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The difficulty of inflation theory is to explain the fluctuations of the price level in short-term. First, we use the relationship between price and money in the traditional quantity equation to derive the inflation equation th at can explain changes of the price level in the long-term and short-term. Then, by analyzing the phase diagram of the single variable and the complex variable in the inflation equation, we find that the fundamental reason of the periodic change of the short-term price is the periodic change of the real interest rate. Because fluctuations of the inflation rate and interest rate are the same in phase, so there is no difference in the business cycle. Finally, this paper analyzes the rise and fall of core price under the influence of money and real interest rate respectively. It lays the foundation for further discussion of the relationship between inflation and unemployment in Phillips curve.

Generally, problems of inflation are regarded as the relation between price and currency, and the price and real output. We need to ask the question that whether currency only has impacts on price or on both price and real output? Friedman concluded that: “Inflation is always and everywhere a monetary phenomenon” [

Categorized by the transaction motive, speculative motive and precautionary motive in Keynesian theory, the demand for money has different models: Diamond built up the iterative model based on the storage functionality of money [

However, statistical data since 1970s have shown that there is no stable functional relation between currency and output as well as currency and interest rate [

Since there is no specific currency variable in Cobb-Douglas function, we cannot deduce the relation among the quantity of money, output and the price lever based only on Cobb-Douglas function. The fundemental function of currency is to increase the exchange efficiency between different kinds of wealth, or in other words, to decrease the transaction cost. Assume the nominal output Y as the exchanging object of currency, then we can get the easiest exchange equation of money:

a M = Y (1)

In which, a refers to the exchange coefficient (varies in terms of different currencies, such as U.S. dollars or Japanese yen), M refers to the quantity of currency or the currency stock, Y refers to the nominal output. To express Equation (1) by growth rate, and a is a constant, then:

d M M = d Y Y , namely M ˙ = Y ˙ (2)

Since there are statistical data on both sides of the equation (take M_{2} as M in the Equation), we can verify the rationality of Equation (2) in reality. As

We have calculated that using M_{2} as the M is a better to use other monetary indices (such as M_{0} or M_{1}) in Equation (2). Therefore, according to statistical data in the United States, we can regard M_{2} as M in our theoretical analysis.

The nominal output Y is the multiplier of price index P and the real output Y r , namely Y = P Y r , then Equation (2) can be expressed as:

d M M = d Y r Y r + d P P , namely M ˙ = Y ˙ r + P ˙ (3)

In the Equation (3), M ˙ can be observed directly from the statistical data, but only one of Y ˙ r and P ˙ is from the statistical data, and the other can be calculated indirectly by the definition Y = P Y r . Friedman and Schwartz states in “A Monetary History of the United States, 1867-1960” that there is a fact but not strict relation between the growth rate of currency and price [

In order to explain the difference of P ˙ and M ˙ in the fluctuation, we define the residual of M ˙ and P ˙ as V ˙ , namely V ˙ = Y ˙ − M ˙ , and observe the direction of fluctuation of V ˙ and M ˙ . As

In fact, V ˙ = Y ˙ − M ˙ equals to d ( ln V ) = d ( ln Y ) − d ( ln M ) = d [ ln ( Y / M ) ] . By neglecting initial conditions, we can rewrite it as V = Y / M . Since Y = P Y r , then:

M V = P Y r , or M ˙ + V ˙ = P ˙ + Y ˙ r (4)

The above equation is the famous Fisher equation with different symbols from the original equation [

From the above derivation process, we can see that Equation (4) is not sacred by people’s imagination, but is an identity defined by the hypothesis V ˙ = Y ˙ − M ˙ .

It is easy to fall into the trap of subjective judgment or cyclic interpretation, as the various explanations based on the identity Y = C + I g + ( X − M ) .

In Equation (4), M and Y r are exogenous variables, in which M is generally determined by the monetary authority, and Y r is theoretically determined by Cobb-Douglas function. We cannot rewrite Equation (4) into Y r = M V / P and regard M and P as reasons for the change of Y r . Otherwise, when V is a constant, the real output can be increased as long as the money supply is increased, as the IS-LM model does. Unfortunately, in traditional macroeconomics, there are many such subjective theories.

