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Communication and Learning: The Bilateral Information Transmission in the Cobweb Model

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Abstract

Communication is a natural activity to gain information and exchange ideas for making proper economic decisions. The mechanism of social interaction remains inadequately understood. Previous studies have investigated the consequences of communication under a one-way information transmission mechanism from more-informed leaders to less-informed followers. This mechanism leads to the “boomerang effect”—a reduction in leaders’ forecasting efficiency due to the information transmission. However, communication should be interactive. This paper devises a two-stage interactive cobweb model with a generalized information diffusion process called the bi-directional information diffusion (BID) process, which allows both information transmission and feedback mechanisms. Numerical analysis and simulations show that the model has multiple equilibria and can be dynamically stable under certain conditions. More importantly, the BID process improves forecast efficiency for all individuals, and the boomerang effect disappears if leaders correctly observe followers’ forecasts to revise their expectations.

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Notes

  1. We thank the referee for introducing a set of critical studies related to communication in the literature of game theory and experimental economics.

  2. Cooper et al. (1992) consider two types of coordination game: cooperative coordination game (CCG) and simple coordination game (SCG). The CCG has a cooperative, dominated strategy for both players, while SCG does not.

  3. Technically speaking, adaptive learning is a process where individuals update parameters of a forecasting rule—-perceived law of motion (PLM)—associated with the stochastic process of the variable in question to reach the rational expectations equilibrium (REE) (Bray, 1982; Evans, 1983). This process of convergence to REE requires a condition called the E-stability condition. The E-stability condition determines the stability of the equilibrium in which the parameters of a PLM adjust to the implied actual parameters operating in the model, called the actual law of motion (ALM).

  4. Kaldor (1934) introduced the cobweb model to explain fluctuations of market equilibrium in the agricultural sector, where the production process is affected by a time lag.

  5. In the literature of information acquisition, the seminal paper by Stigler (1961) introduces an information searching model where an imperfectly informed consumer acquires an optimal level of information based on the cost of searching (Stigler, 1962; McCall, 1965; Nelson, 1970). Similarly, Aidt (2000) argues that some voters choose to be less informed due to the high cost of understanding or consuming information, representing the opportunity cost of disentangling the relevant information from unrelated or inaccurate information.

  6. See Honkapohja and Mitra (2003) for an example.

  7. Lucas (1972) derives the reduced form of Eq. (1) based on a monetary model consisting of aggregate supply function, aggregate demand function, and a monetary policy rule.

  8. GLW (2011) empirically investigate the existence of the boomerang effect using surveyed inflation expectations data in the United States between 1978 and 2000. The authors extend the applied statistical work of Granato and Krause (2000) by dividing the inflation expectation survey data into different educational categories. They find a long-run positive relationship between the variance of observational errors from the less-educated group and the mean square error (MSE) of the more educated group’s expectations. The empirical result confirms the existence of the boomerang effect. GLW (2011: 389) also state that “[o]ne implication of this finding pertains to economic policy and economic volatility: because policymakers have more information than the public, the boomerang effect can lead policymakers to make inaccurate forecasts of economic conditions and conduct erroneous policies which contribute to economic instability.”

  9. Kandel and Zilberfarb (1999) find that people do not interpret existing information identically. They show that the hypothesis of identical-information interpretation is rejected using Israeli inflation forecast data.

  10. We assume the following values for the parameters: \(\gamma =2\), \(\mu =0.9\), and \(\sigma _{\eta }^{2}=1\).

  11. The parameters in the simulated model are: \(\beta =0.75,\) \(\gamma =2,\) \(\mu =0.9,\) \(\sigma _{x}^{2}=4\), \(\sigma _{v}^{2}=4,\) \(\sigma _{u}^{2}=4,\) and \(\sigma _{\eta }^{2}=1.\)

References

  • Aidt, T. S. (2000). Economic voting and information. Electoral Studies, 19(2–3), 349–362.

    Article  Google Scholar 

  • Arifovic, J. (1994). Genetic algorithm learning and the cobweb model. Journal of Economic Dynamics and Control, 18(1), 3–28.

    Article  Google Scholar 

  • Babe, R. E. (2018). Communication and the transformation of economics: Essays in information, public policy, and political economy. Routledge.

    Book  Google Scholar 

  • Boussard, J.-M. (1971). A model of the behavior of farmers and its application to agricultural policies. European Economic Review, 2(4), 436–461.

    Article  Google Scholar 

  • Branch, W. A., & Evans, G. W. (2007). Model uncertainty and endogenous volatility. Review of Economic Dynamics, 10(2), 207–237.

