Technology-Skill Complementarity

Nancy L. Stokey

University of Chicago

November 2, 2015

Preliminary and incomplete Tech-Skill Compl 107

Abstract

This paper develops a model in which technology and human capital are com-

plements in production, so the labor market produces assortative matching between

firms and workers: firms with higher productivity employ higher quality workers and

pay higher wages. Thus, wage differentials across firms have two sources: differences

in firm productivity and differences in labor quality.

I am grateful to Daron Acemoglu, Marios Angeletos,Gadi Barlevy, Marco Bassetto,

Paco Buera, Ariel Burstein, Joe Kaboski, and Jesse Perla, for useful comments.

1

1. OVERVIEW

This paper develops a model in which technology and human capital are com-

plements in production, so the labor market produces assortative matching between

firms and workers: firms with higher productivity employ higher quality workers and

pay higher wages. Thus, wage differentials across firms have two sources: differences

in firm productivity and differences in labor quality.

The goal of the model is to assess the sources of wage inequality and differences in

firm size. Empirically, it is not easy to distinguish the importance of technology and

human capital differences in generating wage differentials across individuals and size

differences across firms. A theoretical framework that incorporates complementarity

between the two may be useful in assessing the importance of each.

There is an extensive literature using search models to study wage determination in

settings with heterogeneous firms, including Burdett and Mortensen (1998), Postel-

Vinay and Robin (2002), Menzio and Shi (2010, 2011), Moscarini and Postel-Vinay

(2013), and Lise and Robin (2013). Relative to this literature, the present paper

micro-founds the surplus function. Here, each firm faces a downward sloping demand

curve for its product. This produces an interaction between the profits generated

by different workers within a firm. Specifically, it determines the quantity of labor

the firm wants to employ, as a function of its productivity. In equilibrium, more

productive firms charge lower prices and produce more output. They may or may not

employ more workers, although they certainly employ higher quality workers.

There is also an extensive literature on the size distribution of firms, going back to

Gibrat (1931), and more recently including Dunne, Roberts and Samuelson (1988),

Ijiri and Simon (1994), Sutton (1997), and Luttmer (QJE, 2007, R.E. Stud 2011).

Relative to this literature, the model incorporates heterogeneous labor. The model is

also related to the model of technology and wage inequality in Jovanovic (1998).

2

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. It may provide a dynamic external benefit, by raising the human capital

of other workers, but it can be employed directly only by its owner. Technology is an

asset that belongs to a firm. The firm can employ multiple workers, and technology

is a “nonrival” input used by all of the workers. The fact that it is a nonrival input,

within the firm, is a feature distinguishes it from both human and physical capital.

How do technology and human capital interact in production? Here they are com-

plementary inputs in a CES production function for intermediates with a substitution

elasticity that is less than unity. In addition, the labor markets are assumed to be

frictionless. The low substitution elasticity means that the market (and efficient) allo-

cation of labor across intermediate goods firms displays positively assortative match-

ing.

The heterogeneous firms produce intermediate goods, which are combined with a

Dixit-Stiglitz aggregator to produce the single final good. Final goods are used for

consumption. (In a dynamic version of the model, studied elsewhere, they are also

use for investment.)

The rest of the paper is organized as follows. Section 2 presents the basic model

and characterizes the equilibrium. In particular, the distribution of wage rates across

workers and the size distribution of firms (employment and revenue) is characterized.

Section 3 looks at two comparative statics exercises. In the first, technology differences

across firms are shut down, to ask how the wage distribution would change if all firms

had a common technology level. In the second, human capital differences across

workers are shut down, to ask how the size distribution of firms would change if all

workers had the same level of human capital.

In section 4, the model is used to ask whether ‘a rising tide lifts all boats’? That

3

is, does a technology improvement that affects only a subset of firms raise wages for

all workers? To answer this question, we look at a change that improves technology

only for firms at the top of the distribution, leaving the others unaffected.

Section 5 looks at incentives to invest. Specifically, it first asks to what extent the

incentive for firms to invest in better technologies is affected by their ability to adjust

the quantity and quality of their workforce. It the looks at a similar question for

workers. To what extent does the incentive to invest in human capital depend on the

ability to switch jobs.

Section 6 looks at a multi-sector version of the model. Outputs of firms in each

sector are used as inputs in producing a sectoral aggregate. These sector aggregates

are in turn used as inputs to produce final goods. The main goal here is to re-

visit the ‘rising tide’ question. The answer is different here. An example shows that

technological improvement for high-end firms in a basically low-tech sector can reduce

wages for the low-skill workers employed there.

Section 7 concludes.

2. BASIC MODEL: PRODUCTION AND PRICING

In this section the technologies and decision problems for differentiated good and

final good producers are described, and the competitive equilibrium is characterized.

A. Final good technology

The final good is produced by competitive firms using differentiated goods as inputs.

All of these inputs enter symmetrically into final good production, but demands for

them differ if their prices differ. Specifically, differentiated good producers are indexed

by their technology level 0 which determines the price for their good. The

final goods sector takes the prices as given.

4

There are technology levels, = 1 ordered so 0 1 2 Let

be the share of differentiated good producers with technologies of level and

be the total mass of firms. Each final good producer has the CRS technology

=

Ã

X=1

(−1)

!(−1)

(1)

where 1 is the substitution elasticity. As usual, the price of the final good is

=

Ã

X=1

1−

!1(1−) (2)

and input demands are

=

µ

¶− all (3)

We will take final output as numeraire throughout the analysis, = 1

B. Differentiated good technology, labor allocation

Producers of differentiated goods use heterogeneous labor, characterized its human

capital level as the only input. Assume that has a continuous distribution. Let

() with density () on ≡ (min max) with 0 min max ≤ ∞ denote the

distribution of human capital across workers, and let denote the size (mass) of the

workforce.

The output of a firm depends on the size and quality of its workforce, as well as

its technology. In particular, if a firm with technology employs workers of various

human capital level, with () ≥ 0 all denoting the number (density) of each

type, then its output is

=

Z()( ) all

where ( ) is the CES function

( ) ≡ £(−1) + (1− )(−1)¤(−1)

∈ (0 1) (4)

5

The elasticity of substitution between technology and human capital is assumed to

be less than unity, and is the relative weight on human capital.

C. Equilibrium

An equilibrium consists of final output price and output functions { }=1for producers of differentiated goods, a wage function () ∈ and a description

of the allocation of labor to to differentiated good producers, that satisfy the usual

optimization and market clearing conditions.

In principle, the allocation of labor could be quite complicated, with firms of a

given type employing workers with human capital in various disjoint intervals,

and with workers of a given human capital level working for firms of different types

. This does not occur in equilibrium, and it is straightforward to see why it does

not.

Since labor markets are competitive, and labor is supplied inelastically, the alloca-

tion of labor is efficient. In addition, a CES production function is log supermodular

if (and only if) the elasticity is less than unity, as assumed here. Hence efficiency

implies positively assortative matching: firms with better technologies employ work-

ers with higher human capital (Costinot, Ecma 2009). Consequently the equilibrium

labor allocation is characterized by cutoff levels {}−1=1 where firm is willing to

hire workers with human capital ∈ [−1 ] with 0 = min and = max

The allocation of labor across firms with the same technology level is indeterminate.

