Monetary rules

The question of whether the Federal Reserve Board should use rules in the conduct of monetary policy is almost as old as the Fed itself. For a brief time in the Fed's history it used a policy-making rule based on monetary aggregates, and today many are suggesting that it used a rule based on the federal funds rate. Other countries have used policy-making rules that are based on explicit inflation targets. While at this moment the Fed is an institution where members vote on monetary policy using their own best judgement, the issues illustrated in discussing the question of rules are still interesting and controversial.

There are several types of policy-making rules. The simplest form is an unconditional rule, such as having the monetary authorities raise the money supply x percent per year, come what may. An alternative approach would base a rule on some target objective, such as stable prices, and have monetary authorities reduce the inflation rate to some specified amount, however the authorities choose to do that. An intermediate approach might be called a feedback rule. Under this approach policy objectives, or targets, might he specified in the rule and the authorities would respond in a regular way to deviations between actual values and the target levels of these variables.

Remarks by Governor E. M. Gramlich on 24^th Annual conference of the eastern economic association. New York, 2/27/98 (with adaptations).

A target rule: inflation targeting

A well-known approach, used in a number of industrialized countries (Canada, the United Kingdom, New Zealand. Sweden, Australia, Finland, Spain and Israel, to name a few) is known as inflation targeting. Rather than having some monetary quantity under the control of the authorities advance x percent per year, the idea of inflation targeting is to move right to the ultimate goal of monetary policy, stable prices - overall price levels should grow no more than y percent per year. Rather than having monetary authorities operate in terms of a simple role, the authorities are simply told to get inflation down, one way or another. In this sense, inflation targeting is a very different type of role. It gives very great discretion to the monetary authorities to pursue one objective, and no ability to pursue any other objective. While inflation targeting would seem to force central banks to become very specific about their policies, in fact the actual inflation targeting strategies have been more flexible. They have usually required the central bank to target between one and three percent inflation. They have also been defined in terms of some version of the, underlying rate of inflation - the overall inflation rate less food and energy prices, the impact of exchange rates, government taxes, and perhaps other clearly exogenous prices. Moreover, the real world inflation targets that have been instituted usually give the central bank an out, if this quarter it wants to worry about exchange rates, output gaps, or other economic goals.

Ben Bernanke and Frederic Mishkin. 1997 (with adaptations).

In the early days, before most countries had central banks, countries operated under the gold standard, which entailed its own set of rules. The world supply of money was determined by the usable gold supply. New gold discoveries would lead to monetary expansions in recipient countries, which would then experience rises in prices and output. Contractions in the supply of usable gold would require contractions in prices and output. If a country on its own over-inflated demand, say by fiscal policy, its demand would spill over to foreigners and its gold would flow out. While the gold standard was in this sense self-regulating, it was not a perfect system. Monetary policy was not set consciously in terms of the economic needs of the country, but by the world gold market. The world gold stock would fluctuate in line with international discoveries, while the stock in particular countries reflected trade flows. There was no automatic provision for money or liquidity to grow in line with the normal production levels in the economy. John Taylor (1998) has shown that this regime was responsible for large fluctuations in real output, much less stability in real output than has been achieved in the post gold standard era. In the gold standard period of 1890-1905, for example, the US economy suffered five major recessions.

Remarks by Governor E. M. Gramlich on 24^th Annual conference of the eastern economic association. New York, 2/27/98 (with adaptations).

Considere uma firma caracterizada por uma tecnologia ou, equivalentemente, suponha que exista um conjunto convexo !$ Y \subset \mathbb{R}^n !$, contendo a origem , que esteja associado à produção dessa firma. Pode-se interpretar !$ Y !$ como o conjunto dos pontos !$ y = (y_1, ... y_n) !$ nos quais a firma pode operar; se !$ y_i \le 0 !$, a firma está usando o bem !$ i !$ como insumo para a produção e se !$ y_i \ge 0 !$, a firma está produzindo o bem !$ i !$. Dado um preço

!$ p \in L _+^{n -1} = \lbrace (p_1, ..., p_n) \in \mathbb{R}^n : p_i \ge 0, \quad \quad1 \le i \le n, \quad e \quad \sum \limits_{ i= 1}^n p_i = 1 \rbrace !$

!$ p.y {= \sum \limits_{i = 1}^n p_i y_i} . !$

Suponha que para o preço !$ p !$, o objetivo da firma seja buscar o conjunto dos níveis de atividade !$ \psi (p) !$ que maximizem o seu lucro. Nesse modelo, !$ \psi : L_+^{n-1} \rightarrow P (Y) !$ é uma correspondência determinada pela relação

!$ P (Y) !$ denota o conjunto das partes de !$ Y !$.

