## Appendix A.   K&S Model with Bequest

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### I. Introduction — framing and research objective
- Automation has accelerated due to computing and information technology improvements and cost reductions; latest breakthrough involves generative, pretrained transformers.
- Offset forces:
  - Working-age population has started shrinking in advanced economies (Spence, 2022).
  - Globally, working-age population expected to grow until about 2040 but share already declining.
  - In China, working-age population expected to shrink by a fifth over the next 30 years.
  - Recurring pandemics (examples: AIDS, Covid-19) can adversely affect labor supply.
- Paper objective:
  - Examine combined welfare effects of automation and lower labor supply using neoclassical growth models with and without intergenerational transfers.
  - Replicate Kotlikoff and Sachs (K&S, 2012) result that a one-time improvement in machine technical efficiency can immiserate future generations under two assumptions:
    - Machines are very good substitutes for unskilled labor, displacing workers and lowering wages economy-wide.
    - No operative intergenerational transfers exist, so owners of capital consume the windfall during their lifetime, reducing saving and investment in physical and human capital.
  - Assess whether government policy (wealth taxes or socializing part of capital stock to finance a universal basic income) is necessary when altruism/selfishness prevents voluntary transfers.

### II. Role of intergenerational transfers and bequests
- Empirical motivation:
  - Private intergenerational transfers are substantial: about half of all households plan to leave estates (Laitner and Juster, 2017).
- Model findings:
  - With operative bequests, intergenerational transfers are positive in equilibrium if altruism exceeds a threshold, mitigating automation’s negative effects.
  - Perfect altruism can pass automation gains to future generations without nationalizing capital.
- Need for heterogeneity:
  - Balanced models should include heterogeneity: some households make efficient bequests, others are bequest-constrained.
  - Paper assesses automation immiseration in:
    - K&S variant with pure altruism.
    - Glomm and Ravikumar (G&R, 1992) schooling-investment framework.
    - Ivanyna, Mourmouras and Rangazas (2018) with two household types: poor (bequest-constrained) and rich (unconstrained).
- Policy implications:
  - Fiscal and educational policies that alleviate financing constraints for human capital can raise welfare.
  - Strengthening altruistic bonds raises bequests and human capital investments and stimulates growth.
  - Government transfers financed by taxing the rich can improve poor welfare and reduce inequality when altruistic links are weak.

### III. Combined shocks: automation plus demographic labor supply decline
- Focus: jump in automation together with reduction in labor supply driven by demographics.
- Key mechanisms and outcomes:
  - Strengthened altruism increases intergenerational transfers and human capital investment.
  - Tax-financed transfers from rich to poor mitigate inequality and improve cross-generational welfare when private transfers are insufficient.

### IV. Relevant literature and quantitative findings
- Selected literature: Gordon (2012); Cowen (2011); Acemoglu and Restrepo (2017, 2018); Sachs and Kotlikoff (2012, 2015); Ford (2015); Freeman (2015); Brynjolfsson and McAfee (2014); Autor (2014, 2015); Bostrom (2014); Graetz et al. (2018); Gaaitzen et al. (2020).
- Quantitative evidence (Graetz et al., 2018):
  - Increasing use of robots raised annual GDP growth by 0.37 percentage points and labor productivity by 0.36 percentage points during 1993-2007 in 17 countries.
  - Those robots did not significantly reduce total employment, although they reduced low-skilled workers’ employment share.

### Automation, employment, and sectoral impacts
- Increased use of robots associated with:
  - positive changes in employment share of non-routine analytic jobs
  - negative changes in share of routine manual jobs
- Pandemic effect:
  - The 2020-22 pandemic accelerated digital transformation; many sectors (including the public sector) lag, sustaining concerns about employment.
- Occupation-level heterogeneity:
  - Occupations with low age-appreciated cognitive ability and low age-depreciated physical ability (examples: production occupations; food preparation and serving related occupations) experience strongest negative employment effect from automation.
  - Occupations with high scores in both (examples: protective service occupations; healthcare practitioners and technical occupations) experience weakest negative effect.

### AI exposure, complementarity/substitution, and cross-country variation
- Pizzinelli and others (forthcoming) extend AI exposure measures to account for complementarity or substitution.
- Worker-level microdata (US, UK, Brazil, Colombia, India, South Africa) show substantial variation in unadjusted AI exposure across countries.
- Findings:
  - AI poses risks of labor displacement via task automation.
  - AI can enhance productivity and complement human labor in high-cognitive occupations.
  - Advanced Economies (AEs) may expect more polarized impacts: greater labor substitution risk and greater productivity benefits.

### Public policy on digital technologies and R&D
- Enhancing R&D and regulatory platforms in private and public sectors is key to improving productivity.
- Concerns about automation’s employment effects persist unless lagging sectors accelerate adoption and supportive policies.

### Education spending and empirical estimates linking education to growth
- Empirical study findings:
  - Fabrizio Carmignani (2016): Using World Bank WDI data on 151 countries for 2000 - 2010, “increase in education expenditure by 1 point of GDP increases GDP growth by 0.9 percentage points”.
  - Gheraia, Zouheyr et al. (2021): For Saudi Arabia 1990-2017, a rise in educational expenditure by 1% leads to increase in economic growth by 0.89%.
  - Yahya, Mohd et al. (2012): For Malaysia 1970-2010, educational expenditure is short-term Granger cause of economic growth and vice versa.
  - Mehmet Mercana et al. (2014): For Turkey 1970-2012, 1% increase in education expenses increases economic growth by 0.3%.

