Reassessing the Role of Capital
in the Dynamics of the Labor Share

Ricardo Gouveia-Mendes

ricardo.mendes@iscte-iul.pt

Iscte – University Institute of Lisbon

July 5, 2025
18th Annual Meeting of The Portuguese Economic Journal

Introduction

Introduction

The Decline of the Labor Share in the US

Introduction

Motivation

• Consensual 6 p.p. decline since the 1980s across developed countries (Karabarbounis 2024)
• The labor share is central to research on inequality
• Concerns about:
  • Living standards of the poor
  • Social stability
  • Economic growth sustainability

Introduction

Literature Focus on the Primary Mechanism

Several declining mechanisms (Grossman and Oberfield 2022)

• Technological progress (Karabarbounis and Neiman 2014; Acemoglu and Restrepo 2020)
• Globalization (Elsby, Hobijn, and Şahin 2013)
• Market Power (Autor et al. 2020; Barkai 2020; De Loecker, Eeckhout, and Unger 2020)
• Labor force composition (Acemoglu and Restrepo 2022; Glover and Short 2020; Grossman et al. 2021)

Grossman and Oberfield conducted an extensive meta-analysis:

1. Technological change, namely the adoption of technologies that substitute labor in production.
2. Globalization, which occurred through the integration of capital markets, reducing the cost of capital for many countries, as well as through developed countries’ access to low-cost labor from other countries, especially China.
3. Market power: Concentration is high in the most productive sectors of the economy, allowing firms to pay employees below their marginal revenue.
4. Labor Force Composition: Population stagnation and aging reduce labor productivity (e.g., reducing returns to experience).

Introduction

Literature Focus on the Primary Mechanism

But…

• Overestimated partial effects for a rather stable historical trend (Harrison 2024)
• Intellectual Property Products capitalization explains it all (Koh, Santaeulàlia-Llopis, and Zheng 2020)

• Grossman and Oberfield (2022): decline three to four times greater, though they do not explain how they arrived at these numbers
• Ann Harrison (2024):
  • omitted variables lead to overestimated partial effects
  • attempt to address simultaneously estimate the partial effects of three of the four mechanisms
• Magnitudes do not align with what might simply be a regression to the mean when considering a large temporal horizon
• Koh et al. (2020):
  • decline in the labor share results exclusively from changes in National Accounts accounting rules
  • Since 1999, Intellectual Property Products have gradually changed from being considered intermediate consumption to being considered a form of capital in the US
  • Their rents were fully allocated to capital income
  • The same change was promoted in OECD countries in 2008
  • Undoing this effect reveals that the labor share exhibits trendless long-term behavior in the US and elsewhere

Introduction

This Paper

• Labor share stability demands offsetting forces
• Investment-embodied technological progress decreases the relative price of investment goods
• The resulting inputs reallocation can act as a countervailing factor of the decline of the labor share
• When there are sectoral differences in capital-output elasticities

Introduction

The Decline of the Relative Price of Investment Goods

Introduction

The Decline of the Relative Price of Investment Goods

• Drivers: equipment and intellectual property Details
• Evidence of investment-embodied technological progress (Solow et al. 1960; Greenwood, Hercowitz, and Krusell 1997; Karabarbounis and Neiman 2014; Hubmer 2023)
• It plays a role on the decline of the labor share when \sigma_{K,L}>1 (Karabarbounis and Neiman 2014; Lawrence 2015)

Introduction

Approach

• Standard two-sector growth model
• Three key assumptions
  • Unitarian capital-labor elasticity of substitution
  • Different capital-output elasticities across sectors
  • Investment-embodied technological progress

• Investment-embodied technological progress or “vintage approach”: Solow’s hypothesis back in the 1960s
  • Technical change arising from the replacement of old vintages capital with newer and more productive
  • This technical change affects the production of consumption goods through investment, which is an endogenous decision, unlike TFP
• Empirical analysis arrived in the late 1990s by Greenwood et al.
• More recently, other papers have found empirical evidence of this type of progress at the aggregate and sectoral levels