Under the thought of the exchanging proportion of currency and real wealth is varying along with time, then Equation (4) is a quantity equation of currency and real wealth at a certain point of time. At time point t, the monetary income collected at t would suffer lose when it is exchanged to real wealth at time point ( t + 1 ) due to the increase of price. In that case, the most rational chose in a inflationary environment should be: in consumption area, to exchange monetary income into real wealth at each point of time; in investment area, to switch investment objects on those object whose nominal revenue could reflects the latest inflation rate. The assumption above indicates that the change of price would affect the velocity of currency flow. We assume the velocity of currency flow is the function of inflation rate, namely V = f ( P ˙ ) . Since:

d V = d f ( P ˙ ) = d f ( P ˙ ) d P ˙ ⋅ d P ˙ P ˙ ⋅ P ˙ , simplifiedas d f ( P ˙ ) = f ′ ( P ˙ ) P ¨ P ˙

then

V ˙ = d V V = d f ( P ˙ ) f ( P ˙ ) = P ˙ f ′ ( P ˙ ) f ( P ˙ ) ⋅ P ¨ (5)

Substitute Equation (5) into (4),

M ˙ = − V ˙ + P ˙ + Y ˙ r = − P ˙ P ¨ f ′ ( P ˙ ) f ( P ˙ ) + P ˙ + Y ˙ r = P ˙ [ 1 − P ¨ f ′ ( P ˙ ) f ( P ˙ ) ] + Y ˙ r ,

therefore

P ˙ = M ˙ − Y ˙ r 1 − f ′ ( P ˙ ) f ( P ˙ ) ⋅ P ¨ (6)

Equation (6) indicates that the change of P ˙ is not only related to ( M ˙ − Y ˙ r ) , but also to f ( P ˙ ) and the growth rate of P ˙ in short-term. In long-term, the average value of f ′ ( P ˙ ) and P ¨ are equal to 0, otherwise there is a contradiction between Equation (6) and P ˙ = M ˙ − Y ˙ r in Equation (3). In that case, we need to verify if the average value of P ¨ or f ′ ( P ˙ ) is equal to 0.

Since we do not know the details of the function V = f ( P ˙ ) , it is impossible to verify the average value of f ′ ( P ˙ ) . However, we could estimate the average value of P ¨ by observing statistical data of P ˙ . As

If the average value of M ˙ and Y ˙ in long-term are the same, then their residual V ˙ should be equal to 0. In short-term, when the direction of fluctuation of V ˙ and P ¨ are the same, then we can assume the changing pattern of V ˙ is determined by P ¨ . Make θ = P ˙ f ′ ( P ˙ ) / f ( P ˙ ) , and according to Equation (5), then V ˙ = θ P ¨ . In

In fact, the value of θ is not important in the long-term, because as long as the long-term average of P ¨ is 0, there is P ˙ = M ˙ − Y ˙ r according to Equation (6). This is the reason that money is always neutral in long-term, but is not clear in the medium and short term.

As

of 0.1 P ¨ is close to that of V ˙ , and the direction of fluctuation of 0.1 P ¨ is consistent with that of V ˙ , but some differences in the range of fluctuation. Substitute V ˙ = θ P ¨ into Equation (4), then:

P ˙ − θ P ¨ = M ˙ − Y ˙ r (7)

Equation (7) indicates that the inflation rate P ˙ is not only affected by exogenous variables M ˙ and Y ˙ r , but also its own change rate P ¨ . More importantly, the variables in Equation (7) have corresponding statistical data in the United States, so that we can verify the results of the inference. Since the relationship between P ˙ and P ¨ can be drawn as a phase diagram, therefore the change of difference between M ˙ and Y ˙ r determines the change of phase diagram P ˙ ~ P ¨ .

In our paper “A Kind of neither Keynesian nor Neoclassical Model (2): The Business Cycle” [

When we analyze the relationship between the variable P and other variables, the other variables should be real variables that are not affected by the price, so that there is no suspicion that the loop is interpreted. P ˙ is not only related to M ˙ but also to Y ˙ r in Equation (7). Theoretically, the periodicity of Y ˙ r is determined by the periodicity of r r , so the fluctuation of P ˙ is likely to be related to r r or r . In

According to time path of P ˙ in

The theoretical analysis of the phase diagram P ˙ ~ θ P ¨ can be seen: the rotation of P ˙ ~ θ P ¨ must be clockwise based on mathematical logic as shown in

Take ( P ˙ − θ P ¨ ) in Equation (7) as a composite variable, then we can draw the phase diagram of P ˙ ~ ( P ˙ − θ P ¨ ) which rotates counterclockwise as

Since the direction of rotation of P ˙ ~ θ P ¨ and P ˙ ~ ( P ˙ − θ P ¨ ) are reverse, P ˙ ~ θ P ¨ and P ˙ ~ ( P ˙ − θ P ¨ ) will be significant differences in the time path. In

The average of P ˙ and the composite variables ( P ˙ − θ P ¨ ) are 0.035 and 0.036 for 1970-2015 respectively, and their trends are also consistent in long- term. As the

In the equation P ˙ − θ P ¨ = M ˙ − Y ˙ r , since the composite variable ( P ˙ − θ P ¨ ) on the left side of the equation changes periodically, the composite variable ( M ˙ − Y ˙ r ) on the right side should also change periodically.

