CAPITULO V: DISCUSION
5.1 Contrastación de los Resultados
5.1.2 Contrastación de los resultados con los referentes bibliográficos
This section describes aspects of the benchmark model omitted from the main body of the paper.
A.1
Industry equilibrium
The production of the consumption good is perfectly competitive and requires a mixture of industry-specific labor Lit and intermediate goods Xit according
to Yt≡ ˆ 1 0 Yitdi, Yit ≡(AitLit)1 −α Xitα, 0< α <1.
Consumption good producers maximize profit taking as given the interme- diate good prices Pit and wages Wit (consumption is the numeraire)
max Xit,Lit Yt− ˆ 1 0 (PitXit+WitLit)di,
which implies equilibrium expressions for the industry-specific wage and an inverse demand curve for intermediate goods
WitLit = (1−α)Yit, Pit =α AitLit Xit 1−α . (A.1)
In equilibrium, Lit =Lsince the supply of industry-specific labor is inelas-
tic.
An investment ofIitconsumption goods attyieldsXi,t+1 =Iitintermediate
goods at t+ 1. The interest rate is 1 +rt+1 (for now, we consider the general case). The intermediate good is produced by a monopolist, who understands equation (A.1). At the time of production, the good’s quality Ai,t+1 is pre- determined. The monopolist chooses Iit to maximize discounted profit
Πi,t+1 ≡ Pi,t+1Xi,t+1−(1 +rt+1)Iit 1 +rt+1 = Pi,t+1 1 +rt+1 −1 Iit,
which in equilibrium implies (after substitution of the inverse demand curve)
Pi,t+1
1 +rt+1
= 1
α,
and investment and profit are linear functions of productivity Ai,t+1:
Iit= α2 1 +rt+1 1−1α Ai,t+1L, (A.2) Πi,t+1 = (1−α)α 1+α 1−α (1 +r t+1) −1 1−α A i,t+1L. (A.3)
The contribution of each industry toward the production of the consump- tion good is driven by the scale of production and technological change. The marginal contribution of each industry at t is
Yit= (AitL)1 −α (Ii,t−1)α = α2 1 +rt 1−αα AitL, (A.4)
and the industry wage
Wit= (1−α)Yit L = (1−α) α2 1 +rt 1−αα Ait. (A.5)
A.2
Aggregation
The linearity of the industry-specific equations and the convenience of the no- tation adopted make aggregation simple. Aggregate (and average) investment and output are
It = α2 1 +rt+1 1−1α At+1L, Yt= α2 1 +rt 1−αα AtL, where At= ´1
0 Aitdi. The average wage is
Wt= (1−α) α2 1 +rt 1−αα At.
It is straightforward to reformulate these aggregate variables in productivity- adjusted terms: it(zH,zL) = It At = α2 1 +rt+1 1−1α (1 + (γ−1)µ(zH,zL))L, yt = Yt At = α2 1 +rt 1−αα L, wt = Wt At = (1−α) α2 1 +rt 1−αα ,
where µ(zH,zL)was defined in Section 2.3. These normalised variables do not
depend on t once we assume a zero risk-free interest rate.
A.3
Condition for a zero risk-free interest rate
The assumption of linear preferences with a zero discount rate ensures that, if r = 0, young agents are willing to lend (including equity to themselves) up to their entire net income, (1−τw)wL. For the interest rate to be zero
in equilibrium, total demand for credit (both equity and external borrowing),
i(zH, zL) +z, must be lower than (1−τw)wL.
Assumption 3 requires that the minimum labor tax needed to implement the first best without taxing consumption, τw = wL1 [s−τπ∗π] (where τ
∗ π < 0)
be small enough, so thati(zH∗, zL∗) +12zH∗ +12zL∗ <(1−τw)wL. The assumption
boils down to the requirement that |τπ∗| and s be not too large.
Although the minimum labor tax needed to finance τπ,pool∗ without taxing consumption, τw = wL1 [s− τπ,pool∗ π], is lower than
1
wL[s−τ ∗
ππ], Assumption
3 is not enough to ensure implementability of the constrained first best, and Assumption 4 is needed. This is for two reasons. First, implementation of the constrained first best imposes an additional lower bound on the labor tax, τw. If τw ≥ wL1 [s−τπ∗π], implementation of the constrained first best
requires a higher labor tax (and lower credit supply) than implementation of the first best. Second, one cannot rule out that total credit demand is higher at the constrained first best than at the first best, i.e. ipool z∗
pool
i(zH∗, zL∗) + 1 2z ∗ H + 12z ∗ L.
