2.1 Biased expectations

The belief wedges, the measures of biased expectations, suggest that individuals can perceive the future economic situation more optimistically or pessimistically than current overall economic conditions.

Definition

The belief wedge is defined as the difference between the mean unemployment rate forecast one year ahead from the Michigan survey and the VAR unemployment rate forecast—henceforth, VAR Wedge. Here, the VAR forecasts are considered the data-generating measure. I also exploit the unemployment forecast from a SPF conducted by the Federal Reserve Bank of Philadelphia as an alternative to the VAR forecast—henceforth, SPF Wedge.Footnote ⁴

\begin{align*} Belief\ Wedge = Expected\ Unemployment\ of\ Michigan\ Survey\ - VAR\ (or\ SPF)\ Forecast \end{align*}

Results

Figure 2 illustrates the changes in two belief wedges over time, which are similar to the findings of Bhandari et al. (Reference Bhandari, Borovicka and Ho2024). Both VAR and SPF Wedges exhibit a noticeable trend of increasing sharply during economic recessions, suggesting over-pessimistic expectations as a result of negative economic shocks, and gradually decreasing post-recession. In particular, during the COVID-19 crisis, these biased expectations are even stronger than in any previous crisis. Overall, these patterns suggest that individuals hold time-varying biased expectations.

Figure 2. Belief wedges. Notes: The belief wedges are computed based on Bhandari et al. (Reference Bhandari, Borovicka and Ho2024) and Mankiw et al. (Reference Mankiw, Reis and Wolfers2003). This paper does not report VAR Wedge after 2020 as the COVID-19 shock causes a structural break in the VAR model.

2.2 Heterogeneous biased expectations

Belief wedges also suggest that biased expectations vary by age. Figure 3 shows the difference in the belief wedge between the age group “under 65” and “over 65,” especially using the SPF Wedge.

\begin{align*} Difference\ in\ Belief\ Wedge = Belief\ Wedge\ Under\ 65 - Belief\ Wedge\ Over\ 65. \end{align*}

Noticeably, the difference increases sharply during recessions and gradually decreases in the post-recession periods. Hence, it can be inferred that young individuals react more sensitively to recent macroeconomic shocks than older individuals, which causes the young to hold more biased pessimistic expectations than the older during and right after economic downturns. The detailed mechanism for the difference in biased expectations by age is discussed in Section 2.3.

Figure 3. Difference in belief wedge between “Under 65” and “Over 65.”

Now, this paper estimates Equation (1) to verify how biased expectations of each age group, i.e., “Over 65” or “Under 65,” differently affect the economy, especially consumption

(1) \begin{align} PCE_t^{detrend} = \alpha + \sum _{h=0}^{1}\beta _hBW^{over65}_{t-h} + \sum _{h=0}^{1}\gamma _hBW^{under65}_{t-h} + e_t \end{align}

where $PCE_t^{detrend}$ is the detrended personal consumption expenditure using Equation (2)

(2) \begin{align} PCE_t^{detrend} = \frac{PCE_t-PCE_t^{trend}}{PCE_t^{trend}} \times 100 \; (\%) \end{align}.

Here, $PCE_t^{trend}$ is the trend of personal consumption expenditure from HP-filtering. Also, $BW^{over65}$ and $BW^{under65}$ indicate the belief wedges of the age group “Over 65” and “Under 65,” respectively.

For data, personal consumption expenditure is from Bureau of Economic Analysis, and the sample period is from 1980Q1 to 2023Q3 for estimation using SPF Wedge and from 1980Q1 to 2019Q4 for estimation using VAR Wedge.

Table 1 provides estimation results and shows only the young individuals’ belief wedge is inversely related to private consumption. Specifically, the coefficient on $BW^{under65}$ has a statistically significant negative value, which means excessively negative expectations about future unemployment result in a reduction in consumption below the trend. However, the coefficients on $BW^{over65}$ are not statistically significant and have positive signs, making it difficult to derive economic meanings.

Table 1. Effects of biased expectations by age group on consumption

Notes: Standard errors in parentheses, and ^***p $\lt$ 0.01, ^**p $\lt$ 0.05, ^*p $\lt$ 0.1.

Through these empirical findings, this paper argues that older individuals are less overly affected by recent experiences, resulting in less biased expectations and subsequently weaker impacts on economic activities. From this perspective, the impact of biased expectations is expected to decrease as the proportion of older individuals increases.

2.3 Mechanisms for heterogeneous biased expectations

This paper explains heterogeneous biased expectations with a mechanism devised in the learning-from-experience literature. Although previous papers suggest that the deviation from rational expectations may be attributed to sticky or noisy information, rational inattention, etc, these assumptions cannot fully support the empirical findings that the expectations of older individuals are less biased than those of younger individuals. However, the learning-from-experience literature provides clear evidence for that. In particular, the literature highlights that the biased expectation formation rule can be well-captured by the constant gain learning algorithm with a different gain parameter by age.

The following are more details about the learning-from-experience literature. According to Malmendier and Nagel (Reference Malmendier and Nagel2011), individuals tend to assign more weight to recent stock returns when forming their expectations, leading to more optimistic beliefs about future stock returns after experiencing higher returns. Similarly, Malmendier and Nagel (Reference Malmendier and Nagel2016) find that people overweight inflation realized during their lifetimes, especially more recent data when predicting future inflation. Especially, these beliefs are heterogeneous between the young and old since young individuals place more weight on recent data than older individuals. Malmendier and Shen (Reference Malmendier and Shen2024) also show that personal experiences of unemployment, particularly recent ones, have long-lasting effects on consumption decisions. However, this beliefs-based channel is weaker in an old cohort than in a younger cohort.

3.1 Firms

There are identical competitive firms of mass one. Each firm produces goods using capital $K_t$ and labor $H_t$ . The production function is

(3) \begin{align} Y_t = (K_t)^\alpha (X_t H_t)^{1-\alpha } \end{align}

where $0\lt \alpha \lt 1$ . Here, $X_t$ denotes the aggregate labor-augmenting technical progress which evolves via

(4) \begin{align} \ln \left(\frac{X_{t+1}}{X_t}\right) = \ln (\chi _{t+1}) = \ln (\overline{\chi }) + u_{\chi, t+1} \end{align}

where $u_{\chi, t}$ indicates an i.i.d. random variable with zero mean and standard deviation $\sigma _{u_{\chi }}$ . The stochastic process for the evolution of the technological shock is assumed known to the agents.

