Homework's Mathematical Methods
leon wengExchange Rates I: The Monetary
Approach in the Long Run
3
Exchange Rates and Prices in the Long Run
Money, Prices, and Exchange Rates in the Long Run
The Monetary Approach
Money, Interest Rates, and Prices in the Long Run
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2
The UIP approximation equation says that the home interest rate equals the foreign interest rate plus the expected rate of depreciation of the home currency.
Uncovered Interest Parity:
Recall
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3
Absolute and Relative PPP:
(3-1)
Purchasing power parity implies that the exchange rate at which two currencies trade equals the relative price levels of the two countries.
Recall
3
4
The Measurement of Money
FIGURE 3-4
The Measurement of Money This figure shows the major kinds of monetary aggregates (currency, M0, M1, and M2) for the United States from 2004 to 2012. Normally, bank reserves are very close to zero, so M0 and currency are virtually identical, but reserves spiked up during the financial crisis in 2008, as private banks sold securities to the Fed and stored up the cash proceeds in their Fed reserve accounts.
The Supply of Money: In practice, a country’s central bank controls the money supply. We make the simplifying assumption that the central bank’s indirectly, but accurately, control the level of M1.
Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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5
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Money/Assets and Their Attributes
An investor’s entire portfolio of assets may include stocks, bonds, real estate, art, bank deposits in various currencies, and so on. All assets have three key attributes that influence demand: return, risk, and liquidity.
An asset’s rate of return is the total net increase in wealth resulting from holding the asset for a specified period of time, typically one year.
The risk of an asset refers to the volatility of its rate of return.
The liquidity of an asset refers to the ease and speed with which it can be liquidated, or sold.
We refer to the forecast of the rate of return as the expected rate of return.
The Demand for Money: What influences the demand?
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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6
Interest rates/expected rates of return: monetary assets pay little or no interest, so the interest rate on non-monetary assets like bonds, loans, and deposits is the opportunity cost of holding monetary assets.
A higher interest rate means a higher opportunity cost of holding monetary assets lower demand of money.
Prices: the prices of goods and services bought in transactions will influence the willingness to hold money to conduct those transactions.
A higher level of average prices means a greater need for liquidity to buy the same amount of goods and services higher demand of money.
Income: greater income implies more goods and services can be bought, so that more money is needed to conduct transactions.
A higher real national income (GNP) means more goods and services are being produced and bought in transactions, increasing the need for liquidity higher demand of money.
What Influences Aggregate Demand of Money?
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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7
We assume money demand is motivated by the need to conduct transactions in proportion to an individual’s income and we infer that the aggregate money demand will behave similarly (known as the quantity theory of money).
All else equal, a rise in national dollar income (nominal income) will cause a proportional increase in transactions and in aggregate money demand.
The Demand for Money: A Simple Model
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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8
Dividing the previous equation by P, the price level, we can derive the demand for real money balances:
The Demand for Money: A Simple Model
Real money balances are simply a measure of the purchasing power of the stock of money in terms of goods and services. The demand for real money balances is strictly proportional to real income.
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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The condition for equilibrium in the money market is simple to state: the demand for money Md must equal the supply of money M, which we assume to be under the control of the central bank.
Imposing this condition on the last two equations, we find that nominal money supply equals nominal money demand:
Equilibrium in the Money Market
and, equivalently, that real money supply equals real money demand:
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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10
Expressions for the price levels in the U.S. and Europe are:
A Simple Monetary Model of Prices
These two equations are examples of the fundamental equation of the monetary model of the price level.
In the long run, we assume prices are flexible and will adjust to put the money market in equilibrium.
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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Plugging the expression for the price level in the monetary model to Equation (3-1), we can use absolute PPP to solve for the exchange rate:
A Simple Monetary Model of the Exchange Rate
This is the fundamental equation of the monetary approach to exchange rates.
(3-3)
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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12
The implications of the fundamental equation of the monetary approach to exchange rates are intuitive.
Suppose the U.S. money supply increases, all else equal. The right-hand side increases (the U.S. nominal money supply increases relative to Europe), causing the exchange rate to increase (the U.S. dollar depreciates against the euro).
Money Growth, Inflation, and Depreciation
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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Now suppose the U.S. real income level increases, all else equal. Then the right-hand side decreases (the U.S. real money demand increases relative to Europe), causing the exchange rate to decrease (the U.S. dollar appreciates against the euro).
Money Growth, Inflation, and Depreciation
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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The U.S. money supply is MUS, and its growth rate is μUS:
Money Growth, Inflation, and Depreciation
The growth rate of real income in the U.S. is gUS:
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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Therefore, the growth rate of PUS = MUS/LUSYUS equals the money supply growth rate μUS minus the real income growth rate gUS. The growth rate of PUS is the inflation rate πUS. Thus, we know that:
Money Growth, Inflation, and Depreciation
The rate of change of the European price level is calculated similarly:
−
(3-4)
(3-5)
When money growth is higher than income growth, we have “more money chasing fewer goods” and this leads to inflation.
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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Combining (3-4) and (3-5), we can now solve for the inflation differential in terms of monetary fundamentals and compute the rate of depreciation of the exchange rate:
Money Growth, Inflation, and Depreciation
(3-6)
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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The intuition behind Equation (3-6) is as follows:
If the United States runs a looser monetary policy in the long run measured by a faster money growth rate, the dollar will depreciate more rapidly, all else equal.
If the U.S. economy grows faster in the long run, the dollar will appreciate more rapidly, all else equal.