Introduction

Five Main Findings

1. Labor share changes are transitional
   
2. Triggered by capital redistribution across sectors
   
3. Require different sectoral capital-output elasticities
   
4. Occur despite a unitary capital-labor elasticity of substitution
   
5. Increases can happen even when r>g (Piketty and Zucman 2014; Piketty 2014)

The Model

The Model

Setup: Households Preferences

Preferences are described by the utility function: U = \sum_{t=0}^{+\infty} \beta^t \frac{\left(C_{t}\right)^{1-\phi}}{1-\phi} \tag{1} with \phi^{-1}\geq 0 and 0<\beta<1

The Model

Setup: Labor and Capital

• Labor supply is exogenous and homogeneous:

L_{t} = L_{0} \left(1+g^{_L}\right)^t, \quad g^{_L}\geq 0 \tag{2}

• Capital is homogeneous and evolves according to:

K_{t+1} = K_{t} (1-\delta) + I_{t}, \quad 0<\delta<1 \tag{3}

The Model

Setup: Production

• Two sectors: consumption (C) and investment (I)
• Both use two inputs: capital (K) and labor (L)
• In each sector j\in\{C,I\}:

Y^{_j}_{t} = \left(K_{t}^{_j}\right)^{\alpha^{_j}} \left(A^{_j}_{t} L_{t}^{_j}\right)^{1-\alpha^{_j}} \tag{4} with 0<\alpha^{_j}<1 and \alpha^{_j}\neq\alpha^{_{-j}}

The Model

Setup: Technological Progress

• Harrod neutral technological progress: A^{_j}_{t} = (1+g^{_{A^j}})^t A^{_j}_0, \quad g^{_{A^j}} \geq 0 \tag{5}
• Investment-embodied technological progress means: g^{_{A^I}} > g^{_{A^C}}

The Model

Setup: Resources Constraints for Inputs and Outputs

K_{t} = K^{_C}_{t} + K^{_I}_{t} \tag{6} L_{t} = L^{_C}_{t} + L^{_I}_{t} \tag{7} Y^{_C}_{t} = C_{t} \tag{8} Y^{_I}_{t} = I_{t} \tag{9}

The Model

Setup: Some Convenient Definitions

• Shares of inputs allocated to sector I:

s^{_K}_{t} \equiv \frac{K^{_I}_{t}}{K_{t}} s^{_L}_{t} \equiv \frac{L^{_I}_{t}}{L_{t}}

• Capital per effective worker:

k_{t} \equiv \frac{K_{t}}{A^{_I}_{t} L_{t}}

The Model

Planner Problem

• A benevolent social planner chooses the path for:
  • Shares of inputs \{s^{_K}_{t},s^{_L}_{t}\}_{t=0}^{+\infty}
  • Capital per effective worker \{k_{t+1}\}_{t=0}^{+\infty}
• That maximize the utility function in Equation 1
• Subject to the resources constraints in Equation 6 to Equation 9
• Given: \beta, \phi, \alpha^{_C}, \alpha^{_I}, \delta, g^{_{A^C}}, g^{_{A^I}}, and g^{_L}, and K_0

The Model

Solution: Planner Problem

• Single solution to s^{_K}_{t}, s^{_L}_{t} and k_{t+1} Details
• Property: \alpha^{_C}=\alpha^{_I}=\alpha \Rightarrow s^{_L}_{t}=s^{_K}_{t}
• With these values we can determine all the quantities
  • Inputs in each sector: K^{_C}_{t}, K^{_I}_{t}, L^{_C}_{t} and L^{_I}_{t}
  • Outputs Y^{_C}_{t} and Y^{_I}_{t}
  • Allocations C_{t} and I_{t}