Firms maximize their profits with factor prices, the real wage $W_t$ and returns to capital $R^k_t$ , as given. The optimality conditions are

(5) \begin{align} W_t = (1-\alpha )K_t^{\alpha }X_t^{1-\alpha }H_t^{-\alpha } = (1-\alpha )\frac{Y_t}{H_t} \end{align}

(6) \begin{align} R^k_t = \alpha K_t^{\alpha -1}(X_t H_t)^{1-\alpha } = \alpha \frac{Y_t}{K_t}\end{align}.

From now on, lowercase letters denote variables normalized with the technology $X_t$ . Then, the normalized production function and optimality conditions are

(7) \begin{align} y_t = (k_t)^\alpha (\chi _t)^{-\alpha }(H_t)^{1-\alpha } \end{align}

(8) \begin{align} w_t = (1-\alpha )(k_t)^\alpha (\chi _t)^{-\alpha }H_t^{-\alpha } = (1-\alpha )\frac{y_t}{H_t} \end{align}

(9) \begin{align} R^k_t = \alpha k_t^{\alpha -1}(\chi _t)^{-\alpha }(H_t)^{1-\alpha } = \alpha \frac{y_t}{k_t} \end{align}.

3.2 Belief updating

Households are assumed to use an economic model to forecast future market-clearing prices. The model relates the wage and returns to capital to the aggregate capital and the distribution of the wealth between young and old households, which are two minimum state variables (MSV) in the model

(10) \begin{align} \hat{R}^k_t = \mu _r + \mu _{rk}\hat{k}_t + \mu _{r\lambda }\lambda _t + e^r_t \end{align}

(11) \begin{align} \hat{w}_t = \mu _w + \mu _{wk}\hat{k}_t + \mu _{w\lambda }\lambda _t + e^w_t \end{align}

(12) \begin{align} \hat{k}_{t+1} = \mu _k + \mu _{kk}\hat{k}_t + \mu _{k\lambda }\lambda _t + e^k_t \end{align}

(13) \begin{align} \lambda _{t+1} = \mu _{\lambda } + \mu _{\lambda k}\hat{k}_t + \mu _{\lambda \lambda }\lambda _t + e^{\lambda }_t \end{align}

where $\lambda _t=\frac{\hat{a}^o_t}{\hat{a}_t}$ represents the distribution of the wealth between young and old households.Footnote ⁶ Here, $a^y_t$ , $a^o_t$ , and $a_t$ are normalized $A^y_t$ , $A^o_t$ , and $A_t$ , which are the assets of aggregate young, old, and total households, respectively, and $\hat{x}$ indicates the log-linearization of the variable $x$ around a balanced growth path (BGP). $e_t$ denotes a regression error or households consider $e_t$ an idiosyncratic disturbance (i.e., a perceived white noise unobserved shock). As in Eusepi and Preston (Reference Eusepi and Preston2011), this paper excludes the technology shock from the household forecasting system since the technology shock is the only disturbance in the model, and thus, households learn quickly, which means they correctly expect market prices right away, if it is included.Footnote ⁷

Equation (10)-(13) can be rewritten in a matrix form as Equation (14)

\begin{equation*} z_t^{\prime} = \begin {pmatrix} \hat {R}^k_t \\ \hat {w}_t \\ \hat {k}_{t+1} \\ \lambda _{t+1} \end {pmatrix},\; x_{t-1} = \begin {pmatrix} 1 \\ \hat {k}_t \\ \lambda _t \end {pmatrix},\; \zeta _{t} = \begin {pmatrix} \mu _{r,t} & \mu _{w,t} & \mu _{k,t} & \mu _{\lambda, t} \\ \mu _{rk,t} & \mu _{wk,t} & \mu _{kk,t} & \mu _{\lambda k,t} \\ \mu _{r\lambda, t} & \mu _{w\lambda, t} & \mu _{k\lambda, t} & \mu _{\lambda \lambda, t} \end {pmatrix},\; e_t^{\prime} = \begin {pmatrix} e^r_t \\ e^w_t \\ e^k_t\\ e^{\lambda }_t \end {pmatrix} \end{equation*}

(14) \begin{align} z_{t} = x^{\prime}_{t-1}\zeta _t + e_t. \end{align}

Under rational expectations, $\mu _{r,t} = \mu _{w,t} = \mu _{k,t} = \mu _{\lambda, t} = 0$ , and the coefficients of the model are time-invariant (i.e., $\mu _{ij,t}=\overline{\mu }_{ij} \quad \text{where} \quad i \in \{r,w,k,\lambda \} \quad \text{and} \quad j \in \{k,\lambda \}$ ). Also, $e^r_t = \overline{\mu }_{r\chi }\hat{\chi }_t$ , $e^w_t = \overline{\mu }_{w\chi }\hat{\chi }_t$ , $e^k_t = \overline{\mu }_{k\chi }\hat{\chi }_t$ , and $e^{\lambda }_t = \overline{\mu }_{\lambda \chi }\hat{\chi }_t$ . In other words, agents in the RE model have a complete understanding of the model and know the individual preferences and technologies of other market participants. Thus, they are aware of the exact relationship among the state variables, technology shocks, and factor prices. Therefore, the technology shocks are included in the forecasting system and the coefficients in Equation (10)–(13) are true and time-invariant. However, under learning, households update the coefficients every period as they observe new data. I employ constant gain recursive least squares estimates as an updating algorithm

(15) \begin{align} \zeta _{t}^i = \zeta _{t-1}^i + g^i(Q^i_t)^{-1}x_{t-1}\underbrace{(z_t - x_{t-1}^{\prime }\zeta ^i_{t-1})}_{\text{Forecast Error}} \end{align}

(16) \begin{align} Q_t^i = Q^i_{t-1} + g^i(x_tx_t^{\prime } - Q^i_{t-1}) \end{align}

where $i \in \{y, o\}$ and $Q^i_t$ is the estimate of the second-moment matrix of regressors.

Equations (15) and (16) imply that the coefficients estimated in the previous period are updated according to the forecast errors produced for the current period. So, households use the previous period’s coefficient estimates when forming expectations, and then, update their coefficients at the end of each period.

