Chapter 5: Price Levels and the Exchange Rate in the Long Run

Learning Objectives and Overview

Definition of the Long Run: In this context, the long run refers to a period of time sufficient for the prices of all goods and services to adjust to market conditions. This adjustment ensures that both the goods/services markets and the money market reach a state of equilibrium.
Role of Prices: Because prices are flexible in the long run, they influence both interest rates and exchange rates.

The Law of One Price (LOOP)

General Principle: The Law of One Price states that in competitive markets, the same good must sell for the exact same price when expressed in a common currency, provided that transportation costs and trade barriers (such as tariffs) are negligible.
The Mechanism of Arbitrage: If a pizza costs $\$20$ at one restaurant and $\$40$ at an identical restaurant across the street, consumers will flock to the $\$20$ location. Entrepreneurs will see an incentive to buy at the cheap location and sell at the expensive one to earn an "easy profit." * Increased demand and decreased supply at the $\$20$ location will drive the price up. * Decreased demand and increased supply at the $\$40$ location will drive the price down. * Prices adjust until they converge to a single value across the market.
International Application: If there is a pizza restaurant in Seattle and one in Vancouver, the LOOP dictates: * $P_{i}^{US} = (E_{\text{US}\$/ ext{CAN}\$}) \times (P_{i}^{CAN})$ * Where $P_{i}^{US}$ is the price of pizza in Seattle, $P_{i}^{CAN}$ is the price in Vancouver, and $E_{\text{US}\$/ ext{CAN}\$}$ is the exchange rate.

Purchasing Power Parity (PPP)

Relationship to LOOP: Purchasing Power Parity is the application of the Law of One Price across countries for all goods and services, or for a representative group ("basket") of goods and services.
General Formula: * $P_{US} = (E_{\text{US}\$/ ext{CAN}\$}) \times (P_{CAN})$ * Where $P_{US}$ is the level of average prices in the United States and $P_{CAN}$ is the level of average prices in Canada.
Price Level Determination: PPP implies that the exchange rate is determined by the ratio of average price levels: * $E_{\text{US}\$/ ext{CAN}\$} = \frac{P_{US}}{P_{CAN}}$
Numerical Example: If the U.S. price level is $\text{US}\,\$200$ per basket and the Canadian price level is $\text{C}\,\$400$ per basket, the exchange rate should be $\frac{\$400}{\$200} = \frac{\$2}{\$1}$ . This implies 2 Canadian dollars have the same purchasing power as 1 U.S. dollar because prices in Canada are twice as high.

Forms of Purchasing Power Parity

Absolute PPP: This version states that exchange rates equal the level of relative average prices across countries. * $E_{\$/€} = \frac{P_{US}}{P_{E}}$
Relative PPP: This version states that the percentage change in the exchange rate between two periods equals the difference in inflation rates between those two countries. * $\frac{E_{\$/€, t} - E_{\$/€, t-1}}{E_{\$/€, t-1}} = \pi_{US, t} - \pi_{E, t}$ * Where $\pi_t$ represents the inflation rate from period $t-1$ to $t$ , calculated as $\frac{P_t - P_{t-1}}{P_{t-1}}$ .

The Monetary Approach to Exchange Rates

Definition: A model that uses monetary factors (money supply and money demand) to predict long-run exchange rate adjustments, based specifically on the absolute version of PPP.
Fundamental Prediction: The price levels across countries adjust so that the quantity of real monetary assets supplied equals the quantity of real monetary assets demanded. * For the U.S.: $\frac{M_{US}^{s}}{P_{US}} = L(R_{\$}, Y_{US})$ * For Europe: $\frac{M_{E}^{s}}{P_{E}} = L(R_{€}, Y_{E})$
Exchange Rate Determination: In the long run, exchange rates are determined by prices, which in turn are determined by the relative supply and demand of real monetary assets in national money markets.

