Capital-labor substitution, structural change and growth

Transcription

1 Capital-labor substitution, structural change and growth Theoretical Economics 12 (2017), Francisco Alvarez-Cuadrado, Ngo Van Long and Markus Poschke McGill University Presented by Sergio Feijoo UC3M Macro Reading Group October 24, 2017

2 Motivation Multisector models with BGP + Kaldor Facts + Kuznets Facts. Related literature: Demand driven: Kongsamut et al. (2001), Boppart (2014),... Supply driven: Ngai and Pissarides (2007), Acemoglu and Guerrieri (2008),... Data: Evidence of sector heterogeneity in elasticity of substitution, capital-labor ratios and factor income shares. Periods of changes in sectoral capital-labor ratios and factor income shares coincide with substantial structural change. This paper Supply-driven structural change (with quasi-balanced growth at the aggregate level) as a consequence of sectoral differences in the elasticity of substitution between capital and labor.

3 Contribution Develop two-sector Solow model where structural change can be purely driven by sector heterogeneity in the elasticity of substitution between capital and labor. Main (new) idea: when K{L and thus w{r increase, it s easier for the flexible sector to substitute capital for labor. Three effects pin down the factor allocations reaction to economic development: Relative price effect, Relative marginal product effect, Factor rebalancing effect.

4 A general model of structural change Economy with a final good Y ptq and intermediate goods, Y s ptq, s 1, 2. ı Y ptq F py 1 ptq, Y 2 ptqq γy 1 ptq ε 1 ε ` p1 γqy 2 ptq ε 1 ε ε 1 ε Assumption 1 9Kptq ` δkptq ` Cptq Y ptq Constant saving rate that is fully invested in every period. Sptq Iptq vy ptq ñ 9 Kptq vy ptq δkptq Y s ptq p1 α s q pa s ptql s ptqq σs 1 σs ı ` α s K s ptq σs 1 σs σs 1 σs L 1 ptq ` L 2 ptq Lptq, K 1 ptq ` K 2 ptq Lptq, 9 A s ptq A s ptq g A s

5 GMST: The static problem Given state variables pkptq, Lptq, A 1 ptq, A 2 ptqq at any t, let k Kptq Lptq, κ K 1ptq Kptq, λ L 1ptq Lptq and R r ` δ, w, p s, P, be, respectively, the rental rate, the wage rate, the prices of the intermediates and the price of the final good. Assumption 2 Sector 2 is more flexible, i.e. ą σ 1. Optimal allocation involves two trade-offs: 1 Optimal balance of resources (or optimal k s in each sector). ñ CONTRACT CURVE 2 Optimal allocation (consumption and production point of view) of resources across sectors. ñ LABOR MOBILITY CONDITION

6 GMST Static: Contract curve Free mobility of capital and labor implies p 1 MP K 1 p 2 MP K 2 R r ` δ (1) p 1 MP L 1 p 2 MP L 2 w (2) Let Ω tκ, λ, k, A 1, A 2 u, then combining both expressions CCpΩq 1 α σ α 2 A 1 ptq σ 1 k 1 1 κ 1 σ 1 1 α 2 α 1 A 2 ptq σ2 1 σ 1 p1 κq 1 p1 λq 1 λ 1 σ 1 1 (3) Lemma 1: Factor rebalancing effect Under Assumption 2, an increase in k shifts the contract curve up in κ, λ space. Besides, a proportional increase in A 1 and A 2 shifts it down.

7 GMST Static: Labor mobility condition Normalize P 1. Under perfect competition, the ratio of input demands for intermediates is given by p 1 γ 1 ˆY2 ptq ε (4) p 2 1 γ Y 1 ptq which combined with the condition for optimal allocation of labor across sectors (2) yields LMpΩq p 1 p1 α 1 q p 2 p1 α 2 q Y 1 ptq 1 σ 1 Y 2 ptq 1 1 p1 λq L 1 λ 1 σ 1 1 A σ 1 ptq σ1 1 σ 1 1 A 2 ptq 1 1 (5) Lemma 2: Shifts in the LM curve For given κ, an increase in k shifts LM up if 1 σ 1 1 ε ɛ 1 ą 1 1 ε ɛ 2, where ɛ s is the elasticity of output w.r.t. capital in sector s. Besides, a proportional increase in A 1 and A 2 shifts it down. Example: CC and LM

8 GMST Static: Slopes and Effects CC has a positive slope. Moreover, if σ 1 1 Factor rebalancing effect: As a factor becomes more abundant (thus cheaper) the more flexible sector uses it more. However, the slope of LM depends on the balance of the relative price and relative marginal product effects. Suppose κ Ò 2 Relative marginal product effect K 1 Ò ñ MP L 1 Ò ñ L 1 Ò 3 Relative price effect K 1 Ò ñ Y 1 Ò ñ p 1 đ p 2 Since p 1 p 2 MP L 1 MP L 2 1 ñ L 1 Ó ` L 2 Ò