Equilibrium determines only the output level, which is the same across firms within a

type. For concreteness we can suppose that each firm employs various human capital

levels in the interval [−1 ] in proportion to their representation in the population,

but this is not required.

Hence the equilibrium conditions are as follows:

6

a. market clearing for differentiated goods: { }=1 satisfy (3), with as in

(1);

b. cost minimization by producers of differentiated goods,

∈ [−1 ]⇐⇒ ∈ argmin

()

( ) all ; (5)

c. optimal price setting by producers of differentiated goods,

=

− 1()

( ) ∈ [−1 ] all ; (6)

d. labor market clearing,

Z

−1( )() = all (7)

The third condition says that the optimal price for firm is the usual markup over

unit cost. The last condition says that total productive capacity of labor with human

capital ∈ (−1 ) is sufficient to produce the output supplied by firms of type .To characterize the thresholds {}−1=1

note that the FOC and local SOC for the

cost minimization problem in (5) is

0()()

− ( )

( )= 0 (8)

00()()

− ( )

( ) 0 ∈ (−1 ) all

Hence the equilibrium wage function is piecewise continuously differentiable, with

kinks at the points {}−1=1

Since firms of types and +1 are both willing to hire workers of type it follows

immediately that+1

=

( )

( +1) = 1 − 1 (9)

Unit cost and price are strictly decreasing in : firms with better technologies set

lower prices. From the demand curve in (3),

+1

=

µ( +1)

( )

¶

= 1 − 1 (10)

7

so the output level is strictly increasing in : firms with better technologies have

higher sales. Since 1 they also have higher revenue and profits.

Combine (7) and (10) to find that {}−1=1satisfyZ +1

()( +1) =+1

µ( +1)

( )

¶ Z

−1()( ) (11)

= 1 − 1

Since 1 the ratio ( +1)( ) is strictly increasing in Therefore, since

0 is given, for any conjectured 1 the sequence {}=2 defined recursively by using(11) is increasing in 1 A competitive equilibrium requires = max Thus a solution

exists and it is unique.

It is useful to define Ψ as “total productivity” of labor at firms of type

Ψ ≡Z

−1( )() = 1 (12)

Then from (7), output for each firm type can be written as the product of a size term

and a productivity term,

=

Ψ = 1 (13)

and (11) can be written in the more symmetric form

( +1)− 1

+1Ψ+1 = ( )

− 1Ψ = 1 − 1 (14)

Firms of type and +1 are both willing to employ workers with skill level =

so

( ) =

− 1()= +1( +1) = 1 − 1 (15)

8

Computing the solution.–

Fix low and high value 1 1 use (11) to compute {}=2 for each, and compute

the errors − max Use linear interpolation to the desired accuracy. For initial

guesses fix 0 small and 1 large, and let

1 = −1 (1) =

E. A Numerical example

Figures A1 - A7 display the results for a numerical example. For the example, the

distributions { }=1 and are chosen to reflect, at least qualitatively, the distri-

bution of firm size by employment or revenue, and the distribution of wages across

workers. To this end, the distribution of technologies is a discrete approximation to

a (truncated) Pareto with a shape parameter that is close to unity, and the distri-

bution of human capital is a (truncated) lognormal. The total numbers for firms and

workers, and (both in millions), are taken from Axtell (2001). The parameters

describing firms and workers are

= 104 min = 1 max = 8 = 5

= 1 2 = 1 min = 04 max = 15 = 100

For the differentiated good technology, the substitution elasticity and the weight on

human capital are both set at one half. For the final goods technology, the substitution

elasticity is = 60 which gives a markup of 20%,

= 05 = 05 = 60

Figure A1 displays labor productivity ( ) as a function of for several values

of with a logarithmic scale for both and The complementarity between and

is reflected in the fact that the curves fan out at higher values for That is, better

technologies–higher –have a proportionately greater effect on the productivity of

workers with higher human capital.

9

Figure A2 displays the allocation of worker types across firm types. The allocation

is perfectly assortative, but the elasticity changes considerably across types, reflecting

relative supply and demand. Since the distribution of firm types is Pareto, there are

many firms at the low end, and relatively few workers. Hence worker type increases

rapidly at the low end: the elasticity is higher than average. Since the distribution of

human capital is lognormal, there are even fewer workers than firms at the high end.

Hence worker type also increases rapidly at the high end.

Figure A3 displays price and unit cost (panel a) and employment, output and

revenue per firm (panels b-d), all of the latter on a log10 scale. Price and unit cost

are decreasing in firm type. Except for a small dip at the bottom end, employment

is strongly increasing in firms type. Firms at the top of the distribution hire about

15 times as many workers as firms at the low end. Hence output and revenue are also

strongly increasing. Output is 235 times higher at the top. Part of this is higher

employment, but an even larger part is labor productivity, which is about 18 times

higher at the top end. Revenue,which reflects both increasing output and decreasing

price, is 95 times larger at the top end.

Figure A4 shows the (Pareto) distribution of firm types, employment per firm,

and total employment by firms types. Although each low-type firm employs few

workers, because they are so numerous, collectively they account for a large share of

employment.

Figure A5 shows the size distribution of firms. The two panels show the right CDFs

for employment and revenue (both log 10). The latter has more curvature than the

empirical distribution, and a much narrow range.

The top panel of Figure 6A show the wage function () (log-log scale), together

with a constant-elasticity function. The bottom panel shows in more detail the differ-

ence between the two. The model delivers a wage function with a decreasing elasticity,

with an employment-weighted mean of about 0.55. The range is from about 0.72 at

10

the bottom, to about 0.35 at the top. Recall that the wage at each firm is proportional

to labor productivity times the differentiated good price. The former is increasing in

the worker’s human capital, in part because workers with higher human capital work

with better technologies. The latter is decreasing, since firms with better technologies

produce more and hence set lower prices.

Figure A7 shows the CDFs and pdfs for human capital and wages, each measured

relative to its mean. The wage distribution is considerably more compressed.

3. COMPARATIVE STATICS

A. Homogeneous firms

To assess the effect of firm heterogeneity on the wage distribution, we can compare

the solution above with one in an economy with the same distribution of human

capital across workers, but homogeneous firms. To focus on wage inequality, we will

choose the technology level for firms so that output of the final good is the same in

both economies. Then the total wage bill is also the same, and only the distribution

of wages across workers changes.