### Analytical framework: The K&S two-period OLG model
- Model structure:
  - Two-period-lived overlapping generations, periods t = 1,2,...
  - Young supply one unit of inelastically supplied unskilled labor, invest part of income in physical capital (machines, M) and human capital (skilled labor, S). Machines and human capital are perfect substitutes in saving portfolios.
  - Gross output Q is a CES production function of economy-wide stocks of M, L, and S:
    - Q = Q(N(uM, L), S)  (A.1)
    - M and L combine in a CES with elasticity ε_ML to produce N; N and S combine in a CES with elasticity ε_SN to produce Q.
    - u is technical efficiency of machinery; rise in u is a pure technical advance.
- Wage response to machine productivity:
  - d ln(Q_L) / d ln(u) = [ε_SN − θ ε_ML] / ε_ML  (A.2)
  - θ = (Q_S * S) / Q (share of skilled labor in the economy)
  - A rise in machines’ productivity reduces the unskilled wage if ε_ML > ε_SN θ /.
- Conditions making immiserizing productivity more likely:
  - ε_ML large (high substitutability of machines and unskilled labor)
  - ε_SN small (low substitutability of intermediate goods and skilled labor)
  - θ high (large share of skilled labor in final output)
- Lifetime budget and utility (selected equations):
  - I_t = L_t W_t = C_t + S v_t  (A.3)
  - I_{t+1} = (∂Q/∂M)_t M_{t+1} + (∂Q/∂S)_t S_{t+1} = D_{t+1}  (A.4)
  - S v_t = M_{t+1} + S_{t+1}  (A.5)
  - R_t = (∂Q/∂M)_t = (∂Q/∂S)_t = 1 + r_t  (A.6)
  - U = β log(C_t) + (1−β) log(D_{t+1})  (A.8)
  - First order condition: β / C_t = (1−β) / (D_{t+1} R_t)  (A.9)
- Cobb-Douglas formulation (K&S baseline):
  - Q(N,S) = A N^{α} S^{1−α}  (A.10)
  - N = L + u^{1} M  (A.11) (machines and labor perfect substitutes)

### K&S numerical experiment and Proposition A1 (immiserating productivity)
- Parameterization used to illustrate dynamics:
  - 훼=0.74 on labor and 0.26 on machines
  - 훽=0.5 on skills
  - Productivity unexpectedly rises by 35% in period t=3 (u_1 increases by factor 1.35)
- Key numerical outcomes:
  - Current wage of unskilled workers (earnings of the young of generation 3) declines from 2.82 to 2.61 due to rise in u_1.
  - Returns on M and S both rise; the old generation (owners of M and S) experiences a boom while the young experience a bust.
  - By period 5, new equilibrium has lower wages, lower skills, lower M, and higher total output than baseline.
  - Ratio W_s / W_l permanently raised from 1 to 1.35.
  - Lifetime utility: generation 3 utility slightly below generation 1; generations 4 and later higher than baseline. Young unskilled workers lose; generation 3 benefits from high returns to M and S.
- Proposition A1:
  - Improvements in automation benefit owners of capital (initial old generation) and lower welfare of subsequent generations of workers.
  - A positive shock in machine efficiency leads to higher consumption and income of the old, lowers wage incomes of unskilled workers, lowers their lifetime consumption and utility, and increases intergenerational inequality.

### Government intervention: One-time tax and transfers (Section B)
- Policy mechanism:
  - Government imposes a one-time tax (wealth levy) on machines M_G at time t=3, taking amount M_G < M.
  - Private machines left: M_P = M − M_G.
  - Government income Q_M M_G split into transfer T = (Q_M − 1) M_G added to wage of the young, and remainder M_G reinvested.
  - Government machines M_G constant after t; government pays transfer T to each young generation and reinvests M_G to maintain its stock.
- Modified budget and equilibrium equations (selected):
  - I_t = L_t W_t + T = C_t + S v_t  (B.1)
  - S v_t = (1−β) I_t  (B.2)
  - M_P,t+1 + S_{t+1} = S v_t; M_P,t+1 + M_G + L_t u^{/} = (α S_{t+1})/(1−α)  (B.3)
- Proposition B1:
  - Government tax on machines and redistribution of the windfall from old to young increases welfare of all generations if the tax is sufficiently large.
  - Productivity shocks accompanied by taxes on machines result in higher utility for all future generations; future generations are no longer immiserated if the government tax is sufficiently large.
- Numerical example:
  - Government imposes a 10% levy on machines owned by the old in period 3.
  - Government transfers T = (Q_M − 1) M_g to the young of generation 3; government reinvests M_G to maintain constant M_G.
  - Example outcomes:
    - Utility at period 1 = 2.37; rises to 2.63 at period 6, leaving generation 3 better off than baseline despite the capital levy.
    - Income and consumption of old drop at period 3 because of transfer from old to young; in period 4 and subsequent periods consumption and income of old increase.
    - Future generations are even better off because of the government transfer program.

### Bequests, altruism, and implications (Section C)
- Model extension: pure altruism where household utility depends on own consumption and utility of children; bequests B_t enter young income:
  - I_t = L_t W_t + B_t = C_t + S v_t  (C.1)
  - I_{t+1} = R_t S v_t = D_{t+1} + (1+ n) B_{t+1}
- Bequest growth multiplier:
  - g = (φ R) / ((1+ n)(1−φ))
  - Bequests grow exponentially if φ > φ_cr = 1 / [1 + R_t / (1+ n)]
- Proposition C1:
  - Bequests increase welfare of the young and decrease inequality between old and young.
  - Future generations are not immiserated if altruism parameter is above the threshold; model can sustain exponential growth in consumption and utility driven by accumulation of machines and skills.
- Numerical experiments:
  - Assume machine productivity grows each year by 1%; over 30 years this results in jump in u_1 by 35%.
  - Parameters: n=0, α=0.74, A=5, β=0.5.
  - Example with selfishness parameter φ=0.3 (> φ_cr): no steady state solution; starting period 3 with productivity growth by 35%, from period 4 economy grows exponentially, income and consumption of young increase, wage of skilled > wage of unskilled at all times.
  - Example with φ < φ_cr: model predicts a new steady state with lower income and consumption of young; temporary shock causes positive bequests in period 4 and economy reaches steady state by period 6 in an example.

### Education, bequests, and public vs private schooling (Section D)
- Extension: bequests plus parental expenditures E_t on children’s education.
- Proposition D1:
  - Individuals with private schooling possess higher income and consumption than those with public schooling.
  - Bequest-constrained poor households may not afford rising educational costs; transfers financed by taxes on machines of rich households with private schooling can partially alleviate this.
  - For selfishness parameter less than critical (old invest more in their young), such transfers are unnecessary.
  - Targeted transfers to poor households with public education increase income and consumption of these households when selfishness parameter larger than critical; not beneficial when it is less.
  - Public schooling provides equal opportunities and decreases population inequality.