Diz-se uma correspondência !$ \varphi !$!$ : X \rightarrow P (Y) !$, em que !$ X \subset \mathbb{R}^m !$ e !$ Y \subset \mathbb{R}^n !$, é semicontínua superiormente (s.c.s) se para !$ x \in X !$e !$ y \in Y !$, e para quaisquer pares de seqüências !$ \lbrace x_k \rbrace !$!$ \subset !$!$ X !$, !$ \lbrace y_k \rbrace !$!$ \subset !$!$ Y !$, tais que !$ y_k \in \varphi (x_k) !$ para todo !$ k \in \mathbb{N}, \quad x_k \rightarrow x \quad e \quad y_k \rightarrow !$!$ y !$, tem-se !$ y \in \varphi (x). !$

Considere uma firma caracterizada por uma tecnologia ou, equivalentemente, suponha que exista um conjunto convexo !$ Y \subset \mathbb{R}^n !$, contendo a origem , que esteja associado à produção dessa firma. Pode-se interpretar !$ Y !$ como o conjunto dos pontos !$ y = (y_1, ... y_n) !$ nos quais a firma pode operar; se !$ y_i \le 0 !$, a firma está usando o bem !$ i !$ como insumo para a produção e se !$ y_i \ge 0 !$, a firma está produzindo o bem !$ i !$. Dado um preço

!$ p \in L _{+}^{n -1} = \{ (p_1, ..., p_n) \in \mathbb{R}^n : p_i \ge 0, \quad \quad 1 \le i \le n, \quad e \quad \sum \limits_{ i= 1}^n p_i = 1 \} !$

!$ p.y {= \sum \limits_{i = 1}^n p_i y_i} . !$

Suponha que para o preço !$ p !$, o objetivo da firma seja buscar o conjunto dos níveis de atividade !$ \psi (p) !$ que maximizem o seu lucro. Nesse modelo, !$ \psi : L_+^{n-1} \rightarrow P (Y) !$ é uma correspondência determinada pela relação

!$ P (Y) !$ denota o conjunto das partes de !$ Y !$.

Diz-se uma correspondência !$ \varphi !$!$ : X \rightarrow P (Y) !$, em que !$ X \subset \mathbb{R}^m !$ e !$ Y \subset \mathbb{R}^n !$, é semicontínua superiormente (s.c.s) se para !$ x \in X !$e !$ y \in Y !$, e para quaisquer pares de seqüências !$ \lbrace x_k \rbrace !$!$ \subset !$!$ X !$, !$ \lbrace y_k \rbrace !$!$ \subset !$!$ Y !$, tais que !$ y_k \in \varphi (x_k) !$ para todo !$ k \in \mathbb{N}, \quad x_k \rightarrow x \quad e \quad y_k \rightarrow !$!$ y !$, tem-se !$ y \in \varphi (x). !$

Se !$ Y !$ é compacto, existe um preço de mercado !$ p^* !$!$ \in L_+^{n-1} !$ e um nível de atividade associado !$ y^* \, \in \, Y !$ que permitem à firma obter o maior lucro possível, isto é, !$ p^*. y^* = max \lbrace p^. z : p \in L_+^{n-1}, z \in Y \rbrace . !$

Considere !$ E !$ um espaço vetorial normado com norma !$ || \bullet || !$ e !$ E^* !$ o seu espaço dual, formado pelos funcionais contínuos !$ f \quad : \quad E \rightarrow \mathbb{R} !$, dotado da norma dual:

!$ E^* !$ será um espaço de Banach se, e somente se, !$ E !$ for um espaço de Banach,

Considere !$ E !$ um espaço vetorial normado com norma !$ || \bullet || !$ e !$ E^* !$ o seu espaço dual, formado pelos funcionais contínuos !$ f : E \rightarrow \mathbb{R} !$, dotado da norma dual:

!$ E !$ seja um espaço de Hilbert e que !$ \lbrace x_n \rbrace \subset \quad E !$ é uma seqüência limitada. Se !$ \lim_{n \rightarrow \infty} \quad f(x_n) = f(x) !$ para todo funcional !$ f \in E^* !$, então !$ \lbrace x_n \rbrace !$ converge para !$ x !$.

Existe uma seqüência !$ [0,1] \times [0,1] !$, tal que !$ (x, y) !$ em !$ [0,1] \times [0,1] !$.

Considere o espaço !$ L^2 (0,1) !$ com medida de Lebesgue e a norma usual !$ || \, \bullet \, || !$. Suponha que !$ T : L^2 (0, 1) \rightarrow \mathbb{R} !$ seja uma aplicação linear contínua e defina !$ \varphi : L^2(0,1) \rightarrow \mathbb{R} !$ por

!$ \varphi (u) = {1 \over 2} || u ||^2 - T (u), \forall \quad u \quad \in L^2 (0,1). !$