On the Optimal Labor Income Share

On the Optimal Labor Income Share Jakub Growiec 1,2 Peter McAdam 3 Jakub Mućk 1,2 1 Narodowy Bank Polski 2 SGH Warsaw School of Economics 3 European Central Bank 7th NBP Summer Workshop Warsaw, June 14, 218 J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 1 / 32

Outline 1 Introduction 2 Model setup 3 Balanced growth path 4 Calibration 5 Quantitative results 6 Conclusion J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 2 / 32

Recent decline in the global labor income share Figure: Global Labor Share: 1975 212 Source: Karabarbounis and Neiman (214). J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 3 / 32

Historical evolution of the labor income share Figure: Historical Labor Share: US (1899-21) & France (1897-21).65.75.85 United States 189 191 193 195 197 199 21.6.7.8.9 1. France 189 191 193 195 197 199 21 Notes: The French data is taken from Piketty and Zucman (214). The US data also taken from Piketty and Zucman (214) over the sample 1929-21; prior to 1929 the labor share is extrapolated using the database by Groth and Madsen (216) which provide compensation of employees and value added data starting in 1898 based on historical source provided by Liesner (1989). J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 4 / 32

Research questions Level of the LIS The LIS has fallen in recent decades (since 198s), across many countries. But is it now undesirably low? Does it require intervention? Crucially, what is the socially optimal labor share? Dynamics of the LIS LIS has fluctuated considerably in the long time frame (e.g., > 11 years, Piketty and Zucman, 214). If the labor share is now falling, yet still above its optimal level then this might be interpreted passively, as a manifestation of recognized fluctuations in factor shares (Mućk et al., 218). Is the decentralized labor share characterized by excessive volatility? J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 5 / 32

Model setup Setup: generalization of Acemoglu (23), capital- and labor-augmenting R&D 1 The model is non-scale: both R&D functions are specified in terms of percentages of population employed in either R&D sector. 2 We assume R&D workers are drawn from the same pool as production workers. 3 We assume more general R&D technologies which allow for mutual spillovers between both R&D sectors (cf. Li, 2) and for concavity in capital augmenting technical change. 4 In contrast to Acemoglu (23), the BGP growth rate in our model depends on preferences via employment in aggregate production. 5 We use normalized CES production functions which, importantly, ensure valid comparative static comparisons in the elasticity of factor substitution. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 7 / 32

Model setup Setup: generalization of Acemoglu (23), capital- and labor-augmenting R&D 1 The model is non-scale: both R&D functions are specified in terms of percentages of population employed in either R&D sector. 2 We assume R&D workers are drawn from the same pool as production workers. 3 We assume more general R&D technologies which allow for mutual spillovers between both R&D sectors (cf. Li, 2) and for concavity in capital augmenting technical change. 4 In contrast to Acemoglu (23), the BGP growth rate in our model depends on preferences via employment in aggregate production. 5 We use normalized CES production functions which, importantly, ensure valid comparative static comparisons in the elasticity of factor substitution. We compare two allocations: 1 Social planner allocation (new). 2 Decentralized allocation (Growiec et al., 218). J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 7 / 32

Two R&D sectors Production functions for new technologies: KATC : LATC : λb = B ( λ 1 ω b x η b l ν ) b b dλb, (1) ( ) λa = A λ φ b x ηa l νa a λ a, (2) where l a and l b are the shares (or research intensity ) of population employed in labor- and capital-augmenting R&D, ν a, ν b (, 1] represent R&D duplication externalities (Jones, 22), the lab equipment term x λ bk λ a captures the technology-corrected degree of capital-augmentation of the workplace, d > captures gradual obsolescence of capital-augmenting technologies. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 8 / 32

