Input-Output and Inflation

I have been a little puzzled by how dominant certain views about inflation are. The quantity theory of money seems to be dominating a lot of circles, in some cases contrary to their own interests and contradicting their claims on other issues. Most descriptions of inflation outside economist circles try very hard to reduce it to a policy mistake (well, you see and hear this in our field too), a one-time event, etc. and I don't think they have bad intentions or an interest in doing so necessarily. "X caused Y" sells! Yet these stories take themselves so seriously that they sometimes seem to forget the definition of inflation and reduce it to changes in demand only.

As always, this is a place where I feel like playing with some data and analytical models to throw out some ideas. Below, I'll run some simulations in a two-sector economy, without considering the accuracy of the analysis.

Production Networks

My starting point will be the commonly used production function in production network studies. There are two goods, two producers, and labor in this economy. Producing the good one (y_1) looks like y_1 = z_1*(l_1)^{alpha_1)*(q_11)^{a_11)*(q_21)^{a21} where z is a productivity shock, l is the labor input, q's are the other inputs: how much i needed for producing j. The exponents then are the relative shares where alpha_1 + a_11 + a_21 = 1, and the opposite indices apply for the second good. Profits for the first producer are given as pi_1 = p_1*y_1 - w*l _1 - p_1*q_11 - p_2*q_21 ; so you pay for all inputs, including the ones you produce.

I will not get into the general equilibrium implications and robustness of the simulations but here's the idea: apply a productivity shock to this economy multiple times and see how the price level acts. To avoid extremes I start with a shock coming from a uniform distribution with u(.5,1) and initially, there's no autoregression. The shares are assumed to be constant throughout but they are also assumed equal as a starting point and w = 1 is assumed for all simulations.

AR(0): Equal Shares and Not Equal Shares

Starting with equal shares, Plot A shows the evolution of prices of two goods

Taking the simple average and then looking at the log differences of that index, Plot B shows the movement of the price level and Plot C shows the inflation rate in this "economy."

What if the requirements in the two sectors were different? Now assume alpha_1 = 1/6, a_11 = 4/6 and a_21 = 1/6 while for the other producer alpha_2 = a_22 = a_12 = 1/3 still. Below are the corresponding prices, price levels, and inflation rates, respectively.

Can we compare inflation rates in two cases? Sure we can:

Just by changing the relative shares of inputs in this two-goods world, I introduced a lot of variation to the system.

AR(1): Equal Shares and Not Equal Shares

Let's assume the productivity shocks are in fact autoregressive, that is, z_1t = .1*z_1t-1 + e_1t. Below are the prices of two goods, price levels, and inflation rates respectively under the equal shares assumption.

What if the previously described not-equal shares were assumed together with the above-described autoregressive process? Below are the prices of two goods, price levels, and inflation rates respectively

Let's compare the inflation rates for the two relative shares under AR(1) assumption:

As shown previously, the not equal shares lead to a lot of variation in the system: propagate, if you want to call it that.

Conclusion

There's not much to say apart from what we all know: "Inflation is an increase in the general price level of goods and services in an economy." There's no explicit reference to many actors you hear getting blamed regularly for inflation in the textbook definition of inflation and in an extremely simple two-good economy that just faces productivity shocks, changes in the price level can easily reach two digits. Considering how small the shocks are and how little the persistence is, and the lack of a trend in the price level, one may want to avoid overly simplifying stories of inflation and realize how easy it can occur.