Money Growth, Inflation, and Depreciation
2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model
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4 Money, Interest Rates, and Prices in the Long Run: A General Model
The trouble with the theory we studied earlier is that it assumes that the demand for money is stable, and this is implausible.
We will now explore a more general model that allows for money demand to vary with the nominal interest rate.
We consider the links between inflation and the nominal interest rate in an open economy.
We then return to the question of how best to understand what determines exchange rates in the long run.
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19
A rise in national dollar income (nominal income) will cause a proportional increase in transactions and, hence, in aggregate money demand (as is true in the simple quantity theory).
A rise in the nominal interest rate will cause the aggregate demand for money to fall.
Dividing by P, we derive the demand for real money balances:
The Demand for Money: The General Model
4 Money, Interest Rates, and Prices in the Long Run: A General Model
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The Demand for Money: The General Model
FIGURE 3-11
Panel (a) shows the real money demand function for the United States. The downward slope implies that the quantity of real money demand rises as the nominal interest rate i$ falls. Panel (b) shows that an increase in real income from Y1US to Y2US causes real money demand to rise at all levels of the nominal interest rate i$.
The Standard Model of Real Money Demand
4 Money, Interest Rates, and Prices in the Long Run: A General Model
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With two relationships, PPP and UIP, we can derive a striking result concerning interest rates that has profound implications for our study of open economy macroeconomics. We use:
Long-Run Equilibrium in the Money Market
(3-7)
Inflation and Interest Rates in the Long Run
and
4 Money, Interest Rates, and Prices in the Long Run: A General Model
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The nominal interest differential equals the expected inflation differential:
All else equal, a rise in the expected inflation rate in a country will lead to an equal rise in its nominal interest rate.
This result is known as the Fisher effect.
The Fisher effect predicts that the change in the opportunity cost of money is equal not just to the change in the nominal interest rate but also to the change in the inflation rate.
The Fisher Effect
(3-8)
4 Money, Interest Rates, and Prices in the Long Run: A General Model
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Rearranging the last equation, we find
Subtracting the inflation rate (π) from the nominal interest rate (i), results in a real interest rate (r), the inflation-adjusted return on an interest-bearing asset.
This result states the following: If PPP and UIP hold, then expected real interest rates are equalized across countries. This powerful condition is called real interest parity.
Real interest parity implies the following: Arbitrage in goods and financial markets alone is sufficient, in the long run, to cause the equalization of real interest rates across countries.
Real Interest Parity
(3-9)
4 Money, Interest Rates, and Prices in the Long Run: A General Model
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In the long run, all countries will share a common expected real interest rate, the long-run expected world real interest rate denoted r*, so
We treat r* as an exogenous variable, something outside the control of a policy maker in any particular country.
Under these conditions, the Fisher effect is even clearer, because, by definition,
Real Interest Parity
4 Money, Interest Rates, and Prices in the Long Run: A General Model
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Evidence on the Fisher Effect
FIGURE 3-12
Inflation Rates and Nominal Interest Rates, 1995–2005 This scatterplot shows the relationship between the average annual nominal interest rate differential and the annual inflation differential relative to the United States over a ten-year period for a sample of 62 countries.
The correlation between the two variables is strong and bears a close resemblance to the theoretical prediction of the Fisher effect that all data points would appear on the 45-degree line.
APPLICATION
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This model differs from the simple model (the quantity theory) by allowing L to vary as a function of the nominal interest rate i.
When nominal interest rates change the general model has different implications from the simple model.
We now examine the forecasting problem for an increase in the U.S. rate of money growth. We learn at time T that the United States is raising the rate of money supply growth from μ to a higher rate μ + Δμ.
The Fundamental Equation Under the General Model
(3-10)
4 Money, Interest Rates, and Prices in the Long Run: A General Model
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Exchange Rate Forecasts Using the General Model
FIGURE 3-14 (1 of 4)
Before time T, money, prices, and the exchange rate all grow at rate μ. Foreign prices are constant. In panel (a), we suppose at time T there is an increase Δμ in the rate of growth of home money supply M.
4 Money, Interest Rates, and Prices in the Long Run: A General Model
An Increase in the Growth Rate of the Money Supply in the Standard Model
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Exchange Rate Forecasts Using the General Model
FIGURE 3-14 (2 of 4)
This causes an increase Δμ in the rate of inflation; the Fisher effect means that there will be a Δμ increase in the nominal interest rate; as a result, as shown in panel (b), real money demand falls with a discrete jump at T.
4 Money, Interest Rates, and Prices in the Long Run: A General Model
An Increase in the Growth Rate of the Money Supply in the Standard Model (continued)
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Exchange Rate Forecasts Using the General Model
FIGURE 3-14 (3 of 4)
If real money balances are to fall when the nominal money supply expands continuously, then the domestic price level must make a discrete jump up at time T, as shown in panel (c).
4 Money, Interest Rates, and Prices in the Long Run: A General Model
An Increase in the Growth Rate of the Money Supply in the Standard Model (continued)
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Exchange Rate Forecasts Using the General Model
FIGURE 3-14 (4 of 4)
Subsequently, prices grow at the new higher rate of inflation; and given the stable foreign price level, PPP implies that the exchange rate follows a similar path to the domestic price level, as shown in panel (d).
4 Money, Interest Rates, and Prices in the Long Run: A General Model
An Increase in the Growth Rate of the Money Supply in the Standard Model (continued)
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