Social planner allocation The social planner maximizes the representative household s utility from discounted consumption, given standard CRRA preferences subject to the budget constraint, the two R&D technologies, the labor-market clearing condition, and the normalized CES aggregate production function: max c 1 γ 1 1 γ e (ρ+n)t dt s.t. (3) k = y c (δ + n)k ζȧ, (4) λ b = B ( λ 1 ω b x η b l ν ) b b dλb, (5) ( ) λ a = A λ φ b x ηa l νa a λ a, (6) 1 = l a + l b + l Y, (7) ( ( ) ξ ( ) ) ξ 1/ξ y λb k λa l Y = π + (1 π ), (8) y λ b k λ a l Y ȧ = gλ a ( π π ) 1/α. (9) J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 9 / 32

Outline of the decentralized allocation The decentralized allocation requires addressing the following points: the representative household, final goods producers, aggregators of capital- and labor-intensive intermediate goods, producers of differentiated intermediate goods, capital- and labor-augmenting R&D firms, the general equilibrium. Key building blocks: Dixit and Stiglitz (1977) monopolistic competition in capital- and labor-augmenting intermediate goods sectors, directed R&D increases the variety of (differentiated) intermediate goods (Romer, 199), infinite duration of patents (although capital-augmenting patents are subject to obsolescence with a rate d > ), free entry to R&D. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 1 / 32

The representative household The representative household maximizes discounted CRRA utility: c 1 γ 1 max 1 γ e (ρ n)t dt (1) subject to the budget constraint: where v = V /L and V = K + p a λ a + p b λ b. v = (r δ n)v + w c, (11) J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 11 / 32

The representative household The representative household maximizes discounted CRRA utility: c 1 γ 1 max 1 γ e (ρ n)t dt (1) subject to the budget constraint: where v = V /L and V = K + p a λ a + p b λ b. The familiar Euler equation: v = (r δ n)v + w c, (11) ĉ = r δ ρ. (12) γ J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 11 / 32

Final goods producers Final goods producers operate in a perfectly competitive environment and face a normalized CES technology: ( ( ) ξ ( ) ) 1 ξ ξ YK YL Y = Y π + (1 π ). (13) Y K Y L J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 12 / 32

Final goods producers Final goods producers operate in a perfectly competitive environment and face a normalized CES technology: ( ( ) ξ ( ) ) 1 ξ ξ YK YL Y = Y π + (1 π ). (13) Y K Y L The first order condition: p K = π Y Y K, p L = (1 π) Y Y L, (14) ( ) ξ where π = π YK Y Y K Y is the elasticity of final output with respect to YK. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 12 / 32

Aggregators of capital- and labor-intensive intermediate goods Differentiated intermediate goods are aggregated according to: ( ) 1 NK ε Y K = XKidi ε. (15) J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 13 / 32

Aggregators of capital- and labor-intensive intermediate goods Differentiated intermediate goods are aggregated according to: ( ) 1 NK ε Y K = XKidi ε. (15) Aggregators operate in a perfectly competitive environment and decide upon their demand for intermediate goods given their prices. For capital-intensive bundles, the aggregators maximize: max X Ki ( NK p K XKidi ε ) 1 ε NK p Ki X Ki di. (16) J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 13 / 32

Aggregators of capital- and labor-intensive intermediate goods Differentiated intermediate goods are aggregated according to: ( ) 1 NK ε Y K = XKidi ε. (15) Aggregators operate in a perfectly competitive environment and decide upon their demand for intermediate goods given their prices. For capital-intensive bundles, the aggregators maximize: max X Ki ( NK p K XKidi ε ) 1 ε NK p Ki X Ki di. (16) Optimization implies the following demand curve: ( ) 1 pki ε 1 1 X Ki = x K (p Ki ) = Y ε K. (17) p K J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 13 / 32

Aggregators of capital- and labor-intensive intermediate goods Differentiated intermediate goods are aggregated according to: ( ) 1 NK ε Y K = XKidi ε. (15) Aggregators operate in a perfectly competitive environment and decide upon their demand for intermediate goods given their prices. For capital-intensive bundles, the aggregators maximize: max X Ki ( NK p K XKidi ε ) 1 ε NK p Ki X Ki di. (16) Optimization implies the following demand curve: ( ) 1 pki ε 1 1 X Ki = x K (p Ki ) = Y ε K. (17) Symmetrically for labor-intensive goods. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 13 / 32 p K