To keep output of the final good unchanged, output of each differentiated good

in the homogeneous firm (HF) economy must be

= −(−1) (16)

and with = 1 (2) implies that the price for each differentiated good is

= 1(−1)

To determined the required technology level, suppose each firm gets workers,

with a pro rata share of all the human capital levels in the workforce. Then the

11

common technology level satisfies

=

Z( )() (17)

The wage function is then

() =− 1

( ) all (18)

Let Ψ denote total labor productivity in the economy with homogeneous firms,

Ψ ≡Z

( )()

It follows from (6) and (18) that a worker with human capital level ∈ (−1 ] whois employed at a firm with technology if firms are heterogeneous, gains or loses as

1 S ()

()

=( )

( )

()

=( )

( )

µ

¶−1=

( )

( )

µΨ

Ψ

¶1 ∈ (−1 ] (19)

There are two forces at work. If then the worker with human capital

∈ (−1 ] becomes individually more productive, which tends to raise his wage,the first term in (19). But the the price of the differentiated good he produces also

changes. If , then output rises and price falls at the worker’s firm, the second

term in (19). Output at his firm can rise because employment rises or because his

co-workers become more productive, or both.

The earlier numerical example provides an illustration of the effects. The required

technology level for the economy with homogeneous firms (HF) is = 26451

which is slightly lower than the employment-weighted mean E[] = 27628 in the

baseline economy. Expected labor productivity also falls slightly.

12

Figure HF1 compares the baseline economy, shown in blue, with the HF economy,

shown in red. Panel (a) displays the obvious: the direct technology effect raises

(lowers) individual productivity for workers in the lower (upper) end of the human

capital distribution. But the same figure shows that average productivity of co-

workers moves in the same direction, for two reasons. For workers at the lower

(upper) end of the distribution, the average human capital of their co-workers rises

(falls), as well as the technology they work with. Both effects push the price of the

worker’s differentiated good in an offsetting direction. Employment changes, shown

in panel (b), also work in an offsetting way: employment rises (falls) at firms with low

(high) baseline technologies. Hence the price effect lowers (raises) wages for workers

in the lower (upper) end of the human capital distribution, as shown in panel (c). In

this example, the net effect is to depress wages at the very low and very high ends of

the human capital spectrum, and to raise wages for those in the middle, as shown in

panel (d).

Figure HF2 shows additional detail about the effect on the wage distribution. Panel

(a) shows more clearly the magnitude of the wage changes across human capital levels.

Workers at the bottom of the distribution suffer large losses–up to 20 basis points,

and those at the top suffer moderately large losses–up to 15 basis points. The larger

group in the middle enjoys a modest gain–up to 6 or 7 basis points.

Panel (b), which displays the change in the CDF for log wages, shows more clearly

the number of winners and losers. Wages fall for about 20% of workers at the bottom

of the distribution and for another 20% at the top. Wages rise for the remaining 60%

in the middle.

This example suggests that aggregate technology changes that compress the tech-

nology distribution across firms will raise wages in the middle of the human capital

distribution more than it raises wages at either end, and could even reduce wages at

the ends. Aggregate changes that increase the variance of the technology distribution

13

would reverse the pattern of relative effects.

B. Homogenous labor

Similarly, to assess the effect of labor heterogeneity on the size distribution of

firms, we can compare the baseline solution with the one in an economy with the

same distribution of productivity across firms, but homogeneous labor. As in the

previous exercise, we will keep the final output level unchanged, here choosing the

common human capital level that keeps unchanged.

Since wages are the only cost for firms, and price is a constant markup over unit

cost, the equilibrium wage in the economy with homogeneous labor is simply labor’s

share of per capita output,

=− 1

Unit cost for each firm is the wage divided by labor productivity, so it also depends

on labor quality . Price is then the usual markup over unit cost, and output is

related to price. In particular,

=

− 1

( ) (20)

=

¡

¢−

The required human capital level can be determined from the output condition

(1), the price normalization (2), or labor market clearing (7). Any of these implies

satisfies

X=1

( )

−1 =³

´−1 (21)

To determine the effect of labor heterogeneity on the size distribution of firms, use

(9), (10) and (20) to find that

+1

T +1

as

¡ +1

¢ ( )

T ( +1)

( ) (22)

14

Since has an elasticity of substitution less than unity, it follows that

+1

T +1

as T all (23)

Thus, with homogeneous labor the disparity in output levels across firms increases

for firms at the lower end of the size distribution, and decreases for firms at the high

end.

Figures HL1-HL2 show results for the numerical example. The level of human

capital in the homogenous labor economy, = 31908 is lower than the mean

E[] = 39895 in the baseline economy. The critical value for technology is crit = 288

Firms with technologies below (above) this level hire labor with human capital below

(above ) in the baseline economy.

The first panel of Figure HL1 shows that labor productivity varies less across firms

in the economy with homogeneous labor, rising (falling) for firms with technologies

below (above) crit The second panel shows that employment across firms varies more

sharply in the economy with homogeneous labor, working to offset the first effect. The

third panel shows that the effect on output is mixed. Output is lower for firms at the

top and bottom of the technology distribution, and higher for those in the middle.

Figure HL2 shows the right CDFs for employment and output. Firm size, by either

measure, is more disperse in the economy with HL. If labor is homogeneous, firms

adjust quantity more strongly, to compensate for the fact that they cannot adjust

quality. Note, too, that the distributions for the economy with HL are closer to the

linear shape–Zipf’s Law–seen in the data.

4. TECHNOLOGY IMPROVEMENTS

Consider technical change that increases the technology level for one type of firm,

by a small increment, with all other technologies unchanged. We are interested in the

effect on the labor allocation, described by the thresholds {}−1=1; the output levels

15

and prices { }=1 for all differentiated goods, and the wage function () It is

helpful to begin with a brief overview of the main forces at work.

Suppose that in the short run workers do not switch firms. All firms produce at

capacity, and prices for differentiated goods are determined by demand. Firms that

experience productivity improvements increase output, so production of the final

good increases. The price change for each differentiated good depends on whether its

supply has increased or decreased relative to supply of the final good. Thus, firms

that experience the technology improvement cut their prices to stimulate demand for

their increased output. Firms with unchanged technologies raise their prices.

With labor immobile, there is no obvious way to allocate a firm’s revenue change

between wages and profits. Here we will assume that workers at every firm continue to

get the share (− 1) of revenue. Under this assumption a worker’s wage dependson his productivity and the price of the good he produces. This outcome corresponds

to a competitive equilibrium where workers can switch firms but not firm types.

With this assumption, in the short run the real wage rises for workers employed at

firms with unchanged technology. At firms that experience the technology improve-

ment, the real wage of a worker increases if and only if the increase in his productivity

outweighs the price cut. But in the long run, labor is mobile and workers switch firms.

The rest of this section analyzes these changes in more detail. Let “hats” denote

proportionate changes induced by the perturbation, ≡ −1 Note that in both

the short run and the long run, the change in output of the final good is a weighted

average of the changes in the output of all the intermediates,

=

X=1

(24)

where the weights

() ≡

all (25)

16

withX

=1

= 1

are the cost shares for the various types of differentiated goods in producing the final

good. Similarly, with the price of the final good fixed at unity, so = 0 the same

weighted average of the price changes for differentiated goods is zero,

0 =

X=1

A. Short run

Suppose that the technical change affects firms of type increasing their technol-

ogy by a small increment = with all other technologies unchanged. Consider

first the short run effects, with labor immobile. With the allocation of labor–both

quantity and quality–unchanged, output increases at firms of type and is un-

changed at firms of all other types.

