## Leseprobe

## Table of Contents

II. Signs and Symbols

1. Introduction

2. Construction of a Model

3. Subgame Perfect Equilibrium

3.1. Proof of unique Equilibrium

3.2. Implications of Equilibrium

4. Placebo reforms

5. Conclusion

III. List of Sources

IV. Appendix A: Shares of payoff in different quarters

V. Appendix B: Mathematical Derivation of Gross Payoff

## II. Signs and Symbols:

Abbildung in dieser Leseprobe nicht enthalten

## 1. Introduction

Nowadays during election campaigns the word “populist” is very common. The idea of politicians making empty promises is nothing special, but are there “empty reforms”, reforms where nothing is actual reformed? What are crucial assumptions for these? Or starting from a more general point: How do agents decide whether to act and about the point in time of this possible action? In which way does the decision maker tradeoff between a positive impact in the long run and a potential worsening of the short term situation? Are there actions without a real effect, some kind of placebo actions?

As a short example for this decision maker’s problem consider following scenario:

*The branch manager of a sales company would like to improve the turnover by increasing the amount of sales of his representatives. He is continuously monitored by his boss who evaluates his decisions. The manager knows that due to an economic upturn the turnover would increase with a high probability anyway. After exploring his options he came to the result that there are three main alternatives for an increase of the turnover:*

*I. One possible option could be an expensive and lengthy staff training. After this staff training the amount of sales will increase with a high probability, but in the short-run the cost of the training and the absence of the representatives during the training could lead to a decrease of the turnover in the short-run. Without absolute certainty about his professional future he cannot take it for granted that the long-run turnover increase will be attributed to him. Maybe he would be forced to step down after the short term turnover decrease and the following increase will be attributed to his successor.*

*II. Another, probably safer option could be holding out a high bonus for sales in the next few months. This decision would lead for sure to a higher turnover in the short term, but the long term effect is uncertain.*

*III. The last option is the easiest one: Knowing that the turnover increases anyway, he could just change something which does not have any effect on the turnover, e.g. changing the t-shirt color of the representatives. Outside observers without the knowledge of the economic upturn could tend to draw a connection between the color of the t-shirts and the turnover increase.*

*The main question is: Which alternative is superior for the company, which is the best one for the manager?*

Ran Spiegler created in his paper “Placebo Reforms” a dynamic model to observe how such issues affect the decisions of agents.

## 2. Construction of a Model

The model contains an economic variable x which evolves dynamically over time. An infinite sequence of policy makers (PMs) observes this variable. The PMs act successive, while each of them chooses an action a out of an finite sum of actions A. In A, there are at least 2 elements, a minimum of one element where the PM acts (called “*active reform ” or “ intervention ”*) and one where he does nothing (called *“ default ”* or *“ inaction ”*). The target of each PM is, considering a constant discount factor δ (Є(0,1)) on the outcome of their actions, to maximize his public evaluation. As a consequence the PM only chooses an active reform if the expected discounted credit is strictly positive.^{1} So in the case of being indifferent between an intervention and an inaction the PM selects the default.

One of the most important assumptions of the model is the public attribution rule. Spiegler does not assume “rational expectations”. Instead in his model the evaluators are not able to fully understand causal relationships between actions and outcomes. More precisely, this means that the public attributes all changes in x which appear between two interventions as and at, while s<t, to player s. In other words: all “changes in x are always attributed to the most recent intervention”.^{2} Thinking for example about football-business in Germany, this is not a very strict or uncommon assumption. After a period of poor performance of a club the head coach is often fired and replaced. If the performance enhances now shortly afterwards, the improvement is highly likely attributed by the fans to the replacement.

The actions of the players may affect the evolution of x. More precisely Spiegler assumes “that xt follows a linear growth trend with independently distributed, transient noise, such that both, the trend and the noise distribution are determined by the latest intervention.”^{3} This means that every action a has a trend slope µ_{a}>0 and a characteristic continuous density function fa. The function is symmetrically distributed around zero with support [-ka,ka] , where F_{a} is the cumulative distribution function (cdf) of f_{a}. Out of fat the noise εt is independently drawn.

With s as the latest period prior to t in the given game path who contains an active reform (a_{s}≠0), we can define b_{t}=a_{s}. If there was no action before period t then b_{t}=0. The first period with an active reform after period t is defined by r(t). If there isn’t any following period with an intervention, then r(t)= ∞.

