VIII. Supply effects and induced bias in innovation

Induced Bias

• Traditionally, one assumes that TP enters the production function in some exogenous way

• What if innovators can ex-ante seek to obtain a given type of technical progress?

• That is what the old “induced bias” literature tries to model

The traditional view

• Innovation augmenting one factor increases the return to the other factor

• Because of such “bottlenecks”, further innovation will be biased in favor of the other factor

• We eventually expect innovation to be “balanced”

• These bottlenecks imply that if one factor is more abundant, innovation will favor the other factor, to compensate

The modern view (Acemoglu)

• There exists market size effects: profits from an innovation depend positively on its market size

• Market size is larger if factor affected by innovation is more abundant

• This effect tends to reinforce existing biases rather than compensate them

Consequences for the distribution of wages

• If market size effects dominate bottleneck effects, then an increase in H/L triggers skilled-biased innovations

• These innovations themselves increase the skill premium,

• Which in turn induces people to accumulate more human capital…

A simple heuristic argument

• Suppose firms could “pay” for technical progress in one dimension or another

• How much are they willing to pay?

• That gives us an idea of where TP is going to take place

• Firms willing to pay more more profits for innovators

The logic

• Production function

• Marginal product conditions

• Cost function

• Willingness to pay per unit of output

The equilibrium MWP for technical progress

• Diferentiating cost functions and substituting MP conditions, we get

Interpretation

• Because of optimality, how I reduce costs is irrelevant at the margin

• Suppose, upon an increase in A, that I reduce L proportionally

• I produce the same, and costs are reduced by wLdA/A = F’1LdA

What happens to MWP when factor endowments change?

• Assume H goes up

• The wage of human capital falls, thus reducing my savings from human capital augmenting technical progress (bottleneck effect)

• But a proportional reduction in H would affect more people, and thus be more profitable (market size effect)

Implications for the induced bias

• Assume the bias in TP adjusts so as to equate MWP across types of TP

• Then the endogenous bias is determined by

• That formula determines the endogenous skilled bias b = B/A as a function of the factor endowment h = H/L

Which effect dominates?

• db/dh > 0 if curvature of f not too large

• Bottlenecks are small if H and L are substitute, market size effects then dominate

• Botlleneck effects are large if complementarities between H and L strong enough

The CES example

Net effect on the skill premium

• While an increase in H/L induces SBTC, skill premium need not go up on net

• For skill premium to go up, the positive effect of induced technical change must be higher than the direct negative effect of a greater H/L

The net effect on the skill premium

• To get an increase in the skill premium, we need a more than proportional response of b to h

• That implies even less complementarity

Computing the skill premium in the CES case

An endogenous innovation model

• Can the preceding analysis be made more rigorous by explicitly taking innovation into account?

• The answer is: yes

• And the analysis and intuition are basically similar

The model’s ingredients

• We need a simple model of endogenous innovation

• Building on Romer, we assume innovation introduces new varieties

• These varieties enter as inputs into the production of aggregate intermediate goods

• Two categories of varieties depending on whether H or L is used

• Two different aggregate intermediate goods

From product diversity to factor productivity:

• As in Romer, the CES aggregate for the intermediate input entails « taste for diversity »

• Hence, an increase in the number of varieties is equivalent to technical progress which increases the efficiency of the relevant factor

The production structure

• Aggregate production function:

• l-aggregate uses l-inputs:

• Similarly for h:

• l-inputs use labor

• Similarly for h

Computing aggregate factor productivity

• By symmetry, all goods of the same kind are produced in the same quantity

• Therefore, yl = L/Nl and yh = H/Nh

• Hence, Yl = AL, Yh = BH, Y = F(AL,BH)

Pricing

• Individual price-setters face constant elasticity

• Using price index for intermediate aggregates, we get

• Profit maximization for the final good allows to recover wages

Profits

• In equilibrium, labor uniformly allocated

• This allows to compute quantities, and thus profits

Patents and innovation

• The value of a patent is given by

• In an interior BGP, Vh = Vl throughout, implying

Profit equalization allows to compute equilibrium bias in

technology

• We get a formula quite similar to the heuristic model:

The effects

• When a factor is in larger supply, any intermediate good which uses that factor will have a larger market (Market size effect) term in H/L

• If a factor is more abundant in efficiency units, its marginal product falls, which reduces the demand for the corresponding intermediate inputs (Bottleneck effect) term in F’1/ F’2

• If a factor is more productive, it means more intermediate goods for that factor, and lower market size term in (B/A)1-η

• If a factor is more productive, its price is higher, which boosts prices and profits for intermediate goods term in (B/A)

The CES case

• Relative profits are equal to

• That determines a stable interior value of b provided

• That value is then given by

When does the supply effect increase the bias?

• Clearly, db/dh > 0 iff γ > 0

• Substitutability market size effects dominate

• Complementarity bottleneck effects dominate

When does the skill premium go up on net?