### 1. Private Education (model specifics)
- Budget constraints:
  - I_t = L_t W_t + B_t + E_t = C_t + S v_t  (D.1)
  - I_{t+1} = R_t S v_{t+1} = D_{t+1} + (1+ n) B_{t+1} + (1+ n) E_{t+1}
- Value function and conjectured Bellman form:
  - V(B_t,E_t) = a + b log(B_t + E_t + m_t)  (D.4)
- Key closed-form coefficients and sharing rule:
  - b = 1 / (1−φ)  (D.8)
  - Like Glomm and Ravikumar model, private agents spend half of their saving on bequests and half on educational expenses.
- Explicit allocations:
  - C_t = β (1−φ)/(1+φ) [I_t + (1+ n)/R_t m_{t+1}]  (D.9)
  - D_{t+1} = (1−β)/β R_t C_t  (D.9)
  - B_{t+1} = E_{t+1} = [2 + (1+ n)(1−β)(1−φ)/φ]^{−1} [I_{t+1} + (1+ n) m_{t+1}] − m_{t+1}/2  (D.9)
- Production with human capital:
  - Q(N_t,H_t) = A N_t^{α} H_t^{1−α}  (D.10)
  - H_t = S_t + E_{t−1}
- Perfect foresight conditions:
  - L_u / (1+ M_{t+1}) = α/(1−α) H_{t+1}  (D.11)
  - M_{t+1} + S_{t+1} = S v_t

### 2. Public Education (model specifics)
- Public schooling financing:
  - E_t = τ S\bar{v}_t = τ ( \bar{M}_{t+1} + S_{t+1} )  (D.12)
- Individual budgets with public schooling:
  - I_t = W_t L + B_t + E_t = C_t + S v_t
  - I_{t+1} = R_t (1−τ) S v_t = D_{t+1} + (1+ n) B_{t+1} + (1+ n) E_{t+1}  (D.13)
- First-order conditions (selected):
  - β / C_t = φ b (1−τ) (1+ n)(B_{t+1}+E_{t+1}+m_{t+1}) / R_t  (D.15)
  - (1−β) / D_{t+1} = φ b (1+ n)(B_{t+1}+E_{t+1}+m_{t+1})  (D.15)
- Closed-form solution under public education (selected):
  - C_t = β (1−φ) [ I_t + (1+ n)(1−τ) R_t / m_{t+1} ]  (D.16)
  - D_{t+1} = (1−β)(1−τ)/β R_t C_t  (D.16)
  - B_{t+1} = φ / (1−β+βφ) ( I_{t+1}/(1+ n) + m_{t+1} ) − E_{t+1} − m_{t+1}  (D.16)
- Proposition D2 and simulations:
  - Public education financed by income taxes on skilled-worker households reduces within-generation inequality and across-generation inequality; private inter-generational transfers can spread technological windfalls without socializing physical capital.
- Numerical experiments (heterogeneous φ):
  - Number of draws: 1,000.
  - Truncated normal distribution of selfishness φ with bounds (0,1).
  - Variance σ = 0.022, 0.012 chosen so CDF of truncated normal is 0.5 at φ_{cr} = 0.14 (public) and 0.08 (private). Only half of households can afford to leave a bequest.
  - For private schooling: consumption and income of young and old grow exponentially.
  - For public schooling: old generation income up to six periods insufficient to leave a bequest; variables reach a new low steady state.
  - K&S model with bequest plus education predicts higher bequest, utility, income, savings, and consumption compared with K&S with bequest alone.
  - Example plotted scenario: mean altruism parameter φ̅ = 0.15 and tax rate τ = 5%.

### Targeted transfers and taxation experiments
- One-time policy experiment:
  - Government imposes a one-time tax on 5% of the machine population at time t = 3.
  - Government transfers constant payment T = 0.05 * (Q_{M−1}) * M to poor public households from period 3 onward, increasing human capital of poor households.
- Targeted transfer scenarios:
  - Scenario 1: payments to households with φ < φ̅ (group 1).
  - Scenario 2: payments to households with φ > φ̅ (group 2).
  - Poor and wealthy households receive same transfer amount; relative change larger for poor households.
- Note: For impatience parameters β < 0.5 poor households can invest more into machines and education compared to consumption and afford higher educational costs.

### Inequality effects of public vs private schooling
- Simulation setup:
  - Number of draws: 1,000.
  - Initial skills distribution: truncated normal with standard deviation σ = 1.
  - Selfishness parameter φ = 0.15 and tax rate τ = 10%.
- Results:
  - Public schooling provides equal opportunity; distributions of human capital, consumption, and income become more homogeneous.
  - Private schooling leads to growth in variance of human capital after a productivity increase.
- Summary implication:
  - Education (private or public) improves well-being across generations. Public education financed by income taxes on skilled households reduces within-generation and across-generation inequality; targeted transfers and taxing richer machine owners can help cover educational shortfalls for the poor.

---

### Appendix B. Replicating Kotlikoff and Sachs (2012)
- Replication setup parameters:
  - L=1, A=10, α=0.6, β=0.5
- Replication results (Table A1; reproduced rows):
  - T 1: u 1, M 1.1, S 1.4, Wl 5.10, Ws 5.10, Sv 2.55, Q 18.12
  - T 2: u 1, M 1.1, S 1.4, Wl 5.10, Ws 5.10, Sv 2.55, Q 18.12
  - T 3: u 10, M 1.1, S 1.4, Wl 2.53, Ws 14.6, Sv 2.04, Q 51.88
  - T 4: u 10, M 0.7, S 0.6, Wl 2.03, Ws 20.3, Sv 1.79, Q 27.72
  - T 5: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
  - T 6: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
  - T 7: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
  - T 8: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
  - T 9: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
- Note: These results reproduce Table 1 (unanticipated case) from Sachs and Kotlikoff paper except those authors claim they obtained results for slightly different parameters: L=1, α=0.5, A=10, β=0.5.

