3.1. Households

There are two regions indexed by i, representing an advanced economy (i = A) and a developing economy (i = D).¹⁷ Each region is populated by a household that lives forever and owns the three factors of production: labor (L), capital (K) and robots (Z). The household owns the firms operating the production technology and a financial asset, which allows it to borrow or save against the other region.

Household preferences are given by the utility function:

where C_i,t is consumption of household i (with i = A, D) in period t, β is the discount factor, and τ determines the inter-temporal elasticity of substitution.

Household i seeks to maximize utility given its budget constraint:

where $I_{i,t}^{K}and{I}_{i,t}^Z$ are investment in capital and robots, $\overline{L_i}$ is the endowment of labor, and (B_-i,t – B_i,t) is the net financial asset holding for region i. Rates of return on capital, robots, and financial assets are given by $r_t^K,r_t^Z,and{r}_t^B$ respectively. The wage rate is given by W_i,t. Finally, Π_i,t represents the profits of firms operating the production technology in the country. The price of the final good is normalized to 1 in every period.

The laws of motion for accumulation of capital and robots are given by:

where δ^k and δ^z are depreciation rates of capital and robot stock.

Maximizing utility subject to the budget constraint and laws of motion for capital and robots yields the standard Euler equation

and a no arbitrage condition that implies the equalization of net rates of return (after accounting for depreciation) for capital, robots and the financial asset¹⁸

3.2. Firms

A representative firm operates the production technology in each region. Inputs are hired in competitive markets. Labor and robots are combined using a CES technology, with the composite then combined with capital using a Cobb-Douglas function to obtain the final output. The production function is given by

where $K_{i,t}^d,Z_{i,t}^d,and{L}_{i,t}$ is the quantity of capital, robots, and labor demanded by the firm. The level of total factor productivity (A_i) is the only parameter allowed to vary across regions. The cost-share parameters α and e, and the elasticity of substitution between labor and robots (σ) is assumed to be the same across the two regions.

The CES technology allows for flexibility, as different values of σ will correspond to different degrees of substitutability between labor and robots. Using this production function, the latest wave of technological innovation can be modeled as an increase in the productivity of robots, b_t.

Solving the profit maximization problem yields the standard first order conditions equating marginal products to factor prices. Dividing the first order condition for robots and capital with that for labor yields

3.3. Equilibrium

A market equilibrium for this model is a set of prices and allocations such that:

• 1. Households choose consumption and holdings of robots, capital, and the financial asset to maximize utility given their budget constraint and the laws of motion for capital and robots.

• 2. Firms maximize profits by choosing the optimal combination of labor, capital, and robots.

• 3. All markets clear.

  • (a) Labor Market. In each country i with i = A,D and for every period t

    $\overline{L_i}=L_{i,t}$

  • (b) Capital Market. In each country i with i = A,D and for every period t

    $K_{i,t}=K_{i,t}^d$

  • (c) Robots Market. In each country i with i = A, D and for every period t

    $Z_{i,t}=Z_{i,t}^d$

  • (d) Good Market. For every period t:

    ${\displaystyle \sum _i(C_{i,t}+I_{i,t}^K+I_{i,t}^Z)={\displaystyle \sum _i{Y}_{i,t}}}$

  • (e) Financial Asset Market. For every period t:

    $B_{A,t}+B_{D,t}=0$

3.4. Calibration of the Initial Steady States

We calibrate our model to study the differential response across regions of an increase in robot productivity for different levels of substitutability between labor and robots. We consider different values for σ, ranging from 1 (i.e., standard Cobb-Douglas production function with three factors of production) to 3 (i.e., robots substitute for labor). Rigorously estimating σ is beyond the scope of this paper. However, putting aside concerns regarding endogeneity and transition dynamics, the data shown in Figure 3 and 4 suggest that σ is greater than one. The slope of the fitted line in these figures can be viewed as a rough estimate for σ, indicating that the elasticity of substitution may lie somewhere between 1.5 and 3. Eden and Gaggl (2018), using a somewhat different nesting than that employed here, conclude that the elasticity of substitution between labor and information-and-communication-technology (ICT) capital has increased rapidly since the late 90s, rising from 2.5 to 3.3. Calibrating their U.S. data to a production function similar to that employed here, Berg et al. (2018) find an elasticity between ICT capital and unskilled labor of 2.1.