The Model

Solution: Decentralized Competitive Equilibrium

• Identical conditions for quantities and conditions for prices, namely:

\begin{align} \frac{1}{q_{t}} \equiv \frac{P^{_I}_{t}}{P^{_C}_{t}} &= \frac{\alpha^{_C}}{\alpha^{_I}} \left(\frac{A^{_C}_{t}}{A^{_I}_{t}}\right)^{1-\alpha^{_C}} \left(k_{t}\right)^{\alpha^{_C}-\alpha^{_I}} \times \notag\\ &\quad \times \left[ \left(\frac{s^{_K}_{t}}{s^{_L}_{t}}\right)^{1-\alpha^{_I}} \bigg/ \left(\frac{1 - s^{_K}_{t}}{1 - s^{_L}_{t}}\right)^{1-\alpha^{_C}} \right] \end{align} \tag{10}

• Notice that: \alpha^{_C}=\alpha^{_I}=\alpha \Rightarrow 1/q_{t} = \left(A^{_C}_{t}/A^{_I}_{t}\right)^{1-\alpha}

Details

The relative price of consumption goods will be higher:

1. the more output in sector I is sensitive to capital relative to sector C
2. the greater the technological progress in sector I relative to sector C
3. the higher the level of capital per effective worker, if output of sector I is more sensitive to capital than output of sector C
4. the higher the K/L ratio of sector C relative to sector I

The Model

Determinants and Long-Run Behavior of The Labor Share

m^{_L}_{t} \equiv \frac{w^{_I}_{t} L_{t}}{Y_{t}} = \frac{\alpha^{_I} (1-\alpha^{_C})+ (\alpha^{_C}- \alpha^{_I}) s^{_K}_{t}} {\alpha^{_I} + \left(\alpha^{_C} - \alpha^{_I}\right) s^{_K}_{t}} \tag{11}

• Notice that: \alpha^{_C}=\alpha^{_I}=\alpha \Rightarrow m^{_L}_{t}=1-\alpha
• A Balanced Growth Path in this economy requires s^{_K}_{t} to be constant over time
• So, the labor share is constant in the long-run Details

The Model

Transition Dynamics of The Labor Share to the BGP Level

• Along a transition path of s^{_K} to the steady state:

\frac{\mathrm{d} m^{_L}_{t}}{\mathrm{d} s^{_K}_{t}} = \frac{\left(\alpha^{_C} - \alpha^{_I}\right) \alpha^{_I}\alpha^{_C}}{\left(\alpha^{_C} s^{_K}_{t} + \alpha^{_I} \left(1 - s^{_K}_{t}\right)\right)^2} \tag{12}

• Hence:
  • if \alpha^{_C}>\alpha^{_I}: m^{_L}_{t} moves in the same direction of s^{_K}_{t}
  • if \alpha^{_C}<\alpha^{_I}: m^{_L}_{t} moves in the opposite direction of s^{_K}_{t}

Calibration

Calibration

Baseline Parameters

Parameter	Value	Source/Calibration target
\beta	0.96	Prescott (1986)
\phi^{-1}	0.65	Vissing-Jørgensen (2002)
\delta	3.8%	1980–2024 average from Feenstra, Inklaar, and Timmer (2015)
g^{_L}	0.9%	1980–2024 g^{_{Pop}} from US Census Bureau (2025)

Calibration

Baseline Parameters

Parameter	Value	Source/Calibration target
\alpha^{_I}	26%	Basu et al. (2013)
\alpha^{_C}	50%	1980–2024 of q_{t}^{-1} and K/Y
g^{_{A^C}}	1.0%	1980–2024 of q_{t}^{-1} and K/Y
g^{_{A^I}}	2.7%	1980–2024 of q_{t}^{-1} and K/Y
k_0	7.1	Initial level of K/Y

BGP values

• We cannot use Fernald 2014 nor Gortz 2018 estimates, because they compute TFP by the change in the relative price, i.e., equal alphas across sectors

Results

Results

Calibration Targets

The fit for these two targets seems good. The decline in the K/Y ratio requires starting the simulation with an excess of capital per effective worker relative to the steady state.