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### Appendix C. Robustness Checks
- Key model identities (selected):
  - I_t = W_t + B_t = C_t + S v_t  (1)
  - S v_t = (1+ n) K_{t+1}
  - I_{t+1} = R_t S v_t = D_{t+1} + (1+ n) B_{t+1}  (2)
- Value function augmented with shadow prices of bequest p_t:
  - V = log(C_t) + β^ log(D_{t+1}) + φ^ p_{t+1} B_{t+1} − p_t B_t  (3)
- First-order and envelope conditions yield recursive relations for p_t and B_t; if B_t > 0:
  - p_{t+1} = (1+ n) φ^ R_t p_t  (5)
- Production technology and factor returns:
  - Q(K,L) = A (K/L)^α L  (6)
  - L = 1
  - W_t = (1−α) A K_t^α  (7)
  - R_t = α A K_t^{α−1}
- Analytical solution (Michel et al. (2004)) constructs variables v_t and B_t with relations:
  - v_{t+1} = 1/(αφ̂) (v_t − 1 − β̂/φ̂)
  - B_t and p_t B_t expressed in terms of v_t (8)
- Steady state and stability:
  - v0 = v_t = 1/(1 − αφ̂) (1 + β̂/φ̂)  (9)
  - Steady state exists for specific initial conditions but is unstable and grows exponentially.
  - Instability driven by multiplier 1/(αφ̂) > 1 because α < 1 and φ̂ ≤ 1.
  - Numerical experiments with initial v ∈ (0.99 v0, 1.01 v0) illustrate instability of steady state with positive bequest.

_italics: Source: wpiea2024011-print-pdf - Appendix A.   K&S Model with Bequest._

### Appendix A.   K&S Model with Bequest ...................................................................................

### Appendix A.   K&S Model with Bequest

### I. Introduction — framing and research objective
- Automation has accelerated in recent decades due to improvements in computing and information technologies and associated cost reductions; machines are performing increasingly complex tasks across industries.
- The latest breakthrough in Artificial Intelligence involves generative, pretrained transformers.
- Offset forces may mitigate automation’s effects:
  - Populations are aging almost everywhere; in advanced economies the working-age population has started shrinking for the first time since World War II (Spence, 2022).
  - Globally, the population of working age is expected to continue to grow until about 2040, but the ratio of the working age population to the total is already declining.
  - In the case of China, the working-age population is expected to shrink by a fifth over the next 30 years.
- Recurring global pandemics can adversely affect labor supply (examples: AIDS pandemic, Covid-19).
- In absence of mass south-north migration, robots may become essential to meet more of the needs of the elderly and reverse declines in aggregate output and welfare.1

- Paper’s objective:
  - Examine combined welfare effects of automation and lower labor supply using neoclassical growth models with and without intergenerational transfers.
  - Replicate Kotlikoff and Sachs (K&S, 2012) result that a one-time improvement in machine technical efficiency can immiserate all future generations under two crucial assumptions:
    - Machines are very good substitutes for unskilled labor, displacing workers and lowering wages economy-wide.
    - No operative intergenerational transfers exist, so owners of capital consume the windfall during their lifetime (a generation, roughly thirty years), reducing saving, depressing investment in physical and human capital, and triggering persistent welfare decline.
  - Assess whether government policy (e.g., wealth taxes or socializing part of capital stock to finance a universal basic income) is necessary to spread automation windfalls across generations when altruism/selfishness prevents voluntary transfers.

### II. Role of intergenerational transfers and bequests
- Empirical motivation:
  - Private intergenerational transfers are substantial: about half of all households plan to leave estates (Laitner and Juster, 2017).
- Model findings:
  - In a version of the K&S model with operative bequests, intergenerational transfers are positive in equilibrium if the strength of altruism exceeds a certain threshold, mitigating negative effects of automation.
  - Perfect altruism (full operative bequests) can pass automation gains to future generations without nationalizing capital—but is an extreme assumption since many families cannot make efficient transfers to their children.
- Need for heterogeneous models:
  - A balanced model should feature heterogeneity within and across generations: some households make efficient bequests, others are in corner solutions (bequest-constrained).
  - The paper assesses automation immiseration in:
    - A K&S variant with pure altruism.
    - Glomm and Ravikumar (G&R, 1992) framework where parents invest in children’s schooling.
    - A model used by Ivanyna, Mourmouras and Rangazas (2018) with two household types: poor (bequest-constrained) and rich (unconstrained).
- Policy implications highlighted:
  - Fiscal and educational policies that alleviate financing constraints for human capital investments and restore equality of opportunity can raise welfare.
  - Strengthening altruistic bonds raises bequests and human capital investments of the young, providing additional stimulus to economic growth.
  - Government transfers financed by taxation of the rich can improve welfare of the poor and reduce inequality within and across generations when altruistic links are weak.

### III. Combined shocks: automation plus demographic labor supply decline
- Analysis focuses on a combined shock: a jump in automation together with a reduction in labor supply driven by demographics.
- Key mechanisms and outcomes:
  - Strengthened altruism increases intergenerational transfers and human capital investment.
  - Tax-financed transfers from the rich to the poor can mitigate inequality and improve cross-generational welfare when private transfers are insufficient.

### IV. Relevant literature (selected mentions and quantitative findings)
- Literature spans pessimistic and optimistic views on automation’s labor effects (authors cited in source: Gordon (2012), Cowen (2011), Acemoglu and Restrepo (2017, 2018), Sachs and Kotlikoff (2012, 2015), Ford (2015), Freeman (2015), Brynjolfsson and McAfee (2014), Autor (2014, 2015), Bostrom (2014), Graetz et al. (2018), Gaaitzen et al. (2020)).
- Quantitative evidence from Graetz et al. (2018):
  - Increasing use of robots raised the annual growth of GDP by 0.37 percentage points and labor productivity by 0.36 percentage points during 1993-2007 in 17 countries.
  - Those robots did not significantly reduce total employment, although they reduced low-skilled workers’ employment share.

- Historical and empirical context emphasized: whether automation replaces labor share and employment through routine-task substitution or increases labor participation by creating high-paying jobs in emerging occupations remains a key open issue.

