## 19 May 2015

### The Nominal Interest Rate and Inflation Determination

The dynamics of inflation in relation to the nominal interest rate are generally assumed to be governed by the liquidity effect. That is, increases in the money supply temporarily decrease the nominal interest rate because of money demand and some form of nominal rigidity. In a perfectly friction-less world, increases in the money supply (particularly increases in the growth or future path of the money supply) cause the nominal interest rate to increase instantly, rather than after the economy returns to its natural level (some economists would say equilibrium, but defining equilibrium as simply a solution to a model makes more sense to me). Of course, explaining this whole thing with math is a whole lot more descriptive, so here I go:

I'm going to assume households have a linear utility function and derive utility from consumption, so $u(c_t) = \ln c_t$ where $c_t$ is consumption. This implies the "consumption Euler equation" that is used to determine inflation given the nominal interest rate.

$$(1) \: i_t = \left(\frac {E_t c_{t+1}}{c_t}\right)\left(\frac {1 + E_t \pi_{t+1}}{\beta} \right) - 1$$

$i_t$ is the nominal interest rate, $\pi_t$ is the rate of inflation, and $\beta$ is the constant discount factor. Assume consumption grows at a constant gross rate "$g$",

$$(2) \: c_t = g c_{t-1}$$

and the interest rate equation becomes

$$(1a) \: i_t = g \left(\frac {1 + E_t \pi_{t+1}}{\beta} \right) - 1$$

Now, if the central bank targets inflation so that it evolves according to

$$(3) \: \pi_t = \bar \pi + \rho (\pi_{t-1} - \bar \pi) + \epsilon_t^\pi$$

where $\bar \pi$ is the "trend" rate of inflation, $0 < \rho < 1$ is the "shock stickiness" parameter, and $\epsilon_t^\pi$ is white noise, and $E_t \epsilon_{t+1}^\pi = 0$, then expected inflation, $E_t \pi_{t+1}$, is defined by

$$(3a) \: E_t \pi_{1+1} = \bar \pi + \rho (\pi_t - \bar \pi)$$

To fill in the model, the final interest rate equation becomes

$$(4) \: i_t = g \left( \frac {\bar \pi + \rho (\pi_t - \bar \pi)}{\beta}\right) - 1$$

This shows essentially what the "Neo-Fisherian" assertion is. The nominal interest rate and inflation rise with each other. In fact,

$$(5) \: \frac {d i_t}{d \pi_t} = \frac {g \rho}{\beta}$$

So, if the rate of inflation increases by 1%, then the nominal interest rate will increase by $\frac {g \rho}{\beta}$%.

Of course no monetarist will be happy until I use the money supply as a determinate of the price level, so assume a cash in advance constraint:

$$(6)\: M_t = P_t c_t$$

where $M_t$ is the money supply and $P_t$ is the price level. Since consumption still grows according to (2), the interest rate equation is redefined to

$$(7)\: i_t = g \left( \frac {E_t P_{t+1}}{P_t\beta} \right)$$

$M_t$ grows at gross rate "$m_t$", so its law of motion is

$$(8)\: M_t = m_t M_{t-1}$$

and, given (6),

$$(9)\: E_t P_{t+1} = \frac{m_{t+1} M_t}{g c_t}$$

Integrating all this back into (1) gives

$$(10)\: i_t = \frac{m_{t+1}}{\beta}-1$$

If the central bank permanently increases $m_t$, which is equivalent to $\pi_t +1$, by 1%, the nominal interest rate will increase by $\frac{1}{\beta}$% ($\frac {d i_t}{d m_t} = \frac {1}{\beta}$).

Basically, absent nominal rigidity, the nominal interest rate and inflation have a positive, even causal, relationship not afforded to them by conventional wisdom. Of course, this is really driven by the way that the money supply interacts with the nominal interest rate. "Neo-Fisherism" is really an incomplete hypothesis because of this. More focus should be given to the effects of open market operations as non-nominal-rigidity ways of explaining the liquidity effect.