Since the money has a capital attribute, it is assumed that the periodicity of M ˙ in the short-term may be similar to the periodicity of the capital change K ˙ in Cobb-Douglas function. By the marginal condition of Y = A K α L β , M P K = ∂ Y / ∂ K = r , there is K = α Y / r , so

d K K = d Y Y − d r r + d α α ,

According to the fundamental equation Δ Y = r Y [

d Y Y = lim Δ Y → 0 Δ Y Y = lim Δ Y → 0 r Y Y = r ,

therefore, d K / K = r − d r / r + d α / α . hypothesis d α / α = 0 , then

K ˙ = d K K = r − d r r = r − r ˙ .

Compared with Y ˙ = r + r ˙ , the front of r ˙ is a negative sign in K ˙ = r − r ˙ , so the rotation of the phase diagram r ~ K ˙ is counterclockwise. This causes the peak of K ˙ to fall behind the peak of r in the time path, as shown in

As guess from

Because the phase diagram r r ~ M ˙ rotates counterclockwise, we can overlap it on the phase diagram r r ~ ( − Y ˙ r ) and then get the phase diagram of the composite variable ( M ˙ − Y ˙ r ) . In this way, the phase diagram P ˙ ~ ( P ˙ − θ P ¨ ) and r r ~ ( M ˙ − Y ˙ r ) can be related according to Equation (7).

The phase diagram r r ~ M ˙ rotates counterclockwise, and the core variable M ˙ ¯ is affected by monetary policies and multiple times of the money supply of

central bank. The central bank supplies basic currency, and M ˙ ¯ comprises of basic currency and other derived currency created by business banks and financial market. The bigger of the monetary multiplier of the basic currency and the derived currency, the higher place on the vertical axis of M ˙ ¯ . In order to facilitate our narration, we assume M ˙ ¯ is determined by the supply of basic money under a constant monetary multiplier even though the multiplier might not be a constant in reality.

Since the phase diagram r r ~ M ˙ and r r ~ ( − Y ˙ r ) both involve r r , we can overlap corresponding points (point a, b, c, d) in these two diagrams and then determine the location, shape and direction of rotation of the phase diagram r r ~ ( M ˙ − Y ˙ r ) as show in

We now could relate the phase diagram P ˙ ~ ( P ˙ − θ P ¨ ) to r r ~ ( M ˙ − Y ˙ r ) . Since P ˙ − θ P ¨ = M ˙ − Y ˙ r , then the direction of rotation of r r ~ ( M ˙ − Y ˙ r ) and P ˙ ~ ( P ˙ − θ P ¨ ) are the same, and points on r r ~ ( M ˙ − Y ˙ r ) (as a, b, c, d) are corresponding to that on P ˙ ~ ( P ˙ − θ P ¨ ) . Since both phase diagrams rotate counterclockwise, then the maximum value of ( M ˙ − Y ˙ r ) and ( P ˙ − θ P ¨ ) locates at the second half of a cycle. We can verify this conclusion in statistical data in

In addition, the cycle of ( P ˙ − θ P ¨ ) and ( M ˙ − Y ˙ r ) fluctuates in accordance with the inflation equation, but also their long-term average should be equal. The statistical data show that the average value of ( P ˙ − θ P ¨ ) and ( M ˙ − Y ˙ r ) are 0.036 and 0.041 for the period 1970-2015. There are two reasons why the average value of ( P ˙ − θ P ¨ ) is slightly smaller than ( M ˙ − Y ˙ r ) . First, the time of the statistical data is not long enough. The impact of money will lag behind for some time to be reflected in P ˙ , especially in the monetary policy of QE (quantitative easing) over the years. Second, there are leakages of the money, such as part of the currency flow to other countries and regions.

Theoretical analysis in

In order to identify all above factors, we call M ˙ ¯ , r ¯ r and P ˙ ¯ as “core variables” which stand for changes of M ˙ , r r and P ˙ due to the migration of phase diagrams r r ~ M ˙ , r r ~ r ˙ r and P ˙ ~ θ P ¨ . M ˙ ˜ , r ˜ r and P ˙ ˜ as “fluctuating variables” which stand for changes of M ˙ , r r and P ˙ due to the fluctuation of phase diagrams. There relations are: M ˙ = M ˙ ¯ + M ˙ ˜ , r r = r ¯ r + r ˜ r , P ˙ = P ˙ ¯ + P ˙ ˜ .

In this way, we can more clearly understand the factors that affect price changes, and get some preliminary inference. For example, with an established phase diagram r r ~ r ˙ r , we can analyze the impact of change of exogenous variable M ˙ ¯ on P ˙ .

As

and make ( M ˙ ¯ ) 1 = Y ˙ ¯ r , then ( M ˙ ¯ − Y ˙ ¯ r ) 1 equals to 0, therefore ( P ˙ ¯ − θ P ¨ ¯ ) 1 = 0 , P ˙ ¯ 1 = 0 .

