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Prospect Theory and sentiment-driven fluctuations

  • Giuseppe Ciccarone ORCID logo EMAIL logo , Francesco Giuli and Enrico Marchetti
Published/Copyright: January 12, 2019
The B.E. Journal of Macroeconomics
From the journal The B.E. Journal of Macroeconomics Volume 20 Issue 1

Abstract

In this paper we aim to present a novel channel through which the volatility of the monetary/financial sector affects the instability of the real macroeconomic variables originated by self-fulfilling market sentiments. To this aim, we insert some elements of Prospect Theory in the preferences of agents living in an overlapping generations economy where consumers’ heterogeneity and firms’ imperfect information on the level of aggregate demand allow market sentiments to affect the equilibrium path of the economy under rational expectations. In this environment, greater heterogeneity in the household’s narrow framing parameter and in the degree of competition in goods markets favor the emergence of self-fulfilling equilibria by exacerbating the coordination problem generated by a pair-wise matching process taking place in the labor market. Furthermore, the more dispersed are agents’ deviations from standard rationality the higher is the volatility of the economy due to sentiment fluctuations. Finally, a higher volatility of the money/financial market, by increasing the effect of Prospect Theory on households’ choices under risk, increases the noise of the signal upon which firms make their hiring decisions; this, in its turn, generates greater variability in market sentiments and hence in real economic activity.

Keywords: prospect theory; self-fulfilling equilibria; sentiments; behavioral macroeconomics
JEL Classification: D84; E03; E32

Acknowledgments

We thank G. Di Bartolomeo, F. Lucidi and W. Semmler for comments. We are especially indebted to B. Liseo for useful suggestions. We also thank two anonymous reviewers and an Associate Editor for observations and suggestions. The usual disclaimer holds. Financial support from the Sapienza Università di Roma, Università degli Studi Roma Tre, Università degli Studi di Napoli Parthenope and Ministero dell’Istruzione, dell’Università e della Ricerca is gratefully acknowledged.

Appendix 1

The optimization problem (14) can be written in this form:

maxNjtGite=11−γNjt(1−1θ)(1−γ){[EΩ(F)(PtYt1θ)]1−γEΩ(F)Hi}−Njt1+ψ1+ψ

and the first order condition is:

(30)Njt1θ+ψ+γ(1−1θ)=(1−1θ){[EΩ(F)(PtYt1θ)]1−γEΩ(F)Hi}.

Hence the two conditional expectations EΩ(F)(PtYt1/θ) and EΩ(F)Hi must be computed, by using the conjectures (11)–(12), together with Hi=e12(1−γ)2σz2P¯−(1−γ)Mt−(1−γ)Bi and Mt=XtMt−1.

As for the first expectation, we have:

E(PtYt1/θ|Sjt)=Y¯P¯Mt−1E(Xt)E(Zt1θ−1|Sjt)

as the shock variable Xt is independent from the signal Sjt. We can now use the standard formula E(Ξ|S)=exp⁡{E[ξ|s]+12var(ξ|s)} for generic log-normal random variables Ξ=exp⁡ξ and S=exp⁡s, where in our case it is ξ=(1θ−1)zt and s=λbi+(1−λ)zt, to obtain:

E(Zt1θ−1|Sjt)=exp⁡{(1θ−1)(1−λ)σz2λ2σb2+(1−λ)2σz2[λbi+(1−λ)zt]+QY2}

where QY=var((1θ−1)zt|s) is a function of λ,θ,σz2 and σb2.

The second expectation E(Hi|Sjt) is equal to:

E(Hi|Sjt)=e12(1−γ)2σz2P¯−(1−γ)Mt−1−(1−γ)E(Xt1−γ)E(Bi|Sjt)

and by adopting the same procedure for the conditional expectationE(Bi|Sjt) we obtain:

(31)E(Bi|Sjt)=exp⁡{λσb2λ2σb2+(1−λ)2σz2[λbi+(1−λ)zt]+QB2}

where QB=var(bi|s) is also a function of λ,θ,σz2 and σb2.

We can use these results in the first order condition (30), together with the production function Yjt = Njt, to recover the equilibrium solution (16) with: QT=(1−γ)QY+QB.

Appendix 2

2.1     Two fundamental equilibria can be singled out in our model. The first one is the fundamental equilibrium under perfect information, when firms do not face uncertainty over the type of worker they will be matched with and the only existing shock is the monetary disturbance Xt. The firm’s optimization problem becomes:

maxNjtGite=[11−γ(PitNjt)1−γHi]−Njt1+ψ1+ψ

where the only difference with respect to equation (14) is the absence of the conditional expectation operator. In such a case, agents conjecture that in equilibrium output is constant and the price level proportional to Mt, that is Yt=Y~ and Pt=P~Mt, respectively. The first order condition is:

Njt1θ+ψ+γ(1−1θ)=(1−1θ)[(PtYt1θ)]1−γHi

Using the conjectures, the production function and the definiton for Hi and Bi in this condition, we obtain:

Yjt=(1−1θ)θ1+θψ+γ(θ−1)(Y~)1−γ1+θψ+γ(θ−1)Biθ1+θψ+γ(θ−1)

Integrating across firms based on Yt=[∫Yjtθ−1θdj]θθ−1 yields:

