日本オペレーションズ・リサーチ学会
2021年 秋季研究発表会

2-E-8

Labor share and corporate investment
05000246

1Tokyo University of Science Haejun JEON
William Paterson University Chuanqian ZHANG
Shenzhen University
Xue CUIIntroduction

Proposition 1 Given demand shock X, the firm
value isMost studies on corporate investment have focused(αif X < XI∗ ,A(XI∗ ) XX∗
only on capital as a production factor. In this study, weI
V (X) =consider both labor and capital as production factors
A(X)if X ≥ XI∗ ,and investigate the impact of labor share on a firm’s
whereinvestment timing and size decisions.
(1−a)bγ1A(X) = φK ∗ 1−abγ X 1−abγ − (δ0 + δ1 K ∗ )2Setupψwith φ :=and α := 12 − σµ2 +abγσ 2µ− 2(1−abγ)
r− 1−abγ
2
A firm’s output at time t is Qt = (Lat Kt1−a )b where q 1µ 22r( 2 − σ2 ) + σ2 .
Lt and Kt denote the amount of labor and capital,
The optimal capital K ∗ := K ∗ (XI∗ ) and investment
respectively, and a ∈ (0, 1) and b ∈ (0, 1) denote the∗∗∗labor share in production and the degree of returns to threshold XI := XI (K ) are determined byscale, respectively. The product’s price is Pt = Xt Qγ−1n (1 − a)bγφ o 1−abγt11−bγ
K ∗ (X) =X 1−bγ ,where γ ∈ (0, 1). The demand shock Xt followsδ1 (1 − abγ)n α(δ + δ K) o1−abγ  1 (1−a)bγ
01dXt = µXt dt + σXt dWt,
XI∗ (K) =1K(α − 1−abγ)φwhere µ and σ are positive constants and (Wt )t≥0 isa standard Brownian motion. A risk-free rate is given and the optimal labor at the timing of investment is∗∗∗∗by a positive constant r. Lumpy investment in capital L := L (XI , K ).
K incurs costs δ0 + δ1 K and labor requires wage w perunit time. The firm’s profit flow after investment is
3.2 Partially adjustable labor

π(Xt , Lt , Kt ) = Pt Qt − wLt = Xt (Lat Kt1−a )bγ − wLt .

3Models and solutions

3.1

Fully adjustable labor

Suppose that the firm can adjust the amount of laborat any time without costs. The optimal labor given
demand shock X and capital K isL∗ (X, K) =1
 abγXK (1−a)bγ  1−abγ

w,and thus, the profit flow given X and K is1(1−a)bγπ(X, L∗ (X, K), K) = ψX 1−abγ K 1−abγ
abγ

1−abγ .
where ψ := (1 − abγ)( abγw )Now suppose that the firm can make incremental adjustment in labor but only with irreversibility. Thatis, once the amount of labor is increased, it cannot be
reduced afterwards.
Lemma 1 The firm’s optimal choice of labor at timet given capital K is1
 (α − 1)rabγ X̄ K (1−a)bγ  1−abγtLˆ∗ (Xt , K) =αw(1 + r)(r − µ)where X̄t := sup0≤s≤t Xs .
Proposition 2 Given demand shock X, the firmvalue is α Â(Xˆ∗ , Lˆ∗ (Xˆ∗ )) X
if X < XˆI∗ ,IIXˆI∗
V̂ (X) = Â(X, Lˆ∗ (X))
if X ≥ Xˆ∗ ,Iwhere Lˆ∗ (X) := Lˆ∗ (X, Kˆ∗ ) andFigure 1a shows that the investment thresholds areconcavewith respect to a. This comes from the conLabγ Kˆ∗wLvexityofaverage product of capital and the concavity
Â(X, L) =X−r−µrof average cost of capital regarding the labor share inα(1−a)bγ α+ χL1−α(1−abγ) Kˆ∗
X − (δ0 + δ1 Kˆ∗ ) production.
α−1 
α(α−1)rabγ
1with χ := w(1+r)α(r−µ)
α(1−abγ)−1 .
The optimal capital Kˆ∗ and investment threshold Xˆ∗(1−a)bγ

7I

6are determined by5XˆI∗ =ˆ∗α(r − µ)( wrL + δ0 + δ1 Kˆ∗ )abγ(1−a)bγ(α − 1)Lˆ∗ Kˆ∗abγ(1−a)bγ−1Xˆ∗(1 − a)bγ I Lˆ∗ Kˆ∗r−µ+ α(1 − a)bγχLˆ∗

1−α(1−abγ),

432α(1−a)bγ−1αKˆ∗XˆI∗ = δ1 ,with the optimal labor at the timing of investment
Lˆ∗ := Lˆ∗ (XˆI∗ , Kˆ∗ ).

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0.9(a) Investment threshold
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Comparative statics

76We adopt the following benchmark parameters forcomparative statics: r = 0.06, µ = 0.02, σ = 0.2,
b = 0.9, γ = 0.1, w = 1, δ0 = 5, δ1 = 10.

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(b) Labor and capital

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Figure 2: Labor share and investment decisions for σ =0.3, b = 1, and γ = 0.3

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(a) Investment threshold
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Figure 2a shows that investment can be made earlier with partially adjustable labor than it would withfully adjustable labor (i.e., XˆI∗ < XI∗ ) when demand isvolatile and price-elastic and labor share is low. Thisfinding is associated with the sensitivity of marginalvalue of capital with regard to demand shocks.
A more detailed illustration on the results and related empirical evidence will be given at the presentation.

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References

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(b) Labor and capital

0.9[1] Abel, A., 1983, Optimal investment under uncertainty, American Economic Review 73, 228-233.

[2] Caballero, R., 1991, On the sign of the investmentuncertainty relationship, American Economic ReFigure 1: Labor share and investment decisions based
view 81, 279-288.
on benchmark parameters2