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\newtheorem{prop}{Proposition}

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\begin{document}
	
\begin{center}
\textbf{Online Appendix for\\ Unexpected Longevity, Intergenerational Policies and Fertility}
\vspace{0.5in}

Jisoo Hwang and Seok Ki Kim 
\end{center}


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\appendix


\begin{figure}
	\includegraphics[width=\linewidth]{try5}
	\caption{Expected and Unexpected Changes in Life Expectancy at Age 65 in the U.S.}
	\label{fig:exp65_USA_5}
	\begin{minipage}[t]{1.0\textwidth}%
		\begin{spacing}{0} \footnotesize \textit{Notes.} $e65_{t}$ is the life expectancy at age 65 calculated using age-specific mortality rates in year $t$. The change in life expectancy at age 65 from year $t-5$ to $t$, the difference between the solid and dashed lines, can be decomposed as $e65_{t} - e65_{t-5} = \{e65_{t} - \widehat{e65}_{t|t-5}\} + \{\widehat{e65}_{t|t-5} - e65_{t-5}\}$. The first term is the unexpected component and the second term is the expected component. The forecast of life expectancy 65 at $t$ made 5 years ago ($\widehat{e65}_{t|t-5}$) is obtained by applying the Lee-Carter model to past 30 years of age-specific mortality rates. Mortality rate data are from the Human Mortality Database. See Section \ref{m-sec:data} for more details.
		\end{spacing}
	\end{minipage}
\end{figure}

\clearpage
\begin{figure}
	\includegraphics[width=\linewidth]{covid_5_1}
	\caption{Expected and Unexpected Changes in Life Expectancy at Age 65 during COVID-19}
	\label{fig:covid_5}
	\begin{minipage}[t]{1.0\textwidth}%
		\begin{spacing}{0} \footnotesize \textit{Notes.} The change in life expectancy at age 65 from year $t-5$ to $t$ are decomposed such that $e65_{t} - e65_{t-5} = \{e65_{t} - \widehat{e65}_{t|t-5}\} + \{\widehat{e65}_{t|t-5} - e65_{t-5}\}$, where the first term is the unexpected component and the second term is the expected component. The forecast of life expectancy 65 at $t$ made 5 years ago ($\widehat{e65}_{t|t-5}$) is obtained by applying the Lee-Carter model to past 30 years of age-specific mortality rates. Mortality rate data are from the Human Mortality Database except South Korea, for which we use data from Statistics Korea. See Section \ref{m-sec:data} for more details.
		\end{spacing}
	\end{minipage}
\end{figure}


\clearpage
\begin{figure}
	
	\centering
	\begin{subfigure}[b]{0.49\textwidth}
		\includegraphics[width=\textwidth]{off_3gen}
		\caption{Result Corresponding to Case II}
		\label{fig:off_3gen}	
	\end{subfigure}
	\begin{subfigure}[b]{0.49\textwidth}
		\includegraphics[width=\textwidth]{off_3gen_l}
		\caption{Larger Increase in Survival Probability}
		\label{fig:off_3gen_l}		
	\end{subfigure}
	\caption{Offspring Size Response to Longevity Shock, Three-generations Model}
	\label{fig:resp_g3}
	\begin{minipage}[t]{1.0\textwidth}%
		\begin{spacing}{0} \footnotesize \textit{Notes.} See \autoref{sec:appendix, 3gen}.
		\end{spacing}
	\end{minipage}	
\end{figure}

\clearpage
\begin{figure}
	
	\centering
	\begin{subfigure}[b]{0.49\textwidth}
		\includegraphics[width=\textwidth]{off_family}
		\caption{Offspring Size}
		\label{fig:off_family}	
	\end{subfigure} \\
	\begin{subfigure}[b]{0.49\textwidth}
		\includegraphics[width=\textwidth]{oldagesp_family}
		\caption{Old-Age Spending}
		\label{fig:oldagesp_family}		
	\end{subfigure}
	\begin{subfigure}[b]{0.49\textwidth}
		\includegraphics[width=\textwidth]{familysp_family}
		\caption{Family Spending}
		\label{fig:familysp_family}		
	\end{subfigure}
	\caption{Response to Longevity Shock, Model with Public Family Spending}
	\label{fig:resp_family}
	\begin{minipage}[t]{1.0\textwidth}%
		\begin{spacing}{0} \footnotesize \textit{Notes.} See \autoref{sec:appendix, family}.
		\end{spacing}
	\end{minipage}	
\end{figure}	