*Source: wpiea2024011-print-pdf - Appendix A.   K&S Model with Bequest.*

### 2015.    The  authors  found  that  increased  use  of  robots  is  associated  with  positive  changes  in  the

### wpiea2024011-print-pdf - 2015.    The  authors  found  that  increased  use  of  robots  is  associated  with  positive  changes  in  the

### Automation, employment, and sectoral impacts
- Increased use of robots is associated with:
  - positive changes in the employment share of non-routine analytic jobs
  - negative changes in the share of routine manual jobs
- The 2020-22 pandemic accelerated digital transformation, but many sectors (including the public sector) are lagging, sustaining concerns about automation’s effects on employment (Spence, 2022).
- Automation negatively affects employment growth holding other factors constant (consistent with Graetz and Michaels (2018) and Phiromswad et al (2022)).
- Disaggregated measures of age-related abilities affect employment growth, while an aggregate age measure does not.
  - Occupations with low scores on both age-appreciated cognitive ability and age-depreciated physical ability (e.g., production occupations; food preparation and serving related occupations) experience the strongest negative effect of automation on employment.
  - Occupations with high scores in both age-appreciated cognitive ability and age-depreciated physical ability (e.g., protective service occupations; healthcare practitioners and technical occupations) experience the weakest negative effect.

### AI exposure, complementarity/substitution, and cross-country variation
- Pizzinelli and others (forthcoming):
  - Propose an extension to standard AI exposure measures to account for AI as either a complement or a substitute for labor (complementarity reflects lower risks of job displacement).
  - Worker-level microdata from US, UK, Brazil, Colombia, India, South Africa reveal substantial variations in unadjusted AI exposure across countries.
  - Findings:
    - AI poses risks of labor displacement via task automation.
    - AI can enhance productivity and complement human labor, especially in occupations requiring high cognitive engagement and advanced skills.
    - Advanced Economies (AEs) may expect a more polarized impact: greater risk of labor substitution but also greater productivity benefits.

### Public policy on digital technologies and R&D
- Enhancing policy—including R&D and regulatory platforms in private and public sectors—to support digital technologies is key to improving productivity.
- Concerns about automation’s employment effects will persist unless lagging sectors (notably the public sector) accelerate adoption and supportive policy frameworks.

### Education spending and empirical estimates linking education to growth
- The authors find government spending on education promotes economic growth; conclusions supported by multiple empirical studies:
  - Fabrizio Carmignani (2016): Using World Bank WDI data on 151 countries for 2000 - 2010, concluded that “increase in education expenditure by 1 point of GDP increases GDP growth by 0.9 percentage points”.
  - Gheraia, Zouheyr et al. (2021): For Saudi Arabia 1990-2017, found that a rise in educational expenditure by 1% leads to an increase in economic growth by 0.89%.
  - Yahya, Mohd et al. (2012): For Malaysia 1970-2010, found economic growth is positively correlated with fixed capital formation, labor force participation and expenditure in education; educational expenditure is the short-term Granger cause of economic growth and vice versa.
  - Mehmet Mercana et al. (2014): For Turkey 1970-2012, using ARDL bounds testing, found that 1% increase in education expenses increases economic growth by 0.3%.

### Analytical framework: The K&S two-period OLG model (A. Analytical Tool)
- Model structure:
  - Two-period-lived overlapping generations, periods t = 1,2,...
  - Young supply one unit of inelastically supplied unskilled labor, invest part of income in physical capital (machines, M) and human capital (skilled labor, S). Machines and human capital are perfect substitutes in saving portfolios.
  - Gross output Q is a CES production function of economy-wide stocks of M, L, and S:
    - Q = Q(N(uM, L), S)  (A.1)
    - M and L combine in a CES with elasticity ε_ML to produce N; N and S combine in a CES with elasticity ε_SN to produce Q.
    - u is technical efficiency of machinery; rise in u is a pure technical advance.
- Wage response to machine productivity:
  - d ln(Q_L) / d ln(u) = [ε_SN − θ ε_ML] / ε_ML  (A.2)
  - θ = (Q_S * S) / Q (share of skilled labor in the economy)
  - A rise in machines’ productivity reduces the unskilled wage if ε_ML > ε_SN θ /.
- Conditions making “im-mesmerizing” productivity more likely:
  - Substitutability of machines and unskilled labor is high (ε_ML large)
  - Substitutability of intermediate goods and skilled labor is low (ε_SN small)
  - The share of skilled labor in final output is high (θ high)
- Lifetime budget and utility:
  - Young income: I_t = L_t W_t = C_t + S v_t  (A.3)
  - Old income: I_{t+1} = (∂Q/∂M)_t M_{t+1} + (∂Q/∂S)_t S_{t+1} = D_{t+1}  (A.4)
  - Savings S v_t = M_{t+1} + S_{t+1}  (A.5)
  - No-arbitrage: R_t = (∂Q/∂M)_t = (∂Q/∂S)_t = 1 + r_t  (A.6)
  - Lifetime utility U = β log(C_t) + (1−β) log(D_{t+1})  (A.8)
  - First order condition: β / C_t = (1−β) / (D_{t+1} R_t)  (A.9)
- Cobb-Douglas final goods formulation (K&S baseline):
  - Q(N,S) = A N^{α} S^{1−α}  (A.10)
  - Intermediate N = L + u^{1} M  (A.11) (machines and labor perfect substitutes)
  - Marginal products and equilibrium conditions lead to analytical solutions for consumption, savings, M and S (A.12–A.15).

### K&S numerical experiment and Proposition A1 (immiserating productivity)
- Parameterization used to illustrate dynamics:
  - 훼=0.74 on labor and 0.26 on machines
  - 훽=0.5 on skills
  - Productivity unexpectedly rises by 35% in period t=3 (u_1 increases by factor 1.35)
- Key numerical outcomes:
  - Current wage of unskilled workers (earnings of the young of generation 3) declines from 2.82 to 2.61 due to rise in u_1.
  - Returns on M and S both rise; the old generation (owners of M and S) experiences a boom in income while the young experience a bust.
  - By period 5, new equilibrium has lower wages, lower skills, lower M, and higher total output than baseline.
  - Ratio of earnings of skilled workers relative to unskilled workers, W_s / W_l, is permanently raised from 1 to 1.35.
  - Lifetime utility: utility of generation 3 slightly below generation 1; generations 4 and later higher than baseline. Young unskilled workers lose from the rise in machine productivity; generation 3 benefits from high returns to M and S.
- Proposition A1:
  - Improvements in automation benefit owners of capital (the initial old generation) and lower the welfare of all subsequent generations of workers.
  - A positive shock in the efficiency of machines leads to higher consumption and income of the old, lowers wage incomes of unskilled workers, lowers their lifetime consumption and utility, and increases intergenerational inequality.