Recall the definition of Ψ in (12). For firms with technology let Ψ denote

the direct effect of the technology perturbation on the firm’s labor productivity,

= Ψ ≡

1

Ψ

Ψ

=1

Ψ

Z

−1( )() (26)

and = 0 for 6= The change in final output is

= 0

The price changes for differentiated goods are

=1

¡ −

¢ all

17

so price falls for firms with technology and rises for all others. Revenue is the sum

of the output and price changes, so

=

1

+

− 1

() 0 all

and all firms enjoy an increase in revenue.

B. Long run

In the longer run, with labor mobile, the changes in {}=1 and must be aug-

mented to account for the impact of changes in the quantity and quality of labor

across firms, changes in the ’s. From (12) and (13), the long run changes in supplies

of differentiated goods are

=1

Ψ

£( )()

0 − (−1 )(−1)

0−1¤+ Ψ

all (27)

The next proposition shows that, to a first-order approximation, the equilibrium

change in the labor allocation has no impact on production of the final good: the

long-run increase is the same as the short-run increase.

Proposition: In the long run, with labor mobile across firms, the change in

output of the final good is, to a first-order approximation, the same as in the short

run, =

Proof: Use (27) in (24) to find that

=

X=1

½1

Ψ

£( )()

0 − (−1 )(−1)

0−1¤+ Ψ

¾ (28)

Since the sum of the last terms is and = it suffices to show that

0 =

X=1

Ψ

£( )()

0 − (−1 )(−1)

0−1¤

=

−1X=1

"

1−

Ψ

( )−+1

1−+1

Ψ+1

( +1)

#()

0

18

=

− 1−1X=1

"

−

Ψ

− +1

Ψ+1

−+1

#()()

0

=

− 1−1X=1

∙( )

Ψ

− +1( +1)

Ψ+1

¸[( )]

−()()

0

where the second line uses (3) and the fact that 0 = 00 = 0 the third uses (15), and

the fourth uses (9). From (14), the term in square brackets in the last line is zero,

for all . ¥

This result is not surprising. The potential difference between the short-run and

long-run effects arises only from the reallocation of labor. Since labor markets are

perfectly competitive, the allocation of labor in the baseline equilibrium, described

by the thresholds {}−1=1 maximizes Hence to a first order approximation small

changes in the allocation of labor across firms–either quantity or quality–have no

effect on final output. An increase (decrease) in raises (lowers) the output of

firms with technology but the effect on final output is exactly offset by decrease

(increase) in the output of firms with technology +1

The changes in the labor allocation do, however, alter the output, prices, revenue,

and profits of differentiated good firms, as well as wages. Let {()}−1=1denote the

solution to (11) or (14) as a function of and let

( ) ≡ ( )

( ) ( ) ≡ ( )

( )

Since 1, it follows that ( ) is strictly increasing in and ( ) is strictly

increasing in

To determine the effect on the labor allocation, described by©0()

ª−1=1

differen-

tiate (14), using (12), to get a system of − 1 linear equations,

= −10−1 +

0 +

0+1 = 1 − 1 (29)

19

where

−1 = − 1Ψ

(−1 )(−1) (30)

=1

Ψ

( )() +1

Ψ+1

( +1)()

+h( +1)− ( )

i

= − 1

Ψ+1

(+1 +1)(+1)

and

−1 = −(−1 ) + Ψ (31)

= ( )− Ψ 0

= 0 otherwise.

Clearly −1 0 Since 1 it follows that ( ) is increasing in so 0

For the sign of note that Ψ is the proportionate change in total labor productivity–

and hence also output–at a firm of type as a direct resultfrom the technology in-

crease, with employment unchanged. The term ( ) is the proportionate change

in own labor productivity for the worker at the upper skill threshold for firm ,

one with = As noted above, is strictly increasing in Since the inte-

gral in Ψ involves values for that are less than and 1 it follows that

Ψ ( ) ( )

The sign of −1 is ambiguous, but for plausible parameters it seems likely to be

positive. Here the skill level −1 is the lowest among workers at the firms receiving the

technology increase. Hence the integrals in Ψ involve −1 so Ψ (−1 )

Hence for sufficiently close to unity and (−1 ) sufficiently far below the average

Ψ for the firm, the term in square brackets in the first line of (31) can be negative.

But firms mark up unit cost by the factor (− 1) so close to unity is implausible.The term in brackets is positive if is large and (−1 ) is not too far below Ψ.

In this case −1 0

20

Special case = 2 = 1.–

Suppose = 2 and = 1 so there are only two technologies and the lower one is

improved. There is one equation in (29), = = 1 to determine 01

= 101 (32)

Since = 1 0 and 1 0 it follows that 01 0 and 1 firms increase employ-

ment.

The intuition is as follows. Before the technology change both types of firms are

willing to employ workers of type 1 the condition in (15). Let and denote

the proportionate output and price changes with the labor allocation unchanged, and

let denote the proportionate change in final output. Since output does not change

at 2 firms, (3) implies

1 =1

¡ − 1

¢ 2 =

1

The increment to the worker’s marginal revenue product at firm 1 minus the incre-

ment at 2 is the productivity effect at 1 plus the difference in the price effects,

(1 1) + 1 − 2 = (1 1)− 11

= (1 1)− 1Ψ1 (33)

Recall that ( 1) is increasing in and that 1 is the highest skill type employed

by firms of type 1 Since 1 = Ψ1 is a weighted average of the increases in

labor productivity across workers at firms of type 1, and 1 it follows that

(1 1) 1 1 The labor productivity of a worker with skill 1 increases

enough at firms with technology 1 that firms of type 2 can no longer compete for

him. Thus, 1 firms retain the workers with = 1 and in addition hire some with

1 who are also better complements with the new, higher technology level. The

threshold 1 rises, reinforcing the original pattern of output and price changes.

21

The change in wages across skill levels, () is continuous. For workers at 1 firms,

the wage change is strictly increasing in

0 () = 1 + ( 1) 1 ∈ (min 1 + 01]

All workers at 2 firms get the same increase,

0 () = 2 2 0 ∈ (1 + 01 max)

Special case = 2 = 2.–

Suppose = 2 and = 2 so the higher technology is improved. As in the previous

example there is one equation in (29), here the one for = − 1 = 1 to determine01

−1 = 101

As discussed above, the sign of −1 is ambiguous, but the plausible case is −1 0

Then 01 0 and 2 firms increase employment. The intuition is similar to the

previous example, with one difference. As before, both types of firms are willing to

employ workers of type 1 before the technology change, so here the analog of (33) is

(1 2) + 2 − 1 = (1 2)− 12

= (1 2)− 1Ψ2 (34)

But here 1 is the lowest skill level employed by firms of type 2 so (1 2)

Ψ2 and for sufficiently close to unity, (1 2) 2 In this case, the labor

productivity of a worker with skill 1 does not increase enough at firms with technology

2 for those firms to continue to compete for him. The price of their product has fallen

too sharply. Thus, workers with = 1 and in addition some with 1 separate

from firms of type 2 and instead go to firms with technology 1 The threshold 1

rises, partially offsetting the original pattern of output and price changes. But if the

range (1 2) is not wide and is large, then the inequality is reversed, and 1 falls.