• The skill premium is given by

• It is increasing in h provided

Discussion

• Again, more substitutability is needed for the SP• The bias must react enough to h• Furthermore, condition more likely when η is

smaller• η is smaller b more reactive to h

– The lower η, the lower the increase in N associated with a given increase in A, the smaller the profit dissipation effect associated with an increase in A

Dynamics

• Assume a fixed supply of researchers• They can pick any kind of innovation at

any point in time• Assume a productivity spillover à la

Grossman-Helpman• At each date, all researchers work in the

most profitable kind of R & D• Unless patent values are equalized, they

are then indifferent

Aggregate research dynamics

The allocation of R & D

db/dt = 0

dΔ/dt = 0

C

b

Δ

Figure 5.7: the dynamics of the technology bias

E

E

D

D

b =B/AFigure 5.8: convergence path of the technology bias

b*

db/dt = 0

dΔ/dt = 0

C’

b

Δ

Figure 5.9: response of the technology bias to an increase in H/L

C

A

B

time

time

Figure 5.10: response of the skill premium to an increase in H/L

b

b

IX. The bundling model

The limitations of neo-classical models

• In neo-classical models, individuals are irrelevant

• They earn the sum of the income of the characteristics they bring to the market

• Where they work and whom they work with does not matter

• We now turn to models where individuals matter

The role of unbundling

• A first mechanism by which individuals may matter is unbundling

• Unbundling means that all the characteristics of the individual must be supplied to the same employer

• We will show that the price of each characteristic then need not be equal across sectors

• One implication is that people will sort themselves into different sectors by different skills

Back to the basic model

• Each worker has a skill s

• Skill determines h(s) and l(s)

• We order skills by comparative advantage so that h(s)/l(s) grows with s

• Workers can’t elect which characteristic they supply

A pseudo-obvious result

• If there is a single homogeneous final good, then each worker earns an income

• Where ω and w are the economy-wide price of H and L

• In the unbundling model, that result is obvious• But is is not in the bundling model• We have to prove that each firm offers the

same return to each characteristic

The firm’s optimization problem

• To complete the proof we need to show that these MPs are equalized across firms• Workers are paid the marginal roduct of their characteristics in the firm where they work

Completing the proof

• In equilibrium, all firms have the same H/L ratio

• Therefore, each marginal product is equalized across firms

• Otherwise, firms with a higher H/L pay more for L and less for H

• But then they attract lower-skilled workers and cannot have a higher H/L ratio

The 2-sector model

• Firms in different sectors sell their output at different prices

• A non unique price may be supported

• Example: sector 1 pays more for H and attracts the higher skilled workers

• It does so not because it has a lower H/L ratio, but because its production is more intensive in H

The model

• Under unbundling, the allocation is determined by standard considerations

• The two FPF interact if both goods are produced

• Each factor price is unique

wFigure 6.1: factor-price equalization in the two-sector model

PFPFA

ω

PFPFB

E

-L/H

wFigure 6.2: full specialization under unbundling

PFPFA

ω

PFPFB

E

-L/H

Can the bundling allocation be an unbundling equilibrium?

• A necessary condition is that there exist an allocation of people which matches H and L in both sectors

• Because people come to a sector with their own endowment of both h and l, an arbitrary allocation is not necessarily feasible

• A feasible allocation must be between the minimal and maximal human capital intensity curves

L

H

Figure 6.3: the allocation of labor and human capital

LBLA

HA

HB

MM

MM’

E

The “lens”

• It defines the set of macro allocations of H and L such that their exists a micro allocation of individuals which yields that macro allocation

• The lowest possible H/L ratio in sector A is obtained by allocating the lowest skilled people there first

• If I want to allocate more labor to A, I must move more skilled people there, and the minimum H/L ratio goes up

The determination of MM and MM’

Equilibrium and the lens

• Any equibrium must lie in the lens, otherwise it cannot be supported by an allocation of people

• Any point in the lens can be supported by an allocation of people

• If the unbundling equilibrium is in the lens, it is also an equilibrium with bundling– Neither firms nor workers have an incentive to

deviate

If bundling is outside the lens:

• Factor prices can’t be equalized across sectors, otherwise one would be outside the lens

• Assume B is more H-intensive• Sector B pays more for H and less for L

than sector A• Workers below a critical skill level work in

A, workers above it work in B sorting• Equilibrium thus lies on MM

L

H

Figure 6.4: Equilibrium when E is outside the lens

LBLA

HA

HB

MMMM’

E

E’

O

O’

Uniqueness?

• As I move up along MM, the marginal worker’s income goes up faster in B than in A:– The marginal worker has more H, which is more

valued in B– Fewer people work in B, whose relative price goes up

• Therefore, at most 1 point where it is the same in both sectors

• If that point does not exist, corner equilibrium exists where everybody works in 1 sector

If E is in the lens, it is the only equilibrium

• Any point F on MM has a lower H/L ratio in A than E

• Its H/L ratio is higher in B

• Thus it has a lower ωB/wB,and a higher ωA/wB,than E

• But then the least skilled want to work in B, and the most skilled in A not an equilibrium

h(s)/l(s)

z(s)/l(s)

Sector B

Sector A

Figure 6.5: The distribution of income under full specialization

The impact of an increase in demand for good b

• Under FPE, we get the usual prediction: ω goes up and w goes down

• Under non FBE, same prediction, but we must move along MM– The effect on wages not only depends on

technology but also on the distribution of skills– If h/l varies little across people, that puts limits

on the inegalitarian effects

L

H

Figure 6.6: The effect of an increase in p under full specialization

MM

MM’

E’

E’’

Triggering segregation

• An increase in the demand for B may move the economy outside the lens

• Sectoral skill segregation emerges• As B bids for workers, it tends to drive ω/w up• The least skilled of B move to A, and the most

skilled of A move to B• At some point, B is no longer able to match its

H/L target, it attracts workers with a lower H/L, and offers a higher ω/w

• An opposite phenomenon occurs in sector A

L

H

Figure 6.7: An increase in p may trigger full specialization

MM

MM’

E

E’’

F