Results

The Transition Path of k_{t+1}

• Initial excess of capital enables high consumption and low investment
  • s^{_K}_0=12.26\%<14.47\% =s^{_K}_*
  • s^{_L}_0=28.44\%<32.5\% = s^{_L}_*
• Then, depreciation and technological progress trigger the transfer of inputs from sector C to sector I Plot shares

Results

The Transition Path of k_{t+1}

• Sector switching always equalizes the Marginal Rate of Technical Substitution across sectors \frac{1-\alpha^{_C}}{\alpha^{_C}} \frac{\alpha^{_I}}{1-\alpha^{_I}} = \frac{1-s^{_K}_{t}}{1-s^{_L}_{t}} \bigg/ \frac{s^{_K}_{t}}{s^{_L}_{t}}

• Both sectors become more capital intensive

Results

The Transition Path of the Labor Share

• Recall that: m^{_L}_{t} \equiv \left(w^{_I}_{t} L_{t}\right)/Y_{t}
• The real wage w^{_I} increases due to higher capital intensity
• Impact on total output Y_{t}\equiv q_{t} Y^{_C}_{t} + Y^{_I}_{t} is unclear
  • Y^{_C}_{t} decreases and Y^{_I}_{t} increases due to input sector switching
  • q_{t}^{-1} decreases (mainly) because g^{_{A^I}}>g^{_{A^C}} Eq. Condition

The real wage goes up and the population grows at a steady rate. However, what happens to aggregate output is unclear.

But as we know, if \alpha^{_C}<\alpha^{_I} the labor share will move in the opposite direction of s^{_K}. However, in our calibration \alpha^{_C}>\alpha^{_I} and so they move in the same direction.

Results

The Labor Share

Results

The Labor Share

• We already knew that m^{_L}_t and s^{_K}_{t} move in opposite directions when \alpha^{_C}>\alpha^{_I}
• The increase in the real wage dominates over the increase in aggregate output
• The labor share increases around +1p.p.
• According to Piketty, this requires r-g<0

Results

Piketty’s r>g Channel is Missing

Concluding Remarks

• Long-run labor share unaffected
• Short-term labor share changes driven by sectoral capital-output elasticities
• An increase in the labor share is expected when:
  • Consumption goods sector has a higher capital-output elasticity than investment goods
  • Capital per effective worker is above the steady state

Concluding Remarks

• Mechanism does not require \sigma_{L,K}>1
• The increase happens despite r>g
• Acts as a countervailing mechanism to the observed decline in labor share over the past four decades

Open Questions

• What does \alpha^{_C}>\alpha^{_I} mean? And what is the exact magnitude of the difference?
• Why was the capital per effective worker above the steady state in the 80s?
• What are the effects of other declining mechanisms?
• How does the welfare distribution change with the relative price?

Appendix

Relative Prices of Investment

By Type of Asset

Introduction

Shares of Investment

By Type of Asset

Introduction

The Capital-Output Ratio in the US

• Capital stock is the black sheep in National Accounts
• Perpetual inventory method vs. balance sheet data
  

First Best Solution

Equilibrium Condition for Variable s^{_K}_{t}

\begin{align} \left(1+g^{_{A^C}}\right)^{(1-\alpha^{_C}) (\phi - 1)} \left(1+g^{_{A^I}}\right)^{1-\alpha^{_C} (1-\phi)} \left(1+g^{_L}\right)^\phi = \\ = \left(\frac{k_{t}}{k_{t+1}}\right)^{\alpha^{_I} - \alpha^{_C} (1-\phi)} \left(\frac{s^{_L}_{t}}{s^{_L}_{t+1}}\right)^{1-\alpha^{_I}} \left(\frac{s^{_K}_{t+1}}{s^{_K}_{t}}\right)^{1-\alpha^{_I}} \times \\ \times \left(\frac{1-s^{_L}_{t+1}}{1-s^{_L}_{t}}\right)^{(1-\alpha^{_C}) (1-\phi)} \left(\frac{1-s^{_K}_{t}}{1-s^{_K}_{t+1}}\right)^{1-\alpha^{_C} (1-\phi)} \times \\ \times \beta \left[\alpha^{_I} \left(k_{t+1}\right)^{\alpha^{_I}-1} \left(\frac{s^{_L}_{t+1}}{s^{_K}_{t+1}}\right)^{1-\alpha^{_I}} +1-\delta\right] \end{align} \tag{13}

