## 3.1  Households

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---

### Introduction and motivation
- Context:
  - New wave of technological change involves "artificial intelligence (AI) and machine learning, robotics, big data, and networks".
  - The paper models the AI/robot revolution as a reduction in the price of “robot capital”, which may substitute directly for labor and complements traditional capital.
- Primary modeling approach:
  - Two-country one-sector neoclassical model with three factors of production: labor, capital, and “robots”.
  - Extension to two types of labor (skilled and unskilled) with developed country relatively well endowed in skilled labor.
  - Robot revolution captured as an increase in the productivity of “robots” (productivity parameter b_t).
- Central qualitative expectation:
  - When robots are close substitutes for labor, automation tends to increase GDP per capita but may increase inequality; real wages may fall during a long transition.

### Three channels through which developing economies could diverge after the robot revolution
- Share-in-production channel:
  - Advanced-country higher TFP implies higher wages and more intense robot use in the advanced economy, producing higher robot shares in income.
  - Higher robot shares lead to much higher GDP growth in the advanced economy than in the developing economy when robot productivity increases.
- Capital-flows channel (transitional effect):
  - Increase in robot productivity induces strong demand for resources in the advanced economy to finance investment in robots and complementary physical capital.
  - Capital flows “uphill” from developing countries to advanced economies during the transition (unless the financial account is closed).
- Terms-of-trade channel (two-sector extension):
  - Two-sector model with one sector relatively intensive in unskilled labor; developing country specializes in unskilled-intensive goods.
  - If robots substitute more closely with unskilled labor, an increase in robot productivity reduces demand for unskilled workers and unskilled wages, lowering the relative price of the unskilled-intensive good and reducing incentives to invest in that sector.
  - For developing countries with low robot shares, this relative-price effect can result in income levels declining in the long run.

### Stylized facts on international robot adoption and wages (data and empirical patterns)
- Data source and scope:
  - Robot data from the International Federation of Robots (IFR); manufacturing sector focus.
  - Robot definition per ISO cited from International Federation of Robotics (2012): “an automatically controlled, reprogrammable, multipurpose manipulator programmable in three or more axes, which may be either fixed in place or mobile for use in industrial automation applications”.
- Cross-country and time trends:
  - Robot adoption rising rapidly in developing countries as well as advanced economies.
  - Middle-income countries' share of the total operational stock of robots rose from "about 5 percent" in "2010" to "almost 24 percent" by "2016".
  - The increase is largely driven by China but is broad-based across regions.
- Robot density and wages relation:
  - Robot density (robots per worker in manufacturing) remains much higher in advanced economies despite faster recent adoption in developing countries.
  - Cross-section evidence (43 countries, "2010"):
    - Positive relation between log real wages and log robots-to-employment ratio.
    - Estimated slope is "about 1.65", indicating robot intensity varies more than proportionally compared to wages in the cross-section.
  - Panel/time-change evidence:
    - Change in log wages vs change in log robot density over available sample period shows a positive relationship with a slope greater than 1, suggestive of a relatively high elasticity of substitution between robots and labor.
- Data notes and exclusions:
  - Figure 3: cross-section of "43 countries" in "2010"; "Venezuela" is excluded from Figure 3.
  - Figure 4: "Greece, Iran and Argentina are excluded from figure 4."

### Model — Households, firms, and equilibrium (selected equations and implications)
- Households:
  - Preferences: ∑_t β^t C_{i,t}^{1−1/τ} / (1−1/τ); β is the discount factor; τ determines inter-temporal elasticity of substitution.
  - Budget constraint: C_{i,t} + I^K_{i,t} + I^Z_{i,t} + (B_{−i,t+1} − B_{i,t+1}) = r^K_t K_{i,t} + r^Z_t Z_{i,t} + w_{i,t} L_i + (1 + r^B_t)(B_{−i,t} − B_i,t) + Π_{i,t}
  - Laws of motion: K_{i,t+1} = (1 − δ_K) K_{i,t} + I^K_{i,t}; Z_{i,t+1} = (1 − δ_Z) Z_{i,t} + I^Z_{i,t}
  - Euler: [C_{i,t+1} / C_{i,t}]^{1/τ} = β (1 + r^B_t)
  - No-arbitrage: r^K_t − δ_K = r^Z_t − δ_Z = r^B_t
- Firms:
  - Production: Y_{i,t} = A_i (K^d_{i,t})^α [ e^{1/σ} L_{i,t}^{(σ−1)/σ} + (1−e)^{1/σ} (b_t Z^d_{i,t})^{(σ−1)/σ} ]^{(1−α)σ/(σ−1)}
  - Robot-labor ratio condition: Z^d_{i,t} / L_{i,t} = ((1−e)/e) b_t^{σ−1} (w_{i,t} / r^Z_t)^σ
  - Labor-capital ratio condition: L_{i,t} / K^d_{i,t} = (1−α)/α * r^K_t / w_{i,t} * [1 + ((1−e)/e)^{1/σ} ( (b_t Z^d_{i,t} / L_{i,t})^{(σ−1)/σ} ) ]
- Equilibrium market-clearing:
  - L_i = L_{i,t}; K_{i,t} = K^d_{i,t}; Z_{i,t} = Z^d_{i,t}
  - Goods: Σ_i (C_{i,t} + I^K_{i,t} + I^Z_{i,t}) = Σ_i Y_{i,t}
  - Financial assets: B_{A,t} + B_{D,t} = 0