    Article  Google Scholar 

  • Branch, W. A., & McGough, B. (2008). Replicator dynamics in a cobweb model with rationally heterogeneous expectations. Journal of Economic Behavior and Organization, 65(2), 224–244.

    Article  Google Scholar 

  • Brandts, J., Cooper, D. J., & Rott, C. (2019). Communication in laboratory experiments. In A. Schram & A. Ule (Eds.), Handbook of research methods and applications in experimental economics. Edward Elgar.

    Google Scholar 

  • Bray, M. (1982). Learning, estimation, and the stability of rational expectations. Journal of Economic Theory, 26(2), 318–339.

    Article  Google Scholar 

  • Camerer, C. F., & Weber, R. A. (2012). Experimental organizational economics. In R. S. Gibbons & J. Roberts (Eds.), The handbook of organizational economics. Princeton University Press.

    Google Scholar 

  • Colucci, D., & Valori, V. (2011). Adaptive expectations and cobweb phenomena: Does heterogeneity matter? Journal of Economic Dynamics and Control, 35(8), 1307–1321.

    Article  Google Scholar 

  • Cooper, R., DeJong, D., Forsythe, R., & Ross, T. (1989). Communication in the Battle of the Sexes game: Some experimental results. Rand Journal of Economics, 20(4), 568–587.

    Article  Google Scholar 

  • Cooper, R., DeJong, D., Forsythe, R., & Ross, T. (1992). Communication in coordination games. Quarterly Journal of Economics, 107(2), 739–771.

    Article  Google Scholar 

  • Crawford, V. P. (2003). Lying for strategic advantage: Rational and boundedly rational misrepresentation of intentions. American Economic Review, 93(1), 133–149.

    Article  Google Scholar 

  • Devenow, A., & Welch, I. (1996). Rational herding in financial economics. European Economic Review, Papers and Proceedings of the Tenth Annual Congress of the European Economic Association, 40(3), 603–615.

    Google Scholar 

  • Ellingsen, T., & Östling, R. (2010). When does communication improve coordination? American Economic Review, 100(4), 1695–1724.

    Article  Google Scholar 

  • Evans, G. W. (1983). The stability of rational expectations in macroeconomic models. In R. Frydman & E. S. Phelps (Eds.), Individual forecasting and aggregate outcomes. Cambridge University Press.

    Google Scholar 

  • Evans, G. W., & Honkapohja, S. (1996). Least squares learning with heterogeneous expectations. Economics Letters, 53(2), 197–201.

    Article  Google Scholar 

  • Evans, G. W., & Honkapohja, S. (2001). Learning and expectations in macroeconomics. Princeton University Press.

    Book  Google Scholar 

  • Evans, G., & McGough, B. (2018). Interest rate pegs in new Keynesian models. Journal of Money, Credit and Banking, 50(5), 939–965.

    Article  Google Scholar 

  • Evans, G., & McGough, B. (2020). Equilibrium stability in a nonlinear cobweb model. Economics Letters, 193, 1091030.

    Article  Google Scholar 

  • Farrell, J. (1987). Cheap talk, coordination, and entry. Rand Journal of Economics, 18(1), 34–39.

    Article  Google Scholar 

  • Farrell, J. (1988). Communication, coordination, and Nash equilibrium. Economics Letters, 27, 209–214.

    Article  Google Scholar 

  • Féménia, F., & Gohin, A. (2011). Dynamic modelling of agricultural policies: The role of expectation schemes. Economic Modelling, 28(4), 1950–1958.

    Article  Google Scholar 

  • Fisher, M., & Robertson, B. (2016). Market expectations of fed policy: A new tool. The Federal Reserve Bank of Atlanta: Notes from the Vault.

  • Fonseca, M. A., & Normann, H.-T. (2012). Explicit vs. tacit collusion—The impact of communication in oligopoly experiments. European Economic Review, 56(8), 1759–1772.

    Article  Google Scholar 

  • Geiger, F., & Sauter, O. (2009). Deflationary vs. inflationary expectations—A new-Keynesian perspective with heterogeneous agents and monetary believes. Working Paper, Nr. 312/2009. Institut fur Volkswirtschaftslehre, Universität Hohenheim.

  • Giannitsarou, C. (2003). Heterogeneous learning. Review of Economic Dynamics, 6(4), 885–906.

    Article  Google Scholar 

  • Granato, J., Guse, E. A., & Wong, M. C. S. (2008). Learning from the expectations of others. Macroeconomic Dynamics, 12(3), 345–377.

    Article  Google Scholar 

  • Granato, J., & Krause, G. A. (2000). Information diffusion within the electorate: The asymmetric transmission of political-economic information. Electoral Studies, 19(4), 519–537.