The gain parameter $g$ allows the deviation from rational expectations. That is, the larger $g$ , the more weight is given to recent data.Footnote ⁸ Here, young and old households have different gain parameters, i.e., $g^y$ and $g^o$ , based on Malmendier and Nagel (Reference Malmendier and Nagel2016)’s empirical findings.Footnote ^{9 ,} Footnote ¹⁰ In particular, young individuals are assumed to put more weight on recent data than old individuals, resulting in a larger gain parameter for the former, i.e., $g^y$ is greater than $g^o$ . Collin-Dufresne et al. (Reference Collin-Dufresne, Johannes and Lochstoer2017) also assume that the gain parameter of the young agents is five times larger than that of the old agents.Footnote ¹¹ Since young and old households have different weighting schemes on the past data, their adaptive expectations about the factor prices can be distinct from each other. These differences between age groups finally produce the novel dynamics of the macro-variables rather than assuming a representative agent.

There is one more assumption worth mentioning. In my model, agents do not know what other agents expect, and thus, their expectations are not affected by others. If young households are aware of old households’ expectations or vice versa, they can internalize the impacts of updating their beliefs on the economy. So, households quickly learn and find the true evolution of wages and interest rates since the household learning behavior does not create forecast errors.

3.3 Households

This paper assumes an economy with overlapping generations following Gertler (Reference Gertler1999), Gali (Reference Gali2021), and Blanchard (Reference Blanchard1985). As explained in Section 3.2, households have heterogeneous expectations depending on their age due to the different rules of forming expectations. Therefore, I index households according to their age, i.e., young (workers) or old (retirees) households. The size of the population is constant and normalized to one. Each individual has the constant probability $\gamma$ of surviving into the following period, independently of their age and economic status. Moreover, each worker faces the constant probability $1-v$ of becoming old and retired permanently. This probability is also independent of their age and economic status. Consequently, the size of young individuals or workers at any time is the constant $\phi = \frac{1-\gamma }{1-v\gamma } \in (0,1]$ and that of old individuals or retirees is $1-\phi = \frac{\gamma (1-v)}{1-v\gamma }$ .

This study adopts the perfect annuity market assumption introduced by Gertler (Reference Gertler1999) and Gali (Reference Gali2021), where agents are insured against the risk of death. More specifically, households have an annuity contract with a perfectly competitive insurance company that issues payments proportional to the household’s financial wealth. Upon death, the household’s wealth is transferred to the insurance company. The surviving households receive all returns in this market.Footnote ¹² Furthermore, an insurance market is introduced to mitigate the risk of income loss during retirement. The complete market assumption makes the model more tractable, especially in terms of aggregating household consumption.

3.3.1 Old households (retirees)

The old household or retiree of cohort ‘ $a$ ’ is the agent who retired ‘ $a$ ’ quarters ago.Footnote ¹³ Each agent maximizes the following Bellman equationFootnote ¹⁴

(17) \begin{align} V^o(A_{a,t}^o, A^o_t, A_t, K_t) = \text{Max} \{ \ln C^o_{a,t} + \gamma \beta \tilde{E}^o_t V^o(A^o_{a+1,t+1}, A^o_{t+1}, A_{t+1}, K_{t+1})\} \end{align}

and the budget constraint for old households is

(18) \begin{align} C_{a,t}^o + \gamma A^o_{a+1,t+1} = R_t A^o_{a,t} + S_t \end{align}

where $\beta$ is a time discount factor, $\tilde{E_t}$ indicates subjective expectations for the future, and the superscript $o$ denotes old or retired households. Also, $C^o_{a,t}$ is consumption, and $A^o_{a,t}$ is the asset that old households hold at the beginning of time $t$ . As mentioned above, only survivors receive all the returns, and households who die are paid nothing. The real interest rate $R_t$ satisfies $R_t = R^k_t + 1 - \delta$ where $\delta$ is the depreciation rate of capital due to the absence of arbitrage between loans and capital. $S_t$ is the social security benefit that old households receive. The state variables { $A^o_t, A_t, K_t$ }—aggregate old household asset, total asset, and total capital—are used when households expect the future wages and returns to capital, which is discussed in Section 3.2 in detail. Households know only their own objectives, constraints, and beliefs as in Eusepi and Preston (Reference Eusepi and Preston2011).

Then, Euler equation is

(19) \begin{align} (C_{a,t}^o)^{-1} = \beta \tilde{E}^o_t [(C_{a+1,t+1}^o)^{-1}R_{t+1}] \end{align}

and the Euler equation and budget constraint are normalized with technology as follows

(20) \begin{align} (c_{a,t}^o)^{-1} = \beta \tilde{E}^o_t [(c_{a+1,t+1}^o)^{-1}\chi ^{-1}_{t+1}R_{t+1}] \end{align}

(21) \begin{align} c_{a,t}^o + \gamma a^o_{a+1,t+1} = R_t a^o_{a,t}\chi _t^{-1} + s_t \end{align}.

To get the aggregate consumption of old households, this paper derives the intertemporal budget constraint (IBC) from the one-period budget constraint, Equation (21). After that, the IBC is log-linearized around a BGP and then I aggregate each cohort’s consumption using Euler equation, Equation (20). Finally, we obtain log-linearized aggregate consumption of old households

(22) \begin{align} \hat{c_t}^o = \; &\eta _{ay}(1-v)\hat{a}_{t-1}^y + \eta _{ao}\hat{a}^o_t + \eta _{r\chi }(\hat{R}_t-\hat{\chi }_t) + \eta _s\hat{s}_t \\ \notag & -\eta ^e_{r}\underbrace{\tilde{E}^o_t\sum ^{\infty }_{h=0}(\gamma \beta )^h\hat{R}_{t+1+h}}_{\text{EPV of Returns to Capital}} +\eta ^e_{s}\tilde{E}^o_t\sum ^{\infty }_{h=0}(\gamma \beta )^h\hat{s}_{t+1+h} \end{align}

where $\eta$ ’s consist of primitive model parameters.Footnote ¹⁵ Note that Equation (22) shows how current and expected variables, such as real interest rates, affect the consumption decisions of old households.