### Calibration of initial steady states (selected parameter values and targets)
- Elasticity of substitution considered: σ ∈ {1, 2, 3}.
  - σ = 1 corresponds to Cobb-Douglas with three factors.
  - Empirical suggestions place σ between 1.5 and 3; Eden and Gaggl (2018) report increases from 2.5 to 3.3; Berg et al. (2018) find elasticity 2.1 for ICT capital and unskilled labor.
- Calibration targets (for each σ):
  - Relative GDP per capita between regions set to 15.
  - Robot share in income in the advanced economy set to 4 percent.
- Table 1 selected parameters:
  - β = 0.96
  - τ = 1
  - δ_K = 0.05
  - δ_Z = 0.05
  - α = 0.35
  - L_D = 1
  - L_A = 1
  - A_A = 1
  - b_0 = 0.1
  - B_0 = 0
- Calibration implications:
  - Calibrated labor-share parameter e is higher for larger σ to maintain a 4 percent robot income share in the advanced economy.
  - Robot share in the developing region declines as σ increases.
  - Relation: (Z_A / L_A) / (Z_D / L_D) = (w_A / w_D)^σ, implying higher wages in the advanced economy lead to higher robot-to-labor ratios, amplified when σ > 1.

### Long-run impact of the robot revolution (doubling b)
- Experiment: robot productivity b doubles in both regions.
- Long-run outcomes by σ:
  - σ = 1: outcomes symmetric; change in GDP identical across regions.
  - σ > 1: advanced economy benefits more because higher initial robot share and higher wages make robot and complementary capital investment more profitable in the advanced economy → divergence.
- Distributional effects:
  - Real wages increase in both regions in the long run, but robot and capital accumulation outpace wage increases, so labor share falls in both regions, more so in the advanced economy.
  - Higher growth in advanced region associated with higher inequality (within and across regions).

### Short-run (transition) dynamics (example with σ = 3 and capital mobility)
- Transition path: robot productivity doubles over four periods (equal-sized increases each period) then remains constant.
- Main transitional dynamics:
  - Developing region may experience an absolute fall in GDP in the short run.
  - Robot stocks increase in both regions; capital stock in the developing economy can fall for an extended period as resources flow to advanced economy to finance robot and capital investment there.
  - Developing region runs a current account surplus (acts as lender), accumulating net foreign assets; final steady state: consumption in developing region may exceed output financed by interest on accumulated assets.
  - Consumption falls in both regions initially; larger and more persistent fall in the developing economy.
  - Wages drop after the robot revolution in both regions when σ = 3, with a larger decline in the developing region during transition; eventually wages recover as capital accumulates, slower in developing region.
  - Labor income share decline is larger in the advanced economy; gross national income composition shifts in developing region toward interest income due to foreign asset accumulation.
- Role of capital account openness:
  - With open capital accounts, developing region lends and finances advanced-region robot expansion; creates sharper short-run divergence in output but allows developing region to earn interest income and benefit in consumption eventually.
  - With closed capital accounts, developing country is insulated from capital outflows; advanced region experiences amplified investment response; developing country misses out on owning some advanced-country robot capital stock and loses long-run consumption benefits from higher robot productivity (consumers in developing country lose almost all long-run consumption benefits).