    Article  Google Scholar 

  • Granato, J., Lo, M., & Wong, M. C. S. (2011). Modeling and testing the diffusion of expectations: An EITM approach. Electoral Studies, 30(3), 389–398.

    Article  Google Scholar 

  • Guse, E. A. (2005). Stability properties for learning with heterogeneous expectations and multiple equilibria. Journal of Economic Dynamics and Control, 29, 1623–1624.

    Article  Google Scholar 

  • Guse, E. A. (2014). Adaptive learning, endogenous uncertainty, and asymmetric dynamics. Journal of Economic Dynamics and Control, 40(2014), 355–373.

    Article  Google Scholar 

  • Honkapohja, S., & Mitra, K. (2003). Learning with bounded memory in stochastic models. Journal of Economic Dynamics and Control, 27(8), 1437–1457.

    Article  Google Scholar 

  • Kaldor, N. (1934). A classificatory note on the determinateness of equilibrium. Review of Economic Studies, 1(2), 122–136.

    Article  Google Scholar 

  • Kandel, E., & Zilberfarb, B.-Z. (1999). Differential interpretation of information in inflation forecasts. Review of Economics and Statistics, 81(2), 217–226.

    Article  Google Scholar 

  • Katz, E., & Lazarsfeld, P. F. (1955). Personal influence. Free Press.

    Google Scholar 

  • Kriss, P. H., & Weber, R. A. (2013). Organizational formation and change: Lessons from economic laboratory experiments. In A. Grandori (Ed.), Handbook of economic organization: Integrating economic and organization theory. Edward Elgar.

    Google Scholar 

  • Lazarsfeld, P. F., Berelson, B., & Gaudet, H. (1944). The people’s choice: How the voter makes up his mind in a presidential campaign. Columbia University Press.

    Google Scholar 

  • Lucas, R. E., Jr. (1972). Expectations and the neutrality of money. Journal of Economic Theory, 4(2), 103–124.

    Article  Google Scholar 

  • Lucas, R. E., Jr. (1973). Some international evidence on output-inflation tradeoffs. American Economic Review, 63(3), 326–334.

    Google Scholar 

  • McCall, J. J. (1965). The economics of information and optimal stopping rules. Journal of Business, 38(3), 300–317.

    Article  Google Scholar 

  • Mitra, S., & Boussard, J.-M. (2012). A simple model of endogenous agricultural commodity price fluctuations with storage. Agricultural Economics, 43(1), 1–15.

    Article  Google Scholar 

  • Mitra, K., Evans, G., & Honkapohja, S. (2019). Fiscal policy multipliers in an RBC model with learning. Macroeconomic Dynamics, 23(1), 240–283.

    Article  Google Scholar 

  • Muth, J. F. (1961). Rational expectations and the theory of price movements. Econometrica, 29(3), 315–335.

    Article  Google Scholar 

  • Muto, I. (2011). Monetary policy and learning from the central bank’s forecast. Journal of Economic Dynamics and Control, 35(1), 52–66.

    Article  Google Scholar 

  • Nelson, P. (1970). Information and consumer behavior. Journal of Political Economy, 78(2), 311–29.

    Article  Google Scholar 

  • Pfajfar, D. (2013). Formation of rationally heterogeneous expectations. Journal of Economic Dynamics and Control, 37(8), 1434–1452.

    Article  Google Scholar 

  • Rabin, M. (1990). Communication between rational agents. Journal of Economic Theory, 51(1), 144–170.

    Article  Google Scholar 

  • Rabin, M. (1994). A model of pre-game communication. Journal of Economic Theory, 63(2), 370–391.

    Article  Google Scholar 

  • Romer, C. D., & Romer, D. H. (2000). Federal reserve information and the behavior of interest rates. American Economic Review, 90(3), 429–457.

    Article  Google Scholar 

  • Schelling, T. C. (1960). The strategy of conflict. Harvard University Press.

    Google Scholar 

  • Stigler, G. J. (1961). The economics of information. Journal of Political Economy, 69(3), 213–225.

    Article  Google Scholar 

  • Stigler, G. J. (1962). Information in the labor market. Journal of Political Economy, 70(5), 94–105.

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to the referee for his/her valuable comments and constructive suggestions on our manuscript.

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Appendices

Appendices

1.1 Appendix 1. The Interactive Model with the n-iterated BID Process

The model presented in Sect. 2 introduces the BID process in an interactive model. The leaders initiate their forecasts, which are observed by the followers for their expectation formation. After the followers form their expectations, the leaders take a second step to observe the followers’ forecasts and use them to revise their expectations. In this appendix, we describe this process in which both leaders and followers exchange their information with each other for n iterations.