For each value of σ, we calibrate a separate initial steady state where we chose e and A_D to match two moments: the relative GDP per capita between the two regions and the robot share in income in the advanced economy $\left(\frac{r^Z{Z}_A}{Y_A}\right)$. The relative GDP per capita is calibrated to be 15, which is the ratio of GDP per capita between the US and low-income countries. The robot share in income in the advanced economy is calibrated to be 4 percent following Berg et al. (2018). The remaining parameters are standard values in the literature (Table 1).

^{Table 1:}

Calibration of One-Sector Model

^{Table 1:}

Calibration of One-Sector Model

Parameter	Description	Value	Source/Reason
β	Discount factor	0.96	Long-run net return rate on capital around 4 percent
τ	Inter-temporal elasticity of substitution	1	Cobb-Douglas utility function
δk	Depreciation rate for capital	0.05	Berg et al. (2018)
δz	Depreciation rate for robots	0.05	Berg et al. (2018)
α	Share of capital in production function	0.35	Berg et al. (2018)
$$\overline{L_D}$$	Stock of labor in the developing economy	1	Normalization
$$\overline{L_A}$$	Stock of labor in the advanced economy	1	Normalization
AA	Total factor productivity in the advanced economy	1	Normalization
b0	Initial robot productivity	0.1	Normalization
B0	Initial asset holdings	0	Symmetry in the initial steady state

View Table

^{Table 1:}

Calibration of One-Sector Model

Parameter	Description	Value	Source/Reason
β	Discount factor	0.96	Long-run net return rate on capital around 4 percent
τ	Inter-temporal elasticity of substitution	1	Cobb-Douglas utility function
δk	Depreciation rate for capital	0.05	Berg et al. (2018)
δz	Depreciation rate for robots	0.05	Berg et al. (2018)
α	Share of capital in production function	0.35	Berg et al. (2018)
$$\overline{L_D}$$	Stock of labor in the developing economy	1	Normalization
$$\overline{L_A}$$	Stock of labor in the advanced economy	1	Normalization
AA	Total factor productivity in the advanced economy	1	Normalization
b0	Initial robot productivity	0.1	Normalization
B0	Initial asset holdings	0	Symmetry in the initial steady state

Figure 5 shows the resulting parameters for different values of the elasticity of substitution. The calibrated value of the labor share parameter (e) is higher for larger σ, so as to maintain a robot share in income of 4 percent in the advanced economy in the initial steady state.

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Parameters

Citation: IMF Working Papers 2020, 184; 10.5089/9781513556505.001.A001

Notes: The figure plots the calibrated values of total factor productivity in the developing economy (A_D) and the labor share parameter (e) of the production function in the initial steady states. These parameters are chosen such that for each σ, GDP per capita in the advanced economy is 15 times that of the developing economy, and the robot share in income in the advanced economy is 4 percent.

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However, the robot share in the developing region declines as σ increases in the initial steady state (Figure 6). For the Cobb-Douglas case, with σ =1, the robot share in the developing economy is the same as in the advanced economy. For higher values of σ, the robot share is lower in the developing region in the initial steady state because the higher elasticity of substitution magnifies the divergence in robot intensity (Z/L ratio) emerging from different wages. To get intuition, dividing equation 6 for the advanced economy with the same equation for the developing economy yields

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Moments in the Initial Steady State

Citation: IMF Working Papers 2020, 184; 10.5089/9781513556505.001.A001

Notes: The top panel of the figure plots the labor share $\left(\frac{w_i\overline{L_i}}{Y_i}\right)$ and the robot share $\left(\frac{r^Z{Z}_i}{Y_i}\right)$ for each steady state i.e. for each value of σ. The bottom left panel plots the robot intensity in the advanced economy $\left(\frac{Z_A/L_A}{Z_D/L_D}\right)$s.

