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

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)