### Government intervention: One-time tax and transfers (Section B)
- Policy mechanism:
  - Government imposes a one-time tax (wealth levy) on machines M_G at time t=3, taking amount M_G < M.
  - Private machines left: M_P = M − M_G.
  - Government income Q_M M_G split into transfer T = (Q_M − 1) M_G added to the wage of the young, and remainder M_G reinvested.
  - Government machines M_G are constant after t; government pays transfer T to each young generation and reinvests M_G to maintain its stock.
- Modified budget and equilibrium equations: (B.1–B.3)
  - I_t = L_t W_t + T = C_t + S v_t  (B.1)
  - S v_t = (1−β) I_t  (B.2)
  - M_P,t+1 + S_{t+1} = S v_t; M_P,t+1 + M_G + L_t u^{/} = (α S_{t+1})/(1−α)  (B.3)
- Proposition B1:
  - Government tax on machines and redistribution of the windfall from old to young increases welfare of all generations if the tax is sufficiently large.
  - Productivity shocks accompanied by taxes on machines result in higher utility for all future generations; future generations are no longer immiserated if the government tax is sufficiently large.
- Numerical example:
  - Government imposes a 10% levy on machines owned by the old in period 3.
  - Government transfers T = (Q_M − 1) M_g to the young of generation 3; government reinvests M_G to maintain constant M_G.
  - Example outcomes:
    - Utility at period 1 = 2.37; rises to 2.63 at period 6, leaving generation 3 better off than baseline despite the capital levy.
    - Income and consumption of old drop at period 3 because of transfer from old to young; in period 4 and subsequent periods consumption and income of old increase.
    - Future generations are even better off because of the government transfer program.

### Bequests, altruism, and implications (Section C)
- Model extension: pure altruism where household utility depends on consumption in two periods and on utility of children; bequests B_t enter young income.
  - Young income: I_t = L_t W_t + B_t = C_t + S v_t  (C.1)
  - Old income: I_{t+1} = R_t S v_t = D_{t+1} + (1+ n) B_{t+1}
- Derived relations (C.2) for B_{t+1}, C_t, D_{t+1}, m_t in terms of parameters including selfishness/altruism parameter 휑 and number of children 1 + n.
- Bequest growth multiplier:
  - g = (φ R) / ((1+ n)(1−φ))
  - Bequests grow exponentially if φ > φ_cr = 1 / [1 + R_t / (1+ n)]
- Proposition C1:
  - Bequests increase welfare of the young and decrease inequality between old and young.
  - Future generations are not immiserated if the altruism parameter is above the threshold; the model can sustain exponential growth in consumption and utility driven by accumulation of machines and skills.
- Numerical experiments:
  - Assume machine productivity grows each year by 1%; over 30 years this results in a jump in u_1 by 35%.
  - Parameters: n=0, α=0.74, A=5, β=0.5.
  - Example with selfishness parameter φ=0.3 (> φ_cr): no steady state solution; starting period 3 with productivity growth by 35%, from period 4 economy grows exponentially, income and consumption of young increase, wage of skilled > wage of unskilled at all times.
  - Example with φ < φ_cr: model predicts a new steady state with lower income and consumption of young; temporary shock in machine productivity causes positive bequests in period 4 and economy reaches steady state by period 6 in an example.

### Education, bequests, and public vs private schooling (Section D)
- Model extension: bequests plus parental expenditures E_t on children’s education.
- Proposition D1:
  - Individuals with private schooling possess higher income and consumption than those with public schooling.
  - Bequest-constrained poor households may not afford rising educational costs; this can be partially alleviated by transfers financed by taxes on machines of rich households with private schooling.
  - For selfishness parameter less than critical (old invest more in their young), such transfers are unnecessary.
  - Targeted transfer to poor households with public education increases income and consumption of these households when selfishness parameter larger than critical; not beneficial when it is less.
  - Public schooling provides equal opportunities and decreases population inequality.

*Source: wpiea2024011-print-pdf - 2015.    The  authors  found  that  increased  use  of  robots  is  associated  with  positive  changes  in  the*

### 1. Private Education

### 1. Private Education

### Private education regime: setup and budget constraints
- Young generation income components: wage earnings, bequest from old generation, and educational expenses financed by old generation.
- Savings of young 푆푣_t invested into machines 푀_t and skills 푆_t. Income of old is return on machines and skills.
- Budget equations:
  - 퐼_t = 퐿_t 푊_t + 퐵_t + 퐸_t = 퐶_t + 푆푣_t  (D.1)
  - 퐼_{t+1} = 푅_t 푆푣_{t+1} = 퐷_{t+1} + (1+푛) 퐵_{t+1} + (1+푛) 퐸_{t+1}

### Value function and optimization
- Augmented value function:
  - 푉(퐵_t,퐸_t) = 훽 log(퐶_t) + (1−훽) log(퐷_{t+1}) + 휑 푉(퐵_{t+1},퐸_{t+1})  (D.2)
- Individual choice problem:
  - max_{퐶_t,퐷_{t+1},퐸_{t+1}} 푉, subject to 퐵_{t+1} ≥ 0, 퐸_{t+1} ≥ 0  (D.3)
- Bellman solution conjecture:
  - 푉(퐵_t,퐸_t) = 푎 + 푏 log(퐵_t + 퐸_t + 푚_t)  (D.4)