22

All of the original workers at 1 firms get the same wage increase, while for workers

at 2 firms, the wage change is strictly increasing in

() = 1 0 ∈ (min 1 + 01)

() = 2 + ( 2) 0 ∈ [1 + 01 max)

In summary, technology improvements increase within-firm wage differentials at the

firm receiving the technology change, and leave them unchanged at other firms. They

increase or reduce between-firm wage differentials as the technology improvement

increases or reduces the between-firm technology differential.

Special case = 3 = 2.–

Suppose there are three types of firms, and those in the middle group get the

technology increase. Then there are two equations in (29) 01 and 02⎛⎝ −1

⎞⎠ =

⎛⎝ 1 1

1 2

⎞⎠⎛⎝ 01

02

⎞⎠

It is straightforward to show that

∆ ≡ 12 − 11 0

Recall that 0 and suppose −1 0 for the reasons described in the previous

example. Then

01 =1

∆(2−1 − 1) 0

02 =1

∆(1 − 1−1) 0

Firms getting the technology increase, those with 2, expand employment by adding

workers at both the upper and lower thresholds.

At firms with technologies 1 and 3 prices rise even more than in the short run,

since their own outputs fall,

=1

= 1 3

23

Hence wages for workers who remain at those firms rise by even more than in the

short run. Wages also rise for workers at 2 firms, and the proportionate changes are

increasing in Thus,

() = 1 ≤ 1

() = ( 2) + 2 ∈ (1 + 01 2 + 02)

() = 3 ≥ 2

Since wages are continuous, workers at 3 firms get the largest increases.

Special case = 3 = 1.–

Suppose there are three types of firms, and those in the lowest group get the

technology increase. In this case there are two equations in (29), for = 1 2 to

determine 01 and 02 Write (29) as⎛⎝

0

⎞⎠ =

⎛⎝ 1 1

1 2

⎞⎠⎛⎝ 01

02

⎞⎠

Since 0

01 =1

∆2 0

02 = − 1∆1 0

Firms getting the technology increase, those with 1 expand employment by adding

workers at both the upper threshold. Firms with technology 2 who are losing those

workers, add workers at the upper threshold.

General case.–

The general case involves the system of equations is = 0 where is a tridi-

agonal matrix, with rows (0 0 −1 0 0) Thus, 0 = −1 and we are

interested in the inverse matrix −1 That inverse has a recursive characterization.

[To be completed.]

24

C. An example

Figures RT1-RT3 display results for the earlier simulated economy. Here the top

5% of firms receive a 20% increase in their productivity, and the next 5% receives

smaller increases. For the smaller increases, () is a cubic function with the four

coefficients chosen to match the required level and derivative conditions at the end-

points. Because larger firms have higher employment, 35% of workers are directly

affected by a productivity increase at their own firm.

Figure RT1 shows the changes in technology, employment, labor quality, and out-

put, all as functions of the original technology level. Employment rises at firms

getting the full increase and at some of those getting a partial increase, and falls

for firm firms getting small increases or none. Since employment is shifting toward

higher-technology firms and the supply of human capital is unchanged, all firms ex-

perience a reduction in the human capital of their workforce. The change is zero

for firms at the top and bottom ends of the technology distribution, and is largest

for firms getting small technology increases. Output rises at firms getting the full

technology increase and at some of those getting a partial increase, and falls for firms

getting small increases or none.

The first panel of Figure RT2 displays the technologies used by workers, across

human capital levels, in the baseline economy and after the technology shift. Technical

change increases the type of technology used by every type of worker, but the shift

at the bottom end of the distribution is very small. The second panel shows wages

across worker types, before and after the technology shift. All workers gain: here the

rising tide lifts all boats. Workers at firms affected by the change gain significantly,

but gains for the rest are small.

Figure RT3 displays the wage gains in more detail. Panel (a) shows the size of

the wages gains, and panel (b) shows the wage CDFs before and after the technology

25

shift. The first panel shows that the wage increase is only about 2% for workers

at firms that are unaffected, but about 10% for workers at the top firm, increasing

approximately linearly across workers at firms getting technology increases. Panel

(b) shows that about 25-30% of the workforce gets a substantial increase.

5. INCENTIVES TO INVEST

Although the model here is static, we can nevertheless look at the incentives to in-

vest, on both sides of the market. This section looks at the incentive for an individual

firm to invest in a technology improvement, and for an individual worker to invest

in an increase in her human capital. For the firm, we will compare the increment to

profits from a technology improvement under two scenarios:

a. no change in size or quality of its workforce;

b. an optimal increase in the size and quality of its workforce.

The goal is to see how much limiting worker mobility reduces the gains from a technol-

ogy improvement. If the difference between (a) and (b) is substantial, then limiting

labor mobility reduces the incentive to invest. Call scenarios (a) and (b) SR and LR.

Suppose the firm’s original technology is and it increases to + In the

short run the firm’s costs do not change, but output rises and price falls. Suppose

that the firm’s labor force consists of skill types ∈ (−1 ) in proportion to theirrepresentation in the population. Then the proportionate change in output is

= Ψ ≡

1

Ψ

Ψ

where Ψ was defined in (12). Then since final output is unaltered, the demand

curve (3) implies

= −1

26

Hence the proportionate change in revenue is

= + =

− 1

Ψ

and the change in profit level is

∆ =− 1

Ψ = (− 1) Ψ

so the proportionate change is

= (− 1) Ψ

In the long run the firm adjusts the size and quality of its workforce. Minimal unit

cost now requires using only workers with skill The new price is a markup over

the new unit cost, and profits before and after are the usual share of revenue, 1.

The wage rate for workers with human capital is unchanged, so the proportionate

change in unit cost, and hence also in price, is

−( ) = −( )( )

=

The resulting proportionate change in revenue and hence also in profits is

= = +

= (− 1) ( )

Thus, the long run increase exceeds the short run increase by

− = (− 1)h( )− Ψ

i (35)

The only gain comes from concentrating its hiring in the most high-skill sunset of

its original workforce. If the interval (−1 ) is small, then the difference in (35) is

also small. This conclusion is another reflection of the envelop theorem. The gains

from higher worker productivity accrue in the short run, from higher output. The

27

additional gains from adjusting the composition of the workforce, thereby reducing

unit cost, are not of the same order. This conclusion reflects the fact that the original

choices for worker quality were optimal.

Similarly, for workers we can ask how much the incentives to invest depend on the

ability to move to a firm with a better technology. The question for workers is more

easily addressed if the distribution of technologies across firms is continuous. The

model is easily adapted to allow that. Even with this change, however, the issue is

slightly more clouded for workers. If a worker is “stuck” at his old firm, for whatever

reason, and the old firm knows this, the worker might not get any wage increase at

all. But suppose that (for some reason) the old firm pays him the usual share of his

revenue marginal product. Then an increase in his human capital by ∆ 0 implies

− =− 1

() { (+∆ )− ( )}

≈ − 1

()∆

− = (+∆)− ()

≈ 0 ()∆

= ()

∆

=− 1

()∆

where the last line uses the fact that price is a markup over unit cost. So the same

conclusion holds for workers: to a first-order approximation, labor immobility does

not reduce the incentive to invest in human capital. Of course, the important caveat

is that the old firm will raise his wage in line with his enhanced productivity.

