Purchasing Power Parity
The theoretical assumption of Purchasing Power Parity starts from the Law of One Price. The Law of One Price in open economy states that, if the market is competitive, no transaction cost and no barriers of trade, then identical products in different countries should be sold at the same prices, adjusted by exchange rate, i.e. under the same currency denomination. Otherwise, there is arbitrage opportunity. In notation,
for pi = price of good i at home country, pi*= price of good i at foreign country, s = exchange rate
For example, the price an ounce of gold quoted at London in GBP should be the same as an ounce of gold quoted at New York in USD times exchange rate of GBP/USD.
Next, we consider a model with two countries. Both of them have the floating exchange rate-regimes and Law of One Price holds for all goods in the two counties. Then, the general price level of home country should be the same as the general price level of foreign country, adjusted by exchange rate. In notation,
for P= general price level at home country, P*= general price level at foreign country
P and P*, the general price level is the weighted average of all prices of goods. So if (1) holds for all goods, (2) will holds. (2) is what we called the absolute Purchasing Power Parity (absolute PPP): the general price level of every country should be the same if adjusted to the same currency. In other words, the exchange rate should be determined by the relative price level of two countries. If you can use $1 of home currency to buy a basket of goods at home country, then the $1 converted to foreign currency should be able to buy the same basket of products in foreign country, i.e. they have the same purchasing power.
The Purchasing Power Parity states that relative price level is a fundamental determinant of exchange rate. An empirical test would like to see whether there is such a relationship in historical data. The PPP hypothesis has be enormously and extensively tested empirically by economists. The extensive tests by economists found very little empirical support to PPP. Exchange rate and the relative price level are unrelated in short run and medium run. In the long run, results found that exchange rate would converge to the theoretical equilibrium value from PPP, but at a very slow rate.
As exchange rate is the relative price of two currencies, it is reasonable to consider the supply and demand of money be an important determinant of exchange rates. Introduction of money supply and money demand, two very fundamental macroeconomic variables, into our models
The monetary approach rests on the quantity theory of money in macroeconomics. Firstly, Money supply (Ms) is a quantity determined by the central bank. In the quantity theory, money is for the purpose of medium of exchange. Money demand of an economy is directly proportional to the general price level and also the quantity of real output. For example, if the general price level is doubled, then the economy would need double amount of money for their transactions. The same idea holds for quantity of real output. Then,
Md = kPy
Portfolio Balance Model
In the monetary model, the global economy is simplified as having goods and money only, and money is the medium of exchange to buy domestic and foreign goods. Exchange rates are determined by the relative demand and supply of money, domestic and foreign.
The portfolio balance model takes a further step from the monetary model that there are investment assets in the global economy for people to hold. People would consider holding money, domestic assets and foreign assets alternatively on their portfolio balance. Then the relative demand and supply of these investment assets would determine the exchange rate.
The portfolio balance model assumes there are three kinds of assets for people to allocate their total wealth: Domestic money (M), domestic bond (B), and foreign bond (FB). Domestic money (M), pays no interest, is a riskless asset. In term of finance, the risk-free rate is zero in this simplified model. Domestic bond and foreign bond are risky assets that payout with, with interest rate rand r* respectively. Then the actual interest rate individual receive from foreign bond is sr*.
The portfolio balance model of exchange rate makes further assumption in line with modern portfolio theory. Domestic bond and foreign bond are not perfect substitutes. Holding domestic and foreign bond together in the portfolio would reduce the unsystematic risk. So people would not simply hold the bond with higher yield only but hold a portfolio of domestic and foreign bonds. Moreover, the individuals, being are risk-averse and so they would hold some portion of riskless asset, the money.
The individuals have a total wealth of W would decide how to allocate them into money, domestic bond and foreign bond respectively based on his risk preference and the returns of different assets, as in modern portfolio theory. He would purchase more of one asset if the return of the asset increases, or if the return of the alternative assets decreases. In summary,
Demand of money = M(r, sr*) is decreasing in r and sr*
Demand of domestic bond = B(r, sr*) is increasing in r and decreasing in sr*.
Demand of foreign bond = FB(r, sr*) is increasing in sr* and decreasing in r.
Total wealth, the supply of various assets, would equal to the demand of various assets, such that
W = M(r, sr*) + B(r, sr*) + BF(r, sr*)