### Two-skill and two-sector extensions (structure, calibration, and effects)
- Production with skills: Y_{i,t} = A_i (K^d_{i,t})^{α_K} (S_{i,t})^{α_S} [ e^{1/σ} L_{i,t}^{(σ−1)/σ} + (1−e)^{1/σ} (b_t Z^d_{i,t})^{(σ−1)/σ} ]^{(1−α_K−α_S)σ/(σ−1)}
- Calibration (selected Table 4 values):
  - αK,T1 = αK,T2 = 0.35
  - αS,T1 = 0.35
  - αS,T2 = 0.25
  - L_D = 0.98
  - S_D = 0.02
  - L_A = 0.70
  - S_A = 0.30
- Rationale:
  - Allows cross-country income differences to arise from differences in human capital endowments.
  - Robots can be complementary to skilled labor and substitute for unskilled labor.
- Quantitative changes:
  - Divergence result persists but is quantitatively smaller when robots substitute only for a subset of labor.
  - Example: when σ = 3, advanced economy long-run growth ≈ 39 percent in one-skill baseline versus about 23 percent in the two-skill-level model.
- Two-sector mechanisms:
  - Direct effect: incentive to accumulate robots and complementary physical capital in both sectors.
  - Indirect (terms-of-trade) effect: decline in demand for unskilled labor lowers unskilled wages; because T2 is unskilled-labor intensive, its relative price falls, reducing incentives to invest in T2 and potentially lowering long-run income for unskilled-intensive (developing) economies.
- Sectoral outcomes:
  - Advanced economy: output of both goods increases; for T2, direct investment incentives can dominate the fall in relative price, raising T2 output.
  - Developing region: output of good T2 declines in equilibrium when σ > 1; with baseline specialization in T2, GDP can fall in absolute terms for large σ.

### Inequality and labor-market outcomes
- For large σ:
  - Skilled wages increase in line with per-capita GDP (robots complement skilled labor).
  - Absolute unskilled wages can fall because robots substitute for unskilled workers and increased robot investment reduces demand for unskilled labor.
  - Fall in unskilled wages is larger in advanced economies due to greater robot investment there.
  - Higher σ implies larger increase in robot share at the expense of unskilled labor share and an expanding skill premium.

### Policy-relevant implications and recommendations
- Core conclusions:
  - Improvements in robot productivity drive divergence between advanced and developing countries: advanced countries benefit more because they make greater use of robots.
  - With capital mobility, transitional capital flows can move from developing to advanced countries, causing transitional declines in developing-country GDP. With capital immobility, developing countries lose the long-run consumption gains associated with transitional accumulation of high-yielding robot capital.
  - In the two-sector, two-skill model, developing countries that are relatively unskilled-intensive can face a permanent terms-of-trade decline and possible absolute declines in GDP.
- Policy messages:
  - There is no silver bullet to avert divergence. Developing countries urgently need to invest in raising aggregate productivity and skill levels so that labor is complemented rather than substituted by robots.
  - Human capital accumulation is crucial to prevent divergence; heterogeneous skill levels imply heterogeneous growth dynamics across developing countries.
  - Continued automation threatens existing complementarity between humans and machines; without rapid productivity and education improvements, demographic dividends (e.g., in Africa) could yield negative rather than positive outcomes.
  - By 2030, more than half of the increase in the global labor force is expected to come from the African continent, raising stakes for policy action to capture potential dividends.

*Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020184-print-pdf.pdf*

### 3.1  Households  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  11

### 3.1  Households

### Key themes from the introduction and motivation
- Context:
  - New wave of technological change involves "artificial intelligence (AI) and machine learning, robotics, big data, and networks".
  - The paper models the AI/robot revolution as a reduction in the price of “robot capital”, which may substitute directly for labor and complements traditional capital.
- Primary modeling approach:
  - Two-country one-sector neoclassical model with three factors of production: labor, capital, and “robots”.
  - Extension to two types of labor (skilled and unskilled) with developed country relatively well endowed in skilled labor.
  - Robot revolution captured as an increase in the productivity of “robots”.
- Central qualitative expectation:
  - When robots are close substitutes for labor, automation tends to increase GDP per capita but may increase inequality; real wages may fall during a long transition.
- Evidence and references cited:
  - Frey and Osborne (2017) estimate that some "47 percent of jobs are subject to replacement by AI."
  - Grace et al. (2018) survey AI researchers who on average expect AI to outperform humans at translating languages by "2024", driving a truck by "2027", and working in retail by "2031".
  - Empirical work (Eden and Gaggl (2018)) distinguishes ICT capital as more substitutable with labor than traditional capital for the United States for 1950-2013.