6. MULTI-SECTOR MODEL

In this section the model is extended to include multiple sectors, each producing

an intermediate that is used in final goods production. The goal is to see if technical

28

change for a subset of firms can reduce wages for workers at firms with unchanged

technology. The main idea is as follows.

Suppose there are two sectors, and Initially sector has a preponderance of

firms with higher technology levels, and hence employs mostly higher-skill workers.

Sector has the reverse pattern. Consider technical change that affects some of the

firms in sector improving their technologies. The technical change reduces the

price of the intermediate produced by sector and raises the price of sector ’s

intermediate.

Consider first the short run, with labor immobile. In sector supply of the sectoral

intermediate rises and price falls, while in sector production of the intermediate

is unchanged and the price rises. In sector wages fall for workers at firms with

unchanged technologies, since the prices for their outputs fall. Wages rise at firms

that get the technical change, since the productivity increase more than offsets the

price cut. In sector wages increase at all firms, since productivity is unchanged

and the price of their sectoral intermediate increases.

Next consider the long run, with labor mobile. Some high-skill workers move from

sector to sector since there are now more firms in that sector with high technology

levels. Low-skill labor moves from firms with lower technologies in sector to firms

with moderate technology levels in sector and in sector But since most of the

firms in sector have high technologies, the low-skill labor that would like to move is

not very productive there. Thus, they may not be able to get back to their old wage

level.

A. Final good and intermediate technologies

The final good is produced by competitive firms using sectoral intermediates as

inputs, and sectoral intermediates are produced by competitive firms using differen-

tiated products as inputs. Both technologies are CES, but the sectoral technologies

29

have a (higher, common) elasticity of substitution.

There is a finite number of sectors, and the final good technology is

=

"X=1

(−1)

#(−1) 0 (36)

where 0 with Σ = 1 are sectoral weights, and is the elasticity of substitution.

The price of the final good is

=

"X=1

1−

#1(1−)

For = 1

=

Y=1

=

"Y=1

µ

¶#−1

In either case, demands for sectoral intermediates are

=

µ

¶

all (37)

We will assume that each sector has basically the same technology for producing

its intermediate, but we will allow the number of firms and the distribution of

technology levels across firms {}=1 to vary across sectors. Thus, the technologyfor each sectoral intermediate is

=

Ã

X=1

(−1)

!(−1)

all (38)

where the substitution elasticity 1 is common across sectors. The prices of

sectoral intermediates are

=

Ã

X=1

1−

!1(1−) all (39)

and demands for differentiated inputs are

=

µ

¶− all (40)

30

We will assume throughout that so differentiated goods within a sector are

more substitutable than sectoral intermediates.

Differentiated goods have the CES technology ( ) as before. For simplicity we

will take and to be the same across sectors.

Notice from (40) that for any differentiated good, demand is increasing in the

quantity of the intermediate being produced in its sector, the term and also

increasing in the price of the sectoral intermediate, the term . But and are

also linked through demand by final goods producers, in (37), where is proportional

to −1 Since this fact implies that a change in sectoral output has a

stronger effect on demand for differentiated goods in that sector through the price

channel than through the direct channel. That is, an increase in reduces price

so sharply that demand falls for a firm producing a differentiated input with

unchanged price That is, is proportional to

1− where 1

B. Wages, labor allocation and differentiated good supplies, prices

Next consider the labor allocation and output levels across firms producing dif-

ferentiated goods. Assume labor is mobile across sectors and let () denote the

wage function. The equilibrium labor allocation is described, as before by thresholds

{}=0 with firms hiring labor in the interval [−1 ) Since labor is mobileacross sectors, the thresholds do not depend on and the wage function is as in (5).

Also as before, the firm’s optimal price is a markup over its cost. Hence price is

as in (6), and does not depend on the firm’s sector. Hence (9) still holds, and needs

no sector subscripts.

The demand that the firm faces does depend on its sector, however, so output and

profits depend on the firm’s sector,

=

µ

¶− = 1 (41)

31

and the output relationships in (10) now hold within each sector,

+1

=

µ( +1)

( )

¶

all (42)

C. Equilibrium

Let be as before. Labor market clearing requires supply to equal the sum

of demand from all sectors, so (??) becomes

Z

−1()( ) =

X=1

= 1 (43)

where as before 0 = min and = max

Notice from (42) that equilibrium output levels have the form

+1

= +1 all

where

+1 =

µ( +1)

( )

¶

= 1 − 1

Hence

1= = 2 (44)

where the ratios act as a set of scaling factors. A firm in sector with technology

chooses an output level that is a scaled version of 1 where the scaling factor

depends on the sector but not on

Hence (43) can be written as

Z

−1()( ) = 1

X=1

= 1 (45)

Combining (42) and (45) we find that the analog of (11) to determine the thresholds

{}−1=1is

Z +1

()( +1) = 1+1

X=1

+1

32

= 1

µ( +1)

( )

¶ X=1

+1

orZ +1

()( +1) =

µ( +1)

( )

¶ Z

−1()( )

P

=1+1P

=1(46)

or( +1)

−P

=1+1Ψ+1 =

( )−P

=1Ψ

Since 1 = 1 for = 1 this is exactly as in (14). For 1 we must find the sectoral

factors {}=2 as well as the ’s.For any candidate values 10 and {}=2 solve (46) for the thresholds {}−1=1

Then use (44) to calculate {}2 all and use (38) to calculate sectoral intermedi-ates {} = 1 . The latter also involve and { } the size and technologydistributions for each sector. Finally, use (37) in (40), evaluated at 1 to find that

market clearing for sectoral intermediates requires

=

1

µ

1

¶

=

µ

1

¶µ

1

¶1− = 2 (47)

The fixed point requirement is that 10 and {}=2 must satisfy (46) and the − 1restrictions in (47).

D. An example

Figures MS1 - MS2 show results for a numerical example. All of the parameter

values, including the total number of firms, and {}=1 the economy-wide tech-nology distribution, from the one-sector example above are retained. Here the firms

are allocated between two sectors, with different sizes and technology distributions.