### First-order conditions and optimal allocations
- Optimality conditions (from partial derivatives and (D.1, D.4)):
  - 훽 / 퐶_t = 2휑푏 (1+푛)(퐵_{t+1}+퐸_{t+1}+푚_{t+1}) / 푅_t  (D.5)
  - (1−훽) / 퐷_{t+1} = 2휑푏 (1+푛)(퐵_{t+1}+퐸_{t+1}+푚_{t+1})  (D.5)
- Log-linear relationship from Bellman substitution:
  - 푏 log(퐵_t+퐸_t+푚_t) = (1+휑푏) log(퐵_{t+1}+퐸_{t+1}+푚_{t+1}) + const  (D.6)
- Relationship between bequest and education (after budget substitution):
  - −푏 log(퐵_t+퐸_t+푚_t) = (1+휑푏) log(퐵_t+퐸_t+퐿_t 푊_t + (1+푛)/푅_t 푚_{t+1}) + const  (D.7)

### Closed-form coefficients and sharing rule
- Undetermined coefficients solution (identical to C.16-17):
  - 푏 = 1 / (1−휑)  (D.8)
  - 푚_t = 퐿_t 푊_t + (1+푛) 푚_{t+1} / 푅_t
- Sharing of saving between bequest and education:
  - Like Glomm and Ravikumar model, private agents spend half of their saving on bequests to their children and half on educational expenses.
- Explicit allocations (solving (D.1, D.5)):
  - 퐶_t = 훽 (1−휑)/(1+휑) [퐼_t + (1+푛)/푅_t 푚_{t+1}]  (D.9)
  - 퐷_{t+1} = (1−훽)/훽 푅_t 퐶_t  (D.9)
  - 퐵_{t+1} = 퐸_{t+1} = [2 + (1+푛)(1−훽)(1−휑)/휑]^{−1} [퐼_{t+1} + (1+푛) 푚_{t+1}] − 푚_{t+1}/2  (D.9)

### Production with human capital
- Modified Cobb-Douglas production:
  - Q(푁_t,퐻_t) = 퐴 푁_t^{훼} 퐻_t^{1−훼}  (D.10)
- Human capital definition:
  - 퐻_t = 푆_t + 퐸_{t−1} (skills level plus education attainment)
- Perfect foresight conditions for machines and human capital:
  - 퐿_u / (1+푀_{t+1}) = 훼/(1−훼) 퐻_{t+1}  (D.11)
  - 푀_{t+1} + 푆_{t+1} = 푆푣_t
- Equations (D.9, D.11) describe dynamics of extended K&S model with private education.

---

### 2. Public Education

### Public schooling financing rule
- Public education expense determined by average tax revenue:
  - 퐸_t = 휏 S\bar{v}_t = 휏 ( \bar{M}_{t+1} + 푆_{t+1} )  (D.12)
- Tax rate 휏 is flat across population; state socializes portion 휏 of machines and skills to pay for public school education expenses.

### Budget constraints with public schooling
- Individual budget constraints:
  - 퐼_t = 푊_t 퐿 + 퐵_t + 퐸_t = 퐶_t + 푆푣_t
  - 퐼_{t+1} = 푅_t (1−휏) 푆푣_t = 퐷_{t+1} + (1+푛) 퐵_{t+1} + (1+푛) 퐸_{t+1}  (D.13)

### Value function and first-order conditions
- Maximization problem:
  - max_{퐶_t,퐷_{t+1}} 푉, subject to 퐵_{t+1} ≥ 0  (D.14)
- Value function:
  - 푉(퐵_t) = 훽 log(퐶_t) + (1−훽) log(퐷_{t+1}) + 휑 [푎 + 푏 log(퐵_{t+1} + 퐸_{t+1} + 푚_{t+1})]
- First-order conditions:
  - 훽 / 퐶_t = 휑푏 (1−휏) (1+푛)(퐵_{t+1}+퐸_{t+1}+푚_{t+1}) / 푅_t  (D.15)
  - (1−훽) / 퐷_{t+1} = 휑푏 (1+푛)(퐵_{t+1}+퐸_{t+1}+푚_{t+1})  (D.15)

### Closed-form solution under public education
- Solutions to (D.13, D.15):
  - 퐶_t = 훽 (1−휑) [ 퐼_t + (1+푛)(1−휏) 푅_t / 푚_{t+1} ]  (D.16)
  - 퐷_{t+1} = (1−훽)(1−휏)/훽 푅_t 퐶_t  (D.16)
  - 퐵_{t+1} = 휑 / (1−훽+훽휑) ( 퐼_{t+1}/(1+푛) + 푚_{t+1} ) − 퐸_{t+1} − 푚_{t+1}  (D.16)
  - 푀_{t+1} = 훼 푆푣_t − (1−훼) 퐿_t 푢 / (1 + 훼 퐸_t\bar)  (D.16)
  - 푆_{t+1} = (1−훼) 푆푣_t + (1−훼) 퐿_t 푢 / (1−훼퐸_t\bar)  (D.16)
- Perfect foresight evolution of skills and machines follows conditions (A.6) and parallels (A.15) of K&S.

### Distributional and policy implications (Proposition D2 and numerical experiments)
- Proposition D2:
  - Inequality within generations is reduced when public education is financed by income taxes levied on skilled workers’ households. Inequality across generations declines. No need to socialize physical capital; private inter-generational transfers can spread technological windfalls. Government investments in education help equalize opportunities.
- Numerical experiments (heterogeneous agents by selfishness parameter 휑):
  - Number of draws: 1,000.
  - Truncated normal distribution of selfishness 휑 with bounds (0,1).
  - Variance 휎 = 0.022, 0.012 chosen so CDF of truncated normal is 0.5 at 휑_{cr} = 0.14 (public) and 0.08 (private). Only half of households can afford to leave a bequest.
  - For private schooling: consumption and income of young and old grow exponentially.
  - For public schooling: old generation income up to six periods insufficient to leave a bequest; variables reach a new low steady state.
  - K&S model with bequest plus education predicts higher bequest, utility, income, savings, and consumption compared with K&S with bequest alone.
  - Example parameter for a plotted scenario: mean altruism parameter 휑̅ = 0.15 and tax rate 휏 = 5%.

