Automation and Aging in General Equilibrium:
AI Capital, Fertility, and the Return to Capital

Let $N_{t}^{w}>0$ denote the mass of young working parents and $N_{t}^{r}>0$ the mass of retirees. The economy has an overlapping-generations structure with endogenous fertility and stochastic survival. Each young household of period $t-1$ gives rise to $n_{t-1}>0$ young households in period $t$, so the working-age cohort obeys

$$N_{t}^{w}=n_{t-1}\,N_{t-1}^{w},$$

(1)

where $n_{t-1}$ is the gross fertility rate of the cohort that was young in period $t-1$. A young household survives into retirement with probability $\gamma_{t}\in(0,1)$, so the mass of retirees is

$$N_{t}^{r}=\gamma_{t}\,N_{t-1}^{w}.$$

(2)

The old-age dependency ratio is therefore

$$\psi_{t}\;\coloneqq\;\frac{N_{t}^{r}}{N_{t}^{w}}=\frac{\gamma_{t}}{n_{t-1}}.$$

(3)

The model allows for an imperfect annuity market (Hansen and İmrohoroğlu, 2008; Ludwig and Vogel, 2010), with the degree of annuitization indexed by $\mu\in[0,1]$: $\mu=1$ corresponds to complete (actuarially fair) annuitization, and $\mu=0$ to the absence of annuity markets.

Let $s_{t}^{w}>0$ denote the savings of a young household in period $t$, which survives into retirement in $t+1$ with probability $\gamma_{t+1}\in(0,1)$. Through mortality pooling, each survivor receives, in addition to its own principal, the annuitized savings of non-survivors, so that retirement resources per survivor satisfy

$$s_{t}^{w}+\mu\,\frac{1-\gamma_{t+1}}{\gamma_{t+1}}\,s_{t}^{w}=\frac{\varphi_{t+1}}{\gamma_{t+1}}\,s_{t}^{w},$$

(4)

where

$$\varphi_{t+1}\coloneqq\gamma_{t+1}+\mu\,(1-\gamma_{t+1})\in[\gamma_{t+1},1]$$

(5)

is the annuity factor. The effective gross return on savings, $R_{t+1}\varphi_{t+1}/\gamma_{t+1}$, is increasing in $\mu$ and collapses to the actuarially fair return $R_{t+1}/\gamma_{t+1}$ when $\mu=1$.

When $\mu<1$, the non-annuitized share $(1-\mu)$ of the savings of agents who do not survive is left as accidental bequests and redistributed equally among surviving retirees. Using $N_{t}^{r}=\gamma_{t}N_{t-1}^{w}$, the bequest received by each retiree is

$$b_{t}\coloneqq\frac{(1-\mu)(1-\gamma_{t})\,R_{t}\,s_{t-1}^{w}}{\gamma_{t}}$$

(6)

where $R_{t}=1+r_{t}>0$ is the gross return on savings. Bequests vanish under full annuitization ($\mu=1$) and are largest when annuity markets are absent ($\mu=0$).

The representative household is young in period $t$ and, if it survives, retired in period $t+1$. It chooses worker consumption $c_{t}^{w}$, the number of children $n_{t}$, savings $s_{t}^{w}$, and retirement consumption $c_{t+1}^{r}$ to maximize expected lifetime utility

$\displaystyle\mathcal{U}_{t}=U\!\left(c_{t}^{w},n_{t}\right)+\gamma_{t+1}\,\beta\,V\!\left(c_{t+1}^{r}\right),$

(7)

where $U:\mathbb{R}_{++}^{2}\to\mathbb{R}$ is the felicity over worker consumption and the number of children and $V:\mathbb{R}_{++}\to\mathbb{R}$ the felicity over retirement consumption, both strictly increasing and strictly concave ($U_{c},U_{n}>0$, $V^{\prime}>0$; $U$ jointly concave, $V^{\prime\prime}<0$). The parameter $\beta\in(0,1)$ is the subjective discount factor and $\gamma_{t+1}\in(0,1)$ is the probability of surviving into retirement, which discounts retirement felicity because $c_{t+1}^{r}$ is enjoyed only in the surviving state. Firms are owned by the working-age generation, which receives dividends $d_{t}$.