Sector 1 is high-tech, with a smaller number of firms 1 but a distribution {1} that

33

puts more weight on higher values for . Sector 2 is low-tech, with a larger number

of firms 2, but a distribution {2} that puts more weight on lower values for .The elasticity in the final goods sector is close to unity, and the two sectors get equal

weight in producing final goods,

= 101 1 = 2 = 12

Figure MS1 shows the number of firms by technology type (scaled pdfs) for each

sector. Sector 2 has the bulk of the low-tech firms, and has none of the firms at the

top of the distribution. In equilibrium, sector 1 produces more output and has a lower

price, but revenue = is the same in both sectors, as it should be for ≈ 1

1 = 134 1 = 191 1 = 0459 1 = 878

2 = 367 2 = 161 2 = 0545 2 = 876

= 5 = 176 = 1

Figure MS2 shows output and employment. The first panel shows employment per

firm, by technology level and sector. Firms with better technologies employ more

workers, as before. Here, firms in (the low-tech) sector 2 employ more workers than

those in (high-tech) sector 1. This is a direct consequence of the higher relative price

for their sectoral output, 2 1 Output per firm, shown in the second panel, has

the same pattern. Total employment and output, in the third and fourth panels, also

reflects the number of firms of each type in each sector, (). In sector 1, total

employment and output are increasing in technology level. In sector 2, employment

is concentrated at the bottom and output is from the bottom and middle of the

distribution.

Figure MS3 shows the division of total employment between sectors, across human

capital levels. All of the workers at the top of the distribution are in (high-tech)

sector 1, and most of those at the bottom are in (low-tech) sector 2. The middle

range is split between the two.

34

Next consider the effect of technical change that improves the productivity of some

firms in (low-tech) sector 2. In particular, suppose the change improves technologies

for middle-level firms, leaving technologies at the low end unchanged. This change

will reduce the relative price 2 of the aggregate produced in sector 2. The question

is whether this will reduce wages at the low end of the human capital distribution.

Most of these low-skill workers are in sector 2. There are few employment options

for them in sector 1, since most of those firms have good technologies, jobs for which

they are unsuited.

Figure RTMS1 shows the shift in the pdf for the sector 2 technologies. The new

pdf moves some firms from middle-range values to the high end. The total number

of firms is unchanged, as are the rest of the two distributions.

Figure RTMS2 shows the changes in sectoral and total employment, by firm type.

These changes are dominated by the shift in the distribution of firm types in sector

2, but there is also a significant decline in employment at the lower end of the type

distribution in sector 2.

The technology change improves the distribution of firm types. For firms with

unchanged technologies, this makes their situation worse. For most workers, their

situation is improved. These effects are shown in the next two figure.

Figure RTMS3 shows the changes, across firm types, in output price, wage paid,

labor productivity, and human capital level employed. Note that these are common

to both sectors. The technology change increases the demand for workers with higher

human capital. But labor supplies have not changed, so firms of every type have to

employ workers with lower human capital, as shown in green. Hence labor productiv-

ity also falls at firms of every type, as shown in blue. Wages also fall at almost every

type of firm, shown in black. The only exception is a small set of firms at the top

of the distribution, where output price rises enough to outweigh the decline in labor

productivity.

35

Figure RTMS4 looks at the effects from the perspective of workers. The first panel

shows the shift in the technology levels at which various worker types are employed.

Since the distribution of human capital has not changed, all workers end up employed

at firms with higher technologies. For workers at the bottom of the skill distribution,

the change is very small, however.

The second panel shows log wage changes. The change is positive over most of

the range, but is negative at the bottom. Since the distribution of human capital is

lognormal with a mean of unity, about half of the workforce gets no gain.

Figure RTMS5 shows the effect of the technology shift on the sectoral allocation

of workers, by skill level. (High-tech) sector 1 gains some workers with lower human

capital, because of the higher sectoral price. But they lose some workers with higher

human capital, as those workers move to firms in (low-tech) sector 2 that benefitted

from the technology change.

7. CONCLUSION

Wage inequality has displayed large and long-lived shifts over the last century (see

Katz and Goldin (2007). Recent increases in wage inequality are probably related, at

least in part, to changes in technology. To guide policies targeted toward inequality,

for example, we need better models connecting wages with the distributions of tech-

nology and skills. Similarly, to guide policies to promote long run growth, we need

to understand the interaction between incentives to invest in technology and skills.

36

REFERENCES

[1] Akerman, Anders, et. al. 2013. Sources of wage inequality, American Economic Re-

view: Papers and Proceedings, 103(3): 214-219.

[2] Alvarez, Fernando E., Francisco J. Buera, and Robert E. Lucas, Jr. 2013. Idea flows,

economic growth, and trade, NBER Working Paper 19667, National Bureau of

Economic Research, November.

[3] Alvarez, Fernando E., Francisco J. Buera, and Robert E. Lucas, Jr. 2008. Models of

idea flows, NBERWorking Paper 14135, National Bureau of Economic Research,

June.

[4] Bailey, Martin Neil, Charles Hulten, David Campbell, Timothy Bresnahan, and

Richard Caves. 1992. Productivity dynamics in manufacturing plants, Brookings

Papers on Economic Activity. Microeconomics, 187-267.

[5] Bartelsman, Eric J. and Mark Doms. 2000. Understanding productivity: lessons from

longitudinal microdata, Journal of Economic Literature, 38(3): 569-594.

[6] Card, David, Jorg Heining, and Patrick Kline. 2013. Workplace heterogeneity and the

rise of West German wage inequality. Quarterly Journal of Economics, 128(3),

967-1015.

[7] Costinot, Arnaud. 2009. An elementary theory of comparative advantage, Economet-

rica 77(4): 1165-92.

[8] Costinot, Arnaud, and Jonathan Vogel. 2010. Matching and inequality in the world

economy, Journal of Political Economy, 118(4): 747-786.

[9] Deaton, Angus, and Christina Paxson. 1994. Intertemporal choice and inequality,

Journal of Political Economy, 102(3): 437-467.

37

[10] Dunne, Timothy, Mark J. Roberts, and Larry Samuelson. 1989. The growth and

failure of U.S. manufacturing plants, Quarterly Journal of Economics, 104(4):

671-698.

[11] Faggio, Giulia, Kjell Salvanes, and John Van Reenen. 2007. The evolution of inequality

in productivity and wages: panel data evidence, NBER Working Paper No.

13351.

[12] Foster, Lucia, John Haltiwanger, and Chad Syverson. 2008. Reallocation, firm

turnover, and efficiency: selection on productivity or profitability? American

Economic Review, 98(1): 394-425.

[13] Goldin, Claudia, and Lawrence F. Katz. 2008. The Race between Education and Tech-

nology, Harvard University Press.

[14] Hakanson, Christina, Erik Lindqvist, and Jonas Vlachos. 2015. Firms and skills: the

evolution of worker sorting, working paper.

[15] Hsieh, Chang-Tai, and Peter J. Klenow. 2014. The life cycle of plants in India and

Mexico, Quarterly Journal of Economics, forthcoming.

[16] Jovanovic, Boyan. 1998. Vintage capital and inequality. Review of Economic Dynam-

ics, 1: 497-530.

[17] Lagakos, David, Benjamin Moll, Tommaso Porzio, and Nancy Qian. 2013. Experience

matters: human capital and development accounting, working paper.

[18] Lucas, Robert E. 1988. On the mechanics of economic development, Journal of Mon-

etary Economics, 22: 3-42.

[19] Lucas, Robert E. 2009. Ideas and growth, Econometrica, 76: 1-19.

38

[20] Lucas, Robert E., and Benjamin Moll. 2011. Knowledge growth and the allocation of

time, NBER Working Paper 17495, National Bureau of Economic Research.