First Best Solution

Equilibrium Conditions for Variables s^{_L}_{t} and k_{t+1} and Transversality Condition

s^{_L}_{t} = \left[\frac{\alpha^{_I}}{1-\alpha^{_I}} \frac{1-\alpha^{_C}}{\alpha^{_C}} \frac{1-s^{_K}_{t}}{s^{_K}_{t}} + 1\right]^{-1} \tag{14} k_{t+1} = k_{t} \frac{1-\delta}{ \left(1 + g^{_{A^I}}\right) \left(1 + g^{_L}\right)} + \frac{ \left(s^{_K}_{t} k_{t}\right)^{\alpha^{_I}} \left(s^{_L}_{t}\right)^{1 - \alpha^{_I}} }{ \left(1 + g^{_{A^I}}\right) \left(1 + g^{_L}\right) } \tag{15} \begin{align} \lim_{t\to +\infty} \frac{\beta^{t}}{C_{t}^{\phi}} \frac{\alpha^{_C}}{\alpha^{_I}} \left(k_{t}\right)^{\alpha^{_C} - \alpha^{_I}} \left(\frac{s^{_K}_{t}}{s^{_L}_{t}}\right)^{1-\alpha^{_I}} \times\\ \times \left(\frac{1-s^{_L}_{t}}{1-s^{_K}_{t}} \frac{A_{t}^{_C}}{A^{_I}_{t}}\right)^{1 - \alpha^{_C}} K_{t+1} = 0 \end{align} \tag{16}

Planner solution

First Best Solution

Equilibrium Conditions for the Original Variables

C_{t} = \left((1-s^{_K}_{t}) K_{t}\right)^{\alpha^{_C}} \left((1-s^{_L}_{t}) A^{_C}_{t} L_{t}\right)^{1-\alpha^{_C}} \tag{17} I_{t} = \left(s^{_K}_{t} K_{t}\right)^{\alpha^{_I}} \left(A^{_I}_{t} s^{_L}_{t} L_{t}\right)^{1-\alpha^{_I}} \tag{18} L^{_I}_{t} = s^{_L}_{t} L_{t} \tag{19} K^{_I}_{t} = s^{_K}_{t} K_{t} \tag{20} L^{_C}_{t} = L_{t} - L^{_I}_{t} \tag{21} K^{_C}_{t} = K_{t} - K^{_I}_{t} \tag{22} K_{t+1} = k_{t+1} A^{_I}_{t+1} L_{t+1} \tag{23}

Planner solution

Decentralized Economy

Firms Problem

• Firms in each sector j\in\{C,I\} maximize profits:

\Pi^{_j}_{t} = P^{_j}_{t} Y^{_j}_{t} - W_{t} L^{_j}_{t} - R_{t} K^{_j}_{t} \tag{24}

• Subject to production technologies in Equation 4
• Given:
  • Price of its own output P^{_j}_{t}
  • Nominal cost rate of inputs: W_{t} and R_{t}

Decentralized Economy

Households Problem

• Households maximize utility in Equation 1
• Subject to a budget constraint:

q_{t} C_{t} + I_{t} \leq w^{_I}_{t} L_{t} + r^{_I}_{t} K_{t} \tag{25}

• Given:
  • Relative price q_{t} \equiv P^{_C}_{t}/P^{_I}_{t}
  • Real returns to inputs: w^{_I}_{t} \equiv W_{t}/P^{_I}_{t} and r^{_I}_{t} \equiv R_{t}/P^{_I}_{t}

Competitive Equilibrium

Definition

• Sequence for:
  • Inputs \{L^{_C}_{t}, L^{_I}_{t}\}_{t=0}^{+\infty} and \{K^{_C}_{t}, K^{_I}_{t}, K_{t+1}\}_{t=0}^{+\infty}
  • Outputs \{C_{t}, I_{t}\}_{t=0}^{+\infty}
  • Real returns to inputs \{w^{_I}_{t}\}_{t=0}^{+\infty} and \{r^{_I}_{t}\}_{t=0}^{+\infty}
  • Relative price \{q_{t}\}_{t=0}^{+\infty}