When young, the household supplies labor, devoting a fraction $\kappa\,n_{t}$ of its time endowment to child-rearing, so that effective hours are $h_{t}=1-\kappa\,n_{t}$ and effective labor income is $w_{t}(1-\kappa\,n_{t})$. When old, surviving households finance consumption out of the annuitized return on their savings and the accidental bequests $b_{t+1}$ left by the non-survivors of their own cohort. The budget constraints are

	$\displaystyle c_{t}^{w}+s_{t}^{w}$	$\displaystyle\leq w_{t}\,(1-\kappa\,n_{t})+d_{t},$		(8)
	$\displaystyle c_{t+1}^{r}$	$\displaystyle\leq\frac{R_{t+1}\,\varphi_{t+1}}{\gamma_{t+1}}\,s_{t}^{w}+b_{t+1},$		(9)
	$\displaystyle\varphi_{t+1}$	$\displaystyle=\gamma_{t+1}+\mu\,(1-\gamma_{t+1}),$		(10)

where $R_{t+1}=1+r_{t+1}>0$ is the gross return on savings, $w_{t}>0$ is the competitive wage, and $\varphi_{t+1}$ is the annuity factor. In line with the literature on the cost of children (Donni, 2015), $\kappa$ denotes the fraction of household time allocated to child-rearing per child, generating a fertility-induced labor-supply wedge.

Because the non-annuitized savings of non-survivors are redistributed to the surviving members of the same cohort, $b_{t+1}=(1-\mu)(1-\gamma_{t+1})R_{t+1}s_{t}^{w}/\gamma_{t+1}$, and the two channels recombine into the actuarially fair return $c_{t+1}^{r}\leq(R_{t+1}/\gamma_{t+1})\,s_{t}^{w}$. Combining the two-period budget constraints then yields the intertemporal lifetime constraint

$$c_{t}^{w}+\gamma_{t+1}\,\frac{c_{t+1}^{r}}{R_{t+1}}\leq w_{t}(1-\kappa\,n_{t})+d_{t}.$$

(11)

Let $\Gamma_{t}=(c_{t}^{w},n_{t},c_{t+1}^{r},\lambda_{t})\in\mathbb{R}^{4}$, where $\lambda_{t}$ denotes the Lagrange multiplier on the household’s consolidated intertemporal budget constraint. Substituting the retirement constraint $c_{t+1}^{r}=(R_{t+1}/\gamma_{t+1})\,s_{t}^{w}$ into the working-period constraint yields the present-value budget, in which the price of retirement consumption is $\gamma_{t+1}/R_{t+1}$. Define the mapping $F:\mathbb{R}^{4}\to\mathbb{R}^{4}$ by the first-order conditions of the household problem:

$\displaystyle F(\Gamma_{t})=\begin{pmatrix}U_{c}(c_{t}^{w},n_{t})-\lambda_{t}\\[4.0pt] U_{n}(c_{t}^{w},n_{t})-\kappa\,\,w_{t}\,\lambda_{t}\\[4.0pt] \beta\,\gamma_{t+1}\,V_{c}(c_{t+1}^{r})-\dfrac{\gamma_{t+1}}{R_{t+1}}\,\lambda_{t}\\[8.0pt] w_{t}(1-\kappa\,n_{t})+d_{t}-c_{t}^{w}-\dfrac{\gamma_{t+1}}{R_{t+1}}\,c_{t+1}^{r}\end{pmatrix},$

(12)

where $U_{c}\equiv\partial U/\partial c_{t}^{w}$, $U_{n}\equiv\partial U/\partial n_{t}$, and $V_{c}\equiv\partial V/\partial c_{t+1}^{r}$. The household optimum is the value $\Gamma_{t}$ such that $F(\Gamma_{t})=0$. Under logarithmic preferences, $U_{c}=1/c_{t}^{w}$, $U_{n}=1/n_{t}$, and $V_{c}=1/c_{t+1}^{r}$.