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The higher wages in the advanced economy translate into higher robot-to-labor ratios. Further more, a higher elasticity of substitution amplifies the effect of wages on the robot-to-labor ratio. For a Cobb-Douglas production function, the robot-to-labor ratio between the two economies is proportional to the wage ratio, which results in the same labor share in the two economies. However, when robots and labor are substitutes (σ > 1), then the Z/L ratio in the advanced economy is more than proportionally higher than the wage differential. The high elasticity of substitution implies that the high wages in the advanced economy result in greater substitution of labor for robots, leading to a higher robot share in output. Thus, when σ > 1, the robot share is lower in the developing economy than in the advanced economy and it declines with σ. Given that the capital share is the same across the two regions and does not vary with σ, the opposite is true for the labor share. The labor share is higher in the developing economy than in the advanced economy when σ is greater than 1, and it increases with σ.

3.5. Long-run Impact of the Robot Revolution

We model the robot revolution as a doubling of robot productivity (6) in both regions, as in Berg et al. (2018). The increase in robot productivity leads to higher GDP in both regions in the long run as households invest more in robots and in capital (which complements robots). However, which region grows more, and whether the developing economy falls further behind the advanced economy, depends crucially on the elasticity of substitution between robots and labor (Figure 7).

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Steady State Comparison: percent changes with respect to initial steady state for different ${\sigma }^{\prime }s$

Citation: IMF Working Papers 2020, 184; 10.5089/9781513556505.001.A001

Notes: The figure plots the percent change in various variables between the initial steady state and the final steady state following a doubling of robot productivity. Note that each σ (plotted on the x-axis) is associated with a different initial steady state calibration as described in section 3.4. Results for the one-sector model with calibration as described in Section 3.4.

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In the Cobb-Douglas case, outcomes are symmetric in the two regions, with the change in GDP being the same, as both regions have the same robot share in output in the initial steady state.

However, when the elasticity of substitution is greater than 1, the developing economy benefits less. In this case, the robot share in output is larger in the advanced economy in the initial steady state (robots are a more important input in production), and so, the associated increase in GDP following a doubling of robot productivity is also larger. Investing in robots, and in complementary traditional capital, following an increase in robot productivity is most profitable where wages are high because they save on the cost of employing expensive workers. Thus, the developing region falls further behind, diverging from the advanced region in GDP and (to a lesser extent) consumption.¹⁹

While real wages increase in the long run in both regions, the change in labor share in output is more pronounced in the advanced economy. When robots easily replace workers, the robot and capital stocks increase by more than wages, leading to a fall in the labor share in both regions. The increase in real wages is stronger in the advanced region, but the increase in per capita GDP is even larger (due to faster robot and capital accumulation), so that the fall in labor share in output is more pronounced than in the developing region. Relatively higher growth in the advanced region is then associated with higher inequality as well.

Thus, the robot revolution may exacerbate income differences between advanced and developing economies if robots substitute workers because robots are used more intensively in advanced economies in the initial steady state.

3.6. Short-run Impact of the Robot Revolution

In this subsection, we explore the implications of the robot revolution during the transition. We assume that robot productivity doubles over four periods (with equal-sized increases each time) and remains constant thereafter.²⁰

While in the steady state we saw that GDP in the developing economy may fall back relative to the advanced economy when σ is greater than one, in the transition there may be a drop in the absolute level of GDP in the developing region as a result of the robot revolution. Figure 8 shows the evolution of key variables along the transition path for different values of σ.

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Transition: GDP per capita, Capital, Consumption, and Savings

Citation: IMF Working Papers 2020, 184; 10.5089/9781513556505.001.A001

Note: DR=Developing Region; AR=Advanced Region. All charts plot the transition path, showing the percent changes with respect to initial steady state, except “Net Financial Assets” which are shown as a percent of world GDP.

