Technology-Skill Complementarity

Technology-Skill Complementarity Nancy L. Stokey University of Chicago October 16, 2015 Becker Conference Stokey (University of Chicago) TSC 10/2015 1 / 35

Introduction This paper studies a simple general equilibrium model with complementarity between technology and human capital. There are two main motivations. 1. Wage inequality has displayed large and long-lived shifts over the last century: see Katz and Goldin (2007). In recent years, wage inequality has grown, in the U.S. and elsewhere. Empirical work suggests that most of this change is an increase in between- rm inequality, with very little increase in within- rm inequality. Changes in technology are an obvious candidate to explain these large shifts in the wage structure. Stokey (University of Chicago) TSC 10/2015 2 / 35

Introduction 2. There is an extensive literature that uses search models to study employment and wages in settings with heterogeneous rms. Relative to this literature, the contribution of the present paper is to micro-found the surplus function. Here each rm faces a downward sloping demand curve for its product. This demand curve determines the quantity of labor the rm wants to employ, as a function of its productivity. Thus, the surplus generated by any worker depends on total employment within the rm. Stokey (University of Chicago) TSC 10/2015 3 / 35

Related literature Wage inequality: Song, Price, Guvenen, et. al. (2015) for the U.S.; Faggio, Salvanes, and van Reenen (2007) for the U.K.; Card, Heining, and Kline (2013) for Germany; Hakanson () for Sweden; Helpman, et al. for Brazil. Search models: Burdett and Mortensen (1998), Jovanovic (1998), Moscarini and Postel-Vinay (2013), Lise and Robin (2013), etc. Stokey (University of Chicago) TSC 10/2015 4 / 35

Introduction The distinction between human capital and technology is not clear. Some would argue that technology is simply a form of human capital. Here, human capital is an asset that belongs to a single worker, who is the only one that can employ it in production. Hence it is a rival input. Technology is an asset that belongs to a rm. The rm can employ multiple workers, and technology is a nonrival input used by all of the workers. The fact that it is nonrival, within the rm, also distinguishes it from physical capital. Stokey (University of Chicago) TSC 10/2015 5 / 35

Introduction Here technology and human capital are inputs in a CES production function. They are complements: the substitution elasticity is less than unity. Labor markets are assumed to be frictionless. The low substitution elasticity means that the market (and e cient) allocation of labor across rms displays positively assortative matching. Stokey (University of Chicago) TSC 10/2015 6 / 35

Outline 1. The model 2. The competitive equilibrium 3. Technical change: does a rising tide lift all boats? 4. A multi-sector extension: revisit the rising tide question 5. Conclusions Stokey (University of Chicago) TSC 10/2015 7 / 35

1. Model: nal goods A single nal good is produced competitively, with CRS, using di erentiated goods as inputs. Producers of di erentiated goods are indexed by technology x j > 0, which determines their price p j. All di erentiated goods enter symmetrically, Y F = N J j=1 γ j y (ρ j 1)/ρ! ρ/(ρ 1), where ρ > 1 is the substitution elasticity, γ j J are shares for j=1 technologies fx j g J j=1, and N is the number (mass) of rms. Stokey (University of Chicago) TSC 10/2015 8 / 35

1. Model: nal goods The price of the nal good is p F 1 = N J j=1 γ j p 1 j ρ! 1/(1 ρ), and demands for di erentiated goods are y j = pj p F ρ Y F, all j. Stokey (University of Chicago) TSC 10/2015 9 / 35

1. Model: di erentiated goods Labor, di erentiated by human capital level h, is the only input. The output of a rm depends on the size and quality of its workforce, as well as its technology. A rm with technology x j that employs workers with various human capital levels, `(h) 0, all h, has output Z y j = `j (h)φ(h, x j )dh, all j, where φ(h, x) is a CES function with elasticity η < 1. Stokey (University of Chicago) TSC 10/2015 10 / 35

1. Model: wages, labor allocation A rm employs labor types h that minimize unit cost w(h)/φ(h, x j ). Since η < 1, e ciency requires positively assortative matching. Hence equilibrium is characterized by cuto levels fb j g J 1 j=1, where workers with h 2 (b j 1, b j ] work for rms of type j, with b 0 = h min, and b J = h max. Hence the equilibrium wage function w(h) satis es w 0 (h) w(h) = φ h (h, x j ) φ(h, x j ), h 2 (b j 1, b j ), all j, with kinks at the points b j, j = 1,..., J 1. Stokey (University of Chicago) TSC 10/2015 11 / 35