1.1.1 The First Iteration of (Basic) BID Process (n = 1)

Similar to the model in Sect. 2, we assume that the leaders initiate their forecasts:

$$\begin{aligned} y_{L,t}^{e_{0}}=hx_{t-1}. \end{aligned}$$
(27)

Then the followers observe the leaders’ initial expectations and form the following expectations:

$$\begin{aligned} y_{F,t}^{e_{1}}&=c\tilde{y}_{L,t}^{e_{0}},\nonumber \\ y_{F,t}^{e_{1}}&=chx_{t-1}+cv_{1,t-1}, \end{aligned}$$
(28)

where \(\tilde{y}_{L,t}^{e_{0}}=y_{L,t}^{e_{0}}+v_{1,t-1}\) and \(v_{t-1}\) is the observational errors for the followers. Finally, to complete the first iteration BID process, the leaders revise their expectations by obtaining the followers’ expectations with observational errors \(\left( u_{t-1}\right) \):

$$\begin{aligned} y_{L,t}^{e_{1}}&=bx_{t-1}+d\tilde{y}_{F,t}^{e_{1}}\nonumber \\ y_{L,t}^{e_{1}}&=\left( b+dch\right) x_{t-1}+d \left( cv_{1,t-1}+u_{1,t-1}\right) , \end{aligned}$$
(29)

where \(\tilde{y}_{F,t}^{e_{1}}=y_{F,t}^{e_{1}}+u_{1,t-1}\).

1.1.2 The Second Iteration of BID Process (n = 2)

Now, we assume that the followers realize that the leaders revise their expectations in Eq. (29). As a result, the followers also further revise their expectations:

$$\begin{aligned} y_{F,t}^{e_{2}}&=c\tilde{y}_{L,t}^{e_{1}}\nonumber \\&=c\left( y_{L,t}^{e_{1}}+v_{2,t-1}\right) \nonumber \\&=\left( bc+dc^{2}h\right) x_{t-1}+dc^{2}v_{1,t-1}+ cv_{2,t-1}+dcu_{1,t-1}, \end{aligned}$$
(30)

where \(\tilde{y}_{L,t}^{e_{i}}=y_{L,t}^{e_{i}}+v_{i+1,t-1},\) for \(i=1,2,\ldots .\) The leaders then revise their expectations according to the followers’ second-stage expectations in Eq. (30):

$$\begin{aligned} y_{L,t}^{e_{2}}&=bx_{t-1}+d\tilde{y}_{F,t}^{e_{2}}\nonumber \\&=\left[ b\left( 1+dc\right) +d^{2}c^{2}h\right] x_{t-1}+d^{2}c^{2} v_{1,t-1}+dcv_{2,t-1}+d^{2}cu_{1,t-1}+du_{2,t-1}, \end{aligned}$$
(31)

where \(\tilde{y}_{F,t}^{e_{i}}=y_{F,t}^{e_{i}}+u_{i,t-1},\) for \(i=1,2,\cdots .\)

1.1.3 The Third Iteration of BID Process (n = 3)

This process will go on for n iterations. To establish a general pattern, we consider the third iteration of the expectation formations for both followers and leaders. In this case, the followers’ expectations can be written as:

$$\begin{aligned} y_{F,t}^{e_{3}}& = {} c\tilde{y}_{L,t}^{e_{2}}\nonumber \\& = {} c\left( y_{L,t}^{e_{2}}+v_{3,t-1}\right) \nonumber \\& = {} \left[ b\left( c+dc^{2}\right) +d^{2}c^{3}h\right] x_{t-1}+d^{2}c^{3}v_{1,t-1}+dc^{2}v_{2,t-1}+cv_{3,t-1}\nonumber \\&\quad+\,d^{2}c^{2}u_{1,t-1}+dcu_{2,t-1}, \end{aligned}$$
(32)

and the leaders’ expectations are:

$$\begin{aligned} y_{L,t}^{e_{3}} &= {} bx_{t-1}+d\tilde{y}_{F,t}^{e_{3}}\nonumber \\& = {} \left[ b\left( 1+dc+d^{2}c^{2}\right) +d^{3}c^{3}h\right] x_{t-1} +d^{3}c^{3}v_{1,t-1}+d^{2}c^{2}v_{2,t-1}+dcv_{3,t-1}\nonumber \\&\quad+d^{3}c^{2}u_{1,t-1}+d^{2}cu_{2,t-1}+du_{3,t-1}. \end{aligned}$$
(33)