Producers of differentiated intermediate goods Producers of capital-intensive intermediate goods are monopolists. Linear production function: X Ki = K i. Capital is rented at the gross rental rate r. The optimization problem is: max (p Ki X Ki rk i ) = max(p Ki r)x K (p Ki ). (18) p Ki p Ki We obtain p Ki = r/ε for all i [, N K ]. This implies symmetry across all capital-intensive intermediate goods. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 14 / 32

Producers of differentiated intermediate goods Producers of capital-intensive intermediate goods are monopolists. Linear production function: X Ki = K i. Capital is rented at the gross rental rate r. The optimization problem is: max (p Ki X Ki rk i ) = max(p Ki r)x K (p Ki ). (18) p Ki p Ki We obtain p Ki = r/ε for all i [, N K ]. This implies symmetry across all capital-intensive intermediate goods. Symmetrically for labor-intensive goods. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 14 / 32

Producers of differentiated intermediate goods Producers of capital-intensive intermediate goods are monopolists. Linear production function: X Ki = K i. Capital is rented at the gross rental rate r. The optimization problem is: max (p Ki X Ki rk i ) = max(p Ki r)x K (p Ki ). (18) p Ki p Ki We obtain p Ki = r/ε for all i [, N K ]. This implies symmetry across all capital-intensive intermediate goods. Symmetrically for labor-intensive goods. Aggregation and market clearing implies: K = NK K i di = NK X Ki di = N K XK Y K = N 1 ε ε K K, (19) and thus we recover the aggregate CES production function. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 14 / 32

Capital- and labor-augmenting R&D firms Capital-augmenting R&D firms maximize: where Q K = A (λ ε constant. max l b 1 ε ω b ( pb Ṅ K wl b ) = max l b ((p b Q K w)l b ), (2) x β l ν b 1 b ) ( ε 1 ε ) is treated by firms as an exogenous J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 15 / 32

Capital- and labor-augmenting R&D firms Capital-augmenting R&D firms maximize: where Q K = A (λ ε constant. max l b 1 ε ω b ( pb Ṅ K wl b ) = max l b ((p b Q K w)l b ), (2) x β l ν b 1 b ) ( ε 1 ε Symmetrically for labor-augmenting R&D firms. ) is treated by firms as an exogenous J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 15 / 32

Capital- and labor-augmenting R&D firms Capital-augmenting R&D firms maximize: where Q K = A (λ ε constant. max l b 1 ε ω b ( pb Ṅ K wl b ) = max l b ((p b Q K w)l b ), (2) x β l ν b 1 b ) ( ε 1 ε Symmetrically for labor-augmenting R&D firms. ) is treated by firms as an exogenous Free entry into both R&D sectors implies w = p b Q K = p a Q L. Purchase of a patent entitles the holders to a per-capita stream of profits equal to π K and π L, respectively. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 15 / 32

Capital- and labor-augmenting R&D firms Capital-augmenting R&D firms maximize: where Q K = A (λ ε constant. max l b 1 ε ω b ( pb Ṅ K wl b ) = max l b ((p b Q K w)l b ), (2) x β l ν b 1 b ) ( ε 1 ε Symmetrically for labor-augmenting R&D firms. ) is treated by firms as an exogenous Free entry into both R&D sectors implies w = p b Q K = p a Q L. Purchase of a patent entitles the holders to a per-capita stream of profits equal to π K and π L, respectively. There is a constant rate d at which production of capital-intensive varieties becomes obsolete. This effect is external to patent holders and thus is not strategically taken into account when accumulating the patent stock. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 15 / 32