### Three channels through which developing economies could diverge after the robot revolution
- Share-in-production channel:
  - Advanced-country higher TFP implies higher wages and more intense robot use in the advanced economy, producing higher robot shares in income.
  - Higher robot shares lead to much higher GDP growth in the advanced economy than in the developing economy when robot productivity increases.
- Capital-flows channel (transitional effect):
  - Increase in robot productivity induces strong demand for resources in the advanced economy to finance investment in robots and complementary physical capital.
  - Capital flows “uphill” from developing countries to advanced economies during the transition (unless the financial account is closed).
- Terms-of-trade channel (two-sector extension):
  - Two-sector model with one sector relatively intensive in unskilled labor; developing country specializes in unskilled-intensive goods.
  - If robots substitute more closely with unskilled labor, an increase in robot productivity reduces demand for unskilled workers and unskilled wages, lowering the relative price of the unskilled-intensive good and reducing incentives to invest in that sector.
  - For developing countries with low robot shares, this relative-price effect can result in income levels declining in the long run.

### Stylized facts on international robot adoption and wages (data and empirical patterns)
- Data source and scope:
  - Robot data from the International Federation of Robots (IFR); manufacturing sector focus.
  - Robot definition per ISO cited from International Federation of Robotics (2012): “an automatically controlled, reprogrammable, multipurpose manipulator programmable in three or more axes, which may be either fixed in place or mobile for use in industrial automation applications”.
- Cross-country and time trends:
  - Robot adoption rising rapidly in developing countries as well as advanced economies.
  - Middle-income countries' share of the total operational stock of robots rose from "about 5 percent" in "2010" to "almost 24 percent" by "2016".
  - The increase is largely driven by China but is broad-based across regions.
- Robot density and wages relation:
  - Robot density (robots per worker in manufacturing) remains much higher in advanced economies despite faster recent adoption in developing countries.
  - Cross-section evidence (43 countries, "2010"):
    - Positive relation between log real wages and log robots-to-employment ratio.
    - Estimated slope is "about 1.65", indicating robot intensity varies more than proportionally compared to wages in the cross-section.
  - Panel/time-change evidence:
    - Change in log wages vs change in log robot density over available sample period shows a positive relationship with a slope greater than 1 (Figure 4), suggestive of a relatively high elasticity of substitution between robots and labor.
- Data notes and exclusions:
  - Figure 3: cross-section of "43 countries" in "2010"; "Venezuela" is excluded from Figure 3.
  - Figure 4: "Greece, Iran and Argentina are excluded from figure 4."

### Modeling implications highlighted in the text
- One-sector model:
  - Long-run: per capita GDP always increases following an increase in robot productivity.
  - Transition: per capita GDP in developing economy can fall due to capital outflows to advanced economies financing robot and capital accumulation there.
  - If financial account is closed, capital does not flow uphill, but the developing region loses the transition benefit of accumulating claims on advanced-country robot capital.
- Two-sector, two-skill extension:
  - Direct effect: increased robot productivity raises incentives to accumulate robots and complementary physical capital (raising incomes).
  - Countervailing relative-price effect: if robots substitute strongly for unskilled labor, unskilled wages and the relative price of unskilled-intensive goods fall, reducing investment incentives in that sector and potentially lowering long-run income for unskilled-intensive (developing) economies.

*Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020184-print-pdf.pdf*

### 3.  A Two-Region Model with Robots

### 3.  A Two-Region Model with Robots

### Households
- Two regions indexed by i, with i = A (advanced) and i = D (developing). Each region populated by a representative household that lives forever and owns labor (L), capital (K) and robots (Z).
- Preferences: ∑_t β^t C_{i,t}^{1−1/τ} / (1−1/τ), where C_{i,t} is consumption, β is the discount factor, and τ determines the inter-temporal elasticity of substitution.
- Household budget constraint (period t):
  - C_{i,t} + I^K_{i,t} + I^Z_{i,t} + (B_{−i,t+1} − B_{i,t+1}) = r^K_t K_{i,t} + r^Z_t Z_{i,t} + w_{i,t} L_i + (1 + r^B_t)(B_{−i,t} − B_i,t) + Π_{i,t}
- Laws of motion:
  - K_{i,t+1} = (1 − δ_K) K_{i,t} + I^K_{i,t}
  - Z_{i,t+1} = (1 − δ_Z) Z_{i,t} + I^Z_{i,t}
- First-order conditions:
  - Euler equation: [C_{i,t+1} / C_{i,t}]^{1/τ} = β (1 + r^B_t)
  - No-arbitrage / equalization of net returns:
    - r^K_t − δ_K = r^Z_t − δ_Z = r^B_t