\clearpage
\begin{landscape}
	\begin{threeparttable}	 
		\centering
		\caption{Effect of Unexpected Longevity, Using Further Lags}
		\label{tab:further lag reg}
		\begin{tabularx}{\linewidth}{Xcccccc}
			\toprule
			&\multicolumn{2}{c}{$\Delta$Total fertility rate}&\multicolumn{2}{c}{$\Delta$Public old-age spending}&\multicolumn{2}{c}{$\Delta$Public family spending}\\
			\cmidrule(l){2-3} \cmidrule(l){4-5}  \cmidrule(l){6-7}
			& 2-period & 3-period & 2-period & 3-period & 2-period & 3-period \\
			\midrule
			
			\addlinespace
			Unexpected change in e65& -6.36* & -5.19* & 14.19** & 13.35* & -14.28** & -19.32***  \\
			& (3.20) & (2.87) & (6.81) & (6.82) & (6.56) & (6.44)  \\
			Expected change in e65& -3.74 & -1.81 & 15.74* & 14.90 & -17.51** & -19.74** \\
			&  (4.26) & (3.17) & (9.00) & (9.09) & (7.81) & (9.03) \\
			L.growth(GDP per capita)& -0.02 & -0.27*** & 0.04 & 0.20* & 0.05 & -0.17 \\
			&  (0.10) & (0.08) & (0.17) & (0.11) & (0.22) & (0.32)  \\
			L.diff(old dependency ratio)& 0.32 & -0.37 & -0.77 & -2.09 & 4.86* & -0.25 \\
			& (0.98) & (0.99) & (1.03) & (1.42) & (2.51) & (2.39)   \\
			L.diff(young dependency ratio)& -0.85** & -0.81 & -0.26 & -0.26 & -1.79 & -1.32 \\
			& (0.38) & (0.49) & (0.74) & (1.10) & (1.25) & (0.93) \\
			L.diff(female LFP rate)& 0.63* & 0.55* &  &  &  & \\
			& (0.36) & (0.28) &  &  &  & \\
			\addlinespace
			Country FE             &         Yes   &         Yes   &         Yes   &         Yes   & Yes & Yes \\
			Year FE             &         Yes   &         Yes   &         Yes   &         Yes   &         Yes & Yes   \\
			Dependent variable mean& -0.39 & -0.50 & 5.83 & 5.63 & 8.59 & 8.34  \\
			N                   &   568 & 442 & 665 & 645 & 665 & 645   \\
			
			\bottomrule
		\end{tabularx}
		\begin{tablenotes}[normal, flushleft]
			\item \footnotesize \textit{Notes.} Unexpected and expected change in life expectancy at age 65 (e65) are defined as in equation \eqref{m-eq:exp65 decompose} with $n=5$. All variables are averaged over five-year periods. L. denotes ten-year lags in ``2-period'' columns and fifteen-year lags in ``3-period'' columns. See Table \ref{m-tab:sum} for variable definitions and data sources. Robust standard errors in parentheses. * $p<0.10$, ** $p<0.05$, *** $p<0.01$
		\end{tablenotes}
	\end{threeparttable}	
\end{landscape}