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When the elasticity of substitution is 3, GDP falls in the developing region in the short-run. While the stock of robots increase in both regions, the capital stock in the developing economy falls for an extended period of time. This is because resources are channeled out of the developing region and transferred to the advanced economy to meet the stronger demand for capital and robots created there by the increase in robot productivity. The developing region acts as a lender, running a current account surplus and financing a stronger response to technological change in the advanced region.

Consumption also falls in both regions in the first few years as resources are diverted to investment, but the fall is larger and more persistent for the developing economy, again reflecting the outflow of resources and the build up of a net asset position with respect to the rest of the world. In the final steady state, consumption in the developing region is greater than output, the difference financed by the interest income on the accumulated assets.

The transition is particularly painful and long for workers in the developing region in terms of wages (Figure 9). For an elasticity of substitution of 3, wages drop after the robot revolution in both regions as firms substitute away from workers and into robots. However, the decline is larger in the developing region, where GDP is falling as capital flow out of the region. Eventually, the capital stock grows enough to compensate for the negative substitution effect and raises wages, but that takes longer in the developing region. In terms of labor income as a share of output, the decline is quite small for the developing region but quite large for the advanced economy. However, even in the developing economy, labor income as a share of gross national income²¹ falls substantially as the build up of foreign assets yields large interest income, indicating that inequality might increase in the developing economy as well.

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Transition. Factor Prices and Labor Shares

Citation: IMF Working Papers 2020, 184; 10.5089/9781513556505.001.A001

Note: DR=Developing Region; AR=Advanced Region. All charts plot the transition path, showing the percent changes with respect to initial steady state, except “Interest Rate” which is shown as a percent.

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Most of the transitional effects on the developing country are due to capital account implications of robot adoption in advanced countries (see Figure 10). In the case of closed economies, the shape of the response is similar across regions but greatly amplified in the advanced region, where there are greater incentives to take advantage of the more productive robots. In the developing country, with low wages and low share of robots in output, the increase in demand for robots is smaller. This leads to a small increase in the interest rate in the developing country (barely noticeable in the plot), in contrast to the large spike in the advanced country. In fact, this spike in the advanced country is only slightly larger than what would have happened under an open capital account, reflecting the small mitigating impact of capital flows from the developing region. A closed capital account thus insulates the developing country from most of the impact of the robot revolution. The developing country misses out on the chance to own some of the advanced-country robot capital stock, with consumers in the developing country losing almost all the long-run consumption benefits from the increase in the productivity of robot capital as they do not get to benefit from the high interest rate.

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Transition with Closed and Open Capital Accounts (σ = 3)

Citation: IMF Working Papers 2020, 184; 10.5089/9781513556505.001.A001

DR=Developing Region; AR=Advanced Region. All charts plot the transition path, showing the percent changes with respect to initial steady state, except “Net Financial Assets” which are shown as a percent of world GDP.

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3.7. Adding Two Skill Levels

In this subsection, we extend our model to include two types of workers, skilled and unskilled. This extension has two key benefits compared to the baseline model. First, in the baseline model, all income differences between the two regions arise due to differences in aggregate productivity (TFP), which then drive the magnitude of the divergence result through the gap in wages. The two-skill-level model allows for a more realistic calibration, where differences in human capital endowments account for part of the income differences between the two regions, and so, the gap in TFP is mitigated.²² Second, and perhaps more important, robots may not substitute for all types of workers in the future. The two-skill-level model allows us to see how results differ when robots are complementary to skilled labor while they substitute for unskilled labor. This assumption, which is critical to the main results of this section, is a plausible and widespread view, as briefly discussed in the introduction (particularly footnote 9).

While the basic divergence result remains qualitatively unchanged, quantitatively the extent of divergence is smaller when robots only substitute for a part of the labor force. Furthermore, this richer model has implications for labor income inequality. For high substitutability between unskilled labor and robots, we find that labor income inequality rises and that the increase is higher in the advanced region.