### Firms
- Representative firm in each region uses production function:
  - Y_{i,t} = A_i (K^d_{i,t})^α [ e^{1/σ} L_{i,t}^{(σ−1)/σ} + (1−e)^{1/σ} (b_t Z^d_{i,t})^{(σ−1)/σ} ]^{(1−α)σ/(σ−1)}
- A_i varies across regions (TFP); α, e, and σ common across regions.
- Robots enter via productivity b_t (modeling technological innovation as increases in b_t).
- First-order condition implications (selected relations):
  - Z^d_{i,t} / L_{i,t} = ((1−e)/e) b_t^{σ−1} (w_{i,t} / r^Z_t)^σ
  - L_{i,t} / K^d_{i,t} = (1−α)/α * r^K_t / w_{i,t} * [1 + ((1−e)/e)^{1/σ} ( (b_t Z^d_{i,t} / L_{i,t})^{(σ−1)/σ} ) ]

### Equilibrium (markets clear)
- Households maximize given budget and laws of motion; firms maximize profits; all markets clear.
- Market clearing conditions (for each i and t):
  - Labor: L_i = L_{i,t}
  - Capital: K_{i,t} = K^d_{i,t}
  - Robots: Z_{i,t} = Z^d_{i,t}
  - Goods: Σ_i (C_{i,t} + I^K_{i,t} + I^Z_{i,t}) = Σ_i Y_{i,t}
  - Financial asset market: B_{A,t} + B_{D,t} = 0

### Calibration of the Initial Steady States
- Elasticity of substitution between labor and robots: consider σ ∈ {1, 2, 3}.
  - σ = 1 corresponds to Cobb-Douglas with three factors.
  - Empirical suggestions place σ between 1.5 and 3; other literature: Eden and Gaggl (2018) report increases from 2.5 to 3.3; Berg et al. (2018) find elasticity 2.1 for ICT capital and unskilled labor.
- Calibration targets (for each σ): choose e and A_D to match
  - Relative GDP per capita between regions set to 15 (ratio of GDP per capita between the US and low-income countries).
  - Robot share in income in the advanced economy set to 4 percent.
- Table 1: Calibration of One-Sector Model (selected parameter values)
  - β = 0.96
  - τ = 1
  - δ_K = 0.05
  - δ_Z = 0.05
  - α = 0.35
  - L_D = 1
  - L_A = 1
  - A_A = 1
  - b_0 = 0.1
  - B_0 = 0
- Key calibrated implications:
  - The calibrated labor share parameter e is higher for larger σ to maintain a 4 percent robot income share in the advanced economy.
  - Robot share in the developing region declines as σ increases. For σ = 1, robot share is the same across regions; for σ > 1, robot share is lower in the developing economy.
  - Relation: (Z_A / L_A) / (Z_D / L_D) = (w_A / w_D)^σ, implying higher wages in the advanced economy lead to higher robot-to-labor ratios, amplified when σ > 1.

### Long-run Impact of the Robot Revolution
- Robot revolution modeled as a doubling of robot productivity b in both regions.
- Long-run results depend critically on σ:
  - If σ = 1 (Cobb-Douglas), outcomes symmetric; change in GDP identical across regions.
  - If σ > 1, advanced economy benefits more: higher initial robot share and higher wages make robot and complementary capital investment more profitable in advanced economy → higher GDP growth there → divergence between advanced and developing economies.
- Distributional effects:
  - Real wages increase in both regions in the long run, but robot and capital accumulation outpace wage increases, so labor share falls in both regions, more so in the advanced economy.
  - Higher growth in advanced region associated with higher inequality (within and across regions).

### Short-run (Transition) Impact of the Robot Revolution
- Transition scenario: robot productivity doubles over four periods (equal-sized increases each period) then remains constant.
- Main transitional dynamics (examples for σ = 3):
  - Developing region may experience an absolute fall in GDP in the short run.
  - Robot stocks increase in both regions; capital stock in the developing economy can fall for an extended period as resources flow to advanced economy to finance robot and capital investment there.
  - Developing region runs a current account surplus (acts as lender), accumulating net foreign assets; final steady state: consumption in developing region may exceed output financed by interest on accumulated assets.
  - Consumption falls in both regions initially; larger and more persistent fall in the developing economy.
  - Wages drop after the robot revolution in both regions when σ = 3, with a larger decline in the developing region during transition; eventually wages recover as capital accumulates, slower in developing region.
  - Labor income share: decline is larger in the advanced economy; however, gross national income composition shifts in developing region toward interest income due to foreign asset accumulation, implying rising inequality there as well.
- Role of capital account openness:
  - With open capital accounts, developing region lends and finances advanced-region robot expansion; this creates sharper short-run divergence in output levels but allows developing region to earn interest income and benefit in consumption eventually.
  - With closed capital accounts, developing country is insulated from capital outflows; advanced region experiences amplified investment response; developing country misses out on owning some advanced-country robot capital stock and loses long-run consumption benefits from higher robot productivity (consumers in developing country lose almost all long-run consumption benefits).