3.3.2 Young households (workers)

The young household or worker of cohort ‘ $b$ ’ is the agent who started to work (was born) ‘ $b$ ’ quarters ago. Each agent maximizes the following Bellman equation

(23) \begin{align} V^y(A^y_{b,t}, A^y_t, A_t, K_t) = & \,\text{Max} \{ \ln C^y_{b,t} + \theta \ln (1-H_{b,t}) + \\\notag & \gamma \beta \tilde{E}^y_t[v V^y(A^y_{b+1,t+1}, A^y_{t+1}, A_{t+1}, K_{t+1}) \\\notag & + (1-v) V^o(A^{yo}_{b+1,t+1}, A^y_{t+1}, A_{t+1}, K_{t+1})]\} \end{align}

and the budget constraint for young households is

(24) \begin{align} C^y_{b,t} + v\gamma A^y_{b+1,t+1} + (1-v)\gamma A^{yo}_{b+1,t+1} = R_tA^y_{b,t} + W_tH_{b,t} - T_t \end{align}

where the superscript $y$ denotes young households or workers, $C^y_{b,t}$ is consumption, and since there are two possible future states, young households save $A^y_{b+1,t+1}$ for staying young and $A^{yo}_{b+1,t+1}$ for retiring next period. Complete asset markets insure young households against the risk of retirement. $W_t$ is the real wage, $H_{b,t}$ is the labor supply, and $T_t$ is the lump-sum tax. The state variables { $A^y_t, A_t, K_t$ }—aggregate young household asset, total asset, and total capital—are used when young households expect the future wages and returns to capital, which is discussed in Section 3.2 in detail.

Young households solve the maximization problem, Equation (23), subject to the budget constraint, Equation (24). Then, the normalized Euler equations, labor supply condition, and budget constraint of young households are as follows

(25) \begin{align} (c_{b,t}^y)^{-1} = \beta \tilde{E}^y_t [(c_{b+1,t+1}^y)^{-1}\chi ^{-1}_{t+1}R_{t+1}] \end{align}

(26) \begin{align} (c_{b,t}^y)^{-1} = \beta \tilde{E}^y_t [(c_{b+1,t+1}^o)^{-1}\chi ^{-1}_{t+1}R_{t+1}] \end{align}

(27) \begin{align} \frac{\theta c^y_{b,t}}{1-H_{b,t}} = w_t \end{align}

(28) \begin{align} c^y_{b,t} + v\gamma a^y_{b+1,t+1} + (1-v)\gamma a^{yo}_{b+1,t+1} = R_ta^y_t\chi ^{-1}_t + w_tH_{b,t} - \tau _t \end{align}.

Finally, we obtain log-linearized aggregate consumption of young households using the same way to get aggregate consumption of old households

(29) \begin{align} \hat{c}^y_t =& \; \psi _{ay}\hat{a}^y_t + \psi _{r\chi }(\hat{R}_t-\hat{\chi }_t) + \psi _w\hat{w}_t-\psi _{\tau }\hat{\tau }_t - \psi ^e_{\tau } \tilde{E}^y_t\sum ^{\infty }_{h=0}(v\gamma \beta )^h\hat{\tau }_{t+1+h} \\ \notag & + \psi ^e_w \underbrace{ \tilde{E}^y_t\sum ^{\infty }_{h=0}(v\gamma \beta )^h\hat{w}_{t+1+h}}_{\text{EPV of Wages}} - \psi ^e_{r}\underbrace{\tilde{E}^y_t\sum ^{\infty }_{h=0}(v\gamma \beta )^h\hat{R}_{t+1+h}}_{\text{EPV of Returns to Capital}} \\ \notag & -(1-v)\gamma \psi ^e_{ro}\underbrace{ \tilde{E}^y_t\sum ^{\infty }_{h=1}(\frac{1-v^h}{1-v})(\gamma \beta )^h\hat{R}_{t+1+h}}_{\text{EPV of Returns to Capital after Retiring}} \\ \notag & +(1-v)\gamma \psi ^e_{s} \tilde{E}^y_t\sum ^{\infty }_{h=1}(\frac{1-v^h}{1-v})(\gamma \beta )^h\hat{s}_{t+h} \end{align}

where $\psi$ ’s consist of primitive model parameters.Footnote ¹⁶ Note that Equation (29) shows how current and expected variables such as the real interest rates and real wages affect the consumption decisions of young households. In particular, the last two terms in Equation (29) appear since young households or workers also consider they can lose their jobs and become retirees at any time in the future with the probability of $1-v$ .

3.4 Government

The government levies lump-sum taxes $T_t$ on young households or workers and consumes $G_t$ . Also, it pays retirees social security benefits $S_t$ each period. Thus, the government satisfies the following budget constraint

(30) \begin{align} \phi T_t= G_t + (1-\phi )S_t \end{align}.

The government maintains its policy variables, $T_t$ , $G_t$ , and $S_t$ , at steady-state levels, which is announced in advance. In particular, the policy announcement is credible, so agents believe there will be no policy changes in the future. These assumptions are based on Mitra et al. (Reference Mitra, Evans and Honkapohja2013).

3.5 Market clearing

Goods and asset market-clearing conditions are as follows:

(31) \begin{align} Y_t = C_t + I_t + G_t \end{align}

(32) \begin{align} C_t = C_t^y + C_t^o \end{align}

(33) \begin{align} K_{t+1} = (1-\delta )K_{t} + I_t \end{align}

(34) \begin{align} A_t = K_t \end{align}

(35) \begin{align} A_t = A_t^y + A_t^o \end{align}.

3.6 Model parameter calibration

Table 2 presents calibrated model parameter values. The RBC model parameters such as $\alpha$ , $\beta$ , and $\delta$ are set as commonly used values in literature like King and Rebelo (Reference King and Rebelo2000), etc. The capital share $\alpha$ is 1/3, and the time discount factor $\beta = 0.995$ corresponds to the annual discount rate of 2 percent. The depreciation rate $\delta = 0.025$ matches to the annual depreciation rate of 10 percent. The proportion of young households $\phi = 0.8$ is set based on the relevant data. Specifically, the average ratio of the population aged 15–64 to the population aged 15 and older from 2014 to 2023 is 80.3 percent. The probability of surviving in the next period $\gamma$ is 0.9959 following Gali (Reference Gali2021)’s calibration in which he uses lifetime expectancy.Footnote ¹⁷ Then, this paper assigns 0.9989 to the probability of staying workers in the following period $v$ to set the young households’ share to 80 percent ( $\phi =0.8$ ). The weight on utility from leisure $\theta$ is 3.39 which satisfies the labor supply condition of young households in the steady state. Most importantly, the gain parameters for old and young households, $g^o=0.010$ and $g^y=0.018$ , are derived from Malmendier and Nagel (Reference Malmendier and Nagel2016) in which they estimate the gain parameter by age using inflation expectation data. For robustness checks, I also employ various gain parameter values, which is discussed in Section 4.1. The standard deviation of the technology shock $\sigma _{u_{\chi }}$ is calibrated to 0.0078 to match the volatility of output in the model to that in data as seen in Table 4. The growth of productivity in the steady state $\bar{\chi }$ is 1.0053 based on Eusepi and Preston (Reference Eusepi and Preston2011), and the labor supply in the steady state $\bar{H}$ is 1/4.