General equilibrium Decentralized equilibrium is the collection of time paths of all the respective quantities: c, l a, l b, k, λ b, λ a, Y K, Y L, {X Ki }, {X Li } and prices r, w, p K, p L, {p Ki }, {p Li }, p a, p b such that: households maximize discounted utility subject to their budget constraint; profit maximization is followed by final-goods producers, aggregators and producers of capital- and labor-intensive intermediate goods, and capital- and labor-augmenting R&D firms; the labor market clears: L a + L b + L Y = (l a + l b + l Y )L = L; the asset market clears: V = vl = K + p a λ a + p b λ b, where assets have equal returns: r δ = π L p a + ṗa p a = π K pb + ṗb p b d; the aggregate capital stock satisfies K = Y C δk ζȧl. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 16 / 32

Balanced growth path Ultimately long-run growth is driven by labor-augmenting R&D: labor is the only non-accumulable factor, and it is complementary to capital along the aggregate CES production function (cf. Uzawa, 1961). Along the BGP of the social planner allocation: g = ˆk = ĉ = ŷ = ˆλ a = A(λ b) φ (x ) ηa (l a) νa. (21) The following variables are constant along the BGP: y/k, c/k, l a, l b, λ b (thus asymptotically there is no capital-augmenting technical change). Along the BGP of the decentralized allocation: g = ˆk = ĉ = ŷ = ŵ = ˆp b = ˆp Li = ˆλ a = A(λ b ) φ (x ) ηa (l a ) νa. (22) The following quantities are constant along the BGP: y/k, c/k, l a, l b, λ b, Y K /Y, Y L /Y. The following prices are also constant along the BGP: r, p a, p K, p L, {p Ki }. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 18 / 32

Comparing equilibria: DA vs. SP 1 The consumption Euler equation: (a) in contrast to the social planner, markets fail to account for the external effects of physical capital on R&D activity via the lab equipment terms (with elasticity η b in the case of capital-augmenting R&D, and η a in the case of labor-augmenting R&D); (b) the markup parameter ε appears in the decentralized allocation due to imperfect competition in the labor- and capital-augmenting intermediate goods sectors. 2 In the Euler equations for l a and l b, the shadow price of physical capital ĉ ρ + n is replaced by its market price r δ = επ y k δ which accounts for markups arising from imperfect competition. 3 In the Euler equations for l a and l b : (a) markets fail to internalize the R&D duplication effects inherent when ν a, ν b < 1; (b) markets fail to account for the external effects of accumulating knowledge on future R&D productivity. These effects are included in the shadow prices of λ a and λ b in the social planner allocation but not in their respective market prices. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 19 / 32

Calibration Parameter Value Source/Target Preferences Inverse Intertemporal Elasticity of Substitution γ 1.75 Barro and Sala-i-Martin (23) Time Preference ρ.2 Barro and Sala-i-Martin (23) Income and Production GDP Per-Capita Growth g.171 geometric average ( ( Population Growth Rate n.153 geometric average ) ) Labor in Aggregate Production l Y, l Y.5394.5 γ+ ρ g 1+.5 γ+ ρ g Capital Productivity z, z.3442 geometric average Consumption-to-Capital u.2199 geometric average Capital Income Share π, π.326 arithmetic average Depreciation δ.6 Caselli (25) Factor Substitution Parameter θ -.75 σ =.7, Klump et al. (212) Net Real Rate of Return r δ.499 r δ = γg + ρ Monopoly Price Markup ε.9793 ε = r π z R&D Sectors R&D Duplication Parameters νa = ν b.75 in text Technology-Augmenting Terms λ a, λ b 1. in text Technology-Augmenting Terms Labor Input in R&D sectors λ b l a = l b Lab-Equipment Term x, x 61.79 x x = l Y ( 1. λ b = λ b z π ) 1 ξ z π.233 l a = l b and l a + l b = 1 l Y ( ( 1 z λ b l Y 1 π z λ b ) ) 1 ξ π ξ 1 π J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 21 / 32