### Adding Two Skill Levels
- Extended production function (skilled and unskilled labor):
  - Y_{i,t} = A_i (K^d_{i,t})^{α_K} (S_{i,t})^{α_S} [ e^{1/σ} L_{i,t}^{(σ−1)/σ} + (1−e)^{1/σ} (b_t Z^d_{i,t})^{(σ−1)/σ} ]^{(1−α_K−α_S)σ/(σ−1)}
  - S_{i,t} = skilled labor; L_{i,t} = unskilled labor.
- Calibration (Table 2 selected values):
  - α_K = 0.35
  - α_S = 0.30
  - L_D = 0.98
  - S_D = 0.02
  - L_A = 0.70
  - S_A = 0.30
- Rationale and implications:
  - Allows part of cross-country income differences to arise from differences in human capital endowments (skilled worker shares) rather than only TFP.
  - Allows robots to be complementary to skilled labor while substituting for unskilled labor.
  - Qualitative divergence result unchanged, but quantitatively smaller when robots substitute only for a part of the labor force.
  - For high substitutability between unskilled labor and robots, labor income inequality rises, and the increase in inequality is larger in the advanced region.

*Source: wpiea2020184-print-pdf - 3.  A Two-Region Model with Robots*

### 3.1 through 3.5. Columns 4 and 5 report results for the model described in Section 3.7 while Columns 6 and

### Sections 3.1–4.2 (Two-Sector Model with Two Skill Levels)

### Calibration and model structure
- Two regions: advanced (i = A) and developing (i = D).
- Two goods: T1 and T2 (both tradable). Good T1 is the numeraire and the only good used for capital and robot accumulation.
- Two types of labor: skilled (S) and unskilled (L). Robots substitute for unskilled and complement skilled in the extended model.
- For each level of σ (ranging from 1 to 3) a separate initial steady state is calibrated.
- Robot share in the initial steady state in the advanced region is maintained at 4 percent for all σ.
- Calibrated parameter comparisons:
  - A_D in baseline (one-skill model) ≈ 0.19; A_D in two-skill-level model ≈ 0.38 (both with TFP normalized at 1 for the advanced economy).
- Table 4 calibrated values (technology and endowments):
  - αK,T1 = αK,T2: Share of capital in production function = 0.35
  - αS,T1: Share of skill in T1 production function = 0.35
  - αS,T2: Share of skill in T2 production function = 0.25
  - L_D: Stock of unskilled labor in the developing economy = 0.98
  - S_D: Stock of skilled labor in the developing economy = 0.02
  - L_A: Stock of unskilled labor in the advanced economy = 0.70
  - S_A: Stock of skilled labor in the advanced economy = 0.30
- Skilled labor share in the developing region = 0.25 (equal to calibrated αS,T2) because, in the baseline calibration, the developing region produces only T2.

### Steady-state and divergence effects
- The divergence effect (advanced region gains more from increased robot productivity) persists in the two-skill and two-sector extensions.
- Quantitative comparisons:
  - When σ = 3: advanced economy long-run growth ≈ 39 percent in the baseline (one-skill) model versus about 23 percent in the two-skill-level model.
- Mechanisms dampening divergence in the two-skill model:
  - Robots substitute only for a subset of the labor force (unskilled), reducing overall impact relative to one-skill model.
  - Robots complement skilled labor which is in fixed supply; higher investment in robots raises skilled wages, which reduces the incentive to invest further.
  - These complementarity and wage feedbacks operate in both regions but more strongly in advanced economies (where robot investment is larger), producing a milder divergence than in the one-skill model.

### Two-sector channels: terms of trade and reallocation
- Two forces determine relative impacts after a doubling of robot productivity:
  - Direct effect: incentive to accumulate more robots and complementary physical capital in both sectors.
  - Indirect (terms-of-trade) effect: decline in demand for unskilled labor lowers unskilled wages; because T2 is unskilled-labor intensive, its relative price falls, reducing incentives to invest in T2 and countervailing the direct effect.
- Sectoral outcomes after robot productivity doubles:
  - Advanced economy:
    - Output of both goods increases in equilibrium.
    - For T2, direct investment incentives dominate the fall in relative price, raising T2 output (a “re-shoring” flavor).
  - Developing region:
    - Output of good T2 declines in equilibrium when σ > 1.
    - Because the developing region produces only T2 in the baseline calibration, GDP in the developing region can fall in absolute terms for large σ.
- Comparative statics:
  - Robot share is lower in the developing region for σ > 1; unskilled labor share is higher in the developing region for those σ.
  - The skilled labor share in the advanced region lies between αK,T1 and αK,T2 since the advanced region produces both goods.