Table 2. Model parameter calibration

4.1 LC model under learning vs LC model under RE

The simulation process for obtaining the impulse responses of macro-variables is as follows. First of all, the 2000-period simulation under RE provides initial steady-state coefficients for the household forecasting model, Equation (10) – (13), and the simulation data are discarded. Then, the coefficients are updated every period in the learning model but keep their initial values in the RE model. Next, an N-period impulse response is obtained with a 1 percent permanent technology shock in the period 2001 and no more shocks after that. Then, this simulation is repeated 2,000 times, and I report the median response of model variables to the technology shock for IRF. The simulation methods are based on the work of Eusepi and Preston (Reference Eusepi and Preston2011) and Mitra et al. (Reference Mitra, Evans and Honkapohja2013).

4.1.1 Effects on aggregate variables

Figure 4 illustrates the responses of output, consumption, investment, and hours to a 1 percent permanent technology shock. Then, the upper left panel reveals that the household sensitivity to recent observations in the learning model generates amplification effects on output but not in the RE model due to no biased expectations under RE. We can interpret this amplification as households become overly optimistic about the future economy after the positive economic shock. Particularly, the output response exhibits a hump-shaped profile as found in Eusepi and Preston (Reference Eusepi and Preston2011) and Cogley and Nason (Reference Cogley and Nason1995).Footnote ¹⁸ The upper right panel shows the consumption response in the learning model is less than that in the RE model immediately after the shock, as households have overly optimistic expectations about the future returns to capital, and thus, increase investment sharply, as depicted in the lower left panel. However, in the long run, households in the learning model consume more than those in the RE model by utilizing their over-savings. The lower right panel suggests an overshooting in labor supply under learning.

Figure 4. Impulse responses of macrovariables to 1 percent permanent technology shock. Notes: The solid and dotted lines denote the median, 25th, and 75th percentile impulse response under learning, respectively. The dashed line indicates the impulse responses under RE. Unshocked BGP signifies the initial balanced growth path prior to the permanent technology shock.

Mechanism

To explicate the mechanisms of the heterogeneous biased expectations in the learning model, I provide a graphical representation in Figure 5. This figure illustrates the young and old households’ expected present values (EPV) of returns to capital and labor after a positive technology shock under learning and RE. For example, the EPV of returns to capital and labor for young households under learning and RE are

\begin{equation*} \text {Under Learning:} \quad \tilde {E}^y_t\sum ^{\infty }_{h=0}(v\gamma \beta )^h \hat {R}_{t+1+h} \quad \text {and} \quad \tilde {E}^y_t\sum ^{\infty }_{h=0}(v\gamma \beta )^h \hat {w}_{t+1+h} \end{equation*}

\begin{equation*} \text {Under RE:} \quad E_t\sum ^{\infty }_{h=0}(v\gamma \beta )^h\hat {R}_{t+1+h} \quad \text {and} \quad E_t\sum ^{\infty }_{h=0}(v\gamma \beta )^h\hat {w}_{t+1+h} \end{equation*}

where each term appears in the young households’ aggregate consumption decision rule. Here, the only difference between learning and RE models is the way of forming expectations, i.e., $\tilde{E}$ and $E$ .

Then, the EPV of market prices in the learning model is significantly different from those in the RE model: i.e., the expected returns to capital are higher, while wages are lower under learning than under RE. This is because households in the learning model have biased expectations due to their overweight on recent observations, which causes forecast errors. In particular, young households have more optimistic expectations about the future returns to capital than old households, primarily because of their relatively higher sensitivity to recent shocks. However, in the RE framework, both young and old households have almost the same expectations since no different weighting schemes and heterogeneous biased expectations exist between the young and old.

Figure 5. Expected present value of returns to capital and labor.

Consumption, investment, and hours

The mechanism above gives further explanations about the dynamics of the macro-variables illustrated in Figure 4. The households’ overly positive expectations regarding future returns to capital result in lower consumption and higher investment right after the shock, followed by an increase in consumption in the future through savings. Thus, households smooth their consumption through investment. Also, young households seek to work more today to offset their overly pessimistic wage expectations and to boost their investment and future consumption.

Oscillations in expectations

As seen in Figure 5, the household learning behavior that places more weight on recent observations results in oscillations in expectations, which can lead to fluctuations in economic activities. The paper argues that these oscillations in expectations can create further volatility in business cycles, a topic that is discussed in Section 4.3 with more details.

Persistence

An important distinction between the learning and RE models is whether persistent effects occur. As shown in Figure 4, the responses in the learning model are more persistent than those in the RE model. This persistence arises from the household learning behavior illustrated in Figure 5. Since households assign more significance to recent data, it takes a considerable amount of time for expectations under learning to approach those under RE. In other words, the persistence in the learning model is caused by adjustments in beliefs. However, in the absence of additional technology shocks, household expectations will converge to rational expectations as forecasting errors gradually diminish.

4.1.2 Effects on variables by age group

Based on the impulse responses depicted in Figure 6, it can be inferred that the biased expectations of young households play a crucial role in the amplified response of macroeconomic variables. The upper panels display the response of consumption by age group in the learning and RE model. As shown in the left panel, young households in the learning model exhibit a greater reaction to the technology shock than old households, owing to their relatively higher sensitivity to the shock. Conversely, the right panel indicates no discernible difference between young and old households’ consumption in the RE model since rational expectations do not generate forecast errors or biased expectations. The lower panels also show that heterogeneous biased expectations cause young and old households to respond differently in terms of saving, particularly with young households engaging in more over-saving.

Figure 6. Impulse responses of consumption and saving to 1 percent permanent technology shock by age group.