Predictions of the Monetary Approach

Money Supply: A permanent rise in the domestic money supply causes a proportional increase in the domestic price level. Through PPP, this translates into a proportional depreciation of the domestic currency.
Interest Rates: A rise in domestic interest rates lowers the demand for real monetary assets. This is associated with a rise in domestic prices, which causes a proportional depreciation of the domestic currency via PPP.
Output Level: A rise in domestic production and income (output) raises the domestic demand for real monetary assets. For a fixed money supply, this leads to a decrease in average domestic prices, causing a proportional appreciation of the domestic currency via PPP.

The Fisher Effect

Definition: Named after Irving Fisher, it describes the relationship between nominal interest rates and inflation.
Derivation: Derived from the interest parity condition: * $R_{\$} - R_{€} = \frac{E^e_{\$/€} - E_{\$/€}}{E_{\$/€}}$
Relationship with PPP: If markets expect relative PPP to hold, then the expected change in exchange rates equals the expected inflation differential: * $\frac{E^e_{\$/€} - E_{\$/€}}{E_{\$/€}} = \pi^e_{US} - \pi^e_{E}$
Conclusion: In the long run, a rise in the domestic inflation rate causes an equal rise in the nominal interest rate on domestic currency deposits (assuming other factors are constant). * $R_{\$} - R_{€} = \pi^e_{US} - \pi^e_{E}$

Flexible-Price Monetary Approach Dynamics

Permanent Increase in Money Supply Growth: * Suppose the U.S. central bank increases the money supply growth rate at time $t_0$ . * The inflation rate increases from $\pi$ to $\pi + \Delta$ . * According to the Fisher Effect, the nominal interest rate R_{\也会从 $R_{\$, 1}$ 增加到 $R_{\$, 2} = R_{\$, 1} + \Delta$ .
Market Adjustments: 1. The rise in nominal interest rates reduces the demand for real monetary assets ( $L(R, Y)$ ). 2. For the money market to maintain equilibrium ( $\frac{M}{P} = L$ ), the price level ( $P$ ) must "jump" upward immediately. 3. To maintain PPP ( $E = \frac{P}{P^*}$ ), the exchange rate ( $E$ ) must also jump (the dollar depreciates immediately). 4. Following this initial jump, the money supply and prices grow at the new, higher rate ( $\pi + \Delta$ ), and the currency continues to depreciate at that same rate.
Comparison to Non-PPP Models: * In long-run models without PPP, price levels adjust slowly, and exchange rates may "overshoot." * In the flexible-price monetary approach, prices adjust instantaneously with inflation expectations, meaning there is no overshooting of the exchange rate.

Graphical Representation of Flexible Price Growth (Four-Quadrant Model)

Lower-Right Quadrant: Shows U.S. money market equilibrium. A rise in the growth rate of money increases the interest rate from $R_{\$, 1}$ to $R_{\$, 2}$ , reducing real money demand.
Lower-Left Quadrant: Graphs the relationship between U.S. real money holdings ( $\frac{M}{P}$ ) and the nominal exchange rate ( $E_{\$/€}$ ). It is a downward-sloping hyperbola based on the equation: * $E_{\$/€} = \frac{P_{US}}{P_{E}} = \frac{M_{US} / (M_{US} / P_{US})}{P_{E}}$ * As real money supply falls from $\frac{M}{P_1}$ to $\frac{M}{P_2}$ , the exchange rate rises from $E_1$ to $E_2$ (depreciation).
Upper-Left Quadrant: A $45$ -degree line used to translate the exchange rate coordinates to the vertical axis of the final quadrant.
Upper-Right Quadrant: The foreign exchange market equilibrium. Higher expected monetary growth shifts the expected euro return schedule outward because it implies faster future dollar depreciation. This leads to a higher nominal dollar interest rate and a depreciated dollar.

Shortcomings of Buying Power Parity (PPP)

Empirical Evidence: There is little empirical support for absolute PPP. Relative PPP performs better but is still a poor predictor of short-to-medium term exchange rates.
Factors preventing LOOP/PPP: 1. Trade Barriers and Nontradables: Transport costs and government restrictions (tariffs/quotas) make trade expensive. Some services (e.g., haircuts) are nontradable and offered only in limited geographic regions. 2. Imperfect Competition: This can lead to "pricing to market," where firms engage in price discrimination, selling the same product for different prices in different markets to maximize profit based on local willingness to pay. 3. Measurement Differences: Average price levels differ because countries measure their representative "baskets" of goods differently.