We consider a production function in region i given by

where S_i,t and L_i,t represent skilled and unskilled labor, respectively.

Our calibration strategy remains broadly unchanged. We calibrate the endowments of skilled workers in the two regions to match the share of workers with greater than secondary education in the US and in low income countries. Of course, the advanced region has relatively more skilled workers compared to the developing region. For the share of skilled labor in the production function, we follow Berg et al. (2018). We continue to calibrate aggregate productivity in the developing region A_D and unskilled labor share parameter e to match relative GDP in the two regions and robot share in income in advanced economies. Our calibration strategy implies that the capital and skilled labor shares are the same in both regions, and the robot share in the initial steady state in the advanced region is maintained at 4 percent for all σ. The new parameters are summarized in Table 4. Other parameters remain as in Table 1.

Table 2:

Calibration for Two-Labor Model

Table 2:

Calibration for Two-Labor Model

Parameter	Description	Value
αk	Share of capital in production function	0.35
αs	Share of skill in production function	0.30
$$\overline{L_D}$$	Stock of unskilled labor in the developing economy	0.98
$$\overline{S_D}$$	Stock of skilled labor in the developing economy	0.02
$$\overline{L_A}$$	Stock of unskilled labor in the advanced economy	0.70
$$\overline{S_A}$$	Stock of skilled labor in the advanced economy	0.30

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Table 2:

Calibration for Two-Labor Model

Parameter	Description	Value
αk	Share of capital in production function	0.35
αs	Share of skill in production function	0.30
$$\overline{L_D}$$	Stock of unskilled labor in the developing economy	0.98
$$\overline{S_D}$$	Stock of skilled labor in the developing economy	0.02
$$\overline{L_A}$$	Stock of unskilled labor in the advanced economy	0.70
$$\overline{S_A}$$	Stock of skilled labor in the advanced economy	0.30

^{Table 3:}

Percent Change in per-capita GDP following Increase in Robot Productivity

Notes: Table shows the percent change in per capita GDP following a doubling of robot productivity for different σ for the three models. Columns 2 and 3 report results for the baseline model described in Sections 3.1 through 3.5. Columns 4 and 5 report results for the model described in Section 3.7 while Columns 6 and 7 report results for the model described in Section 4.

^{Table 3:}

Percent Change in per-capita GDP following Increase in Robot Productivity

	Model: 1 Sector and 1 Skill Level		Model: 1 Sector and 2 Skill Levels		Model: 2 Sector and 2 Skill Levels
	Developing Economy	Advanced Economy	Developing Economy	Advanced Economy	Developing Economy	Advanced Economy
σ = 1 (Cobb-Douglas)	4.7	4.7	4.7	4.7	4.6	4.7
σ = 2	0.9	14.5	0.7	12.6	-0.8	11.2
σ = 3	0.1	38.9	0.1	23.5	-4.0	19.5

Notes: Table shows the percent change in per capita GDP following a doubling of robot productivity for different σ for the three models. Columns 2 and 3 report results for the baseline model described in Sections 3.1 through 3.5. Columns 4 and 5 report results for the model described in Section 3.7 while Columns 6 and 7 report results for the model described in Section 4.

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^{Table 3:}

Percent Change in per-capita GDP following Increase in Robot Productivity

	Model: 1 Sector and 1 Skill Level		Model: 1 Sector and 2 Skill Levels		Model: 2 Sector and 2 Skill Levels
	Developing Economy	Advanced Economy	Developing Economy	Advanced Economy	Developing Economy	Advanced Economy
σ = 1 (Cobb-Douglas)	4.7	4.7	4.7	4.7	4.6	4.7
σ = 2	0.9	14.5	0.7	12.6	-0.8	11.2
σ = 3	0.1	38.9	0.1	23.5	-4.0	19.5

Notes: Table shows the percent change in per capita GDP following a doubling of robot productivity for different σ for the three models. Columns 2 and 3 report results for the baseline model described in Sections 3.1 through 3.5. Columns 4 and 5 report results for the model described in Section 3.7 while Columns 6 and 7 report results for the model described in Section 4.