### Targeted transfers and taxation experiments
- Revenue-raising via taxing poor households may be insufficient; taxing rich households can subsidize poor households’ educational costs.
- One-time policy experiment:
  - Government imposes a one-time tax on 5% of the machine population at time 푡 = 3.
  - Government transfers a constant payment T = 0.05 * (Q_{M−1}) * M to poor public households from period 3 onward, increasing human capital of poor households.
- Targeted transfer scenarios:
  - Scenario 1: payments to households with 휑 < 휑̅ (group 1).
  - Scenario 2: payments to households with 휑 > 휑̅ (group 2).
  - Poor and wealthy households receive the same transfer amount; relative change due to transfer is larger for poor households (they are much better off than wealthy ones by the relative effect).
- Note: For impatience parameters 훽 < 0.5 poor households can invest more into machines and education compared to consumption and afford higher educational costs.

### Inequality effects of public vs private schooling
- Simulations with initial heterogeneity in skills (same selfishness parameter 휑 across agents):
  - Number of draws: 1,000.
  - Initial skills distribution: truncated normal with standard deviation 휎 = 1.
  - Selfishness parameter 휑 = 0.15 and tax rate 휏 = 10%.
- Results:
  - Public schooling provides equal opportunity; distribution of human capital, consumption, and income tends to become homogeneous.
  - Private schooling leads to growth in variance of human capital after productivity increase.

### Summary implication
- Education (private or public) provides additional means to improve well-being of younger and older generations and the economy. Public education financed by income taxes on skilled households reduces within-generation inequality and across-generation inequality; targeted transfers and taxing richer machine owners can help cover educational shortfalls for the poor.

_italics: Source: wpiea2024011-print-pdf - 1. Private Education_

### Appendix B.   Replicating Kotlikoff and Sachs (2012)

### Appendix B.   Replicating Kotlikoff and Sachs (2012)

### Replication setup and parameters
- Parameters used in the K&S model run:
  - 퐿=1, 퐴=10, 훼=0.6, 훽=0.5

### Replication results (table reproduced)
- Table A1. Replication of Sachs and Kotlikoff results.
- Columns: T, u, M, S, Wl, Ws, Sv, Q
- Rows:
  - T 1: u 1, M 1.1, S 1.4, Wl 5.10, Ws 5.10, Sv 2.55, Q 18.12
  - T 2: u 1, M 1.1, S 1.4, Wl 5.10, Ws 5.10, Sv 2.55, Q 18.12
  - T 3: u 10, M 1.1, S 1.4, Wl 2.53, Ws 14.6, Sv 2.04, Q 51.88
  - T 4: u 10, M 0.7, S 0.6, Wl 2.03, Ws 20.3, Sv 1.79, Q 27.72
  - T 5: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
  - T 6: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
  - T 7: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
  - T 8: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
  - T 9: u 10, M 0.6, S 0.5, Wl 2.03, Ws 20.3, Sv 1.79, Q 22.66
- Variable definitions:
  - 푇 is the time
  - 푢 is the productivity parameter
  - 푀 is the machines
  - 푆 is the skills
  - 푊푙 is the earnings of young
  - 푊푠 is the earnings of old
  - 푆푣 is the savings of young
  - 푄 is the Cobb-Douglas productivity function
- Note: These results reproduce Table 1 (unanticipated case) from Sachs and Kotlikoff paper except those authors claim that they obtained their results for slightly different set of parameters: 퐿=1, 훼=0.5, 퐴=10, 훽=0.5

### Appendix C.   Robustness Checks

### Model setup and key identities
- Young generation income:
  - 퐼푡 = 푊푡 + 퐵푡 = 퐶푡 + 푆푣푡          (1)
- Savings of young generation invested in capital:
  - 푆푣푡 = (1+푛) 퐾푡+1
- Income of older generation from proceeds of savings of young:
  - 퐼푡+1 = 푅푡 푆푣푡 = 퐷푡+1 + (1+푛) 퐵푡+1         (2)
- Value function augmented with shadow prices of bequest 푝푡:
  - 푉 = 푙표푔(퐶푡) + 훽^ 푙표푔(퐷푡+1) + 휑^푝푡+1 퐵푡+1 − 푝푡 퐵푡       (3)
- First-order conditions yield:
  - 퐶푡 = (1+푛) 휑^ 푝푡+1 푅푡            (4)
  - 퐷푡+1 = (1+푛) 훽^ 휑^ 푝푡+1
- Maximizing the Lagrange function with respect to 퐵푡 gives:
  - 푝푡+1 = (1+푛) 휑^ 푅푡 푝푡, if 퐵푡 > 0           (5)
  - 푝푡+1 < (1+푛) 휑^ 푅푡 푝푡, if 퐵푡 = 0

### Production technology and factor returns
- Cobb-Douglas production function:
  - 푄(퐾,퐿) = 퐴 (퐾/퐿)^훼 퐿           (6)
- Labor supply assumption:
  - 퐿 = 1
- Wage rate and interest factor:
  - 푊푡 = (1−훼) 퐴 퐾푡^훼             (7)
  - 푅푡 = 훼 퐴 퐾푡^(훼−1)

### Analytical solution used for robustness checks (Michel et al. (2004))
- Defined variables and relations:
  - 푣푡 = 퐴 푝푡 퐾푡^훼
  - 푣푡+1 = 1/(αφ̂) (푣푡 − 1 − β̂/φ̂)
  - 퐵푡 = (α − β̂(1 − αφ̂)/φ̂ + β̂) 퐴 퐾푡^훼           (8)
  - 푝푡 퐵푡 = 훼 푣푡 − β̂/φ̂
- Steady state solution for specific initial conditions:
  - 푣0 = 푣푡 = 1/(1 − αφ̂) (1 + β̂/φ̂)          (9)

### Key findings from robustness analysis
- The analytical steady state solution (equation (9)) exists for a specific set of initial conditions.
- The steady state solution is unstable and grows exponentially in time.
- Instability is driven by the multiplier 1/(αφ̂) > 1 because 훼 < 1 and 휑̂ ≤ 1.
- Numerical experiments using initial conditions in the range 푣 ∈ (0.99푣0, 1.01푣0) illustrate instability of the steady state with positive bequest (phase diagram of capital and bequest trajectories computed according to equations (8)).

*Automation, Labor and Welfare: The Role of Bequests and Education — Working Paper No. WP/2024/011*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024011-print-pdf.pdf_