An equilibrium is a vector $\Gamma_{t}^{*}=(c_{t}^{w},n_{t},c_{t+1}^{r},\lambda_{t})\in\mathbb{R}^{4}$ that solves the first-order system $F(\Gamma_{t}^{*})=\mathbf{0}$, i.e. $\Gamma_{t}^{*}\in F^{-1}(\mathbf{0})$, where $\lambda_{t}$ is the Lagrange multiplier associated with the household’s intertemporal budget constraint. Assume $F\in C^{1}(\mathbb{R}^{4},\mathbb{R}^{4})$ and that the Jacobian matrix $D_{\Gamma}F(\Gamma_{t}^{*})$ is invertible. Then, by the Implicit Function Theorem, $\Gamma_{t}^{*}$ is an isolated regular zero of $F$ and the equilibrium is locally unique. Indeed, any equilibrium-preserving perturbation $d\Gamma\in\mathbb{R}^{4}$ must satisfy, to first order, $D_{\Gamma}F(\Gamma_{t}^{*})\,d\Gamma=\mathbf{0}$, that is, $d\Gamma\in\ker\!\big(D_{\Gamma}F(\Gamma_{t}^{*})\big)$; since the Jacobian is invertible, $\ker\!\big(D_{\Gamma}F(\Gamma_{t}^{*})\big)=\{\mathbf{0}\}$, which confirms that the equilibrium is isolated.

The implicit function theorem characterizes the household block from the single $C^{1}$ system $F(\Gamma_{t}^{*})=0$ with $D_{\Gamma}F(\Gamma_{t}^{*})$ non-singular; the formal comparative statics (Proposition A.1, Corollary A.1, and Proposition A.2) are collected in Appendix A.1. Fertility acts as a time wedge on labor income, $w_{t}(1-\kappa\,n_{t})$: a higher $n_{t}$ contracts resources and lowers saving, $\partial s_{t}^{w}/\partial n_{t}<0$, whereas greater longevity strengthens the life-cycle saving motive, $\partial s_{t}^{w}/\partial\gamma_{t+1}>0$. Both shocks depress consumption in both phases of life ($dc_{t}^{w},dc_{t+1}^{r}<0$), and fertility falls with longevity, $dn_{t}/d\gamma_{t+1}<0$, as higher life expectancy raises the opportunity cost of child-rearing. Together, these results map demography and longevity into saving, consumption, and fertility through one coherent equilibrium mechanism.

The economy comprises perfectly competitive final-good producers and a continuum of monopolistically competitive intermediate-good firms indexed by $i\in[0,M_{t}]$, where $M_{t}>0$ denotes the endogenous mass of active firms. In the spirit of macroeconomic models that tie productive activity to demographic structure (Bielecki et al., 2018), we link the extensive margin of production to demographics by letting the mass of firms scale with total population $N_{t}=N_{t}^{w}+N_{t}^{r}$. Because the level of $M_{t}$ is not separately identified from the scale of the economy, we normalize the firm-to-population ratio to one, so that $M_{t}=N_{t}$ and the number of active firms co-moves one-for-one with population through the entry–exit mechanism described below. Product variety is thus endogenous to demographics: population dynamics shape not only factor supplies but also the number of operating firms, and hence the range of varieties available in the economy.

Saving is intermediated by perfectly competitive, risk-neutral investment funds. Each period, the representative fund collects current saving $s_{t}^{w}$, repays the gross return $R_{t-1}s_{t-1}^{w}$ on past saving, and owns the two capital stocks, traditional capital $k_{t}$ and AI capital $a_{t}$, which it rents to intermediate firms at rates $R_{t}^{k}$ and $R_{t}^{a}$ while financing investment $i_{t}^{k}$ and $i_{t}^{a}$. Its net cash flow, rebated to households, is

$$d_{t}^{f}=s_{t}^{w}-R_{t-1}s_{t-1}^{w}+R_{t}^{k}k_{t}+R_{t}^{a}a_{t}-i_{t}^{k}-i_{t}^{a}.$$