Figure 7 denotes the response of output to a 1 percent permanent technology shock in the learning model by the proportion of young households in the economy $\phi$ : 0.7 (=70 percent), 0.8 (=80 percent), and 0.9 (=90 percent). Consistent with previous results, a decrease in the proportion of young households, which corresponds to an increase in the share of the old population, results in a smaller response of output. Therefore, it can be concluded that the amplification effects of biased expectations become weaker in an aging society since population aging leads to a higher proportion of old households who exhibit lower sensitivity to recent shocks.

Figure 7. Impulse response of output by proportion of young households ( $\phi$ ).

Gain parameters

To ensure the robustness of the results, this paper explores how the response of output to the positive technology shock varies as the gain parameters change, which is shown in Figure 8. When the gain parameters of young and old households, i.e., $g^y$ and $g^o$ , are zero, then the response under learning is identical to the one under RE. The zero gain parameters imply no biased expectations in the economy. Thus, the household expectations under learning match rational expectations. The left panel also illustrates that the response of output increases as the gain parameters rise. Larger gain parameters signify households react more sensitively to recent shocks, leading to more significant amplification effects. The right panel suggests an increase in the gain parameter of young households and a decrease in that of old households result in a larger response than the opposite case. This is because young households account for a larger share of the population and have biased expectations not only about future returns to capital but also future wages.

Figure 8. Impulse response of output under learning by gain parameter.

4.2 LC model under learning vs RA model under learning

I compare the LC learning model featuring heterogeneous beliefs and adaptive expectations with the RA learning model incorporating adaptive learning and homogeneous beliefs to offer a more comprehensive perspective. In particular, by comparing the RA and LC learning models, I show the life-cycle assumption better explains the data in the real world.

4.2.1 Representative agent learning model

I modify some parts of the LC learning model to obtain the RA learning model.

Firms

The firm sector in the RA learning model is assumed to be the same as in the LC learning model.

Households

An infinitely-lived representative agent maximizes the following Bellman equation

(36) \begin{align} V(A^{RA}_{t}, K^{RA}_{t}) = \text{Max} \{\ln C^{RA}_{t} + \theta \ln (1-H^{RA}_{t}) + \beta \tilde{E}^{RA}_t[V(A^{RA}_{t+1}, K^{RA}_{t+1})]\} \end{align}

and their budget constraint is

(37) \begin{align} C^{RA}_{t} + A^{RA}_{t+1} = R^{RA}_tA^{RA}_{t} + W^{RA}_tH^{RA}_{t} - T^{RA}_t \end{align}

where $\tilde{E}^{RA}_t$ is the subjective expectation or belief by the representative agent, $C^{RA}_{t}$ is consumption, $A^{RA}_{t}$ is the asset, $W^{RA}_t$ is the real wage, $H^{RA}_{t}$ is the labor supply, and $T^{RA}_t$ is the lump-sum tax.

Then, this paper obtains log-linearized aggregate consumption of the representative agent using the intertemporal budget constraint and Euler equation

(38) \begin{align} \bar{c}\hat{c}^{RA}_t = \; & \frac{(1-\beta )\bar{a}}{(1+\theta )\beta }\hat{a}^{RA}_t + \Big [\bar{c}-\frac{\bar{w}}{(1+\theta )}\Big ](\hat{R}^{RA}_t-\hat{\chi }^{RA}_t) \\\notag & + \frac{(1-\beta )\bar{w}}{(1+\theta )}\hat{w}^{RA}_t - \frac{(1-\beta )\bar{\tau }}{(1+\theta )}\hat{\tau }^{RA}_t \\\notag & - \frac{\beta (\bar{w}^-\bar{\tau })}{(1+\theta )}\tilde{E}^{RA}_t\sum ^{\infty }_{h=0}\beta ^h[\hat{R}^{RA}_{t+1+h}-\hat{\chi }^{RA}_{t+1+h}] \\\notag & + \frac{(1-\beta )\beta \bar{w}}{(1+\theta )}\tilde{E}^{RA}_t\sum ^{\infty }_{h=0}\beta ^h\hat{w}^{RA}_{t+1+h} \\\notag & - \frac{(1-\beta )\beta \bar{\tau }}{(1+\theta )}\tilde{E}^{RA}_t\sum ^{\infty }_{h=0}\beta ^h\hat{\tau }^{RA}_{t+1+h} \end{align}.

For the comparison, log-linearized aggregate consumption in the LC learning model, which is the summation of Equation (22) and (29), is as follows

(39) \begin{align} \bar{c}\hat{c}_t = \; & \Big [\frac{(1-\gamma \beta )(1-v)}{\beta }+\psi ^{^{\prime}}v\Big ]\bar{a}^y\hat{a}^y_t + \frac{(1-\gamma \beta )}{\beta }\bar{a}^o\hat{a}^o_t \\\notag & + \Big [\bar{c}-\frac{\psi ^{^{\prime}}\phi \beta \bar{w}}{(1-v\gamma \beta )}\Big ](\hat{R}_t-\hat{\chi }_t) + \psi ^{^{\prime}}\phi \beta \bar{w}\hat{w}_t - \psi ^{^{\prime}}\phi \beta \bar{\tau }\hat{\tau }_t \\\notag & - \Big [\frac{\psi ^{^{\prime}}\phi v \gamma \beta ^2(\bar{w}-\bar{\tau })}{(1-v\gamma \beta )}\Big ]\tilde{E}^y_t \sum ^{\infty }_{h=0}(v\gamma \beta )^h[\hat{R}_{t+1+h}-\hat{\chi }_{t+1+h}] \\\notag & + \psi ^{^{\prime}}\phi v \gamma \beta ^2\bar{w}\tilde{E}^y_t\sum ^{\infty }_{h=0}(v\gamma \beta )^h\hat{w}_{t+1+h} \\\notag & - \psi ^{^{\prime}}\phi v \gamma \beta ^2\bar{\tau }\tilde{E}^y_t\sum ^{\infty }_{h=0}(v\gamma \beta )^h\hat{\tau }_{t+1+h} \end{align}

where

\begin{equation*} \psi ^{^{\prime}}=\frac {(1-v\gamma \beta )(1-\gamma \beta )}{(1+\theta )\beta (1-\gamma \beta )+(1-v)\gamma \beta ^2} \end{equation*}.

