Fisher equation

The Fisher equation describes in economics named after Irving Fisher relationship between nominal interest rates, real interest rate and expected inflation rate. The Fisher equation states that the nominal interest rate corresponds approximately to the sum of the real interest rate and expected inflation:

Where the nominal interest rate in the current period, the real interest rate and the expected inflation rate for the coming period, respectively.

Background

In order to understand the economic background of the Fisher equation, it is helpful to consider the following thought experiment.

An example with perfect foresight

Have an economic operator 100 euros available that he wants to create for a year. The world is free of surprises, that is, the future development of economic variables is known to all players ( perfect foresight ). Irving has different ways to create the 100 euros. One possibility is to lend the money at an interest rate of. If the interest rate as 4 % (), then he receives in one year its 100 euros plus Euro interest rates so that he has a total of EUR.

Another possibility for Irving, it invests 100 EUR in a profit- promising project, for example, the cultivation of wheat. We assume that one unit of wheat now costs 1 euro and that by sowing and cultivation of the field results in a yield increase of 3 %, so that in a year 103 units of wheat can be harvested.

Which of the two alternatives is better? That depends on how the price for a unit will develop wheat. Due to the perfect foresight is now known that a unit of wheat will cost no more 1 euro, but 1.02 euros in a year. So we go from a rate of price change ( inflation ) of 2% () from. It follows that Irving in one year, the 103 units of wheat for 103 units of wheat times 1.02 euros per unit wheat, ie can sell for about 105 euros (just there are 105.06 euros ). It is therefore advantageous to invest the money in the wheat and not to give.

Rational traders recognize this connection and give under the circumstances no money for 4% interest, but rather invest it in growing wheat. Actors who need money will now offer a higher interest rate to find someone to lend them money. A balance is only achieved when both alternatives after a year produce the same yield. As long as one of the two alternatives promises a higher yield than the other, no one will be willing to choose the alternative. This leads to adaptation processes, such as the rise in interest rates just described for investments. Other adjustment processes are also conceivable. As long as the income from wheat is higher than that of an investment, more and more actors will invest in the cultivation of wheat. This increases the wheat supply in the coming period, so that the price of wheat no longer by 2%, but due to the larger range increases by a smaller percentage in the coming period. If the inflation rate just 1 %, the result is again the equilibrium described by the Fisher equation: both alternatives offer an interest rate of 4%. This 4% contact with the wheat yield increase of 3 % ( real interest rate ) plus 1% price increase ( inflation ) together.

The future is uncertain

Of course, today no one knows exactly how much will be the price of wheat in a year. Therefore, an expectation about we must in the current period t be formed, how high the price of wheat will be in a year, and what that means for the inflation rate. This expected rate of inflation can then be used to compare the two alternatives described above, and the result is shown above the Fisher equation as a characterization of the economic balance between the nominal interest rate, real interest rate and expected inflation rate.

The ex-post real interest rate

The real interest rate and the inflation expectations of economic agents are in contrast to the nominal interest rate no observable quantities. You still want the height of the real interest rate in a given period t determine, one can approximately consider the so-called ex-post real interest rate. This results from the Fisher equation, if the expected inflation rate is replaced by the actual inflation rate, which is, however, only ex post, ie later from period t 1, knows:

It is assumed that there are no systematic expectation errors on the inflation rate. Alternatively survey values ​​for the expected inflation rate can be used.

Precise formulation of the Fisher equation

The exact version of the Fisher equation can be derived, in which the income of an investment and the expected return of a real investment are equated:

Here refers to the price of real goods ( wheat in the example above ) in the current period and the corresponding price in the following period. The superscript indicates that this is an expectation. Together with the definition of the rate of inflation,

Follows the exact Fisher equation

The approximate version is obtained by multiplied out the right side

And the multiplication neglected:

Both the real interest rate and the inflation rate are measured here as a decimal fraction, ie an expected inflation rate of 2 % per cent corresponds, so that the cross product for realistic magnitudes of the real interest rate is negligibly small.

The approximate version is mainly used for illustrative and theoretical representations. The exact version should always be used for practical calculations.