\clearpage
\section{Proofs}\label{sec:appendix, proofs}
\subsection{Interior Solution}\label{sec:app_inner}
We omit time denotation in this section and let $p'$ be the expectation of next period's survival rate. Unless noted otherwise, tax parameter $\tau$ is equal to $0$ without loss of generality. The utility maximization of the young with inactive public pension the next period can be rewritten as
\begin{align*}
	\max_{c^y, n, c^o} \log (c^y) + \gamma \log(n) + \beta p' \log (c^o).
\end{align*}
The budget constraint becomes
\begin{align*}
	w - f n - \frac{p'}{1+r}c^o - c^y \geq 0.
\end{align*}
The first order conditions are
\begin{align*}
	c^y &: \frac{1}{c^y} = \lambda \\
	n &: \frac{\gamma}{n} = f \lambda \\
	c^o &: \beta  \frac{1}{c^o} = \lambda \frac{1}{1+r}.
\end{align*}
Plug in the conditions above into the budget constraint,
\begin{align*}
	w - f \frac{\gamma}{f} c^y - c^y - \frac{p'}{1+r} [ c^y(1+r)\beta ] = 0.
\end{align*}
Reorganizing in terms of $c^y$,  we define 
\begin{align*}
	c^{y*} :=  \frac{w}{1+\gamma+{p'}{\beta}}.
\end{align*}

Let $p'=p_b$ before the increase in survival probability. We assume that agents are at an interior solution ($c^y \in (0,\bar{c^y})$) with inactive public pension before the increase. Agents choosing an interior solution implies that the following two conditions hold.

First, $c^{y*}$ should be less than the threshold level of consumption $\bar{c^y}$, 
\begin{align}
	\label{eq:app_1st}
	c^{y*} < \bar{c^y}.
\end{align}

Second, the utility from the former term in (\ref{m-eq:two}) should be larger than the latter. The maximum utility from the former can be attained with $c^{y*}$. We denote the indirect utility function $U_1(p')$ and it is defined as
\begin{align*}
	U_1(p') = \log(\frac{w}{1+\gamma+\beta p'}) + \gamma \log(\frac{\gamma}{f} \frac{w}{1+\gamma+\beta p'}) + \beta p' \log(w\frac{1+r}{p'}[1-\frac{1+\gamma}{1+\gamma+\beta p'}]).
\end{align*}
An interior solution with $p'=p_b$ implies that the indirect utility function of the latter term in (\ref{m-eq:two}) has a negative slope in young consumption around $p_b$. Thus, the maximum utility-generating $c^y$ in the domain of the latter term in (\ref{m-eq:two}) is $\bar{c^y}$. Denote this indirect utility function $U_2(p')$ and it is defined as 
\begin{align*}
	U_2(p') = \log(\frac{1}{1+\gamma}(w-\theta\frac{p'}{1+r})) + \gamma \log(\frac{\gamma}{f(1+\gamma)}[w-\theta\frac{p'}{1+r}]) + \beta p' \log(\theta + \bar{\eta}).
\end{align*}
The second condition can be rewritten as
\begin{align}
	\label{eq:app_2nd}
	U_1(p_b) > U_2(p_b).
\end{align}


\subsection{Proof of Proposition 1}
\begin{proof}
	$\theta$ governs not only the threshold of savings for public pensions but also the speed at which $\bar{c}$ decreases with $p$. Thus, we need an assumption on the upper bound of $\theta$. We assume that the parameters are such that they satisfy the condition
	\begin{align*}
		\frac{\beta (1+r) w}{\beta+\sqrt{(1+\gamma)^2+(1+\gamma)\beta}} > \theta \\
		\Leftrightarrow \frac{\beta^2}{1+\gamma+\beta} > \frac{(1+\gamma)\theta^2}{[(1+r)w-\theta]^2}.
	\end{align*}	
	
	Note that the two indirect utility functions above are twice differentiable. Since $U_1(p_b) > U_2(p_b)$ and $U_1(p_e) = U_2(p_e)$, 
	\begin{align*}
		U_1(p_e) - U_1(p_b) = \int_{p_b}^{p_e} U'_1(x)dx < \int_{p_b}^{p_e} U'_2(x)dx = U_2(p_e) - U_2(p_b).
	\end{align*}	
	This implies that there exists $\hat{p} \in [p_b, p_e]$ such that 
	\begin{align}
		\label{eq:app_phat}
		U_1'(\hat{p}) < U_2'(\hat{p}). 
	\end{align}	
	