1. Model: di erentiated good prices, output levels Price is the usual markup over unit cost, p j = ρ w(h) ρ 1 φ(h, x j ), h 2 (b j 1, b j ], all j. Since x j and x j+1 are both willing to hire workers b j, p j+1 = φ(b j, x j ) p j φ(b j, x j+1 ), y j+1 y j = φ(bj, x j+1 ) φ(b j, x j ) ρ, j = 1,..., J 1. Firms with higher x j have lower cost and price, p j+1 < p j. and they have higher output, revenue, pro ts, y j+1 > y j. The labor allocation across rms with the same technology x j is not entirely pinned down, but all have the same price and output. Stokey (University of Chicago) TSC 10/2015 12 / 35

2. Competitive equilibrium De ne Ψ j as total labor productivity at rms of type j, Ψ j Z bj b j 1 φ(h, x j )g(h)dh, j = 1,..., J. Labor market clearing requires LΨ j = Nγ j y j, j = 1,..., J. (LMC) CE is characterized by fb j g J 1 j=1 satisfying (LMC) and y j+1 y j = with b 0 = h min and b J = h max. A solution exists and it is unique. φ(bj, x j+1 ) ρ, j = 1,..., J 1, φ(b j, x j ) Stokey (University of Chicago) TSC 10/2015 13 / 35

2. Competitive equilibrium: an example The distribution of rm types in the computed example is continuous. h has a (truncated) lognormal distribution, with parameters (µ h, σ h ). x has a (truncated) Pareto distribution, with shape parameter λ F. The parameters are ω = 0.5, η = 0.5, ρ = 6, λ F = 1.04, x min = 1, x max = 8, N = 5, µ h = 1, σ h = 1, h min = 0.4, h max = 15, L = 100. Stokey (University of Chicago) TSC 10/2015 14 / 35

2.5 Fig A1. labor productivity (h,x), various x-values, log-log scale 2 ln( (h,x)) 1.5 1 0.5 (blue) x = 5.6 0 (red) x = 3.3 (black) x = 1.8-0.5 (green) x = 1.1-1 -1-0.5 0 0.5 1 1.5 2 2.5 3 ln(h)

2.2 Fig A2: unit cost and price, p F = 1 2 1.8 1.6 1.4 1.2 1 0.8 1 2 3 4 5 6 7 8 120 100 = 6 F = 1.04 employment per firm, by x-type = 0.5 = 0.5 80 h = 1 h = 1 60 hmin = 0.4 hmax = 15 40 xmin = 1 xmax = 8 20 0 1 2 3 4 5 6 7 8 x

3 Fig A3: allocation of workers to firms, log-log scale 2.5 2 1.5 ln(h) 1 0.5 0-0.5-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ln(x) 7 elasticity of labor allocation h(x) 6 5 employment-weighted average elast = 1.8379 4 3 2 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ln(x)

2.5 Fig A4: wage function and CE function, log-log scale 2 wage function in red 1.5 CE function in dotted black 1 0.5 0-1 -0.5 0 0.5 1 1.5 2 2.5 3 ln(h) 0.75 elasticity of wage function 0.7 0.65 employment-weighted average elast = 0.55 0.6 0.55 0.5 0.45 0.4 0.35-1 -0.5 0 0.5 1 1.5 2 2.5 3 ln(h)

3. Technology improvements Can a technology improvement dx k = ε > 0 reduce wages for some workers? Or does a rising tide lift all boats? Questions: 1. What are the short run (SR) e ects on outputs y j, Y F, and prices p j while labor is immobile? 2. What are the long run (LR) e ects, when labor adjusts? 3. What are the LR e ects on employment, wages? Stokey (University of Chicago) TSC 10/2015 16 / 35

3. Technology improvements Let hats denote proportionate changes from the perturbation. For both SR and LR, the change in nal output is Ŷ F = J ν j ŷ j, j 1 where the weights are expenditure shares ν j Nγ j Y F p j y j, all j, with J ν j = 1. j 1 Stokey (University of Chicago) TSC 10/2015 17 / 35

3. Technology improvements: Short Run In the SR output changes only through the direct e ect of technology, ŷ SR k = ˆΨ SR k > 0, and ŷ SR j = 0, for j 6= k. Final output changes by Ŷ SR F = ν k ŷ SR k > 0. The price changes for di erentiated goods (with p F = 1 xed) are ˆp SR j = 1 ρ so ˆp k < 0 and ˆp j > 0, j 6= k. Ŷ SR F ŷj SR, all j, Stokey (University of Chicago) TSC 10/2015 18 / 35