By bringing this all together a formal definition of the stochastic process is possible. For

every t>0:^{4}

Abbildung in dieser Leseprobe nicht enthalten

This process, its initial condition (x^{0},ε^{0} ) and the past actions with their resulting realizations of x are general knowledge of the PMs. Hence after his action each player can also spot the realization ε^{t}.^{5}

Under the assumption of non-satiation the preferences of the policy makers are given by their payoffs. The payoff of payer t is defined as:

Abbildung in dieser Leseprobe nicht enthalten

By investigating the function we can immediately see some of the desired properties:

- If the player chooses the default he gets a certain payoff of 0. => No action, no payoff

- If he intervenes, his payoff equals the change of the variable x from the point in time of his action ([Abbildung in dieser Leseprobe nicht enthalten]) up until the next action of a subsequent PM ([Abbildung in dieser Leseprobe nicht enthalten]). In case that b(t)≠0 and player t chooses an action at for which (µa,fa) = (µb(t),fb(t)) there is no change in progression of x. But, according to the payoff function, there is a clear difference in the expected payoff for player t. If he would have chosen the default then his payoff would be equal to zero with certainty. In contrast by choosing at he gets the credit for the evolution of x up until the next subsequent active reform. In this scenario his action could be seen as a placebo reform, “taken solely for the purpose of claiming credit for observed changes in x.”^{6}

The discount factor δ plays obviously an important role in the function too. It shows how much the policy maker cares about the payoff in the following periods, in a way it captures his preferences. A discount factor close to zero means that the policy maker is interested in a high payoff in the short term; later payoffs are weighted much lesser then the first ones. On the other hand, a discount factor close to one implies a far-seeing player who has strong concerns on future evaluation.^{7}

As a very simplifying example consider following:

*In a certain play path the economic variable has a randomly drawn trend (normal distributed with an expected value of 1 and a variance of 1) and a randomly drawn temporary noise (distributed normally with an expected value of 0 and a variance of 0,1). The player acts in period 0 and for 20 periods no other action follows (r(t)=20), so the progression of x is known. ^{8} According to this the only unknown variable left in the policy makers payoff function is the discount factor.*

*In the following table the distribution of the complete expected payoff among the quarters is shown for different discount factors. As claimed above, for a relative high discount factor the expected payoff is allocated consistent between the quarters, while for a relative low discount factor nearly the whole expected payoff consists of the first-quarter payoff.*

Abbildung in dieser Leseprobe nicht enthalten

Table 1: relative quarterly share of complete payoff for different discount factors^{9}

For analyzing purpose of the model two more functions are necessary. The first one is the riskiness function R associated with action a. The riskiness of any action a is defined by:

Abbildung in dieser Leseprobe nicht enthalten^{10}

Helpful properties of the function are:

- Ra(ε) - ε is a non-negative-valued and strictly decreasing function

- Ra(ε) - ε ≤ 0,5*(ka - ε) for all ε (binding at ε = ka , because Ra(ka) = ka).^{11}

The second function is a tradeoff function. It compares the expected trend of action a with its riskiness, weighted by the discount factor of the player.

Abbildung in dieser Leseprobe nicht enthalten

At last Spiegler defines the noise realization ε* as the solution to the maximization problem:

Abbildung in dieser Leseprobe nicht enthalten

The proof that ε* is a unique solution is quite simple. While *Π* is strictly decreasing function for an increasing ε, the right side of the equation is strictly increasing in ε. Consequently the two sides have to have a unique intersection.^{12}

**[...]**

^{1} Cf. Spiegler (2012), p. 2.

^{2} Spiegler (2012), p. 2.

^{3} Spiegler (2012), p. 6.

^{4} Cf. Spiegler (2012), p. 6.

^{5} Cf. Spiegler (2012), p. 6.

^{6} Spiegler (2012), p. 7.

^{7} Cf. Spiegler (2012), p. 7.

^{8} See Appendix A

^{9} Calculated with the random data out of Appendix A

^{10} Cf. Spiegler (2012), p. 7.

^{11} Cf. Spiegler (2012), p. 7.

^{12} Cf. Spiegler (2012), p. 8.