Competitive Equilibrium

Definition

• So that:
  • Firms solve their optimization problem
  • Households solve their optimization problem
  • Input and output markets clear according to Equation 6 to Equation 9

Competitive Equilibrium

Solution

• Same equilibrium conditions as in the First Best Solution for inputs and outputs
• Equilibrium conditions for real returns to inputs:

r^{_I}_{t} = \alpha^{_I} \left(\frac{s^{_L}_{t}}{s^{_K}_{t}}\right)^{1-\alpha^{_I}} \left(k_{t}\right)^{\alpha^{_I}-1} \tag{26}

w^{_I}_{t} = (1-\alpha^{_I}) A^{_I}_{t} \left(\frac{s^{_K}_{t}}{s^{_L}_{t}}\right)^{\alpha^{_I}} \left(k_{t}\right)^{\alpha^{_I}} \tag{27}

Relative price

Balanced Growth Path

Condition for s^{_L}_{*}

• Assume that k_{t} = k_{*} and s^{_K}_{t} = s^{_K}_{*}, for some t
• Then, a solution for the Planner Problem exists if and only if:

s^{_L}_{*} = \left[\frac{\alpha^{_I}}{1-\alpha^{_I}} \frac{1-\alpha^{_C}}{\alpha^{_C}} \frac{1-s^{_K}_{*}}{s^{_K}_{*}} + 1\right]^{-1} \tag{28}

Balanced Growth Path

Conditions for s^{_K}_{*} and k_{*}

Confirming our guess about the stationarity of s^{_K} and k:

\begin{align} s^{_K}_{*} &= \left[ \frac{ \left(1+g^{_{A^C}}\right)^{(1-\alpha^{_C}) (\phi - 1)} \left(1+g^{_{A^I}}\right)^{1-\alpha^{_C} (1-\phi)} \left(1+g^{_L}\right)^\phi }{ \beta \alpha^{_I} \left(\left(1 + g^{_{A^I}}\right) \left(1 + g^{_L}\right) - (1-\delta)\right) } - \right. \notag\\ &\quad \left.- \frac{1-\delta}{ \left(1 + g^{_{A^I}}\right) \left(1 + g^{_L}\right) - (1-\delta) } \right]^{-1} \end{align} \tag{29}

k_{*} = \left( \frac{ \left(s^{_K}_{*}\right)^{\alpha^{_I}} \left(s^{_L}_{*}\right)^{1 - \alpha^{_I}} }{ \left(1 + g^{_{A^I}}\right) \left(1 + g^{_L}\right) - (1-\delta) } \right)^\frac{1}{1- \alpha^{_I}} \tag{30}

Balanced Growth Path

Growth Rates of the Original Variables

\begin{aligned} g^{_{K^I}} = g^{_{K^C}} = g^{_K} = g^{_I} &= \left(1+g^{_{A^I}}\right) \left(1+g^{_L}\right) - 1 \\ g^{_{L^I}} = g^{_{L^C}} &= g^{_L} \\ g^{_C} &= \left(\frac{1+g^{_{A^I}}}{1+g^{_{A^C}}}\right)^{\alpha^{_C}} \left(1+g^{_{A^C}}\right) \left(1+g^{_L}\right) - 1 \\ g^{_{r^I}} &= 0 \\ g^{_{w^I}} &= g^{_{A^I}} \\ g^{_q} &= \left(\frac{1+g^{_{A^I}}}{1+g^{_{A^C}}}\right)^{1-\alpha^{_C}} -1 \end{aligned}

Labor share

Balanced Growth Path

Values Resulting from Calibration

Variable	Value
s^{_K}_*	14.47%
s^{_L}_*	32.50%
k_*	5.54
g^{_C}_*	2.76%
g^{_I}_*=g^{_Y}_*=g^{_K}_*=g^{_{K^C}}_*=g^{_{K^I}}_*	3.62%
g^{_{w^I}}_*	2.70%
g^{_{r^I}}_*	0.00%

Calibration

Simulation of the Core Variables

What’s happening?

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