(22)

The fund chooses $\{i_{t}^{k},i_{t}^{a},k_{t+1},a_{t+1}\}_{t\geq 0}$ to maximize the expected present value of net cash flows,

$$\max\;\mathbb{E}_{t}\sum_{s\geq 0}\Lambda_{t,t+s}\,d_{t+s}^{f},\qquad\Lambda_{t,t+s}=\prod_{j=1}^{s}\frac{1}{R_{t+j-1}},$$

(23)

where, being competitive and risk neutral, the fund discounts at the gross risk-free rate. Following Christiano et al. (2005), capital accumulation is subject to investment adjustment costs,

	$\displaystyle k_{t+1}$	$\displaystyle=(1-\delta_{k})\,k_{t}+\left[1-\frac{\phi^{k}}{2}\left(\frac{i_{t}^{k}}{i_{t-1}^{k}}-\vartheta_{t-1}\right)^{2}\right]i_{t}^{k},$		(24)
	$\displaystyle a_{t+1}$	$\displaystyle=(1-\delta_{a})\,a_{t}+\left[1-\frac{\phi^{a}}{2}\left(\frac{i_{t}^{a}}{i_{t-1}^{a}}-\vartheta_{t-1}\right)^{2}\right]i_{t}^{a},$		(25)

where $\delta_{k},\delta_{a}\in(0,1)$ are depreciation rates and $\phi^{k},\phi^{a}>0$ govern the adjustment costs. The reference $\vartheta_{t-1}$ is the gross growth rate of investment, which on the balanced growth path equals population growth, $\vartheta_{t}=n_{t}$, so that the bracketed terms equal one and adjustment costs vanish in steady state. The multipliers $q_{t}^{k}$ and $q_{t}^{a}$ on the two laws of motion are the shadow prices of installed physical and AI capital (Tobin’s $q$), and coincide with the asset prices $q_{k},q_{a}$ reported in the impulse responses.

Writing gross investment growth as $g^{x}_{t}\equiv i^{x}_{t}/i^{x}_{t-1}$, $x\in\{k,a\}$, the first-order conditions deliver a no-arbitrage condition that equates the expected gross returns on the two assets, together with the investment Euler equations:

	$\displaystyle R_{t}$	$\displaystyle=\frac{\mathbb{E}_{t}\!\big[R^{k}_{t+1}+(1-\delta_{k})q^{k}_{t+1}\big]}{q^{k}_{t}}=\frac{\mathbb{E}_{t}\!\big[R^{a}_{t+1}+(1-\delta_{a})q^{a}_{t+1}\big]}{q^{a}_{t}},$		(26)
		$\displaystyle q^{k}_{t}\Big[1-\tfrac{\phi^{k}}{2}\big(g^{k}_{t}-\vartheta_{t-1}\big)^{2}-\phi^{k}g^{k}_{t}\big(g^{k}_{t}-\vartheta_{t-1}\big)\Big]=1-\phi^{k}\,\mathbb{E}_{t}\!\Big[\tfrac{q^{k}_{t+1}}{R_{t}}\,(g^{k}_{t+1})^{2}\big(g^{k}_{t+1}-\vartheta_{t}\big)\Big],$		(27)
		$\displaystyle q^{a}_{t}\Big[1-\tfrac{\phi^{a}}{2}\big(g^{a}_{t}-\vartheta_{t-1}\big)^{2}-\phi^{a}g^{a}_{t}\big(g^{a}_{t}-\vartheta_{t-1}\big)\Big]=1-\phi^{a}\,\mathbb{E}_{t}\!\Big[\tfrac{q^{a}_{t+1}}{R_{t}}\,(g^{a}_{t+1})^{2}\big(g^{a}_{t+1}-\vartheta_{t}\big)\Big].$		(28)

Financial-market clearing requires that household saving fully finance the value of the capital stock carried into the following period,

$$s_{t}^{w}=q_{t}^{k}k_{t+1}+q_{t}^{a}a_{t+1}.$$

(29)

The zero-profit condition for the competitive funds then determines the gross return on saving $R_{t}$, which the no-arbitrage condition equalizes across the two productive assets.