1.1.4 The nth Iteration of BID Process

According to Eqs. (28), (30), and (32), we find that the expectations for the followers with n iterations are:

$$\begin{aligned} y_{F,t}^{e_{n}} & = {} c^{n}d^{n-1}hx_{t-1}+b\sum _{i=1}^{n-1} d^{i-1}c^{i}x_{t-1}\nonumber \\&\quad+\sum _{i=1}^{n}d^{n-1}c^{n+1-i}v_{i,t-1}+\sum _{i=1}^{n-1} d^{n-i}c^{n-i}u_{i,t-1}, \end{aligned}$$
(34)

for \(n>1.\) If \(n=1,\) we have \(y_{F,t}^{e_{1}}=chx_{t-1}+cv_{1,t-1}\) presented in Eq. (28). Similarly, we can use Eqs. (29), (31), and (33) to obtain the expectations for the leaders with n iterations, which can be expressed as:

$$\begin{aligned} y_{L,t}^{e_{n}}& = {} d^{n}c^{n}hx_{t-1}+b\sum _{i=1}^{n}d^{i-1}c^{i-1}x_{t-1}\nonumber \\&\quad+\sum _{i=1}^{n}d^{n+1-i}c^{n+1-i}v_{i,t-1}+\sum _{i=1}^{n} d^{n+1-i}c^{n-i}u_{i,t-1}, \end{aligned}$$
(35)

for \(n\ge 1.\)

1.1.5 The Market Expectations and Actual Law of Motions

Suppose that the proportion of followers is \(\mu \) and that of leaders is \(1-\mu \). The market expectations \(E_{t-1}^{n}y_{t}\) can be computed as the weighted average of followers’ and leaders’ expectations from Eqs. (34) and (35):

$$\begin{aligned} E_{t-1}^{n}y_{t} & = {} \mu y_{F,t}^{e_{n}}+\left( 1-\mu \right) y_{L,t}^{e_{n}} \end{aligned}$$
(36)
$$\begin{aligned} E_{t-1}^{n}y_{t} & = {} \left( A+B\right) x_{t-1}+V_{t-1}+U_{t-1,} \end{aligned}$$
(37)

where the components of the coefficient on \(x_{t-1}\), A and B, can be written as:

$$\begin{aligned} A & = {} c^{n}d^{n-1}h\left[ \mu +d\left( 1-\mu \right) \right] ,\\ B & = {} b\left\{ \left[ 1+\mu \left( c-1\right) \right] \sum _{i=1}^{n-1}d^{n-1}c^{n-1}+\left( 1-\mu \right) d^{n-1}c^{n-1}\right\} , \end{aligned}$$

and the composition of noise terms can be expressed as follows:

$$\begin{aligned} V_{t-1} & = {} \left[ \mu +\left( 1-\mu \right) d\right] \left[ \sum _{i=1}^{n} d^{n-1}c^{n+1-i}v_{i,t-1}\right] ,\\ U_{t-1} & = {} \left[ \mu +d\left( 1-\mu \right) \right] \sum _{i=1}^{n-1} d^{n-i}c^{n-i}u_{i,t-1}+\left( 1-\mu \right) du_{n,t-1.} \end{aligned}$$

By substituting the market expectations (Eq. 37) into the cobweb model (1), we have derive the following ALM:

$$\begin{aligned} y_{t}=\left[ \beta \left( A+B\right) +\gamma \right] x_{t-1}+\beta U_{t-1}+\beta V_{t-1}+\eta _{t}, \end{aligned}$$
(38)

where \(A=c^{n}d^{n-1}h\left[ \mu +d\left( 1-\mu \right) \right] ,\) \(B=b\{\left[ 1+\mu \left( c-1\right) \right] \sum _{i=1}^{n-1} d^{n-1}c^{n-1}+\left( 1-\mu \right) d^{n-1}c^{n-1}\},\) \(V_{t-1}=\left[ \mu +\left( 1-\mu \right) d\right] \left[ \sum _{i=1}^{n}d^{n-1}c^{n+1-i}v_{i,t-1}\right] ,\) and \(U_{t-1}=\left[ \mu +d\left( 1-\mu \right) \right] \sum _{i=1}^{n-1}d^{n-i}c^{n-i}u_{i,t-1}+\left( 1-\mu \right) du_{n,t-1.}\)

1.2 Appendix 2. Derivations of T-map and MEE Under the Basic BID Process

In this appendix, we describe the derivations of the T-map for the model with the basic BID process. Note that the form of each group of individuals’ PLM is inconsistent with the ALM. However, Evans and Honkapohja (2001) show that a particular PLM is associated with a projected ALM, which is an ALM projected onto the same class of a particular PLM. It is also “the best description of the process within the permitted class of PLMs considered” (Evans and Honkapohja, 2001: 322). We defined the projected ALM in our model as follows (GGW 2008:350):

Definition For Type-j individuals, where \(j\in \left\{ L,F\right\} \), the Type-j projected ALM is \(T_{j}\left( \phi \right) 'z_{j,t-1}+\epsilon _{t},\) where \(T_{j}\left( \phi \right) '\) is from the linear projection of Eq. (20) on \(z_{j,t-1,}\) \(z_{j,t-1}\) is the information set used in PLM for each type j, and \(\phi \) is a vector representing the parameters used in each PLM.