### Inequality and labor-market outcomes
- The model predicts increases in labor income inequality for large values of σ:
  - Skilled wages increase in line with per-capita GDP (robots complement skilled labor).
  - Absolute unskilled wages can fall for high σ because robots substitute for unskilled workers and increased robot investment reduces demand for unskilled labor.
  - The fall in unskilled wages is larger in advanced economies due to greater robot investment in those regions.
- In both regions, higher σ implies a larger increase in the robot share at the expense of unskilled labor share and an expanding skill premium (skilled wages rise more than unskilled wages).

### Policy-relevant implications and conclusions
- Core conclusions under the model’s assumptions:
  - Improvements in robot productivity drive divergence between advanced and developing countries: advanced countries benefit more because they make greater use of robots.
  - With capital mobility, transitional capital flows can move from developing to advanced countries, causing transitional declines in developing-country GDP. With capital immobility, developing countries lose the long-run consumption gains associated with transitional accumulation of high-yielding robot capital.
  - In the two-sector, two-skill model, developing countries that are relatively unskilled-intensive can face a permanent terms-of-trade decline and possible absolute declines in GDP.
- Policy and development messages emphasized:
  - There is no silver bullet to avert divergence. Developing countries urgently need to invest in raising aggregate productivity and skill levels so that labor is complemented rather than substituted by robots.
  - Human capital accumulation is crucial to prevent divergence; heterogeneous skill levels imply heterogeneous growth dynamics across developing countries.
  - Continued automation threatens existing complementarity between humans and machines; without rapid productivity and education improvements, demographic dividends (e.g., in Africa) could yield negative rather than positive outcomes.
  - By 2030, more than half of the increase in the global labor force is expected to come from the African continent, raising stakes for policy action to capture potential dividends.

*Source: wpiea2020184-print-pdf (Sections 3.1–4.2).*

### References

### wpiea2020184-print-pdf - References

### References: scope and thematic coverage
- Bibliography emphasizes automation, artificial intelligence, robots, labor markets, wages, structural transformation, and economic growth. Key recurring authors and works include Acemoglu and Restrepo (multiple papers 2018–2019), Agrawal, Gans, and Goldfarb (2019), Brynjolfsson and McAfee (2014), Frey and Osborne (2017), Graetz and Michaels (2018), Korinek and Stiglitz (2019), and multiple IMF, World Bank, NBER, OECD, and IFR reports.
- Selected numerically specific entries (as cited):
  - IMF Departmental Paper Series (18/18) — Abdychev et al. (2018).
  - National Bureau of Economic Research technical reports — Acemoglu and Restrepo (2018a), Nordhaus (2015), Sachs and Kotlikoff (2012), Shiller (2019).
  - Journal articles with volume/issue and page ranges preserved: Journal of Human Capital 12(2), 204–232; American Economic Review 108(6), 1488–1542; Journal of Economic Perspectives 33(2), 3–30; Review of Economics and Statistics 100(5), 753–768; Technological Forecasting and Social Change 114, 254–280; Journal of Artificial Intelligence Research 62, 729–754; Journal of Monetary Economics 97, 117–148.
  - Working Paper 20/98, International Monetary Fund — Bakker et al. (2020).
  - IFR technical report (2012) on World robotics industrial robots.

### Appendix A: Solving transitions — algorithm preserved verbatim
- Overview: The transition is solved with the following algorithm:
  1. Guess the whole sequence of interest rates for the financial assets {r^B_t}_{t≥0}.
  2. Recover all other prices. Because of no arbitrage between financial assets, robots, and capital, recover rental rates for capital and robots, {r^K_t}_{t≥0} and {r^Z_t}_{t≥0}. Using the rental rates, the cost function for the final good, and the fact that the price of the final good is normalized to 1 in every period, recover wages in each region {w_{A,t}}_{t≥0} and {w_{D,t}}_{t≥0}.
  3. Recover input and output levels. Based on the prices for all inputs, recover ratios of robots and labor from the firm’s first order condition. Using the fact that the stock of labor is constant, recover the stock of capital and robots in each region {K_{i,t}}_{t≥0} and {Z_{i,t}}_{t≥0}. Using the production function, find GDP in each period for each region {Y_{i,t}}_{t≥0}. Using the rules of motion for the stock of capital and robots, find investment paths {I^K_{i,t}}_{t≥0} and {I^Z_{i,t}}_{t≥0}.
  4. Given interest rates and prices, solve for the optimal consumption path for each household as follows:
     - (a) Guess initial consumption level, C_{i,0}.
     - (b) Recover the rest of the consumption path using the Euler equation from the household’s maximization problem {C_{i,t}}_{t≥0}.
     - (c) Compute implicit path of financial assets holdings using the budget constraint, {B_{i,t}}_{t≥0}.
     - (d) Iterate on initial consumption, so that financial asset holdings remain constant in the final steady state.
  5. Adjust sequence of interest rates {r^B_t}_{t≥0} to clear the global financial assets market each period.