9 GMST Static: (Out of the) General Case Development, i.e. increases in k or in A s shift both CC al LM curves. Therefore, the final effect on κ and λ may be ambiguous. Baseline model: A dominant factor rebalancing effect ε σ 1 1, α 1 α 2 α, ą 1 Baseline Special Case 1: Differences in factor intensity (AG) σ 1 1, α 1 ą α 2, ε ă 1 Special Case 2: Differences in productivity growth (NP) σ 1 1, α 1 α 2 α, A 1 A 2, ε ă 1

10 GMST Static: General Case Let ą σ 1 (both CC and LM shift up), and analyse increase in k in 3 situations: 1 ɛ 2 slightly bigger than ɛ 1. Factor rebalancing effect dominates. 2 Modify α 2 s.t. ɛ 2 substantially bigger than ɛ 1. Relative price effect dominates. 3 Diminish to obtain same effect as in 2. Figure: Parameters: (Left) γ 0.5, ε 0.5, σ 1 1, 1.2, α 1 0.3, α 2 0.4, A 1 A 2 1; (Middle) α ; (Right) 1.05.

11 GMST Static: General Case For common As, the relative input price is given by w w AR and the sectoral capital to effective labor ratios by k s Ks As Ls Then from free mobility of capital and labor it can be shown that ˆ w αs σs k s pw q 1 αs Figure: (Left) Left calibration from previous figure; (Middle) Middle

12 GMST: The dynamic problem Let χptq Kptq AptqLptq (6) Focusing on the Baseline parametrization, combining (1), (3), (4) and (6) obtain χpκq pγp1 αqq 1 p1 κq 1 σ 1 1 pκ αγq pκp1 γp1 αqq αγq 1 1 Proposition Detail Given an initial condition χp0q χ 0, the CE path satisfies the differential equation 9κ vbπpκq pδ ` g a ` nq Hpκq

13 GMST: The dynamic problem Define 9L s ptq L s ptq n sptq, Proposition 9K s ptq K s ptq z sptq, 9Y s ptq Y s ptq g sptq, There exists a unique (nontrivial) CGP that satisfies πpκ SS q δ ` g A ` n vb λ SS γp1 αqκss κ SS αγ χ SS pγp1 αqq 1 9Kptq Kptq zptq, p1 κ SS q 1 σ Y ptq Y ptq gptq pκ SS αγq pκ SS p1 γp1 αqq αγq 1 1 g SS z SS g1 SS g2 SS z1 SS z2 SS n ` g A ; n n SS 1 n SS 2

14 Summary Develop two-sector Solow model where structural change can be purely driven by within sector heterogeneity in the elasticity of substitution between capital and labor. Characterize equilibrium, static and dynamic properties of the model. (Not in the slides) Use the model to analyse the structural change out of agriculture in the United States.

15 Contract curve and labor mobility condition Back Figure: Parameters: γ 0.5, ε 0.5, σ 1 1, 1.2, α 1 0.3, α 2 0.4, A 1 A 2 1.

16 GMST Static: Baseline Back Baseline: A dominant factor rebalancing effect ε σ 1 1, α 1 α 2 α, ą 1 CCpΩq LMpΩq 1 k 1 A 2 ptq σ2 1 γ 1 γ γ 1 γ 1 1 ˆ Y2 ptq A 2 ptqlptq κ p1 κq 1 p1 λq λ 1 1 p1 λq 1 p1 αqp1 λq 1 λ ` α 1 1 ˆp1 κqk A 2 ptq σ2 1 p1 λq 1 λ

17 GMST Static: Baseline Back Proposition Assume ε σ 1 1, α 1 α 2 α, and ą σ 1 1. B ln κ B ln k B ln κ p1 q B ln A 2 Gpκqκ ă 0, B ln λ B ln k B ln λ ˆ α λpκq B ln A 2 1 α κ 2 B ln κ B ln λ 0 B ln A 1 B ln A 1 where 1 Gpκq p1 λpκqq ` 1 j ˆλpκq λpκq κ 2 ˆ α 1 α p 1q Gpκq ą 0, 1 ` κ ` j 1 p1 κq

18 GMST Dynamic: Baseline Back θpκq κ p1 γp1 αqq αγ j 1 γ p1 γq 1 B pγp1 αqq p1 αqγ 1 γ πpκq κ σ2 γ 1 θpκq γp1 αq 1 Hpκq p1 κq 1 αγ 1 ˆ j 1 p1 γ ` αγqpκ αγq ` σ2 θpκq 1 pκ αγqθpκq