It not only indicates P ˙ ¯ and M ˙ ¯ change to the same direction, but also the central bank could make the core inflation rate P ˙ ¯ = 0 by controlling M ˙ ¯ (notice: P ˙ ¯ = 0 is different from P ˙ = 0 ). Since P ˙ = P ˙ ¯ + P ˙ ˜ , and P ˙ ˜ fluctuates around P ˙ ¯ , the value of P ˙ might be less or more than 0.

Even though we know that there is relation between the phase diagram r r ~ r ˙ r and P ˙ ~ θ P ¨ in

In order to eliminate the fluctuation of P ˙ or make P ˙ ≡ 0 in a system in which currency are medium of exchange, the government could take two measures: firstly, to restrain the fluctuation of r r ; secondly, to control M ˙ ¯ and therefore cause the change of P ˙ ¯ to offset the change of P ˙ ˜ . The first measure is usually applied by planned economy, and the second one need to make sure the change of M ˙ ¯ would not cause the change of r ¯ r , otherwise there would be unexpected consequences.

As shown in

r r ~ Y ˙ r moves downward along the slash line, and the phase diagram r r ~ ( M ˙ − Y ˙ r ) moves from r r ~ ( M ˙ − Y ˙ r ) 0 to r r ~ ( M ˙ − Y ˙ r ) 1 . Correspondingly, the phase diagram P ˙ ~ ( P ˙ − θ P ¨ ) and P ˙ ~ θ P ¨ move upward and rightward, respectively. This means that the decline in real interest rates will cause the price level to rise. Even if the increase of M ˙ ¯ and the increase of r ¯ r occur at the same time, the change of P ˙ ¯ and M ˙ ¯ are in the same direction, the change of P ˙ ¯ and r ¯ r are the opposite direction, so P ˙ ¯ would not necessarily rise.

We can analyze the change of P ˙ by the inflation equation as in

Although the inflation equation does not tell us “whether currency only has impacts on price or on both price and real output?” However, according to statistics in the United States, the change of P ˙ ¯ usually cause the change of r ¯ r , especially when P ˙ ¯ change to a significant extent. The final answer may require an analysis of the microeconomic mechanism based on the inflation equation.

・ The production function: Y = A K α L β . A marginal condition: r = ∂ Y / ∂ K = α Y / K .

・ The quantity equation: M = a Y . Identity: Y = P Y r .

・ Residual: V ˙ = Y ˙ − M ˙ , among them, V ˙ = [ P ˙ f ′ ( P ˙ ) / f ( P ˙ ) ] P ¨ = θ P ¨ , θ = 0.1 .

・ The Inflation equation: P ˙ − θ P ¨ = M ˙ − Y ˙ r .

・ The decomposition of variables: M ˙ = M ˙ ¯ + M ˙ ˜ , r r = r ¯ r + r ˜ r , P ˙ = P ˙ ¯ + P ˙ ˜ .

・ By transforming of the residual V ˙ , we use the analysis method of phase diagram to explain the short-term and long-term changes of the price in one equation: in the long-term, the average of P ¨ is 0, so P ˙ is determined by M ˙ and Y ˙ r ; in the short-term, since P ¨ is not 0, P ˙ is determined by the phase diagram of the complex variable ( M ˙ − Y ˙ r ) . Since these conclusions can be verified by statistical data, so they challenge the traditional theory of “demand for money”. From the perspective that money is wealth, everyone wants more money than existing ones. From the perspective that money is the medium of exchange, with the popularity of non-currency payment, people realize that their demand for transactions is not a demand for trading medium (such as currency), but to transfer the real wealth of others into their own accounts through monetary policy, or to pass on financial crisis to others. Therefore, the “demand for money” analysis is difficult to explain the changes in inflation.

・ Although this paper explains the factors that affect inflation, it is not able to answer whether money will affect real output at the same time as it affects prices. Omitting the fluctuation factor, if P ˙ ¯ increases with M ˙ ¯ increases, will r ¯ r also increase. Because the change of P ˙ ¯ is the result of the change of r ¯ r and M ˙ ¯ , if the change of M ˙ ¯ will cause the change of r ¯ r by P ˙ ¯ , then r ¯ r is the reason of change of P ˙ ¯ , and is affected by the change of P ˙ ¯ . This logic trap may be an important reason for the unclear role of money in the economic system. We will clarify this problem further by analyzing the Phillips curve in the next paper.

Zhan, M.A. and Zhan, Z. (2017) A Kind of Neither Keyne- sian Nor Neoclassical Model (3): The Deci- sion of Inflation. Open Access Library Jour- nal, 4: e3333. http://dx.doi.org/10.4236/oalib.1103333