[21] Machin, Stephen, and John Van Reenen. 2007. Changes in wage inequality, Center

for Economic Performance Special Paper No. 18.

[22] Neal, Derek, and Sherwin Rosen. 2000. Theories of the distribution of earnings, Chap-

ter 7. Handbook of Income Distribution, ed. by A.B. Atkinson and F. Bour-

guignon, Elsevier Science.

[23] Perla, Jesse, and Chris Tonetti. 2014. Equilibrium imitation and growth, Journal of

Political Economy, 122(1): 52-76.

[24] Perla, Jesse, Chris Tonetti, and Michael Waugh. 2014. Equilibrium technology diffu-

sion, trade, and growth, working paper, U. of British Columbia.

[25] Rossi-Hansberg, Esteban, and Mark L. J. Wright. 2007. Establishment size dynamics

in the aggregate economy, American Economic Review 97(5): 1639-1666.

[26] Song, Jae, David J. Price, Fatih Guvenen, Nicholas Bloom, and Till von Wachter.

2015. Firming Up Inequality, working paper.

39

-1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

2.5Fig A1. labor productivity (h,x), various x-values, log-log scale

ln(h)

ln((h,x))

(blue) x = 5.5537

(red) x = 3.2657

(black) x = 1.7534

(green) x = 1.147

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-1

-0.5

0

0.5

1

1.5

2

2.5

3Fig A2: allocation of workers to firms, log-log scale

ln(x)

ln(h

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21

2

3

4

5

6

7elasticity of labor allocation h(x)

ln(x)

employment-weighted average elast = 1.8379

1 2 3 4 5 6 7 8

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Fig A3: unit cost and price, pF = 1

1 2 3 4 5 6 7 80.8

1

1.2

1.4

1.6

1.8

2

2.2employment per firm, log10 scale

1 2 3 4 5 6 7 80.5

1

1.5

2

2.5

3

3.5output per firm, log10 scale

1 2 3 4 5 6 7 80.5

1

1.5

2

2.5

3revenue per firm, log10 scale

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4Fig A4: distribution of firms, by x-type

x

1 2 3 4 5 6 7 80

20

40

60

80

100

120employment per firm, by x-type

x

1 2 3 4 5 6 7 80

2

4

6

8

10total employment, by x-type

x

0.8 1 1.2 1.4 1.6 1.8 2 2.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Fig A5: firm size (empl), right CDF

employment (log10)

0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1firm size (sales), right CDF

revenue (log10)

= 6 F = 1.04

= 0.5 = 0.5

h = 1

h = 1

hmin = 0.4 hMax = 15

xmin = 1 xMax = 8

-1 -0.5 0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5Fig A6: wage function and CE function, log-log scale

ln(h)

wage function in red

CE function in dotted black

-1 -0.5 0 0.5 1 1.5 2 2.5 3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75elasticity of wage function

ln(h)

employment-weighted average elast = 0.55029

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Fig A7: CDF for relative HC and relative wages

relative human capital in blue

relative wages in red

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35pdf for relative HC and relative wages

wage distr. is more compressed

each has mean = 1

-0.5 0 0.5 1 1.5 2 2.5-1

-0.5

0

0.5

1

1.5

2

2.5FigHF1: (a) labor prod. (ln-ln scale)

ln(h)

HF in red, baseline in blue

dotted lines are means

ln()

ln(HF)

-0.5 0 0.5 1 1.5 2 2.5

2

2.5

3

3.5

4

4.5

(b) ln employer size

ln(h)

ln(l)

ln(L/N)

-0.5 0 0.5 1 1.5 2 2.5

-1.5

-1

-0.5

0

0.5

1

1.5

(c) interm. good price (ln-ln scale)

ln(h)

ln(pHF)

ln(pj)

-0.5 0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5(d) wages (ln-ln scale)

ln(h)

ln(w)

ln(wHF)

-1 -0.5 0 0.5 1 1.5 2 2.5 3-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1Fig HF2: wage changes across workers (ln-ln scale)

ln(h)

ln(w)

-0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CDF for ln(wage)

ln(w)

HF in red, baseline in blue

dotted lines are means

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

1.5

2

2.5

3Fig HL1: labor productivity (ln-ln scale)

ln(x)

green is HL

blue is baselineln(HL)

ln()

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

2

3

4

5

6employment per firm (ln-ln scale)

ln(x)

ln(el)

ln(elHL)

-0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Fig HL2: firm size (empl), right CDF

green is HL

blue is baseline

employment (log10)

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1firm size (output), right CDF

output (log10)

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2Fig RT1: change in log technology

ln(x)

20% increase in x

for top 5% of firms

smaller increases

for next 5% of firms

0 0.5 1 1.5 2-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

change in log labor quality

ln(x)

0 0.5 1 1.5 2-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25change in log employment

ln(x)

35% of workers are

directly affected

0 0.5 1 1.5 2

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

change in log output

ln(x)

final output increases

by 4.7 %

-1 -0.5 0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5Fig RT2: technology across human capital types

ln(h)

ln(x

)

blue is baseline

red is after tech. change

-1 -0.5 0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5wages (ln-ln scale)

ln(h)

ln(w

)

-1 -0.5 0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Fig RT3: log wage changes

ln(h)

ln

(w)

35% of workers

directly affected

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CDF of ln real wages

before in blue

after in red

ln(w)

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6MS1: distr of firm types, bys sector

pdf

sector 1 is high-tech

sector 2 has more firms

n2v

n1v

N1 = 1.34, N2 = 3.67

Y1 = 191, Y2 = 161

P1 = 0.46, P2 = 0.55

1 2 3 4 5 6 7 80

20

40

60

80

100

120

140Fig MS2: employment per firm

sector 2 (blue)

is low-tech

P2/P1 = 1.1886

1 2 3 4 5 6 7 80

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08total employment

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200output per firm

1 2 3 4 5 6 7 80

0.02

0.04

0.06

0.08

0.1total output

0 5 10 150

0.05

0.1

0.15

0.2

0.25

Fig MS3: distribution of workers across sectors

h

green, total

red, sector 1

1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25Fig RTMS1: pdfs for sectoral technologies

x

red is sector 1

blue is sector 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-10

-8

-6

-4

-2

0

2

4

6

8x 10-3 Fig RTMS2: change in sectoral and total employment

ln(x)

red, sector 1

blue, sector 2

green, total

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05Fig RTMS3: changes in price, labor prod, wage, HC type

ln(x)

log

chan

ge

red, output price

black, wage

blue, labor prod, total

human capital level

-1 -0.5 0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5Fig RTMS4: technology across human capital types

ln(h)

ln(x

)

baseline in black

w/ change in green

-1 -0.5 0 0.5 1 1.5 2 2.5 3-0.01

0

0.01

0.02

0.03

0.04

0.05log wage changes

ln(h)

ln

(w)

0 5 10 150

0.05

0.1

0.15

0.2

0.25

Fig RTMS5: distribution of workers across sectors

h

green, total

red, sector 1