The equation of exchange and the inflation rate

The equation of exchange states the money stock times by the velocity of money equals price level times by real income. In other words:

MV=PY

In this equation, all these four variables are so closely related that it is hard to tell how a movement in one variable influences the other.
Let's take implicit derivatives with respect to M.

d(MV)/dM=d(PY)/dM
=> V+M*(dV/dM)=Y*(dP/dM)+P*(dY/dM)

It looks messy. Why not times both sides by dM?

V*dM+M*dV=Y*dP+P*dY

It still looks messy. Why not divide both sides by MV? Wait! Note MV=PY. Therefore, why not divide the left hand by MV and the right hand by PY? Those denominators are basically equal, so it does not change the result.

(V*dM+M*dV)/MV=(Y*dP+P*dY)/PY
=> dM/M+dV/V=dP/P+dY/Y

Aha, the rate of change in money stock plus the rate of change in velocity equals the rate of change in price level plus the rate of change in real income. If you want to know the percentage changes, then just time each term by one hundred.

Let's substitute lower case m, v, p, and y for the rates of changes in money stock, velocity, price level, and real income, respectively.

m+v=p+y

Solve the equation for p. Note that p is the rate of change in the price level or inflation rate.

p=m+v-y

The inflation rate is determined by the rate of change in money stock plus the rate of change in price level less real income.

An increase in money stock can cause inflation only if neither the velocity nor the real income falls.

Can you see a run-away inflation? I see about three percent of inflation in this measure. It is greater than the BLS CPI inflation rate but I measured the price level for all items in real income not only consumption goods. Anyways, it is very different from the story the actual inflation should be much higher and government must be hiding the stuff, isn't it?

The arithmetic mean and geometric mean in the CPI calculation

Aha, the damn BLS bureaucrats have rigged the CPI calculation to help the Fed and Obama. They changed the formula for the CPI. When? 1999. In 1999, the BLS must have anticipated the economic crisis, Obama's election, and the Fed QEs since 2008. What a surprise!
Besides some non-factual criticisms, such as "prices of oil and food are excluded" or "rental equivalence lowers the CPI", what indeed interests me is the introduction of geometric mean that replaced arithmetic mean in 1999. By the way, prices of fuel oil and food are included in CPI calculation and rental equivalence actually has tended to raise the CPI. Crude oil price is not included in the CPI because we do not consume; housing prices are very volatile compared to rents, so rental equivalence has had an effect to raise the CPI overall.
Anyway, regarding arithmetic mean and geometric mean again. An arithmetic mean is what we usually call an average. It is the summation of all observations and divided by the number of them. A geometric mean is the n-th order root of multiplication of all observations. Mathematically, it is true that an arithmetic mean is always not less than the geometric mean of the same observations. In that sense, the BLS critics may not be all wrong. However, their arguments seem to be based on misunderstanding.
Let's imagine that I purchase 10 cheese burgers every week. There are five brands: MD, BK, AW, W, and DQ. Initially, they have an identical price, 5 dollars. They are close enough to be considered perfect substitutes. All of sudden, MD raises its price to 6 dollars while the other brands do not. Then, I will buy 10 cheese from the other brands. My consumption per burger is still 5 dollars. However, the arithmetic mean price of cheese burger will be 5.2 dollars. The mean price is skewed as much as 20 cents in this case. If we take geometric mean, it will be about 5.19. It is still skewed but better than the arithmetic mean. Since perfect substitutes are hardly observed in real life, the geometric mean price likely captures substitution effects much better.
Actually, computing geometric mean prices into the CPI has lowered the index about by not greater than 0.3%, according to the BLS: http://www.bls.gov/opub/mlr/2008/08/art1full.pdf
Isn't it quite different from Internet conspiracy theories?

The Fisher Equation

Irving Fisher constructed one of the most important equations in economics. And it is very simple.

r = i - inflation rate

r refers to the real interest rate; and i to the nominal interest rate. Whenever an inflation occurs, the real interest falls. Given that a firm's investment decision is made based on the real interest rate, a falling real interest rate leads firms to invest more. Therefore, lowering nominal interest rate or raising the price level can stimulate the economy in the short run. However, this equation can be rewritten as followed.

i = r + inflation rate

This equation shows that the inflation will eventually raise the nominal interest rate. Therefore, in the long run, the real interest rate may not be affected, and then the inflation cannot stimulate the economy.
Wait, will the inflation raise the nominal interest rate?
Yes. A rising nominal interest rate reflects inflation or loose money. If money is too loose, the nominal interest rate likely rises. Conversely, a falling nominal interest rate reflects deflation or tight money. If money is too tight, the nominal interest rate likely falls. Not my original idea but Milton Friedman's.

http://www.hoover.org/publications/hoover-digest/article/6549

The Federal Reserve has kept the nominal interest rate close to zero for almost 4 years. Some have argued it is an easy money policy showing the low Federal funds rate and huge growth of the monetary base. The truth is entirely opposite. Borrowing from Friedman, the Fed policy was "too little too late."

The U.S. unemployment rate was the worst through the mid 2010. That is the period where the growth rate of money supply measured in money zero maturity drastically fell. One may still argue it is still the positive growth, so that it still represents rising money supply. However, in the meantime, the falling velocity of money not only offset but dragged the growth of money supply below the level required to restore full employment output.

 
The result seems to be NGDP lower than its potential.

Mathematics in economics

Why do we need to learn math besides a requirement for the profession? I don't think math is really necessary in economics as long as I can elaborate my arguments without it. As long as my arguments don't show any gap in my understanding of things happening in the world, in addition, I don't think math is necessary. Using math in economics and other sciences should not be considered bragging. Rather, it is a confession of our ignorance of our research subjects. Mathematical modeling shows the limit of our knowledge. For example, a simple linear demand function that consists of price, quantity and slope coefficient show I'm not sure about other influential factors, such as income, substitutes or compliments. It, of course, shows that I may acknowledge and take into consideration those factors in follow-up research. Unlike ordinary belief, we may need math because we are not as smart as we think.

Lower money demand means greater money demand.

When we draw a MD-MS model, we unconsciously assume lowered money demand curve represents weaker and smaller money demand. However, I start thinking this perception may be wrong. When we-by "we" I mean an economy, not an individual-have lower money demand curve, the velocity of money is lower and the ratio of money to our nominal income is actually greater because V=(P*Y)/M. It means we do not want to spend our nominal income; instead, we want to hoard cash or deposit a large portion of it in the bank. Therefore, the current state of money demand is we have too strong money demand, not weak one. It is tight money, not easy money. Since we want to hold our money so tight, the opportunity cost of money, a.k.a. Interest rate, likely falls or unlikely rise. In other words, our zero interest rate is not low enough.