Slow Convergence in Economies with Organization Capital

Transcription

1 . Slow Convergence in Economies with Organization Capital Erzo G.J. Luttmer University of Minnesota June 23, 207 SED 207 at the University of Edinburgh

2 related papers. Firm Growth and Unemployment Chicago Fed seminar, November 5, Slow Convergence in Economies with Heterogeneous Firms Minneapolis Fed Working paper 699, March Slow Convergence in Economies with Organization Capital revision of (), focussing on one of the two models in that paper 4. Further Notes on Micro Heterogeneity and Macro Slow Convergence June 206, see includes a counterexample to the idea explored here 2

3 millions a very steady but slow recovery from a sharp recession 25 employment hours (index) unemployment the working age population rose from 95M to 205M over this period 3

4 this is to be expected the key idea firm employment size at time t and age a k t,a = e (g δ k)a, conditional on survival flow of new firms is f random exit at the rate δ f > 0 cohort distributions non-stationary have to think about aggregation cross-sectional age distribution is exponential with right tail e δ fa implied size distribution Pr [k t,a k] = e δ f ln(k)/(g δ k ) = k ζ, ζ = δ f g δ k > aggregate dynamics dk t = (δ g)k t dt + fdt, δ = δ f + δ k mean reversion rate δ g = ( ) δ f = δ f /(g δ k ) vanishingly small in the Zipf limit ζ ( ) δ f ζ 4

5 firm count and size data Pareto tail in US, ζ = δ f g δ k. large firm exit rate of 2% per annum in US, δ f = 0.02, g δ k = exogenous mean reversion rate ( speed = δ f (g δ k ) = ) δ f = ζ ( ) implied half life half life = ln(2) speed = ln(2) ( ) = 38 years

6 high and low quality capital a two-type version of this idea firm capital replicated at rates g H > δ k and g L < g H flows of new firms are f H and f L high quality firm becomes low quality firm at a rate λ, aggregate dynamics [ ] [ ] [ ] KL,t (δ gl ) λ KL,t d = dt + 0 (δ + λ g H ) K H,t K H,t a steady state requires δ = δ f + δ k > g L and δ + λ > g H. the tail index of the firm size distribution will be { δ f ζ = min [g L δ k ] +, δ } f + λ. g H δ k [ fl f H ] dt plausible firm age-size distributions if ζ = (δ f + λ)/(g H δ k ) see Luttmer 20, Review of Economic Studies Zipf limit: ζ implies slow aggregate convergence 6

7 number of establishments what large firms are like firm employment size from Models of Growth and Firm Heterogeneity Luttmer, Annual Reviews of Economics, 200 data compiled by the Small Business Administration interpretation: firms really grow by replicating organization capital 7

8 employment some suggestive evidence WalMart, 2 Home Depot, 35 Microsoft, 33 FedEx, 9 Intel, 5 Dell, 26 Tyson Foods, BestBuy, unemployment

9 different shocks, different recessions firms mothball organization capital in some recessions would expect V-shaped aggregate outcomes not the focus of this paper firms destroy organization capital in other recessions for example a bursting bubble can lead to a sudden abandonment of low-quality organization capital see my The Stolper-Samuelson Effects of a Decline in Aggregate Consumption, Minneapolis Fed working paper 703 (203) will argue in detail here: should expect slow recovery 9

10 a diffi culty for standard models with ln(c) J(L).5 K t C t I t L t w t years (could get L/ K > 0 on impact with a low K/L elasticity of substitution) 0

11 negative wealth effect a problem in any model with recession = destruction of capital solution: abandon the fiction that there is only one type of labor a Roy model workers. aggregate supply relatively elastic substantial population with low surplus from participation 2. can only produce consumption 3. need managerial supervision managers. aggregate supply relatively inelastic high surplus, work all the time 2. can supervise workers, or 3. create new organization capital without the help of workers

12 the model make the entry flows f t and firm growth rates g t endogenous a competitive version of Luttmer 20, Review of Economic Studies simplify by taking the firm growth rate g t to be deterministic technology for producing homogeneous consumption C t = F (K t, min {L t, M t }) simplification in these slides: F is Cobb-Douglas faster convergence if elasticity of substitution below that is, if the capital share is high when K t / min{l t, M t } is low organization capital accumulation dk t = (g(m t ) δ)k t dt + f(n t )dt implied aggregate use of managerial services M t = M t + m t K t + n t 2

13 relation to Cass-Koopmans/standard RBC Cass-Koopmans is dk t = (g(l t ) δ)k t dt C t, L t = K t l t. only one type of curvature while here we have C g,t = D2 g(l t )l t Dg(l t ) dk t = (g(m t ) δ)k t dt + f(n t )dt, C t = F (K t, min{l t, M t }), M t = M t + m t K t + n t, L t = L t two types of curvature C F,t = D 22F (, l t )l t D 2 F (, l t ) C g,t = D2 g(m t )m t Dg(m t ) 3

14 factor supplies large household with preferences ( ) U(c, l, m) = e ρt [h c ln(c t (h)) + h u ( l t (h) m t (h))] dψ(h) dt 0 where h = (h c, h u, h v, h w ) and h v h w = quality of managerial services = quality of labor services implied factor supplies at prices (v t, w t ) = (ṽ t, w t )/C t L t = L(v t, w t ) = h w ι [w t h w > max {h u, v t h v }] dψ(h) M t = M(v t, w t ) = h v ι [v t h v > max {h u, w t h w }] dψ(h) where C t is aggregate consumption factor market clearing L t = L t M t + m t K t + n t = M t 4