Belief updating

The representative agent is assumed to use an economic model to forecast future wages and returns to capital. The model relates the wage and returns to capital to the aggregate capital which is the unique state variable in the RA model

(40) \begin{align} \hat{R}^{k,RA}_t = \mu ^{RA}_r + \mu ^{RA}_{rk}\hat{k}^{RA}_t + e^{r,RA}_t \end{align}

(41) \begin{align} \hat{w}^{RA}_t = \mu ^{RA}_w + \mu ^{RA}_{wk}\hat{k}^{RA}_t + e^{w,RA}_t \end{align}

(42) \begin{align} \hat{k}^{RA}_{t+1} = \mu ^{RA}_k + \mu ^{RA}_{kk}\hat{k}^{RA}_t + e^{k,RA}_t \end{align}

where $e_t$ denotes a regression error.

Gain parameter

The gain parameter for the representative agent is calibrated as $g^{RA}=0.0164$ [= $g^o\times (1-\phi )+g^y\times \phi =0.010\times 0.2+0.018\times 0.8$ ] based on the Malmendier and Nagel (Reference Malmendier and Nagel2016)’s estimation.

4.2.2 Main differences between LC and RA learning model

Differences in model

The LC and RA learning models have similar aggregate consumption equations, Equation (38) and (39), but with some notable differences. First, the LC learning model has one more state variable: other than aggregate capital ( $\hat{k}_t$ ), the distribution of wealth between young and old households ( $\hat{a}^y_t$ and $\hat{a}^o_t$ ) is also necessary to determine aggregate consumption. Thus, there is one more explanatory variable in the household forecasting system in the LC model. Second, Table 3 compares the coefficients and time discount factors on the expected terms from both LC and RA learning models and suggests that the differences in consumption behavior between the two models mostly stem from whether households consider the probability of surviving ( $\gamma$ ) and staying workers ( $v$ ). In the LC learning model, households know they may not survive into the next period or retire in the following period, which leads to greater discounting of the future and so smaller coefficients and time discount factors on the expected terms. On the other hand, the representative household in the RA learning model places relatively greater weight on the expectations about market prices as they do not consider the possibility of death or retirement in the future.

Table 3. Coefficients and time discount factors on expected terms in aggregate consumption

Results

Figure 9 displays the impulse responses in the LC and RA learning models. Despite their overall similarities, the responses in the RA learning model are found to be 1.5 to 2 times larger than those in the LC learning model. This is because the representative household places much more weight on their overly optimistic expectations after a positive shock, in comparison to households in the LC model. However, this poses a challenge as the excessive over-reactions of the representative household do not align with the actual data observed in the real world. Section 4.3 presents quantitative results about this issue.

Figure 9. Impulse responses of output, consumption, investment, and hours to 1 percent permanent technology shock in LC and RA learning model.

4.3 Effects on volatility of business cycles

This paper utilizes the LC learning model to examine the impact of population aging on the recent reduction in business cycle fluctuations.

Simulation method

The 2000-period simulation under RE presents initial steady-state coefficients for the household forecasting system, Equation (10)–(13), and then the simulated data are discarded. After that, the 300-period simulation is conducted to match the sample size for the U.S. data from 1947Q1 to 2019Q4. Here, I do not include the COVID-19 period data due to unprecedentedly high volatility. Finally, I calculate the volatility of business cycles using this 300-period simulated data.

Results

There are several findings from the relative standard deviations of the macroeconomic variables provided by the data and model simulations in Table 4. First, the amplification effects in the LC learning model produce macroeconomic variables’ variations that are almost identical to the data when the proportion of young households $\phi$ is at its baseline value of 0.8 (=80 percent).Footnote ¹⁹ However, the RA learning model generates more substantial fluctuations due to the extensive over-response of the representative household as seen in Section 4.2.Footnote ²⁰ This is because the representative household does not consider the probability of death and losing a job, and so they discount over-expectation less than households in the LC model, which triggers extensive over-responses. Second, the LC learning model shows that the fluctuations in business cycles decrease as the population ages. For instance, when the proportion of young households $\phi$ decreases from 0.9 (=90 percent) to 0.7 (=70 percent), the relative standard deviation of output decreases from 1.77 to 1.49.Footnote ²¹ Lastly, the RE models, regardless of the RA or LC assumption, can only explain half of the volatility of the macro-variables seen in the data.

Table 4. Relative standard deviations of macro-variables from model simulations

Notes: All macro-variables are logged and detrended with the HP filter. $Pr$ stands for productivity.

Implications

The findings above imply a meaningful role of heterogeneous biased expectations and population aging in the recent moderation of business cycle fluctuations. The stylized facts suggest population aging is associated with a reduction in the fluctuations of the macro-variables. Table 5 indicates that the volatility of macroeconomic variables has decreased sharply in recent years. Specifically, the standard deviation of GDP, Consumption, and Investment has declined 74.2 percent, 59.6 percent, and 57.9 percent respectively from the 1980s to 2010s. Moreover, during the same period, the population has aged fast. The ratio of the population aged 65 and over to the population aged 15 and over in the U.S. has risen from 14.6 percent in 1999 to 20.2 percent in 2019 according to OECD.stat. In line with these facts, the LC learning model simulations also show a 10 percentage point increase in the old population ratio leads to about a 16 percent decrease in output volatility.Footnote ²²

Table 5. Standard deviations of annual growth rates of macro-variables from U.S. Data

Notes: This table is the extended version of the table in Stock and Watson (Reference Stock and Watson2003). The parenthesis ( ) indicates the volatility excluding the data for the Great Recession from 2007Q4 to 2009Q2. I exclude the COVID-19 period data due to unprecedentedly high volatility.

It is a novel channel that the increase in the proportion of old individuals who are relatively less sensitive to recent observations contributes to the lower volatility of business cycles. Literature lists the improved monetary policy, regulatory changes, financial market innovation, etc. as the causes of the decline in macroeconomic volatility since the 1990s. Jaimovich and Siu (Reference Jaimovich and Siu2009) also claim that demographic changes account for 1/5 to 1/3 of the moderation of the US economy. However, they mainly point out the channel that the low volatility of old individuals’ employment and hours worked leads to the Great Moderation.