The projected ALM for the followers can be obtained by computing the following projections according to Eqs. (8) and (15):

$$\begin{aligned} E\left[ FE_{t}^{F}\tilde{y}_{L,t}^{e_{0}}\right]&=0 \end{aligned}$$
(39)

where \(FE_{t}^{F}\) represents followers’ forecast error at time t, which is defined as:

$$\begin{aligned} FE_{t}^{F}=y_{t}-\left( T_{c}\left( hx_{t-1} +v_{t-1}\right) \right) , \end{aligned}$$
(40)

and

$$\begin{aligned} y_{t}=T_{c}\left( hx_{t-1}+v_{t-1}\right) +\epsilon _{t}, \end{aligned}$$

is the projected ALM associated with the PLM of followers. The followers’ forecast error is:

$$\begin{aligned} FE_{t}^{F} & = {} \left[ \beta \left( 1-\mu \right) b+\gamma +\left( \left( \beta \mu +\beta \left( 1-\mu \right) d\right) c-T_{c}\right) h\right] x_{t-1}\nonumber \\&\quad+\left( \left( \beta \mu +\beta \left( 1-\mu \right) d\right) c-T_{c}\right) v_{t-1}+\beta \left( 1-\mu \right) du_{t-1} +\eta _{t}. \end{aligned}$$
(41)

By inserting (41) into (39), we consider the first component of the projected ALM for followers, that is \(E\left( FE_{t}^{F}hx_{t-1}\right) \):

$$\begin{aligned} E\left( FE_{t}^{F}hx_{t-1}\right) =h\left[ \beta \left( 1-\mu \right) b+\gamma +\left( \left( \beta \mu + \beta \left( 1-\mu \right) d\right) c-T_{c}\right) h\right] \sigma _{x}^{2} \end{aligned}$$
(42)

The second component \(E\left( FE_{t}^{L}v_{t-1}\right) \) can be written as:

$$\begin{aligned} E\left( FE_{t}^{F}v_{t-1}\right) =\left( \left( \beta \mu +\beta \left( 1-\mu \right) d\right) c-T_{c}\right) \sigma _{v}^{2} \end{aligned}$$
(43)

Combining both components (42) and (43), we have \(E\left( FE_{t}^{F}\left( hx_{t-1}+v_{t1}\right) \right) \) presented as:

$$\begin{aligned} h\left[ \beta \left( 1-\mu \right) b+\gamma + \left( \left( \beta \mu +\beta \left( 1-\mu \right) d\right) c-T_{c}\right) h\right] \sigma _{x}^{2} +\left( \left( \beta \mu +\beta \left( 1-\mu \right) d\right) c-T_{c}\right) \sigma _{v}^{2}=0. \end{aligned}$$
(44)

Now we let

$$\begin{aligned} a_{1}=\frac{\sigma _{x}^{2}}{\sigma _{v}^{2}} \end{aligned}$$

be the ratio of important information \(x_{t-1}\), and unimportant information \(v_{t-1}\) for the followers. Then we rewrite Eq. (44) as:

$$\begin{aligned} h\left[ \beta \left( 1-\mu \right) b+\gamma +\left( \left( \beta \mu +\beta \left( 1-\mu \right) d\right) c-T_{c}\right) h\right] a_{1}+\left( \beta \mu +\beta \left( 1-\mu \right) d\right) c-T_{c}=0. \end{aligned}$$
(45)

This gives us the T-mapping for c:

$$\begin{aligned} T_{c}=\frac{h\left[ \beta \left( 1-\mu \right) b+\gamma +\left( \left( \beta \mu +\beta \left( 1-\mu \right) d\right) c\right) h\right] a_{1}+\left( \left( \beta \mu + \beta \left( 1-\mu \right) d\right) c\right) }{1+a_{1}h^{2}}. \end{aligned}$$
(46)