In equilibrium, all markets clear, firms maximize profits, and households maximize utility. Household saving is channeled entirely into investment in the two capital stocks, physical capital and AI capital, so that aggregate investment $z_{t}$ is the only vehicle for saving. By symmetry across intermediate firms, $k_{t}(j)=k_{t}$, $a_{t}(j)=a_{t}$, and $h_{t}(j)=h_{t}$ for all $j\in[0,N_{t}]$, where the common per-firm input levels satisfy

$$k_{t}\coloneqq\frac{1}{N_{t}}\int_{0}^{N_{t}}k_{t}(j)\,dj,\quad a_{t}\coloneqq\frac{1}{N_{t}}\int_{0}^{N_{t}}a_{t}(j)\,dj,\quad h_{t}\coloneqq\frac{1}{N_{t}}\int_{0}^{N_{t}}h_{t}(j)\,dj.$$

Labor-market clearing equates aggregate labor demand $N_{t}h_{t}$ to aggregate effective labor supply: each of the $N_{t}^{w}$ workers supplies $(1-\kappa\,n_{t})$ units of time, so that

$\displaystyle N_{t}\,h_{t}=(1-\kappa\,n_{t})\,N_{t}^{w}.$

(30)

Goods-market clearing allocates output between consumption and investment. In per-worker terms, with the old-age dependency ratio $\psi_{t}=N_{t}^{r}/N_{t}^{w}$, the aggregate resource constraint is

$\displaystyle y_{t}=c_{t}+z_{t},$

(31)

where aggregate consumption per worker weights retirees by their relative mass,

$\displaystyle c_{t}=c_{t}^{w}+\psi_{t}\,c_{t}^{r},$

(32)

and total investment combines physical and AI investment,

$\displaystyle z_{t}=i_{t}^{k}+i_{t}^{a}.$

(33)

Formally, artificial intelligence productivity $\phi_{t}$ and the survival probability of retirees $\gamma_{t}$ follow stationary AR(1) processes:

	$\displaystyle\phi_{t}$	$\displaystyle=(1-\rho_{\phi})\bar{\phi}+\rho_{\phi}\phi_{t-1}+\varepsilon_{t}^{\phi},\qquad\rho_{\phi}\in(0,1),$		(34)
	$\displaystyle\gamma_{t}$	$\displaystyle=(1-\rho_{\gamma})\bar{\gamma}+\rho_{\gamma}\gamma_{t-1}+\varepsilon_{t}^{\gamma},\qquad\rho_{\gamma}\in(0,1),$		(35)

where $\bar{\phi}$ and $\bar{\gamma}$ denote steady-state levels, and $\varepsilon_{t}^{\phi}$ and $\varepsilon_{t}^{\gamma}$ are i.i.d. shocks capturing innovations in technology and longevity risk. Shocks to $\phi_{t}$ represent exogenous improvements in AI efficiency that raise the marginal productivity of AI capital, while shocks to $\gamma_{t}$ capture changes in expected longevity that reshape intertemporal saving and consumption decisions. The model is used to compute impulse responses and transitional dynamics, highlighting the joint propagation of technological progress and demographic risk through general equilibrium channels.

By substituting for traditional physical capital, AI capital triggers broader equilibrium effects that propagate through labor markets and household decisions. Proposition 2.1 and the global comparative statics of an AI expansion (Proposition A.3, formalized in Appendix A.3) together show that artificial intelligence operates through both factor reallocation and general equilibrium adjustment. At the firm level, AI substitutes for labor and traditional capital by automating production tasks. At the aggregate level, the resulting gains in productivity raise wages, saving, and lifetime consumption; because the income effect dominates, higher wages modestly raise fertility despite the increased opportunity cost of child-rearing. Artificial intelligence thus reshapes not only the structure of production but also demographic dynamics and intertemporal allocation.