Comparing BGPs: DA vs. SP Table: BGP Comparison under the Baseline Calibration (1) (2) (3) (4) DA SP Baseline C D Piketty σ =.7 σ = 1 σ = 1.25 Variable Symbol ξ =.43 ξ = ξ =.2 Output growth rate g.171.339.425.581 Consumption-to-capital ratio u.2199.1628.118.856 Capital productivity z.3442.371.2832.2743 Employment in production l Y.5934.4385.416.3854 Employment in labor augmenting R&D l a.233.2575.2447.224 Employment in capital augmenting R&D l b.233.34.3393.396 Relative Share l a /l b 1.847.7212.5735 Labor income share 1 π.6739.7854.6739.5243 Relative to DA (%) 1 π DA 1 π 1.1655.222 Capital income share π.3261.2146.3261.4757 Net real rate of return r δ.499.59.323.74 Capital augmenting technology λ b 1. 2.3696 3.3162 5.26 Lab equipment x 61.79 173.3363 342.782928.9625 J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 23 / 32

Long-run equilibrium LIS: DA vs. SP (1) Figure: Dependence of Equilibrium Labor Share on Model Parameters 1 1 1 1.8.8.8.8.6.6.6.6.4.4.4.4.2.2.2.2-1.5-1 -.5.5 Parameter ξ.1.2.3.4 Parameter ρ.2.4.6.8 1 Parameter ν a.2.4.6.8 1 Parameter ν b 1 1 1 1.8.8.8.8.6.6.6.6.4.4.4.4.2.2.2.2 1 2 3 4 5 Parameter γ.2.4.6.8.1 Parameter δ -.5.5 1 Parameter η a -.4 -.2.2.4.6 Parameter η b Notes: 1 π on vertical axis; corresponding parameter support on the horizontal axis. Social planner allocation (dashed lines), decentralized equilibrium (solid lines). The vertical dotted line in each graph represents the baseline calibrated parameter value. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 24 / 32

Long-run equilibrium LIS: DA vs. SP (2) Figure: Dependence of Equilibrium Growth on Model Parameters.15.5.8.8.1.4.6.6.3.4.4.5.2.2.2-1.5-1 -.5.5 Parameter ξ.1.1.2.3.4 Parameter ρ.2.4.6.8 1 Parameter ν a.2.4.6.8 1 Parameter ν b.8.4.15.15.6.4.2.35.3.25.2.1.5.1.5 1 2 3 4 5 Parameter γ.15.2.4.6.8.1 Parameter δ.2.4.6.8 Parameter η a -.2 -.1.1.2.3 Parameter η b Notes: The real economic growth rate g on vertical axis; corresponding parameter support on the horizontal axis. Social planner allocation (dashed lines), decentralized equilibrium (solid lines). The vertical dotted line in each graph represents the baseline calibrated parameter value. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 25 / 32

Note: J. Growiec, Social P. McAdam, planner J. Mućk allocation (.) (dashed On the lines), Optimaldecentralized Labor Income Share equilibrium (solid Warsaw, lines). June 14, The 218vertical 26 / 32 Long-run equilibrium LIS: DA vs. SP (3) Consumption-capital ratio, c/k.25.28 Labor-augmenting R&D, l a Capital-augmenting R&D, l b.5.2.26.24.45.4.15.22.35.1.2.18.3.25.5-1.5-1 -.5.5 Parameter ξ.16-1.5-1 -.5.5 Parameter ξ.2-1.5-1 -.5.5 Parameter ξ.36 Output-capital ratio, y/k 1 4 Lab equipment, x=λ b k / λ a 15 Capital augmentation, λ b.34.32 1 3 1.3 1 2 5.28.26-1.5-1 -.5.5 Parameter ξ 1 1-1.5-1 -.5.5 Parameter ξ -1.5-1 -.5.5 Parameter ξ 1 Labor share, 1-π.12 Growth rate, g.2 Net rate of return, r-δ.8.1.8.15.1.6.6.4.4.2.5.2-1.5-1 -.5.5 Parameter ξ -1.5-1 -.5.5 Parameter ξ -.5-1.5-1 -.5.5 Parameter ξ