3. Technology improvements: long run In the long run rms adjust the quantity and quality of labor, but... Proposition: To a rst-order approximation, the reallocation of labor across rms has no e ect on output of the nal good, Ŷ LR F The proof uses the Envelop Condition. Since labor markets are competitive, the original (CE) allocation maximizes nal output. Hence for a small perturbation to technologies, reallocating labor has no rst-order e ect on nal output. But it does a ect individual di erentiated good outputs and prices, and it a ects wages. = Ŷ SR F. Stokey (University of Chicago) TSC 10/2015 19 / 35

3. Technology improvements: long run Let dx k = ε, and let fb j (ε)g J 1 j=1 be the thresholds. Di erentiate the CE condition to characterize the b 0 j s. The signs depend on ρ ˆφ x (b k 1, x k ) ˆΨ k, ρ ˆφ x (b k, x k ) ˆΨ k. The reasoning is illustrated by looking at two special cases, two technologies, J = 2. Either x 1 or x 2 is improved. The size of the price decline ˆp k is proportional ˆΨ k /ρ. Before the change, both x 1 and x 2 are willing to employ h = b 1. Stokey (University of Chicago) TSC 10/2015 20 / 35

3. Technology improvements: long run If x k = x 1, then h = b 1 is the highest skilled worker at his rm, so his so his productivity rises more than the average for the rm, ˆΨ 1 < ˆφ x (b 1, x 1 ) < ρ ˆφ x (b 1, x 1 ). Hence in the long run, x 1 rms expand employment, b 0 1 = ρ ˆφ x (b k, x k ) ˆΨ k positive terms > 0, reinforcing the original pattern of price changes. All workers get wage increases, ŵ(h) = ˆp 1 LR + ˆφ x (h, x 1 ) > 0, at x 1, ŵ(h) = ˆp 2 LR > 0, at x 2, reinforcing the original pattern of price changes. Stokey (University of Chicago) TSC 10/2015 21 / 35

3. Technology improvements: long run If x k = x 2, then h = b 1 is the lowest skilled worker at his rm, so his productivity rises less than the average for the rm, ˆΨ 2 > ˆφ x (b 1, x 2 ). Nevertheless, since ρ > 1, if the gap is not too large, then ˆΨ 2 < ρ ˆφ x (b 1, x 1 ), so b1 0 > 0, and x 2 rms expand employment, reinforcing the original pattern of price changes. As before, all workers get wage increases. Stokey (University of Chicago) TSC 10/2015 22 / 35

3. Technology improvements Conjecture: The same logic holds for J > 2. Wages may rise for all workers even if the condition above fails. Increasing the supply of some di erentiated inputs increases the demand for all others, through the e ect on nal output Y F. Hence their prices are bid up, and wages rise. Stokey (University of Chicago) TSC 10/2015 23 / 35

3. Technology improvements: a simulation Suppose the top 10% of rms are a ected. The top 5% of rms get a 20% increase in productivity. The next 5% get smaller increases (to keep the distribution smooth). Because rms at the top hire more labor, about a third of the workforce is directly a ected. Stokey (University of Chicago) TSC 10/2015 24 / 35

0.2 Fig RT1: change in log technology 0.18 0.16 0.14 0.12 0.1 0.08 0.06 20% increase in x for top 5% of firms smaller increases for next 5% of firms 0.04 0.02 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ln(x) 0.25 change in log employment 0.2 0.15 0.1 35% of workers are directly affected 0.05 0-0.05-0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ln(x)

3 Fig RT2: technology across human capital types 2.5 ln(x) 2 1.5 red is after tech. change blue is baseline 1 0.5 0-1 -0.5 0 0.5 1 1.5 2 2.5 3 ln(h) 0.1 log wage changes 0.09 0.08 ln(w) 0.07 0.06 0.05 0.04 35% of workers directly affected 0.03 0.02 0.01 0-1 -0.5 0 0.5 1 1.5 2 2.5 3 ln(h)

4. A multi-sector extension Can the answer to the rising tide question be reversed? Consider a model with two tiers in production. In the upper tier sectoral aggregates are used to produce nal goods, and lower tiers, one for each sector, di erentiated goods are used to produce the aggregates. Each tier uses a CES aggregator, and the lower tiers have a higher elasticity of substitution. The price e ects of a limited technical shift are quite di erent in this setting. Stokey (University of Chicago) TSC 10/2015 26 / 35

4. A multi-sector extension Suppose the nal good technology is Cobb-Douglas, σ = 1, Y F = S s=1 Y θ s s, S θ s = 1. s=1 The price of nal output is P F 1 = " S s=1 Demands for sectoral intermediates are θs P s θs # 1. Y s = Y F θ s P F P s, all s. Stokey (University of Chicago) TSC 10/2015 27 / 35