	Also from the assumption, 
	\begin{align*}
		U_1''(p') = - \frac{\beta^2}{1+\gamma+\beta p'} < - \frac{(1+\gamma)\theta^2}{[(1+r)w-\theta p']^2} = U_2''(p')
	\end{align*}
	for $p' \leq 1$. This implies that 
	\begin{align}
		\label{eq:app_dd}
		U_1'(p') = U_1'(\hat{p}) + \int_{\hat{p}}^{p'} U_1''(x)dx < U_2'(\hat{p}) + \int_{\hat{p}}^{p'} U_2''(x)dx = U_2'(p')  
	\end{align}	
	for any $p'>\hat{p}$. Note that this holds for $p'\geq p_e$ because  $\hat{p} \leq p_e$.
	
	From Equations (\ref{eq:app_phat}) and (\ref{eq:app_dd}), therefore, for any $p' \geq p_e$,
	\begin{align*}
		U_1(p') = U_1(p_e) + \int_{p_e}^{p'} U_1'(x)dx < U_2(p_e) + \int_{p_e}^{p'} U_2'(x)dx = U_2(p').
	\end{align*}
	
\end{proof}

\subsection{Proof of Proposition 3}
\begin{proof}
	Suppose that tax is levied at time $t$. By definition, $p_e$ satisfies $U_{1}(p_{e})-U_{2}(p_{e})=0$, i.e.,
	\begin{align*}
		&\log(\frac{w-\tau}{1+\gamma+\beta p_e}) + \gamma \log(\frac{\gamma}{f} \frac{w-\tau}{1+\gamma+\beta p_e}) + \beta p_e \log((w-\tau)\frac{1+r}{p_e}(1-\frac{1+\gamma}{1+\gamma+\beta p_e})) \\
		&- \log( \frac{1}{1+\gamma}(w-\tau-\theta\frac{p_e}{1+r}) ) - \gamma \log(\frac{\gamma}{f(1+\gamma)} (w-\tau-\theta\frac{p_e}{1+r})) - \beta p_e \log(\theta + \bar{\eta}) = 0.
	\end{align*}
	Let us denote this function as $F(p_e,\tau) = 0$. Its partial differentiation with respect to $\tau$ is 
	\begin{align*}
		F_\tau &= -(1+\gamma+\beta p')\frac{1}{w-\tau}+(1+\gamma)\frac{1}{w-\tau-\theta\frac{p'}{1+r}} \\
		&= -\frac{1}{c^{y*}} + \frac{1}{\bar{c^y}}.
	\end{align*}
	Since $c^{y*} < \bar{c^y}$ around $p_e$,
	\begin{align*}
		F_\tau < 0.
	\end{align*}
	
	From the proof of Proposition 1, the partial differentiation of $F$ with respect to $p_e$ is negative. The implicit function theorem implies that 
	\begin{align*}
		\frac{dp_e}{d\tau} = -\frac{F_\tau}{F_{p_{e}}} < 0.
	\end{align*}
\end{proof}

\subsection{Proof of Proposition 4}
\begin{proof}
	Suppose that agents change their survival probability expectation from $p_b$ to $p_u$. From the definition of $p_u$ as in equation (\ref{m-eq:pu}), $p_u$ is a function of $p_b$.  If we express $\bar{c^y}$ in terms of $p_b$, it equals the optimal consumption under $p_b$
	\begin{align*}
		\bar{c^y}|_{p'=p_u} =  \frac{w}{1+\gamma+{p_b}{\beta}}.
	\end{align*}
	Note that $\bar{c^y}$ is the minimum consumption level which can activate pension under a certain survival probability expectation and $p_u$ is also the minimum survival probability which activates pension if agents save according to $p_b$. 
	