Relation to Piketty s Fundamental Laws of Capitalism Two Fundamental Laws of Capitalism (Piketty, 214): (i) the degree of capital deepening K/Y goes up whenever the economic growth rate g goes down, (ii) the capital share π goes up whenever the growth rate g goes down. Here, all three variables are endogenous. 1 Taking Piketty s claims together logically implies that K/Y and π are positively correlated: capital and labor are gross substitutes (σ > 1), Rognlie (215); Oberfield and Raval (218). 2 We find that: when households become more patient (ρ goes down) or more willing to substitute consumption intertemporally (γ goes down), only law (ii) holds: the growth rate g goes up, the K/Y ratio goes up, and the capital share π goes down; when the lab equipment exponent ν b in capital augmeting R&D goes up, both laws are verified: the growth rate g goes down, the K/Y ratio goes up, and the capital share π goes up. As the elasticity of substitution σ goes up, the optimal growth rate g goes up hand in hand with the capital share π and the K/Y ratio. In such case, both of Piketty s laws are violated. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 27 / 32

Relation to Piketty s laws (continued) 3.8 Capital-output ratio, K/Y.35 Capital share π.5 Growth rate g 3.6 3.4.3.4 3.2.25.3 3 2.8.2.2 2.6.1.2.3.4 Parameter ρ.15.1.2.3.4 Parameter ρ.1.1.2.3.4 Parameter ρ 7 Capital-output ratio, K/Y.35 Capital share π.8 Growth rate g 6.3.6 5.25.4 4.2 3.15.2 2.5 1 1.5 2 2.5 Parameter γ.1.5 1 1.5 2 2.5 Parameter γ.5 1 1.5 2 2.5 Parameter γ 4 Capital-output ratio, K/Y.4 Capital share π.8 Growth rate g 3.35.6.3 2.4.25 1.2.2.5 1 Parameter ν b.15.5 1 Parameter ν b.5 1 Parameter ν b J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 28 / 32

Oscillatory convergence to the BGP Table: Dynamics Around the BGP Baseline C D Piketty σ =.7 σ = 1 σ = 1.25 ξ =.43 ξ = ξ =.2 Allocation DA SP DA SP DA SP Pace of convergence 6.3% 4.2% 5.8% 3.7% 5.2% 2.9% (% per year) Length of full cycle 52.6 76.7 79.8 83.2 144. 1.3 (years) Labor share cyclicality + + Amplitude of 1 π relative to y/k 62.% 48.% N/A N/A 28.% 44.% Note: computed as 1 e rr where rr < is the real part of the largest stable root; computed as 2π/ir where ir > is the imaginary part of two conjugate stable roots (if they exist). N/A denotes not available/applicable. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 29 / 32

Conclusion Endogenous growth theory tends to suggest that the socially optimal level of activity dominates the decentralized outcome. In this paper, we have confirmed this statement using a micro-founded, calibrated two-sector R&D endogenous growth model. For the labor share we found that if σ < 1 then the decentralized labor share is indeed socially suboptimal. The difference, moreover, is large, around 17% (11 pp.). Effectively, the only parameter which reverses this ordering is the elasticity of substitution σ. Both allocations are characterized by cycles. There is some (mixed) support for the claim that the decentralized equilibrium is likely to feature greater labor share volatility compared to the social optimum. J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 31 / 32

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Thank you for your attention. Commercial break Please read my paper: The Digital Era, Viewed From a Perspective of Millennia of Economic Growth. Fun read (though the ideas are serious) Comments welcome! J. Growiec, P. McAdam, J. Mućk (.) On the Optimal Labor Income Share Warsaw, June 14, 218 32 / 32

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