5.1 Population aging and effects of government spending

Figure 10 illustrates the impact of a government spending shock on output, where the government spending is 5 percent of the steady-state output level and is financed by an equal amount of lump-sum tax increase.Footnote ²³ Here, the government spending shock is a one-time surprise shock. Specifically, normalized government spending $g_t$ evolves via

\begin{equation*} g_t = \begin {cases} \bar {g} + 0.05\bar {y}, & \text {for} \; t=1 \\ \bar {g}, & \text {for} \; t=2,3,4,\cdot \cdot \cdot \\ \end {cases} \end{equation*}

where $\bar{y}$ and $\bar{g}$ are the steady-state values for output and government spending.

Then, the key takeaway is that the responsiveness of output to the government spending shock is inversely related to the proportion of old households in the economy. Since old households have relatively lower sensitivity to the recent government spending shock, the amplification effects become weak in an aging society, which is consistent with the findings from the technology shock analysis shown in Figure 7.

I also note that after government spending, the learning model generates an output response that is the opposite of the response from the RE model. The government spending shock decreases consumption and investment and increases hours worked, causing a rise in returns to capital and a decline in wages. In the learning model, households overreact to increased capital returns and raise investment dramatically. However, in the RE model, households know returns to capital will decrease immediately, so they reduce investment. Therefore, the output responses show opposite reactions.

Figure 10. Impulse response of output to government spending shock by proportion of young households ( $\phi )$ .

Figure 11 provides the cumulative government spending multipliers in the first, second, and third year after the shock using the impulse responses in Figure 10. The cumulative government spending multiplier ( $M$ ) is defined as

(43) \begin{align} M_T = \frac{\sum \limits ^T_{i=0} (\beta \gamma )^i(y_{t+i} - \bar{y})}{g_t - \bar{g}} \end{align}

where $T$ is the cumulation period. Overall, the government spending multiplier declines about 10 percent when the old population ratio rises 10 percentage points. This result is consistent with previous studies, which empirically and theoretically show that the government spending multiplier is low in an aging society [e.g., Basso and Rachedi (Reference Basso and Rachedi2021) and Honda and Miyamoto (Reference Honda and Miyamoto2020)]. However, my model offers a novel factor for the lower government spending multiplier: i.e., heterogeneous biased expectations between the young and old.

Figure 11. Cumulative government spending multiplier by proportion of young households ( $\phi )$ .

5.2 Welfare analyses of government spending

This paper investigates the welfare effects of government spending in the LC learning and LC RE model by conducting welfare experiments and finds the implications for an aging society.

5.2.1 Welfare experiments

This paper conducts welfare experiments under the following scenarios to study the welfare effects of government spending. Specifically, during 500 periods, a random technology shock with mean zero and standard deviation $\sigma ^u$ takes place every period in the LC learning and LC RE model.Footnote ²⁴ Then, I assume two economies in each LC learning and LC RE model. In the first economy, the government counteracts negative technology shocks by increasing its spending.Footnote ²⁵ However, in the second economy, the government does not respond to the technology shocks. By comparing the welfare of the population in these two economies, we can estimate the welfare effects of the government spending policy.

In this setting, government spending shocks can be beneficial to households by reducing the adverse effects of negative technology shocks. In particular, it can be mitigated more in the learning model due to the amplification effects of biased expectations.

Equivalent variations

Following literature such as Hunt (Reference Hunt2021), this paper estimates the welfare effects of government spending using equivalent variations (EV) which are calculated by:

(44) \begin{align} \text{RE:}\quad \sum ^{T}_{t=1}(\beta \gamma )^{t-1}U(C^{NonGov}_{RE,t}(1+\Delta ^{Gov}_{RE}))=\sum ^{T}_{t=1} (\beta \gamma )^{t-1}U(C^{Gov}_{RE,t}) \end{align}

(45) \begin{align} \text{Learning:}\quad \sum ^{T}_{t=1}(\beta \gamma )^{t-1}U(C^{NonGov}_{Learning, t}(1+\Delta ^{Gov}_{Learning}))=\sum ^{T}_{t=1}(\beta \gamma )^{t-1}U(C^{Gov}_{Learning, t}) \end{align}

where $C_t^{NonGov}$ and $C_t^{Gov}$ indicate consumption in the economy without and with the government spending policy, respectively. Also, the welfare experiment length $T$ is 500. The interpretation of Equation (44) and (45) is that the EV equates the utility of household in the first economy with the government spending policy to the utility of household in the second economy without government spending plus a fraction $\Delta$ . In other words, for example, the household in the second economy without government spending needs to consume $\Delta ^{Gov}\%$ more every period to have the same welfare as the household has in the first economy with government spending. As a result, $\Delta ^{Gov}_{RE}\%$ and $\Delta ^{Gov}_{Learning}\%$ represent the welfare effects of government spending in the LC RE and LC learning model.

5.2.2 Welfare effects of government spending

Table 6 provides details of the welfare effects of government spending in the LC learning model, denoted by ( $\Delta ^{Gov}_{Learning}\%$ ), across varying proportions of young households. As the proportion of old households increases, both young and old households experience a decline in welfare due to the weakened amplification effects. However, young households experience a more significant decline in their welfare compared to that of old households. With an increase in the proportion of old households, fewer individuals have a high degree of optimism triggered by government spending. Consequently, the economy exhibits relatively poor performance after the government spending shock. Then, young households respond more strongly to this weaker economic performance by reducing consumption and increasing labor supply, while old households are less responsive. Thus, the welfare of young households deteriorates more significantly as the population ages.

Table 6. Equivalent variations under learning by proportion of young households ( $\phi$ )

Table 7 presents the welfare effects of government spending in the LC learning and LC RE models ( $\Delta ^{Gov}_{Learning}\%$ and $\Delta ^{Gov}_{RE}\%$ , respectively). The welfare effects are more significant in the LC learning model compared to the LC RE model since the biased optimism induced by government spending amplifies the output effects, resulting in an overall improvement in households’ welfare. However, this learning behavior only benefits young households, while it reduces the welfare of the old households. Young households or workers in the LC learning model increase labor supply and investment, leading to higher future consumption. This excessive consumption, funded by savings, causes an improvement in young households’ welfare, despite the disutility caused by the increased labor supply. However, the young’s over-consumption crowds out the consumption of old households, causing it to fall below the RE level, resulting in a reduction in the old’s welfare.

Table 7. Equivalent variations under learning and rational expectation (RE)

Notes: In this experiment, $\phi$ is 0.8. The last column in bold shows the welfare effects of the household expectations under learning relative to under RE.