	If agents would choose the former term in equation (\ref{m-eq:two}) over the latter term when expected survival probability increases from $p_b$ to $p_u$, then $p_u < p_e$. From the assumed condition
	
	\begin{align*}
		&\Big( \frac{1+\gamma+\beta p_b}{1+\gamma+\beta p_u} \Big)^{(1+\gamma)} > \Big( \frac{(\theta+\bar{\eta})(1+\gamma+\beta p_u)}{(1+r)\beta w} \Big)^{\beta p_u} \\[20pt]
		\Longleftrightarrow {   }&U_1(p_u) = \log(\frac{w}{1+\gamma+\beta p_u}) + \gamma \log(\frac{\gamma}{f} \frac{w}{1+\gamma+\beta p_u}) + \beta p_u \log(\frac{w(1+r)\beta}{1+\gamma+\beta p_u}) > \\
		&\log(\frac{w}{1+\gamma+\beta p_b}) + \gamma \log(\frac{\gamma}{f} \frac{w}{1+\gamma+\beta p_b}) + \beta p_u \log(\theta + \bar{\eta})= U_2(p_u).
	\end{align*}
	Since $U_1(p_u) > U_2(p_u)$, $p_u < p_e$.
	
\end{proof}


\section{Three-generations Model}\label{sec:appendix, 3gen}
This section shows that the main result, Case II, can be replicated in a modified model with an additional generation. 
We introduce a middle-aged $(m)$ generation between the young $(y)$ and the old $(o)$, where generation is denoted by superscripts. Young agents become middle-aged with certainty, while middle-aged agents become old with survival probability $p$. The middle-aged generation receives wage, is subject to taxation, and makes consumption-savings decision, but unlike the young, it does not make a fertility choice. We relax the log utility function by assuming CRRA (Constant Relative Risk Aversion) utility function. 

The old generation solves the same utility maximizing problem as in the two-generations model (equation (2) in the paper). They consume the total returns from savings plus public pensions if they are eligible. Their optimal choices are 
\begin{align*}
	c_t^o = (1+r)s_{t-1}^m + \eta_t,
\end{align*}
where the intergenerational public transfer $\eta_t$ can be either $\bar{\eta}$ (a fixed amount) or 0 depending on the size of total returns from savings as in the main model.

Given this, the middle-aged generation solves the following utility maximizing problem:
\begin{align*}
	\max_{c_t^m} \frac{(c_t^m)^{1-\lambda}}{1-\lambda} + \beta p_{t+1|t}^e \frac{(c_{t+1}^o)^{1-\lambda}}{1-\lambda}
\end{align*}
s.t.
\begin{align*}
	c_t^m + s_t^m \leq w-\tau_t - (1+r)s_{t-1}^y,
\end{align*}
where $p_{t+1|t}^e$ is expectation at $t$ about $t+1$ survival probability.

The young generation’s problem is as follows: 
\begin{align*}
	\max_{c_t^y, n_t} \frac{(c_t^y)^{1-\lambda}}{1-\lambda} + \gamma \frac{(n_t)^{1-\lambda}}{1-\lambda}  + 	\beta \frac{(c_{t+1}^m)^{1-\lambda}}{1-\lambda} + \beta^2 p_{t+2|t}^e \frac{(c_{t+2}^o)^{1-\lambda}}{1-\lambda}
\end{align*}
s.t.
\begin{align*}
	c_t^y + fn_t + s_t^y \leq w-\tau_t,
\end{align*}
where $p_{t+2|t}^e$ is expectation at $t$ about $t+2$ survival probability.

Tax revenue from the young and the middle generations finance pension payments to the old generation. The government budget constraint is: 
\begin{align*}
	p_t \eta_t  = \tau_t n_{t-2}(1+n_{t-1})
\end{align*}
where $n_{t-1}$ and $n_{t-2}$ are the size of the young and middle generation, respectively. 