^{Table 4:}

Calibration for Two-Sector Model

^{Table 4:}

Calibration for Two-Sector Model

Parameter	Description	Value
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
$$\overline{L_D}$$	Stock of unskilled labor in the developing economy	0.98
$$\overline{S_D}$$	Stock of skilled labor in the developing economy	0.02
$$\overline{L_A}$$	Stock of unskilled labor in the advanced economy	0.70
$$\overline{S_A}$$	Stock of skilled labor in the advanced economy	0.30

View Table

^{Table 4:}

Calibration for Two-Sector Model

Parameter	Description	Value
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
$$\overline{L_D}$$	Stock of unskilled labor in the developing economy	0.98
$$\overline{S_D}$$	Stock of skilled labor in the developing economy	0.02
$$\overline{L_A}$$	Stock of unskilled labor in the advanced economy	0.70
$$\overline{S_A}$$	Stock of skilled labor in the advanced economy	0.30

Accounting for differences in skill across countries leads to lower differences in calibrated TFP. In particular, while A_D was calibrated to a value of about 0.19 in the baseline model, the calibrated value in the two-skill-level model is almost twice as high at about 0.38 (both with respect to a TFP normalized at 1 for the advanced economy).

As in the baseline model with only one type of labor, the divergence effect emerges in the long run (Figure 11). The advanced region experiences a much larger increase in per capita GDP for large values of σ than the developing region.

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Steady State Comparison (percent changes with respect to initial steady state)

Citation: IMF Working Papers 2020, 184; 10.5089/9781513556505.001.A001

Notes: The figure plots the percent change in various variables between the initial steady state and the final steady state following a doubling of robot productivity. Note that each σ (plotted on the x-axis) is associated with a different initial steady state calibration as described in section 3.4. Results for the one-sector with two types of labor described in Section 3.7.

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Quantitatively, the divergence effect is smaller than in the one-skill-level model reflecting a lower TFP gap across countries and a lower share of unskilled labor/robots. For example, when σ = 3, the advanced economy grows by about 39 percent in the baseline model while only growing by about 23 percent in the two-skill-level model (Table 3). The reason is that robots only substitute for a subset of the labor force and total factor productivity is not so different. An increase in robot productivity leads to greater investment in robots (and complementary physical capital). However, as robots complement skilled labor which is in fixed supply, this greater investment also raises skilled wages which reduces the incentive to invest. A similar effect emerges in the developing economy, but is weaker in magnitude, leading to a milder divergence. This dampening force did not exist in the baseline model.

This version of the model also predicts increases in labor income inequality for large values of σ. Skilled wages in both the economies increase in line with per-capita GDP as skilled labor complements robots. However, the absolute level of unskilled wages can fall following an increase in robot productivity for high values of σ. This is because robots substitutes for unskilled workers, and the increase in robot investment following an increase in robot productivity reduces the demand for unskilled workers. The fall in unskilled wages is larger in advanced economies as there is greater investment in robots in this region, thus reducing unskilled labor demand by more.

4.1. Calibration of the Initial Steady States of Two-Sector Model

Our calibration strategy follows the approach in subsection 3.4. For each level of a, ranging from 1 to 3, we calibrate a separate initial steady state where we choose e_T1, e_T2 and A_D to match three moments: the relative GDP per capita between the two regions and a robot share of 4 percent in the advanced economy in each sectors. Both sectors are assumed to have the same elasticity of substitution between unskilled workers and robots.

As in subsection 3.7, we calibrate the endowments of skilled workers in the two regions to match the share of workers with greater than secondary education in the US and in low income countries. Furthermore, we assume that the main difference between the two sectors is the relative importance of skilled and unskilled labor in the production function. In particular, the Tl sector has a skilled share which is 10 percentage points higher compared to the T2 sector. Table 4 summarizes the values of the additional parameters we calibrate relating to technology in the two sectors and the endowment of the two types of labor in each region. We maintain the same calibrated values as in Table 1 for common parameters across the two models.

