There's been some discussion here on Noah's recent blog post on Milton Friedman's legacy, and while I'm sure there are people here who can evaluate the whole post better than I can, there's one bit that stuck out to me:

This is the idea that nominal interest rates should be set at zero. It generally contradicts the k-percent money growth rule, unless setting rates at zero just happens to make the money supply grow at a constant rate (how much of a PR badass are you if you can get your name on two mutually contradictory rules?). Interestingly, the Friedman Rule is based on Neo-Fisherism - it says that rates should be at zero because in the long run, low interest rates are deflationary and mild deflation is what we want.

Noah's explanation of the Friedman rule as Neo-Fisherism is backwards. The argument isn't that low interest rates cause deflation, which is good, but rather that deflation causes low nominal interest rates, which are good. In a cash-in-advance economy, if people want to purchase goods tomorrow, they have to set aside some of their income today as money. Since money pays zero nominal interest (usually), people would rather hold assets that pay some positive nominal return, but because of the cash-in-advance constraint, they have to hold some money. The idea behind the Friedman rule is to eliminate the deadweight loss, commonly called shoe leather costs, associated with having to hold money instead of assets with positive returns. If money pays the same returns as other assets, then shoe leather costs are zero. Maybe if Noah had paused and asked himself "why did Friedman think mild deflation is what we want?" he wouldn't have made this mistake.

The other mistake Noah makes is to claim that the Friedman rule contradicts the k% rule. It may be that Noah's stuck thinking in an NK framework, but in a monetarist framework, you don't look at interest rates to judge the stance of monetary policy (as Sumner so often points out), but rather the money supply. In monetarist models, the central bank sets the money supply, which determines inflation through the quantity theory of money, which determines nominal interest rates through the Fisher effect. Noah argues that this doesn't work out in reality (see points 2 and 3 in his post), but I'll leave that for others here to evaluate. Another reason it doesn't quite make sense to think of the Friedman rule as a nominal interest rate peg is that Friedman argues in his third most cited work (Google Scholar citation count) that monetary policy "cannot peg interest rates for more than very limited periods." That would suggest that the Friedman rule is more of a motivation for bringing inflation down rather than a strict policy rule like the k% rule.

Now, because it's a Sunday, I'm bored, and I'm a nerd, let's work through a simple cash-in-advance model to show how this all works. I'll use TeX All the Things to typeset some of the math. The model is from chapter 8.3 of Doepke, Lenhert, and Sellgren (1999)

The economy consists of a continuum of identical infinitely lived households who produce using labor l_t in the production function y_t = l_t, but cannot consume what they produce. Instead, they must purchase their consumption c_t with cash m_t, which must be set aside one period before they spend it. They can borrow or lend b_t to each other at the nominal interest R_t with a transversality constraint, and have perfect foresight. The price of the consumption good is P_t. The central bank uses helicopter drops v_t to set the money supply, where $m^s_{t+1} = m_t + v_t$.

The HH chooses $\{c_t, l_t, b_t, m_t^d\}$ in all periods to maximize lifetime utility $\sum_{t=0}^\infty \beta^t(\log(c_t) + \log(1 - l_t))$, facing the following constraints:

• Production function: $y_t = l_t$

• Budget constraint: $P_t c_t + m_{t+1} + b_{t+1} \le P_t y_t + (1 + R_t)b_t + m_t + v_t$

• Cash-in-advance: $P_t c_t \le m_t$

In equilibrium, markets clear:

• Goods: c_t = y_t

• Bonds: b_t = 0

• Money: $m^s_t = m^d_t$

Set up the Lagrangian and solve. If you want more detail for the math, check the textbook chapter I linked.

Suppose the central bank follows a k% rule. In equilibrium, everything is constant over time except the price level (inflation is constant over time). In particular, $1 + r = (1 + R) / (1 + \pi) = 1/\beta$ and $c = \beta / (1 + k + \beta)$. Now consider the social planner's problem. It's pretty easy to show that the solution is c = 1/2 and l = 1/2. Use these equations to show that the central bank can set k such that R = 0 and the equilibrium is the same as the social planner's solution

Is this a plausible model of the economy? Maybe not. But it shows that these concepts--the Friedman rule, the k% rule, and the QTM--are not mutually contradictory, and it makes the reasoning behind the Friedman rule clear.

Grade: D