Yt=(1−1θ)θ1+ψθ+γ(θ−1)(Y~)1−γ1+θψ+γ(θ−1)[∫(Biθ1+θψ+γ(θ−1))θ−1θdj]θθ−1

The assumption of log-normal distribution of Bi implies that:

∫Biθ−11+ψ+γ(θ−1)dj=exp⁡12(θ−11+ψθ+γ(θ−1))2σb2

so we have:

Yt=(1−1θ)θ1+ψθ+γ(θ−1)(Y~)1−γ1+ψθ+γ(θ−1)e12θ(θ−1)[1+θψ+γ(θ−1)]2σb2

Equating coefficients gives

Yt=Y~=(1−1θ)1ψ+γexp⁡(θ−12(ψ+γ)[1−γ+θ(ψ+γ)]σb2)

In this equilibrium, sentiments cannot play any role and the presence of σb2 in Y~ is only due to integration across firms. Money is not neutral as far as levels are concerned (σb2 is affected by Γ and hence by σx2), but it cannot have any influence on the second-order moments of real output.

The other fundamental equilibrium obtains when the signal firms receive does not provide any information on the type of worker they will be matched with. In such situation firms base their expectations on the knowledge of the distribution function of the workers’ type contained in Ω(F) and their optimization problem writes:

maxNjtGite=EΩ(F)[11−γ(PitNjt)1−γ(Hi)]−Njt1+ψ1+ψ

The first order condition is:

Njt1θ+ψ+γ(1−1θ)=(1−1θ)[E(PtYt1θ)]1−γEΩ(F)(Hi)

Analogously to the above case, make the conjectures Yt=Y∗ and Pt=P∗Mt and use them together with the production function and the definition for Hi=(P∗)−(1−γ)Mt−(1−γ)Bi in the first order condition to obtain:

Yjt1θ+ψ+γ(1−1θ)=(1−1θ)(P∗)1−γ(Mt)1−γ(Y∗)1−γθEΩ(F)((P∗)−(1−γ)Mt−(1−γ)Bi)

which may be written as:

Yjt1θ+ψ+γ(1−1θ)=(1−1θ)(Y∗)1−γθEΩ(F)(Bi)

Contrary to the previous fundamental equilibrium, the output of a firm now depends on the expectation about the worker’s type, which is now simply given by EΩ(F)(Bi)=eσb2/2. This implies that:

Yjt=(1−1θ)θ1+θψ+γ(θ−1)(Y∗)1−γ1+θψ+γ(θ−1)eθ2[1+θψ+γ(θ−1)]σb2

Integrating over firms to obtain the aggregate output gives:

Yt=(1−1θ)θ1+θψ+γ(θ−1)(Y∗)1−γ1+θψ+γ(θ−1)eθ2[1+θψ+γ(θ−1)]σb2

and, by comparing coefficients, the final equilibrium value is:

Yt=Y∗=(1−1θ)1ψ+γexp⁡(σb22(ψ+γ))

As in the perfect information case, money is still neutral for the second-order moments of real variables, but it can affect the level of output. It is worth noticing that Y∗≠Y~. The reason is that in the perfect information case the j − th firm does not need to form expectations on the type of worker it will be matched with, and so its output Yjt does not depend on σb2. In the present case, even in the absence of useful information provided by the signal, the j − th firm still needs instead to form a rational expectation on the type of worker (i.e. EΩ(F)(Bi)=eσb2/2), which affects both individual and aggregate output.

2.2     Consider the case in which agents make the conjecture yt=y¯+zt and it is λ>λ¯=θ1+θ+θψ+γ(θ−1). This inequality can be rewritten in this way:

(32)θ(1−λ)−[1+γ(θ−1)+θψ]λ<0.

From (32) the following inequality can be obtained:

[θ(1−λ)−[1+γ(θ−1)+θψ]λθ(1+ψ)(1−λ)2]λσb2σz2<0<1.

Assume that it is σz > 0; by considering the inequality with upper bound at 1, the previous inequality can be written in this form:

θ(1−λ)λσb2−λ2σb2−γ(θ−1)λ2σb2<θ(1−λ)2σz2+θψ[λ2σb2+(1−λ)2σz2]

and, by adding and subtracting the quantity γ(θ−1)(1−λ)2σz2+(1−λ)2σz2 to the left hand side, the inequality reduces to:

θ(1−λ)λσb2−(1−γ)(θ−1)(1−λ)2σz2<1+γ(θ−1)+θψ[λ2σb2+(1−λ)2σz2]

and hence to:

θ(1−λ)1+γ(θ−1)+θψ[λσb2+(1−γ)(1−θθ)(1−λ)σz2λ2σb2+(1−λ)2σz2]<1.

This shows that the coefficient of zt in (19) is smaller than one and hence that the absolute value of the difference between the actual yt from (19) and yt=y¯+zt is always positive.

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Published Online: 2019-01-12

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  35. Ramsey income taxation in a small open economy with trade in capital goods
https://doi.org/10.1515/bejm-2017-0118
Keywords for this article
prospect theory; self-fulfilling equilibria; sentiments; behavioral macroeconomics

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  18. The macroeconomic impact of shocks to bank capital buffers in the Euro Area
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  34. Animal spirits in an open economy: an interaction-based approach to the business cycle
  35. Ramsey income taxation in a small open economy with trade in capital goods
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