With the three-generations model, we perform simulations of an increase in survival probability at period 5 in both expected and unexpected economies. A key difference with the main model is the timing of information arrival in the expected case. Because it now takes two periods (instead of one) for the young to become old, the information of an expected increase in survival probability arrives at period 3 rather than at period 4. The series of actual survival probability is
\begin{align*}
	p_1 = p_2 = p_3 = p_4 = \underline{p} < \bar{p} = p_5 = p_6 = ... 
\end{align*}

The series of survival probability expectations are 
\begin{align*}
	\begin{array}{l}
		\text{Expected longevity:  } \begin{cases}
			$(middle) $ p_{2|1}^e = p_{3|2}^e  = p_{4|3}^e  = \underline{p} < \bar{p} = p_{5|4}^e = p_{6|5}^e  = ... \\
			$(young) $ p_{3|1}^e = p_{4|2}^e  =  \underline{p} < \bar{p} = p_{5|3}^e = p_{6|4}^e =p_{7|5}^e  = ...
		\end{cases} \\
		\text{Unexpected longevity: } \begin{cases}
			$(middle) $ p_{2|1}^e = p_{3|2}^e  = p_{4|3}^e = p_{5|4}^e = \underline{p} < \bar{p} = p_{6|5}^e  = ... \\
			$(young) $ p_{3|1}^e = p_{4|2}^e  = p_{5|3}^e = p_{6|4}^e = \underline{p} < \bar{p} = p_{7|5}^e  = ...
		\end{cases}
	\end{array}
\end{align*} 

Despite the inclusion of an additional generation, we get similar results to Case II in the main model. Figure \ref{fig:off_3gen} shows offspring size (consumption) of the young from the three-generations model which corresponds to Case II, assuming $\lambda=2$.\footnote{We get qualitatively similar results with the log function.} In the expected economy, young and middle-aged agents reduce their consumption beforehand from period 3 to prepare for the upcoming increase in survival probability at period 5 so they need not receive pensions when old. In the unexpected economy, however, old agents at period 5 suddenly become poor and eligible for public pensions due to the longevity shock, and this creates a tax burden for the young and the middle generations at period 5. Young and middle-aged agents choose not to depend on pensions, however, but to increase savings. As a result, the pension system is not activated when they become old; they do not pass the burden of old-age support to their children's generation. Both generations reduce consumption (for young agents, the offspring size as well) more dramatically than in the expected economy with only the life-cycle effect. 

The response to longevity gains can be more diverse in the three-generations model. The middle-aged generation (but not the young) at period 5 may decide to rely on pensions when the longevity shock is large. Figure \ref{fig:off_3gen_l} illustrates such a case. When there is a large, unexpected increase in survival probability at period 5, middle-aged agents at period 5 choose to rely on pensions rather than increasing savings to prepare for the change in longevity. In this case, like the old agents, middle-aged agents pass the burden of old-age support to their children's generation. The public pension continues to be active at period 6. Young agents at period 6 need to finance public pensions, like the young agents at period 5. Thus, offspring size falls below what the life-cycle effect suggests at both periods 5 \textit{and} 6. The young generation at period 5, however, does not depend on pensions. They know of the increase in survival probability earlier in life than their parents' and grandparents' generations. They have two periods of consumption-savings choices to make before retirement, which they can adjust to prepare for longevity. They choose to increase savings unlike the previous generations given unexpected longevity, and hence the burden is not passed down to their descendants.

\section{Model with Public Family Spending}\label{sec:appendix, family}
This section shows that the main result, Case II, can also be replicated in a model with public family spending. To introduce public family spending within our framework, we modify the model by changing how the government levies tax. Most other assumptions and notations remain as in Section \ref{m-sec:model}. There are two generations---young and old---in the economy and only the young receive income $w$ and pay taxes. 

Now we assume that young agents pay lump-sum tax $T$. It is predetermined and fixed. Government uses tax revenue for two purposes: old-age spending and family spending. Old-age spending is a means-tested public pension $\eta_{t+1}$ as in Section \ref{m-sec:model}. Family spending is a childcare subsidy, which reduces the cost associated with childrearing. $b_t$ denotes the subsidy per child. We assume that the size of old-age spending is determined first. If public pension is not active, therefore, all tax revenue would be used for family spending. With an active public pension, however, family spending must be reduced to support old-age spending because tax revenue per young agent is constant.

