A mathematical proof that negative interest rate policy can neutralize any instance of inflation in nominal terms

Note: although the concept of the proof is simple, that a negative rate lowers nominal balances in accounts, and thus serves as a nominal tax offsetting the real tax of inflation, until the valuation of all accounts is reduced to the market determined stable level, although we are just using that simple principle, it is still a work in progress.

Importantly, the final version of the proof, I hope, will demonstrate this robustly via an accounting isomorphism that certain outcomes are equivalent, despite nominal differences, both in terms of relative price structure and relative credit/debit positions of accounts, and incentives for economic agents.

= A mathematical proof that negative interst rate policy can neutralize any instance of inflation in terms of unit of account function(but not store of value function)

Okay, this is just a really quick note. The claim is that negative interest rate policy can neutralize any inflation in nominal terms. What an interest rate does, is separate the unit of account and store of value functions.

First, we need to define a few variables.

• Let g_debt : The total public debt, regardless of account type(reserves, treasuries, cash)
• Let cpi_basket_price : the price of a basket of goods used to measure cpi, in dollars
• Let g_value : The number of cpi baskets match the total public debt.
• Let unit_dollar_value = 1/cpi_basket_price = g_value/g_debt
• Let g_nominal_rate = the nominal yield on government issued accounts
• Let g_surplus_real = g_value(t) - g_value(t-1) The increase in public debt value, in real cpi basket terms
• Let g_deficit_nominal = g_debt(t) - g_debt(t-1) The increase in the nominal public debt.

We will suppose that all govt issued currency accounts offer the same rate of interest, whether cash, reserves, bonds, or other securities.

Under these assumptions, the following equation measures the change in public debt:

• g_debt(t) = (1 + g_nominal_rate(t-1)) * g_debt(t-1) + g_deficit_nominal(t)

We can then define two more variables g_real_passive, and g_real_active

* g_new_surplus_real(t) = g_deficit_nominal(t) * g_real_active(t)

And then we know:

* g_value(t) = g_value(t-1)*g_real_passive(t-1) + g_deficit_nominal(t) * g_real_active(t)

Note that we also have:

* g_value(t) = g_value(t-1) + g_surplus_real(t-1)