Next, we want to consider the linear projection for the leaders’ PLM before they have the followers’ expectations. Using (7) and (15), we have the following linear projection of initial forecasts for the leaders:

$$\begin{aligned} E\left[ FE_{t}^{L_{0}}x_{t-1}\right] =0, \end{aligned}$$
(47)

where:

$$\begin{aligned} FE_{t}^{L_{0}} & = {} y_{t}-T_{h}x_{t-1}\nonumber \\ & = {} \left[ \beta \left( 1-\mu \right) b+\gamma + \left( \beta \mu c+\beta \left( 1-\mu \right) dc\right) h-T_{h}\right] x_{t-1}\nonumber \\&\quad+\,\beta \mu cv_{t-1}+\beta \left( 1-\mu \right) dcv_{t-1}+\beta \left( 1-\mu \right) du_{t-1}+\eta _{t}, \end{aligned}$$
(48)

and

$$\begin{aligned} y_{t}=T_{h}x_{t-1}+\epsilon _{t} \end{aligned}$$
(49)

is the projected ALM associated with the PLM of the leaders before having the followers’ expectations. Equation (47) gives us the T-mapping for h:

$$\begin{aligned} T_{h}=\beta \left( 1-\mu \right) b+\gamma +\left( \beta \mu c +\beta \left( 1-\mu \right) dc\right) h. \end{aligned}$$
(50)

Finally, we consider the linear projected ALM for the leaders’ PLM after observing the followers’ expectations. According to Eqs. (12) and (15), this linear projection is written as follows:

$$\begin{aligned} E\left[ FE_{t}^{L_{1}}\left( x_{t-1},chx_{t-1}+cv_{t-1} +u_{t-1}\right) \right] =0, \end{aligned}$$
(51)

where:

$$\begin{aligned} FE_{t}^{L_{1}} & = {} \left[ \beta \left( 1-\mu \right) b+ \gamma +\beta \mu ch+\beta \left( 1-\mu \right) dch-T_{b} -T_{d}ch\right] x_{t-1}\nonumber \\&\quad+\,\left( \beta \mu +\beta \left( 1-\mu \right) d-T_{d}\right) cv_{t-1}+\left( \beta \left( 1-\mu \right) d-T_{d}\right) u_{t-1}\nonumber \\&+\eta _{t}, \end{aligned}$$
(52)

and

$$\begin{aligned} y_{t}=T_{b}x_{t-1}+T_{d}\left( chx_{t-1}+cv_{t-1}+u_{t-1}\right) +\epsilon _{t} \end{aligned}$$
(53)

is the projected ALM associated with the PLM of the leaders after observing the followers’ expectations. Combining the first component of (51) and (52) gives the following T-mapping for b:

$$\begin{aligned} T_{b}=\beta \left( 1-\mu \right) b+\gamma +\beta \mu ch+ \beta \left( 1-\mu \right) dch-T_{d}ch. \end{aligned}$$
(54)

To obtain the T-mapping for d, we combine the second component of (51) and (52), we have:

$$\begin{aligned} 0 & = {} \left[ \left( ch\right) ^{2}\left( \beta \mu + \beta \left( 1-\mu \right) d-T_{d}\right) +ch \left( \beta \left( 1-\mu \right) b+\gamma -T_{b}\right) \right] \sigma _{x}^{2}\nonumber \\&\quad +c^{2}\left( \beta \mu +\beta \left( 1-\mu \right) d-T_{d}\right) \sigma _{v}^{2}+\left( \beta \left( 1-\mu \right) d-T_{d}\right) \sigma _{u}^{2}. \end{aligned}$$
(55)

We get the following expression by dividing \(\sigma _{v}^{2}\) both sides in (55):

$$\begin{aligned} 0&= {} \left( ch\right) ^{2}\left( \beta \mu +\beta \left( 1-\mu \right) d-T_{d}\right) a_{1}+ch\left( \beta \left( 1-\mu \right) b+\gamma -T_{b} \right) a_{1}\\&\quad +c^{2}\left( \beta \mu +\beta \left( 1-\mu \right) d-T_{d}\right) + \frac{\left( \beta \left( 1-\mu \right) d-T_{d}\right) }{a_{2}}, \end{aligned}$$

where:

$$\begin{aligned} a_{2}=\frac{\sigma _{v}^{2}}{\sigma _{u}^{2}}, \end{aligned}$$

which is the ratio of important information \(v_{t-1}\) to unimportant information \(u_{t-1}\) for the leaders. One can now solve the T-mapping for d:

$$\begin{aligned} T_{d}\left( \left( ch\right) ^{2}a_{1}+c^{2}+\frac{1}{a_{2}}\right) &= \left( ch\right) ^{2}\left( \beta \mu +\beta \left( 1-\mu \right) d\right) a_{1}\\&\quad +\,ch\left( \beta \left( 1-\mu \right) b+\gamma -T_{b}\right) a_{1}\\&\quad +\,c^{2}\left( \beta \mu +\beta \left( 1-\mu \right) d\right) + \frac{\left( \beta \left( 1-\mu \right) d\right) }{a_{2}} \end{aligned}$$

and obtain

$$\begin{aligned} T_{d}=\frac{A}{B}, \end{aligned}$$
(56)

where:

$$\begin{aligned} A & = {} \left( ch\right) ^{2}\left( \beta \mu +\beta \left( 1-\mu \right) d\right) a_{1}+ch\left( \beta \left( 1-\mu \right) b+\gamma -T_{b}\right) a_{1}\\&\quad+\,c^{2}\left( \beta \mu +\beta \left( 1-\mu \right) d\right) +\frac{\left( \beta \left( 1-\mu \right) d\right) }{a_{2}} \end{aligned}$$

and

$$\begin{aligned} B=\left( c^{2}\left( h^{2}a_{1}+1\right) +\frac{1}{a_{2}}\right) . \end{aligned}$$

In equilibrium, the PLM of \(x_{t-1}\) for the leaders’ initial expectation in Eq. (7) should be equivalent to that for the leaders’ revised expectations in Eq. (13) such that \(T_{h}=T_{b}+T_{d}ch.\) As a result, we can, therefore, solve for the mixed expectations equilibrium (MEE) in this model by equating the ALM and a fixed point of the T-map (Guse, 2005). The MEE can be derived according to the following the (unprojected) ALM:

$$\begin{aligned} y_{t}&\,=\left( \beta \bar{b}\left( 1-\mu \right) +\gamma + \beta \mu \bar{c}\bar{h}+\beta \left( 1-\mu \right) \bar{d} \bar{c}\bar{h}\right) x_{t-1}\\&\quad+\beta \mu \bar{c}v_{t-1}+\beta \left( 1-\mu \right) \bar{d}\bar{c}v_{t-1}+\beta \left( 1-\mu \right) \bar{d}u_{t-1}+\eta _{t} {,} \end{aligned}$$

where

$$\begin{aligned} \left( \begin{array}{c} \bar{b}\\ \bar{c}\\ \bar{d}\\ \bar{h} \end{array}\right) =T\left( \begin{array}{c} \bar{b}\\ \bar{c}\\ \bar{d}\\ \bar{h} \end{array}\right) =\left( \begin{array}{c} T_{\bar{b}}\\ T_{\bar{c}}\\ T_{\bar{d}}\\ T_{\bar{h}} \end{array}\right) {.} \end{aligned}$$
(57)

By simplifying \(T_{b}\) in Eq. (54), \(T_{c}\) in Eq. (46), \(T_{d}\) in Eq. (56), and \(T_{h}\) in Eq. (50), we have:

$$\begin{aligned} \bar{b}= \frac{\gamma +\bar{c}\bar{h}\left( \beta \mu +\bar{d}\left( -1+\beta -\beta \mu \right) \right) }{1+\beta \left( -1+\mu \right) } \end{aligned}$$
(58)
$$\begin{aligned} \bar{c} = 1+\frac{\bar{h}-\bar{h}\beta -\gamma }{\bar{h}\beta \mu } \end{aligned}$$
(59)
$$\begin{aligned} \bar{d} = \frac{\bar{c}^{2}\beta \mu a_{2}}{\left( 1+\beta \left( -1+\mu \right) \right) \left( 1+\bar{c}^{2}a_{2}\right) } \end{aligned}$$
(60)
$$\begin{aligned} \bar{h} = \beta \left( 1-\mu \right) \bar{b}+\gamma +\left( \beta \mu \bar{c}+\beta \left( 1-\mu \right) \bar{d}\bar{c}\right) \bar{h} \end{aligned}$$
(61)

Now by plugging Eqs. (58), (59), and (60) into Eq. (61), we can first obtain the expression of \(\bar{h}\), which is a quintic equation. It indicates that the model has up to five real steady states. Given certain numerical parameters for the model, we can, therefore, solve for the MEE of \(\bar{b},\) \(\bar{c}\), and \(\bar{d}\) from Eqs. (58), (59), and (60), respectively. After we obtain the MEE, we can also compute the MSE for leaders’ and followers’ forecasts from Eqs. (25) and (26), respectively.

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Guse, E., Wong, M.C.S. Communication and Learning: The Bilateral Information Transmission in the Cobweb Model. Comput Econ 60, 693–723 (2022). https://doi.org/10.1007/s10614-021-10163-0

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