15 equilibrium in the consumption sector clearing the labor market elasticity α = (βv t + w t )L (v t, w t ) w t = w(v t ) βv t βv E w,v,t = t +w t + E L,v,t w t < E L,v,t βv t +w t + E L,w,t E L,w,t negative if E L,v,t < 0 small or βv t /(βv t + w t ) substantial implied residual supply of managerial services S(v t ) = M(v t, w(v t )) βl(v t, w(v t )) elasticity E S,v,t > 0 these curves shift with K t if F is not Cobb-Douglas things are nice if F has an elasticity of substitution in (0, ] 5

16 incentives to produce new capital assume f is strictly concave (as is g) implicit fixed factor that limits entry scientists, experts, lawyers, land to create or replicate that is the question, v t = q t Df(n t ) v t = q t Dg(m t ) the price q t = q t /C t of organization capital satisfies (ρ + δ)q t = α K t entry and replication co-move + q t [g(m t ) Dg(m t )m t ] + Dq t everyone an entrepreneur when K t far below the steady state could grow like Japan, Taiwan, South Korea, China,... not so when K t close to steady state and size distribution Zipf 6

17 steady state supply curve for managerial services S(v) = M(v, w) βl(v, w), α = (βv + w)l(v, w) demand curve for managerial services parameterized by m (v, D(v)) = (q[m]dg(m), mk[m] + n[m]), where n[m], K[m] and q[m] solve K[m] = q[m] = f(n[m]) δ g(m), = Df(n[m]) Dg(m) ( α)/k[m] ρ + δ [g(m) Dg(m)m] the only key assumption is Df(0) = 7

18 the Zipf limit in a steady state with positive entry K = f(n) δ g(m), = Df(n) Dg(m) (m + l)k + n = M l K = L factor market clearing conditions force K <, and thus δ > g(m) can show M, L g(m) δ, ζ = δ f g(m) δ k assume g(m) > δ is in fact possible for m large enough use Df(0) = to rule out steady state with f(0) = 0 8

19 v managerial services: steady state supply and demand S(v) D(v) lim Λ D(v/Λ)/Λ 0 0 9

20 speed of convergence only one state variable, speed = ρ 2 + (ρ 2) 2 + D, where D = DK t q t Dq t K t DK t K t ( ρ DK ) t. K t in the Zipf limit g(m) δ where D = (ρ + δs g )δs g +E S,v + E S,v +E S,v C g S g = Dg(m)m, C g = D2 g(m)m g(m) Dg(m), E S,v = DS(v)v S(v) no convergence if g t did not respond to state of economy 20

21 a special case: E S,v = 0 in the Zipf limit, this yields D = (ρ + δs g )δs g, and thus speed = ρ 2 + (ρ 2) 2 + (ρ + δsg )δs g = δs g δ = 0.0 implies half life (7, ) more generally, when g(m) < δ, speed = DK t K t = δ g(m) + δs g = ( ) max {(g(m) δ)k + f(n) : mk + n S} m,n ( ) δ f + δs g ζ δ f = 0.02 and ζ =. implies, varying S g (0, ) 6.8 years ln(2) ( ) < half life < ln(2) ( ) 38 years

22 relation to Cass-Koopmans organization capital Cobb-Douglas consumption sector with residual managerial services supply elasticity E S,v ) 2 + (ρ + δs g )δs g speed = ρ 2 + ( ρ 2 +E S,v + E S,v +E S,v C g in the g(m)/δ limit Cass-Koopmans with Frisch labor supply elasticity E, where speed = ρ 2 + (ρ 2 ) 2 + g(m) savings rate = g(m) δ δ C g (ρ + δs g)δs g S g +E + E +E C g = ( + ρ ) S g δ > (!) 22

23 a benchmark example parameters discount and depreciation rates ρ = 0.04, δ f = 0.02, δ k = 0.08 Roy factor supply with Fréchet distributions (i.e. Logit) where Ψ = 0.9 Ψ A + 0. Ψ B Ψ A (h u,, h w ) = exp ( A u h σ u Ψ B (h u, h v, h w ) = exp ( B u h θ u with σ = 5 and θ =.3 CES consumption sector with elasticity of substitution = 0.5 capital share = 0.2 Cobb-Douglas f(n) and g(m), with S f = 0.8, S g = 0.4 ) A w h σ w B v h θ v B w h θ w ) 23

24 a benchmark example results choice probabilities at equilibrium factor prices all A B employed workers managers average managerial earnings 0 average worker earnings half life 0 years for comparison: (US data) employment-population ratio recovery civilian % of gap age % of gap 24

25 I C employment q K trajectories for K 0 < K and K 0 > K years 25

26 C employment q K CES versus Cobb-Douglas CES, ε = 0.5 Cobb Douglas years 26

27 a proposal abandon Cass-Koopmans one-sector growth model as a benchmark replace with Uzawa-Prescott-Visscher-Roy two sector, joint production, more than one task easy to produce a typical recovery evidence on firm size distribution suggests slow aggregate convergence easily consistent with rapidly growing firms key parameters labor and residual managerial services supply elasticities factor shares and elasticities of substitution for K and min{l, M} in the consumption sector for K and M in the replication sector parameters of the entry technology not important near the Zipf limit no need for the fun stuff (nominal non-neutralities, credit and labor market frictions) to do all the heavy lifting 27