The corollary underscores the central role of the elasticity of substitution $\sigma$. The two regimes are separated by the threshold $\rho=1-\alpha$ (equivalently $\sigma=1/\alpha$), at which $\partial^{2}Y_{t}/\partial H_{t}\partial a_{t}$ changes sign: AI is labor-augmenting for $\rho<1-\alpha$ (Proposition 2.2) and labor-replacing for $\rho>1-\alpha$. The baseline calibration ($\sigma=2<1/\alpha\approx 3.03$) lies in the complementarity regime, so the labor-replacing case merely delimits the parameter space.¹¹1In that regime an AI expansion lowers the marginal product of labor, contracts labor demand, and depresses the wage; in the perfect-substitute limit the rental of AI capital, tied to the wage through $R_{t}^{a}=\phi\,w_{t}$, falls alongside it. The effect on consumption is ambiguous: output still (weakly) rises, but the fall in labor income works against it, so the sign of $\partial C_{t}/\partial a_{t}$ depends on the strength of the induced investment response. Throughout, technological change interacts with demographic decisions through the labor-supply adjustments induced by the time cost of children.

The time-cost channel linking fertility to effective labor supply is formalized in Lemma A.1, and a two-capital decomposition of growth in Proposition A.4; both are collected in Appendix A. The latter decomposes the growth effect of AI into a positive productivity channel and two offsetting channels, consumption crowding-out and capital dilution; the net effect is positive precisely when next-period capital is more AI-elastic than current capital. Under complementarity and a sufficiently strong productivity response this condition holds and AI raises long-run growth, whereas otherwise the offsetting channels dominate. More broadly, the section shows that artificial intelligence jointly reshapes production, labor-market outcomes, demographic behavior, and growth, underscoring the tight interaction between technological change, household decisions, and macroeconomic development.

We adopt logarithmic functions,

$\displaystyle u(c_{t}^{w},n_{t})=\log(c_{t}^{w})+\log(n_{t}),$

(36)

$\displaystyle v(c_{t+1}^{r})=\log(c_{t+1}^{r})$

(37)

which deliver interior solutions, unit intertemporal elasticity, and a multiplicative valuation of fertility, as is standard in macro-demographic models.

Table 3.1 presents the baseline calibration. The parameters are divided into two categories. The first category comprises conventional values from the macroeconomic literature: the capital share $\alpha=0.33$, the household discount factor $\beta=0.96$, the physical-capital depreciation rate $\delta_{k}=0.08$, and the elasticity of substitution across intermediate goods $\xi=10$, implying a steady-state markup of approximately eleven percent.

AI capital exhibits a higher depreciation rate than physical capital ($\delta_{a}=0.12$), reflecting the accelerated obsolescence of the equipment and software in which it is embodied. The persistence parameters for the two driving processes, $\rho_{\phi}=0.85$ and $\rho_{\gamma}=0.90$, and the adjustment-cost parameters $\phi_{a}=3.00$ and $\phi_{k}=2.50$, govern the speed at which the economy absorbs each disturbance.

The second category is calibrated to ensure that the deterministic steady state reproduces a limited number of demographic and technological targets. The time cost per child, $\kappa=0.1949$, is calibrated to deliver a steady-state fertility rate of $n_{ss}=2$ children per household; given the long-run survival probability of retirees $\gamma_{ss}=0.93$, this yields an old-age dependency ratio $\psi_{ss}=\gamma_{ss}/n_{ss}=0.465$. The steady-state productivity of AI capital is normalized to $\phi_{ss}=1.20$, and the elasticity of substitution between labor and AI capital is governed by the CES parameter $\rho=0.50$.

We analyze each shock through its impulse response functions, reported in Figures 3.1 and 3.2. To keep responses of very different magnitude visible, the two disturbances are shown on a dual vertical axis: the left-hand scale (solid blue line, circular markers) corresponds to the AI technology shock $\phi_{t}$, and the right-hand scale (dashed vermillion line, square markers) to the longevity shock $\gamma_{t}$. All variables are expressed as percentage deviations from the steady state over a horizon of twenty periods following the shock. Figure 3.1 collects the responses of the main aggregates, while Figure 3.2 reports factor prices and demographic variables. Table 3.2 previews the qualitative predictions that the impulse responses confirm: the two shocks share the same sign on quantities but carry opposite signs on the returns to capital and on fertility, which is the exact imprint of the demand-versus-supply asymmetry.

