# Can inflation in a context of fixed production of good and services be calculated?

According to,

Money and inflation

To summarize, the money supply is important because if the money supply grows at a faster rate than the economy's ability to produce goods and services, then inflation will result.

Suppose you know your production of good and services hasnt grown in a year, suppose the amount of money was in an optimal relation with the amount of good and services (if such thing exists), but your government decides to duplicate the amount of circulating money, can you calculate how much inflation will you have after a period of time? (I believe I heard it takes some months to reflect the impact of money emission in the prices of good and services). Is there an equation or something to calculate this?

You can certainly calculate it, but there isn't a single model to calculate it and depending on the model you use you would need information on more variables.

For example, using very basic model such as the Fisher model of money market equilibrium given by:

$$MV=PY \tag{*}$$

where $$M$$ is quantity of money, $$V$$ velocity of money, $$P$$ price level and $$Y$$ real output. Solving for $$P$$, log-linearizing *, and taking time derivative gives us:

$$\frac{\dot{P}}{P} = \frac{\dot{M}}{M} + \frac{\dot{V}}{V} - \frac{\dot{Y}}{Y}$$

where $$\frac{\dot{P}}{P}$$ is the (continuous) growth rate of the price level (i.e. inflation $$\pi$$ for $$\frac{\dot{P}}{P} > 0$$), $$\frac{\dot{M}}{M}$$ is the growth rate of money supply (let's call it $$m$$), $$\frac{\dot{V}}{V}$$ is growth rate of velocity (let's call it $$v$$), and finally $$\frac{\dot{Y}}{Y}$$ is the real growth rate of the economy (let's call it $$y$$).

Using the model above you can calculate inflation as:

$$\pi = m + v - y$$

If the production of goods and services did not changed then $$y=0$$ so you need to just find $$m$$ and $$v$$.

There are more complex models that could be used to account for some delay in transmission and other important factors. There are too many models to fit in a single SE answer. Good starting point for a review of various different ways of modelling inflation would be Walsh Monetary Theory and Policy.