### Quantity Theory of Money and Rational Expectations: Systems of Equations and Mathematical Expectation

This post will develop a simple rational expectations model for the level of output and price in the economy.  Although, the mathematics might not seem simple, in a later post expectations about multi-period price levels will complicate the construct. In this post the Quantity Theory of Money along with the Friedman supply function with rational expectations will be exploited to gain an understanding of price and output fluctuations in the economy.

## DESCRIBING THE PIECES OF THE MODEL

QUANTITY THEORY OF MONEY

The Quantity Theory of Money describes the relationship between the money supply, velocity of money, price level and output in the economy.  The supply of money multiplied by the velocity is equal to the nominal output in the economy.  If we assume that the velocity is equal to one and take the natural logarithm of this expression we ge the following derivation

The logarithmic expression of this equation captures the relationship between the growth of money and prices and income in the economy.

FRIEDMAN SUPPLY CURVE WITH RATIONAL EXPECTATIONS

The slope of the Friedman short-run supply curve can be expressed as a ratio of the vertical change in prices divided by the horizontal increase in output.  Using this slope a formula for the price level can be derived

Changes in the price level are equivalent to changes in the expectations in prices and a multiple of the amount of output being produced which is larger than the long-run equilibrium output.

MONEY SUPPLY WITH RANDOM ERROR TERM

The growth of the money supply is assumed to fluctuate around its long-term average of m bar. The fluctuations are taken to be in the form of a white-noise error term with a mean of zero and a constant variance.

## SOLVING THE RATIONAL EXPECTATIONS MODEL

The goal is to solve for the price and output level of the economy by simplifying and combining the Quantity Theory of Money, Friedman Supply Function with Rational Expectations and the Money supply functions.  In order to accomplish this, the expectations of the growth in the price level and the money supply must be taken into account.  The first task is going to be taking the expectation of the price level

The expectations operator is linear hence the distribution of it to the left hand side variables.  The expectations of price expectations are equal to the original price expectations.  Since, the full-employment level of output growth is a constant the expectation of a constant is the constant itself. The next step is to substitute the money supply function and the price function from the Friedman Supply curve into the Quantity Theory of Money equation and take the expectation of the Quantity Theory of Money in this form

This states that the expected price level is equal to the average growth rate in the money supply minus the equilibrium growth rate of the output function in the economy. The expected price level can now be placed back into the Friedman Supply Function with Rational Expectations to help derive the time path of the growth in the price level. Another substitution would include the time path of the output growth as a function of the average growth in the money supply minus the price level plus a white-noise error term.

Combining like terms and doing some simple algebra manipulations yields the final price equation

The general price level, in this case, is increases as the average growth rate of the money supply exceeds the growth rate of the equilibrium level of economic output plus an error term.  The error term is multiplied by the coefficient relating the divergence from equilibrium output and the price level.

Using the Quantity Theory of Money and the Money Supply we can derive the path of output of the economy. Make that quick substitution and combining like terms yields the output function