasoning.
(b) Derive an expression for r in terms of π * , Y* , and %∆Et(e)+1 .
(c) Combine your answer to part (b) with the IS curve to obtain an equation tells us what domestic output will be, taking exchange rates and the state of the foreign economy as given.
(d) Suppose that the domestic country maintains an exchange rate peg, and this entails setting Et equal to E , which is chosen by the domestic central bank. If the peg is stable, then currency traders expect the the exchange rate to be equal to the par rate tomorrow, i.e., Et(e)+1 = Et = E.
i. If inflation goes up in the foreign country, holding all else fixed, then what’s going to happen to domestic output? Explain how you know mathematically, and provide a brief economic explanation in words.
ii. If output goes up in the foreign country, holding all else fixed, then what’s going to happen to domestic output? Explain how you know mathematically, and provide a brief economic explanation in words.
(e) Now, consider what happens when there’sa speculative attack in which currency traders expect the currency to depreciate,i.e., Et(e)+1 < Et = E. What will this do to domestic output, holding all else fixed? Explain how you know mathematically, and provide a brief economic explanation in words.
(f) Suppose that, at date t, central bankers in the home country reach a consensus to devalue their currency, but there’sa debate within the central bank about how to do so. All the central bankers in the home country agree that the par rate should be dropped from E to E′ , and everyone expects the foreign central bank to keep the foreign interest rate at i* for the foreseeable future. Proposal A is to announce at date t that the new par rate will be going into effect immediately: Et = E′ < E = Et−1 . Proposal B is to give some advance notice by announcing at date t that the par rate will be reduced one period in the future: Et+1 = E′ < E = Et. Based on the model, do you think that one proposal is better than the other, and if so, which one? Explain your reasoning.