Finally, we show that the tax rates mentioned in Assumption 4 always exist, so that the constrained first best is always implementable. Leteτw,pool = wL1 [s− τπ,pool∗ π]be the rate of labor taxation required to financeτπ,pool∗ without taxing consumption. There are two possible cases. If eτw,pool > τw, then the vector
τπ,pool∗ ,τew,pool,0
0
implements the constrained first best (since it satisfies the definition ofτpool∗ ) without taxing consumption. Ifeτw,pool ≤τw, then the vector
τπ,pool∗ , τw+,−(τw+−eτw,pool)
wL c
0
, with arbitrarily small, implements the constrained first best (since it satisfies the definition ofτpool∗ ) without taxing consumption (since it implies a subsidy on consumption).
A.4
Discussion of Assumptions
The model has a few assumptions that deserve discussion, namely (1) technol- ogy enters as a variable in the production of consumption goods rather than intermediate goods, and (2) labor is industry-specific yet is not an input into intermediate good production.
In most models of imperfect competition, labor is not an input in the consumption good production function. By modeling innovations as labor productivity-augmenting changes in the consumption good production func- tion, the optimal investment decision of the monopolist becomes linear in technology. The linearity of the model substantially simplifies the aggregate and dynamic properties of the economy. For a textbook treatment of this technique, see ?.
The assumption of industry-specific labor is very important for tractability considerations. In a model with financial market imperfections, entrepreneurial net-worth is an important determinant of investment in technological research. In a common labor market wherein all entrepreneurs are born with equivalent labor endowments, entrepreneurs earn equivalent labor incomes. On the other hand, since innovation profits are linear in productivity and an innovating entrepreneur has monopoly power, the benefit to research is larger for en- trepreneurs in high versus low productivity industries. Thus, entrepreneurs with low net-worth in high productivity industries should be more constrained
than entrepreneurs with high net-worth in low productivity industries. Since net-worth is equivalent to labor income in the model, and labor income is equivalent across entrepreneurs in a common labor market, indeed entrepreneurs would be heterogeneously affected by financial market imperfections if we as- sumed a common labor market. To avoid this channel, which is not of first order importance to the themes of this paper and hence not worth the added complexity, we assume that the labor market for each industry is fragmented. In this way, net-worth scales with the gains from research in any given indus- try. One last assumption is crucial to ensure that both the financial market imperfections symmetrically affect entrepreneurs and that the model has nice convergence properties: the contribution of some fixed research effort to the probability of innovation decreases inverse-linearly with the productivity of the entrepreneur’s industry. Otherwise, an entrepreneur whom did not face suffi- ciently steep increasing costs to research would expand his research effort until his probability of innovation approached one (at infinity), whereas he would invest a negligible share of net-worth on research, implying that the financial market imperfections would have no effect in steady state. If the increase in research costs were too steep, the entrepreneur’s desired research effort would approach zero, and there would not exist a [positive] balanced growth path.
B
Proofs
We begin by establishing a few preliminary results, and we then move to the proofs of the lemmas and propositions in the main text.
B.1
Preliminary results
Theorem 1. If zˆH(τπ)≤(1−τw)w, there only exists a separating equilibrium
in which zJ = ˆzJ(τπ) (any combination of equity and external financing being
possible).
Proof. Since the opportunity cost of equity financing is zero, the high types have enough net worth, and the minimum rate they can be offered on external financing is a 1
Hµ(z), the high types would never select a research effort different
from zˆH(τπ). Furthermore, they would never take on external financing at
a rate greater than a 1
Hµ(z). This last fact implies that a pooling equilibrium
does not exist. As argued in footnote 10, at any separating equilibrium, the low types must select zˆL(τπ). Then, there only exists a separating equilibrium
in which zJ = ˆzJ(τπ). If an entrepreneur of type J borrows any money at
such equilibrium, this must be at a rate a 1
Jµ(ˆzJ(τπ)). Then, the entrepreneur
is indifferent as to the amount borrowed, and any combination of equity and external financing is possible.
The following results focus on the case zˆH(τπ)>(1−τw)w.
Theorem 2. Let zsep and zsep be the minimum and maximum solutions to
g
npvL(z) =npvL(ˆzL), (B.1)
or, if no solutions exist, zsep =zsep = ˆzH. Then, for (1−τw)w <zˆH:
1. It iszsep >(1−τw)wif(1−τw)w6= ˆzL, zsep = (1−τw)wif(1−τw)w= ˆzL.
2. As(1−τw)wincreases from 0toω ∈(ˆzL,zˆH), zsep continuously increases
from a value in (0,zˆL) to zˆH, while zsep continuously decreases from a
value above zˆH to zˆH; they are both constant in (1−τw)w and equal to
ˆ
3. There exists zˇ ∈ (ˆzL,zˆpool), such that npvH(zsep) 5 npvpoolH (ˆzpool) ⇐⇒ zsep 5zˇ.
4. npvH(z
sep)5npvHpool(ˆzpool) ⇐⇒ npvL(ˆzL)5npvLpool(ˆzpool).