The model is driven by two structural shocks, each calibrated to a standard deviation of $1\%$ and uncorrelated with the other: an AI technology shock ($\varepsilon_{\phi}$) and a longevity shock ($\varepsilon_{\gamma}$). We summarize its second-order properties through theoretical moments and persistence, the unconditional variance decomposition, and contemporaneous cross-correlations; the supporting tables are provided in the Appendix (Tables C.1–C.3).

Volatility is concentrated in prices and demographic ratios rather than in real quantities: the real interest rate ($\sigma=0.0151$), the dependency ratio $\psi$ ($0.0141$), the capital returns $R^{a}$ ($0.0133$) and $R^{k}$ ($0.0104$), and fertility $n$ ($0.0125$) are the most volatile, whereas output, consumption, capital, and investment fall between $0.0005$ and $0.0045$. The accumulation variables are highly persistent ($\rho(1)\!\geq\!0.97$ for $k$, $a$, $i^{k}$, $i^{a}$, and $w$), which accounts for the slow, hump-shaped impulse responses documented above, while asset prices and workers’ consumption revert more quickly ($\rho(1)\in[0.65,0.80]$). The variance decomposition delivers the central finding: longevity dominates, explaining ninety to one hundred percent of the variance in nearly every variable: output ($95.8\%$), consumption ($98\%$), capital ($99.7\%$), fertility ($99.9\%$), and the dependency ratio ($100\%$); whereas the AI shock matters only for capital pricing, accounting for $75.5\%$ of the variance in $R^{a}$, $18.8\%$ of $R^{k}$, $18.5\%$ of $q^{a}$, and $9.3\%$ of the real interest rate. Since the two shocks share a common standard deviation, these shares reflect the model’s transmission mechanism rather than differences in shock size. The correlation structure indicates a single dominant factor in the real block: output, consumption, capital, AI capital, investment, the wage, and saving co-move strongly ($0.8$ to $0.99$), the real interest rate moves countercyclically as higher saving depresses its equilibrium value, and $R^{a}$ is only weakly tied to the real block ($-0.27$), being governed by the longevity-independent AI shock. Fertility is the mirror image, negatively correlated with every real aggregate and perfectly with hours worked ($-1.00$), a direct expression of the household’s time-allocation trade-off between market work and children. Taken together, these second-order statistics reinforce the general equilibrium intuition obtained from the impulse response analysis.

A noteworthy feature of the correlation matrix is that worker consumption, $c^{w}$, is the only consumption variable that moves negatively with output ($\mathrm{corr}(c^{w},y)=-0.32$). This pattern is a direct signature of the saving-supply mechanism, not an anomaly. The longevity shock accounts for most of the volatility in both $c^{w}$ and $y$ (Table C.2), prompting workers to compress $c^{w}$ to finance capital deepening, thereby increasing $y$. Thus, $c^{w}$ and $y$ move in opposite directions under the dominant shock. The same logic explains the negative comovement of $c^{w}$ with $s^{w}$, $k$, $a$, $i^{k}$, $i^{a}$, $h$, and prices of installed capital, $q^{k}$ and $q^{a}$; all move with output in response to the saving-supply channel. In addition, $c^{w}$ moves positively with fertility, $n$, since the longevity shock depresses both. In a standard one-agent RBC model with only TFP shocks, consumption would be uniformly procyclical. Thus, the negative correlation here highlights the source of fluctuations in an OLG economy dominated by a saving-supply force. Figure 3.3 summarizes this division of labor between the two disturbances: the longevity shock accounts for nearly the entire forecast-error variance of the quantity block and of the real interest rate, while the AI shock is concentrated in the pricing of capital, where it explains the bulk of the variance of the return to AI capital.