### Appendix B: Data sources, construction, and imputation details
- Wage data: primary sources are the Global Wage Report (ILO) and the Conference Board.
  - Conference Board: nominal data in local currency converted by the source using the average annual exchange rate; converted here to real compensation using the US CPI with 2010 as the base year; also converted to real compensation in local currency using countries’ CPI. Wage defined as hourly compensation costs for all employees in manufacturing and includes:
    - Direct pay (Pay for time worked and directly-paid benefits).
    - Employer social insurance expenditures.
    - Labor-related taxes (taxes on payrolls or employment).
  - EU country value exclusions (years prior to which data is not considered):
    - Finland, Italy, Netherlands, Portugal, Spain – 1999
    - Slovakia – 2009
    - Estonia – 2011
- Global Wage Report Data: used to add more developing economies; mostly manufacturing sector but some broader definitions. Data processing specifics preserved:
  - Data in local currency gives gross average monthly wages; converted to USD using annual average exchange rate and deflated using US CPI; also converted to real compensation in local currency using countries’ CPI.
  - Data for 11 countries is taken from the Global Wage Report data from the ILO.
  - Country-specific notes (exact verbatim):
    - Malaysia: two wage series provided, one for the manufacturing and one for the economy as a whole.
    - Indonesia: two series provided, one relevant for the manufacturing sector; manufacturing series ends in 2014 and is not imputed for 2015-2016 for which national series available.
    - Chile: three series available. All combined to form one. Sector coverage indicates the series exclude agriculture.
    - Hong Kong: one series provided, and this excludes agriculture.
    - Iran: nominal wage is used. While an index is also provided for manufacturing it is not used. Note that they both indicate a similar growth trajectory.
    - Thailand: no information provided on sector coverage.
    - Vietnam: three series provided of which the most complete one is taken, and value for 2008 is imputed assuming a linear growth trend between 2007 and 2009. No information on sector coverage.
    - Venezuela: data ends in 2013. After that, another series for an index is provided up to 2014 but growth rates and trends are different; therefore another year is not imputed. No information on sector coverage.
    - Colombia: missing value for 2001 imputed assuming a linear growth trend between 2000 and 2002. No information on sector coverage.
    - Peru: the most complete series is selected. Missing value for 1996 imputed assuming a linear growth trend between 1995 and 1997. No information on sector coverage.
    - Egypt: two series are combined (essentially the same but have missing data in the first few or last few years). This series includes agriculture.
- Labor data: sources include OECD, 10-Sector Groningen database, World KLEMS and ILO. Employment data generally covers the 2000s up to 2011/2012 mostly. The data is for the manufacturing sector except for ILO which exists for industry (a broader definition that includes manufacturing).
- Robot Stock: data from IFR Robotics providing number of robots (operational stock) across countries, sectors and time.
  - Observed zeros are actual zeros — implying that zero robots have been reported.
  - Mostly, the data begins to take on positive values after 2004/2005, especially for developing economies.
  - A minor imputation is made to address a data anomaly for North America: IFR lumps USA, Canada and Mexico into one category till 2010. To address this, the ratios for 2011 for each country are applied to historical data.
  - Footnotes preserved verbatim:
    - 31 This is a proprietary database. Recent papers by Acemoglu and Restrepo this dataset has been used extensively to study demographics and employment in the context of automation in the US.
    - 32 Consulted IFR data representative.

### Appendix C: Figure and table notes
- Figure A.1: Real Wages and Robot Density
  - Note: Data on robots from IFR. Data on employment and wages from multiple sources. See appendix for details.
- Table A.1: Details on Data Selection (table referenced but contents not reproduced here).

*Source: wpiea2020184-print-pdf - References (IMF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020184-print-pdf.pdf_