As in the one-sector model, the robot share varies endogenously in the developing economy (Figure 12). In particular, the robot share is lower in the developing region for values of σ greater than 1, while the unskilled labor share is higher. On the other hand, the robot share in output is the same in the advanced economy for different values of σ because that is one of the targets of our calibration strategy.

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Moments in the Initial Steady State

Citation: IMF Working Papers 2020, 184; 10.5089/9781513556505.001.A001

Notes: The top panel of the figure plots the the robot share and unskilled labor share for each steady state i.e. for each value of σ. The bottom left panel plots the skilled labor share.

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The skilled labor share in this model is the weighted average of the skilled labor shares in both sectors and does not depend on σ. The skilled labor share in the developing region is 0.25, which is the calibrated value of α_s,T2 because the region only produces the unskilled-labor intensive good (T2) in equilibrium for our baseline calibration. Since the advanced region produces both goods, its skilled labor share lies between the values of α_k,T1 and α_k,T2.

4.2. Long-run Impact of the Robot Revolution in the Two-Sector Model

In this model, a combination of two forces determine the relative impact of an increase in robot productivity on the two regions—a direct effect (akin to the one-sector model) and a second effect due to changes in relative prices.

After a doubling of robot productivity, the direct effect is an incentive to accumulate more robots and complementary physical capital in both the sectors, similar to the one-sector model of Section 3.²⁵

However, in addition to this direct effect, the two-sector model also has an indirect effect working through changes in relative prices. The increase in robot productivity decreases the demand for unskilled labor, especially when robots and unskilled labor are highly substitutable (i.e., when σ is large), thus lowering unskilled wages. As good T2 uses unskilled labor (and robots) more intensively, this results in a decline in the relative price of good T2.²⁶ This decline in relative price implies less incentive to invest in robots and capital in the Tl sector, thus countervailing the direct effect.²⁷

Following a doubling of robot productivity, the same divergence effect seen in the one-sector model emerges when σ > 1, with the increase in per capita GDP being much larger in the advanced region (Figure 13). This is driven by the direct effect which is larger in the advanced economy when σ > 1—the intuition for this result is the same as in the one sector model, as the robot share in output is larger in the initial steady state in the advanced region.

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Steady State Comparison (percent changes with respect to initial steady state)

Citation: IMF Working Papers 2020, 184; 10.5089/9781513556505.001.A001

Notes: The figure plots the percent change in various variables between the initial steady state and the final steady state following a doubling of robot productivity. Note that each σ (plotted on the x-axis) is associated with a different initial steady state calibration as described in Section 3.4. Results for the two-sector model with calibration as described in Section 4.1.

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However, while per capita GDP always increased in the long run in the one-sector model following an increase in robot productivity, in the two sector-model income levels can actually decline in the developing economy for large values of σ due to the effects arising from changes in relative prices. Considering how output in both the sectors is affected can provide intuition for this result:

• In the advanced economy, output of both goods increases in equilibrium. For good Tl this is simply reflecting the direct effect. For good T2, there are two forces. While the direct effect provides incentive to invest in more robots and capital, the decline in relative price of T2 reduces the incentive to invest in this sector. In equilibrium, the direct effect is larger in the advanced economy, thus resulting in higher output of good Tl in equilibrium.²⁸

• In the developing region, output of good T2 declines in equilibrium. As in the advanced economy, the direct effect provides incentive to invest in more robots. However, the direct effect is comparatively small in the developing region when σ > 1 because robot share in output is small in the initial steady state (as in the one-sector model). Thus, in the developing region, the effect from a decline in relative price of T2 dominates, and output of good T2 falls. Furthermore, as the developing region only produces good T2, GDP in the region falls.

Income inequality also increases substantially in both regions when robots and unskilled labor are highly substitutable. The greater the σ, the greater the increase in the robot share at the expense of unskilled labor share. But also, the skill premium expands as skilled wages increase by more than unskilled wages, which even drops for large degrees of substitutability.