Price Levels in Rich vs. Poor Countries

Observation: Price levels are positively related to real income per capita. A dollar converted to local currency goes much further in a poor country than a rich one.
Balassa-Samuelson Theory: Assumes labor productivity in tradable goods is much higher in rich countries than poor countries, but productivity in nontradables is similar. Since prices for traded goods are equalized internationally, higher productivity in rich countries leads to higher wages across all sectors. High wages in the nontradable sector (which has lower productivity growth) lead to higher prices for services in rich countries.
Bhagwati-Kravis-Lipsey Theory: Focuses on factor endowments. Rich countries have higher capital-labor ratios, making the marginal productivity of labor (and thus wages) higher. Since nontradables are labor-intensive relative to tradables, they are more expensive in high-wage (rich) countries.

The Real Exchange Rate Approach

Definition: The real exchange rate (q_{$/€}) is the relative value (price or cost) of goods and services across countries.
Formula: * $q_{\$/€} = \frac{E_{\$/€} \times P_{E}}{P_{US}}$
Interpretation: * Real Depreciation of the Dollar: A rise in q_{$/€}. This means the dollar's purchasing power of European products falls relative to its power over U.S. products. U.S. goods become cheaper relative to European goods. * Real Appreciation of the Dollar: A fall in q_{$/€}. The dollar's purchasing power of European products rises. U.S. goods become more expensive/valuable relative to European goods.
General Nominal Exchange Rate Equation: * $E_{\$/€} = q_{\$/€} \times \frac{P_{US}}{P_{E}}$

Factors Influencing the Real Exchange Rate

Relative Demand: An increase in relative demand for U.S. products causes a real appreciation (a fall in q_{$/€}). U.S. exports become more expensive, and imports become cheaper, reducing relative quantity demanded back toward equilibrium.
Relative Supply: An increase in relative supply of U.S. products (due to productivity gains) causes a real depreciation (a rise in q_{$/€}). U.S. goods become less expensive to encourage the market to absorb the increased supply.

Impact of Real Factors on Nominal Exchange Rates

Relative Demand Increase: Leads to real appreciation and a nominal appreciation of the currency.
Relative Supply Increase: The effect is ambiguous. * The real exchange rate effect causes depreciation. * However, increased output increases real money demand ( $L(R, Y)$ ), which decreases the domestic price level ( $P$ ), tending toward nominal appreciation. * The final result depends on which effect dominates: $E_{\$/€} = q_{\$/€} \uparrow \times \frac{P_{US} \downarrow}{P_{E}}$ .

Nominal Interest Rate Differentials (Generalized)

The difference in nominal interest rates is the sum of the expected rate of real depreciation and the difference in expected inflation rates: * $R_{\$} - R_{€} = \frac{q^e_{\$/€} - q_{\$/€}}{q_{\$/€}} + (\pi^e_{US} - \pi^e_{E})$

Real Interest Rates and Real Interest Parity

Real Interest Rate Definition: Inflation-adjusted interest rates representing the quantity of goods savers earn or borrowers forgo. * $r^e = R - \pi^e$
Real Interest Parity: The difference in expected real interest rates between countries equals the expected change in the real exchange rate. * $r^e_{US} - r^e_{E} = \frac{q^e_{\$/€} - q_{\$/€}}{q_{\$/€}}$

Questions & Discussion

Law of One Price for Hamburgers: The slides reference McDonald's (the "Big Mac Index" concept) as a real-world test for the Law of One Price, noting differences in prices and operations (e.g., 24-hour locations) across markets like Japan and the U.S.
Trend in Japan-U.S. Data: Figure 16-3 (Slide 40) shows the Yen/Dollar exchange rate and the Japan-U.S. price level ratio from 1980-2019, illustrating how exchange rates can deviate significantly from price ratios in the medium term but often follow similar long-term trajectories.