4. A multi-sector extension Each sector has its own set of di erentiated inputs fy sj g. The shares γ sj J j=1 and number of rms N s can vary across sectors. The technologies and prices for sectoral intermediates are as before, Y s = J N s j=1 P s = J N s j=1 γ sj y (ρ sj γ sj p 1 sj ρ 1)/ρ! ρ/(ρ 1),! 1/(1 ρ), all s, Key assumption: ρ > σ. Goods within a sector are more substitutable than are intermediates across sectors. Equilibrium conditions: similar to the earlier model. Stokey (University of Chicago) TSC 10/2015 28 / 35

4. A multi-sector extension Demands for di erentiated inputs are so y sj is increasing in Y s and in P s. ρ psj y sj = Y s, all j, s, P s But Y s, P s are also linked through demand by nal goods producers, so y sj = Y s θ ρ s p ρ sj Ys Y F ρ/σ. With ρ > σ, Y s has a stronger e ect through price than directly. An increase in Y s reduces price P s so sharply that demand y sj falls. Stokey (University of Chicago) TSC 10/2015 29 / 35

4. A multi-sector extension: an example Final output Y F = Y1 1/2 Y2 1/2. 3 technology levels and 3 skill levels. Firms: N 1 = N 2 = 1, and all rms in sector 2 have x = x H. among rms in sector 1, share γ 2 (0, 1) have x M, Labor supplies: L H = 1, L M = γ, L L = 1 γ. and (1 γ) have x L. Each rm employs one worker, and x j employs h j, so y j = φ j = φ (h j, x j ), j = L, M, H. Sector-level aggregates are: h (ρ 1)/ρ Y 1 = (1 γ) y L + γy Y 2 = y H. i (ρ 1)/ρ ρ/(ρ 1) M Stokey (University of Chicago) TSC 10/2015 30 / 35

4. A multi-sector extension: a theoretical example Since Y 2 = y H, prices are Y1 1/2, p 2H = 1 2 y H yh 1/2 1/ρ y1j, j = L, M. p 1j = 1 2 Y 1 Y 1 Wages are proportional to revenue product, w H = ρ 1 ρ p 2H y H, w j = ρ 1 ρ p 1jy 1j, j = L, M. Stokey (University of Chicago) TSC 10/2015 31 / 35

4. A multi-sector extension: a theoretical example Consider technical change that increases x M. Then Ŷ 1 = νŷ M, where ν 2 (0, 1) is the cost share for x M goods. All wages increase if ρ > 2, ŵ H = ˆp 2H = 1 > 0, 2Ŷ1 1 1 ŵ L = ˆp 1L = Ŷ 1, ŵ L < 0 () ρ > 2, ρ 2 1 1 1 Ŷ 1 + ŷ M ρ 2 ρŷm ŵ M = ˆp 1M + ŷ M = 1 1 = ρ 2 νŷ M + ρ 1 ŷ M, ρ ŵ M > 0 () ρ + (ρ 2) (1 v) > 0. Stokey (University of Chicago) TSC 10/2015 32 / 35

4. Multi-sector model: a numerical example Keep all of the parameters from before, including N, L, G. Two sectors, S = 2. For nal goods σ = 1, θ 1 = θ 2 = 1/2. Sector 2 is high-tech. The technology shift a ects rms in sector 1, in the middle range of x. Stokey (University of Chicago) TSC 10/2015 33 / 35

1.6 1.4 1.2 1 0.8 MS1: distr of firm types, bys sector sector 1 (blue) is low-tech sector 2 (red) is high-tech 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 x 140 employment per firm 120 100 80 60 40 20 0 1 2 3 4 5 6 7 8 x distribution of workers across sectors 0.25 0.2 0.15 0.1 N1 = 3.67, N2 = 1.34 Y1 = 161, Y2 = 191 P1 = 0.55, P2 = 0.46 0.05 0 0 5 10 15 h

0.25 Fig RTMS1: technology shift, sector 1 0.2 0.15 sector 2 is unchanged 0.1 0.05 0 1 2 3 4 5 6 7 8 x

2.5 Fig RTMS2: technology across human capital types 2 baseline in black 1.5 w/ change in green ln(x) 1 0.5 0-1 -0.5 0 0.5 1 1.5 2 2.5 3 ln(h) 0.05 log wage changes 0.04 0.03 ln(w) 0.02 0.01 0-0.01-1 -0.5 0 0.5 1 1.5 2 2.5 3 ln(h)

5. Conclusions To understand long-run changes in wage inequality, we need better models connecting wage rates to changes in technology at the level of rms and industries. Stokey (University of Chicago) TSC 10/2015 35 / 35