With after-tax income $w-T$, young agents decide how much they would spend on their own consumption $c_t$, savings $s_t$, and their children $(f-b_t)n_t$ where $n_t$ is the size of offspring and $f$ is the cost per unit of offspring. The subsidy directly reduces the cost associated with childrearing and can thus raise fertility. The means-tested pension for the old is assumed as in equation \eqref{m-eq:eta}. The problem for the old generation does not change; their optimal choice can be represented by equating both sides of the budget constraint, \eqref{m-eq:bc_o}.

\begin{align*}
	c_{t+1}^o=\frac{(1+r)s_t}{p_{t+1}}  +\eta_{t+1}.
\end{align*}

Given this, the young generation’s problem is as follows:

\begin{align*}
	U_t(c^y_t, n_t, c^o_{t+1} ) = \log (c^y_t) + \gamma \log (n_t) + \beta p_{t+1}^e \log (c^o_{t+1}),
\end{align*} 
s.t.
\begin{align*}
	c^y_t + (f-b_t)n_t + s_t \leq w-T.
\end{align*}


At each period t, government spending consists of two parts. One is funding the old-age pension $\eta_t {p_t}/{n_{t-1}}$ and the other is family spending $b_t n_t$. They are financed by tax revenue from the young generation $T$. The government budget constraint becomes

\begin{align*}
	T=b_t n_t+ \frac{\eta_t p_t}{n_{t-1}} .
\end{align*}


The size of the two government expenditures are determined in the following order. We assume that old-age spending is determined first and then the remaining tax revenue is assigned to family spending. With an inactive public pension $(\eta_t=0)$, therefore, all tax revenue is spent on family spending $(T=b_t n_t)$. If public pension becomes active $(\eta_t=\bar{\eta})$, family spending decreases because tax revenue is fixed $(T>b_t n_t)$.

With the modified model, we perform simulations of an increase in survival probability at period 5 in expected and unexpected economies. We assume the same timing of information arrival as in Section \ref{m-sec:model}. The series of actual survival probability is

\begin{align*}
	p_1 = p_2 = p_3 = p_4 = \underline{p} < \bar{p} = p_5 = p_6 = ... 
\end{align*}


The series of survival probability expectations are
\begin{align*}
	\begin{cases}
		\text{Expected longevity:  } p_2^e = p_3^e  = p_4^e = \underline{p}  < \bar{p} = p_5^e  = p_6^e  = ... \\
		\text{Unexpected longevity: } p_2^e = p_3^e  = p_4^e  = p_5^e = \underline{p} < \bar{p} = p_6^e  = ...
	\end{cases}
\end{align*} 

We get results similar to Case II of the main model. Figure \ref{fig:off_family} shows offspring size (consumption) of the young from the modified model. The patterns are very similar to the main result in the paper; expected longevity decreases offspring size a period earlier but unexpected longevity decreases fertility more severely. From period 6, offspring size is the same in the two economies. 

The logic is analogous to the main model except that the “policy effect” from increased tax burden in the unexpected economy is replaced with that from reduced childcare subsidy. In the expected economy, young agents at period 4 can prepare for longevity in advance by saving more and having fewer children. With unexpected longevity, however, old agents become poor enough to receive public pensions. 

Figure \ref{fig:oldagesp_family} shows that public pension is active only in the unexpected economy at period 5. This tightens the government budget constraint and family spending is forced to decrease to support the pension system. In Figure \ref{fig:familysp_family}, family spending falls only in the unexpected economy at period 5. Old-age spending crowds out family spending when there is unexpected longevity shock. Reduced childcare subsidy puts extra downward pressure on young agents’ fertility on top of the “life-cycle effect,” similar to the additional tax burden channel in the main model. 

\end{document} 