Question

Suppose the initial exchange rate is \(\$ 4 / \mathfrak{E}\). After 10 years, the US price level has risen from 100 to 300 and the UK price level has risen from 100 to 200 . What nominal exchange rate would preserve purchasing power parity?

Short answer

The new nominal exchange rate is \( \$ 6 / \mathfrak{E} \).

Step-by-step solution

Identify Purchasing Power Parity (PPP) Formula

Purchasing power parity (PPP) suggests that exchange rates should adjust so that identical goods cost the same in two different countries. The formula for PPP adjustment is \( \frac{P_{US}}{P_{UK}} \), where \( P_{US} \) and \( P_{UK} \) are the price levels in the US and UK, respectively, after the change.

Calculate New US and UK Price Levels

The US price level has increased from 100 to 300, and the UK price level has increased from 100 to 200 after 10 years. So, \( P_{US} = 300 \) and \( P_{UK} = 200 \).

Apply Price Levels to PPP Formula

Using the PPP formula \( \frac{P_{US}}{P_{UK}} \), substitute the values to find: \[ \frac{300}{200} = 1.5 \]. This ratio represents the change needed in the exchange rate to maintain parity.

Calculate the New Exchange Rate

Starting with the initial exchange rate of \( \\( 4 / \mathfrak{E} \), adjust by the PPP ratio. The new exchange rate is calculated as:\[ 4 \times 1.5 = 6 \]. Thus, the new nominal exchange rate is \( \\) 6 / \mathfrak{E} \).

Key concepts

Understanding the Nominal Exchange Rate

The nominal exchange rate is the rate at which one currency can be exchanged for another. It is the price of one currency in terms of another currency. For example, if it costs you \( 4 \) US dollars to buy 1 unit of a foreign currency (let's call it \( \mathfrak{E} \)), then the nominal exchange rate is \( 4 / \mathfrak{E} \). This rate fluctuates based on supply and demand for various currencies in the foreign exchange markets.

It's important to note that the nominal exchange rate does not consider the differences in price levels or cost of living between the two countries. It's simply a market rate determined by factors like trade flows, interest rates, and economic stability. In international finance, this rate helps determine how much of a foreign good you could buy with your domestic currency.

However, to make real-world purchasing comparisons, especially over time, we look at the adjusted exchange rate, considering these price level changes.

The Role of Price Levels

Price levels indicate the average prices of goods and services in an economy at a given time. They can be seen as a measure of inflation within a country. In our problem, we have the US price level increasing from 100 to 300, while the UK price level rises from 100 to 200 over 10 years. This means each basket of goods that once cost 100 units in both countries now costs 300 units in the US and 200 units in the UK.

Price levels matter because they influence the purchasing power of a currency within its home country and abroad. As the price level increases, the currency buys fewer goods and services domestically, indicating inflation. Hence, when price levels change significantly between countries, it affects relative purchasing power and requires an adjustment to the nominal exchange rate if purchasing power parity is to be maintained.

In our scenario, the US experienced more inflation than the UK, leading to a necessary adjustment in the exchange rate.

Adjusting the Exchange Rate to Maintain PPP

Exchange rate adjustment is crucial to ensuring two countries maintain purchasing power parity (PPP). PPP suggests that, in the absence of transportation and transaction costs, identical goods should have the same price in any two countries when exchange rates are taken into account. To achieve this, exchange rates must adjust if price levels change in either country.

Using our exercise, we calculated the PPP adjustment factor as the ratio of the US to UK price levels: \( \frac{300}{200} = 1.5 \). This tells us that the US currency must depreciate, or alternatively, the foreign currency \( \mathfrak{E} \) must appreciate, by 1.5 times to maintain parity.

We then apply this ratio to the initial nominal exchange rate of \( 4 / \mathfrak{E} \), multiplying it by the PPP adjustment factor to get the new exchange rate: \( 4 \times 1.5 = 6 \). Thus, the new nominal exchange rate is \( 6 / \mathfrak{E} \), allowing identical goods to be priced equally in both countries when considering their respective currencies.