Can the government talk cheap? Communication, announcements, and cheap talk.

• Author: Conlon, John R.

1. Introduction

   A great deal of attention has recently been focused on communication in economic contexts. While some models of credible announcements have been based on reputational forces or penalties imposed by third parties, as in Sobel |21~ and Cothren |4~, several recent papers have addressed the possibility of communication through "cheap talk," that is, talk in situations with no explicit penalties for deception. Perhaps the most prominent application of cheap talk in the literature is the recent model of Federal Reserve announcements by Stein |22~.(1)

   This paper will show that cheap talk equilibria of the sort modeled by Stein and others often depend in a fundamental way on implausible discontinuities in public responses to government announcements. That is, cheap talk equilibria are frequently only possible if certain infinitesimally small changes in government announcements are capable of causing large, discontinuous changes in public expectations and behavior.

   Intuitively, if public expectations are a continuous function of government announcements, then the government can "fine tune" these expectations. The government will therefore often be tempted to deviate from the cheap talk equilibrium announcements in order to manipulate expectations. This causes the equilibrium to unravel.

   Discontinuous public reactions sometimes prevent these equilibria from unraveling because they convert continuous choice problems, which allow manipulative fine tuning, into discrete choice problems, where such fine tuning is impossible. These discontinuities are implausible, however, and this may often rule out cheap talk as a realistic model of governmental policy announcements.(2)

   While attention in the next two sections is focused on Stein's paper for concreteness, a similar criticism also applies to some, though not all, of the other cheap talk models in the literature. As will be argued below, cheap talk models seem to fall into two major categories:

   (a) models which, like Stein's, depend upon discontinuous reactions to convert continuous into discrete choice problems, and

   (b) models in which cheap talk plays essentially a coordination role, usually in some sort of intrinsically discrete choice setting.

   In models of type (b), the announcer cannot fine tune reactions since reactions are discrete by assumption. Cheap talk is therefore frequently more plausible in this case. However, if the discrete choices in models of type (b) are simply used as an approximation to a continuous choice reality, then cheap talk in such models may still depend implicitly on implausibly discontinuous reactions. In such cases, the arguments in this paper are still relevant.

   Section II below describes Stein's model, and section III draws attention to the discontinuities required in public expectations. Section IV discusses the plausibility of these discontinuities. Section V then discusses other cheap talk models in the literature, and section VI concludes. An appendix generalizes the discussion in section III.

2. Stein's Model

   The details of Stein's model are unimportant, since a variety of different models can lead to the same class of policy dilemmas. However, to keep the discussion concrete, the following gives a general idea of how Stein models the Federal Reserve's policy problem. Stein begins with a two period model in which the Fed has target interest rates with normalized values of zero in both periods, and target exchange rates of T in both periods. The public knows the target interest rate of zero, but does not know the target exchange rate T.

   The Fed has one policy instrument in Stein's model, the second period money supply |M.sub.2~. A high money supply in period 2 pushes the exchange rate up but pushes the interest rate down. The Fed therefore chooses the second period money supply |M.sub.2~ = T/2 to balance off second period interest rate and second period exchange rate targets. However, the Fed wants the public in period 1 to expect |M.sub.2~ to equal T, since this will cause the first period exchange rate to equal the target level of T. Specifically, the Fed would like to manipulate |Mathematical Expression Omitted~ to minimize |Mathematical Expression Omitted~ where |Mathematical Expression Omitted~ is the public's first period expectation of |M.sub.2~. Equivalently, the Fed would like to manipulate |T.sup.e~ to minimize |(|T.sup.e~ - 2T).sup.2~ where |T.sup.e~ is the public's expectation regarding the exchange rate target.

   The important thing in this model is that the Fed is planning on a money supply of |M.sub.2~ = T/2, but it wants the public to expect |M.sub.2~ to equal T. Or, expressed in terms of exchange rate targets, if the Fed's exchange rate target is T, it wants the public to believe that its target is 2T. The Fed's dilemma then becomes, how can any announcement it makes in period 1 be credible, given that it has an incentive to deceive the public, and no penalty for doing so?

   Stein argues that vague, but only vague announcements will be credible. His argument actually shows less, however. Specifically, he shows that if the Fed is somehow restricted to a certain discrete set of permissible announcements, then it will have an incentive to choose the accurate announcement, so its announcements will be believed. Thus, Stein's argument must assume some mechanism which will restrict the Fed to this discrete set of announcements. In the next section it is shown that the Fed will only restrict itself to this discrete set of announcements if it is compelled to do so by public expectations which are a discontinuous function of Fed announcements.

   First, however, we summarize Stein's solution. Suppose possible exchange rate targets for the Federal Reserve board are uniformly distributed along the interval |Mathematical Expression Omitted~. Also, suppose that |Mathematical Expression Omitted~ is partitioned using |Mathematical Expression Omitted~, with

   |a.sub.i + 1~ = 6|a.sub.i~ - |a.sub.i - 1~. (1)

   Then Stein shows that the Fed will honestly report the interval into which its target T will fall. That is, if the Fed is restricted to make announcements of the form:

   "T is in the interval ||a.sub.i~, |a.sub.i + 1~~" for some i, or

   "T is in the interval ||-a.sub.i~, |-a.sub.i - 1~~" for some i, (2)

   then it will announce the correct interval.(3) Equilibria of this kind are called "partition equilibria" |5~.

   The intuitive logic of this result is as follows. If the Fed is forced to choose from the discrete set of announcements in equation (2), then any deviation from the truth in a given direction will push expectations too far in that direction, so the Fed prefers telling the truth to lying. As Stein puts it, "if the Fed wants to lie, it has to tell big lies, rather than small ones. And . . . such big lies can be less attractive than telling the truth" |22, 38~. The question remains, however, what prevents the Fed from deviating from the announcements in (2), and so, telling small lies? This is the issue addressed in the next section.

   Stein does not himself solve equation (1). However, using standard methods for solving difference equations |20~, it can be shown that

   |Mathematical Expression Omitted~

   (this formula can easily be checked by substitution into (1); also, it is easy to see that |a.sub.0~ = 0 and |Mathematical Expression Omitted~).(4)

3. A Closer Look at the "Cheap Talk" Equilibrium

   It is now shown that the sort of mechanism modeled by Stein requires the public's expectations to be a discontinuous function of government announcements. That is, the cheap talk equilibrium breaks down entirely if small differences in government announcements can cause only small differences in public expectations.(5) For concreteness, we focus on the Stein model of Federal Reserve announcements. The general case is examined in the Appendix.

   The key point is that, to determine whether an equilibrium is self enforcing we must, in the spirit of Kreps and Wilson's |13~ sequential equilibria, indicate what public expectations will be "off of the equilibrium path." That is, we must indicate, not only what the public does if the Fed chooses one of the intervals indicated in (2) above, but also what the public would do if the Fed made some other announcement. Stein's equilibrium is then shown to be self enforcing only if the public's reactions to any other announcement are so undesirable to the Fed, that the Fed would never choose any announcements other than those indicated in equation (2). Such undesirable reactions, in turn, are shown to depend upon beliefs which are implausibly discontinuous.(6)

   Suppose that the Fed makes an announcement of the form "our exchange rate target T is between a and b" (or a |is less than or equal to~ T |is less than or equal to~ b, or T |is an element of~ |a, b~). To consider all possible announcements, not just "equilibrium" announcements, we must let a and b vary continuously. Finally, let the public's expectation of T given the...