Proof. Recall g npvL(z) = aLµ(z)γπ(1−τπ)− aL aH z− aH −aL aH (1−τw)w, npvL(ˆzL) = aLµ(ˆzL)γπ(1−τπ)−zˆL, and npvL(ˆz L) is a positive number. npvg L
(z) is a concave function of z, which is non-positive atz = 0, reaches a unique maximum atz = ˆzH, and is negative
forzlarge enough. It also decreases linearly in(1−τw)w. It is easy to see that,
if (1−τw)w = 0, it is npvg L (ˆzL) > npvL(ˆzL), implying npvg L (ˆzH) > npvL(ˆzL); if (1−τw)w = ˆzL, it is npvg L (ˆzL) =npvL(ˆzL), implying npvg L (ˆzH)> npvL(ˆzL); and if (1−τw)w = ˆzH, it is npvg L
(ˆzH) < npvL(ˆzL). Then, there exists ω ∈
(ˆzL,zˆH) such that, if (1− τw)w < ω, equation (B.1) admits two solutions,
0 < zsep < zˆH < zsep < ∞; if (1−τw)w = ω, it admits only one solution, zsep = zsep = ˆzH; and if (1−τw)w > ω, it admits no solutions (which, by
definition, still implies zsep = zsep = ˆzH). Furthermore, zsep is smaller than
ˆ
zL for (1−τw)w = 0, increases with (1−τw)w and goes to zˆH as (1−τw)w
goes to ω; zsep is bigger than zˆH for (1−τw)w= 0, decreases with (1−τw)w
and goes to zˆH as (1 −τw)w goes to ω. Point 2 follows. Next, note that
g npvL((1−τw)w) = npvL((1 −τw)w). Then, npvg L ((1−τw)w) < npvL(ˆzL) if (1−τw)w6= ˆzL,npvg L ((1−τw)w) = npvL(ˆzL)if(1−τw)w= ˆzL, Point 1 follows
from the fact that (1−τw)w < zˆH, and npvg
L
(z) reaches a maximum at zˆH.
Next, the expression
npvH(zsep)−npvpoolH (ˆzpool) (B.2)
is continuously increasing inzsep. To see this, start fromnpvg
L
(zsep) =npvL(ˆzL).
After multiplying both sides by aH
aL and re-arranging, this can written as
aHµ(zsep)γπ(1−τπ)−zsep =aHµ(ˆzL)γπ(1−τπ)− aH aL ˆ zL+ aH −aL aL (1−τw)w (B.3)
where the LHS is equal to npvH(z
sep). Then, (B.2) can be written as
aHµ(ˆzL)γπ(1−τπ)− aH aL ˆ zL+ aH −aL aL (1−τw)w− aHµ(ˆzpool)γπ(1−τπ)− aH a zˆpool+ aH −a a (1−τw)w ,
which is increasing in (1−τw)w and thus zsep. Next, (B.2) is negative for zsep = ˆzL, positive for zsep = ˆzpool. To see the former, note that, by point 1, zsep = ˆzL implies (1−τw)w=zsep. Then
npvHpool(ˆzpool)> npvHpool(zsep) =npvpoolH ((1−τw)w) =npvH((1−τw)w) =npvH(zsep).
To see the latter, note thatnpvpoolH (ˆzpool)< npvH(ˆzpool) =npvH(zsep). Point 3
follows. Finally, supposenpvH
pool(ˆzpool)5npvH(zsep). After replacingnpvH(zsep)
from equation (B.3) and multiplying both sides by aL
aH, this can be written as
aLµ(ˆzpool)γπ(1−τπ)− aL
a [ˆzpool−(1−τw)w]−(1−τw)w5
aLµ(ˆzL)γπ(1−τπ)−zˆL,
which is the same as npvL
pool(ˆzpool)5npvL(ˆzL). Point 4 follows.
Theorem 3. If and only if zsep ≥ zˇ (where zˇwas defined in Theorem 2) the
following is a PBE:
(a) Lenders believe that those who contribute(1−τw)w in equity and invest z ∈ ((1−τw)w, zsep] are high types; those who contribute (1−τw)w in
equity and invest z > zsep have equal probability of being high types or
low types; and everybody else are low types. They then offer rate a 1
Hµ(z)
to the first group, rate aµ1(z) to the second, and rate a 1
Lµ(z) to the third.
Low types invest zˆL (any combination of equity and external financing
being possible). High types invest zsep (contributing (1−τw)w in equity).
If and only if zsep ≤zˇ, the following is a PBE:
(b) Lenders have the same beliefs and strategy as in (a). Both high and low types invest zˆpool (contributing (1−τw)w in equity).
Proof. Let %J represent type J’s preferences, and bJ ≥ 0 be the amount
of borrowing that J could have contributed in equity, but did not. Then,
bJ ∈[0,min ((1−τw)w, zJ)], and the pair(zJ, bJ)uniquely identifies J’s action.
Consider (a) first. Facts AI.-AIII.i. below prove that (a) is a PBE if
zsep ≥zˇ. Fact AIII.ii. proves that it is not a PBE if zsep <zˇ. AI. For every
action that borrowers could play, the lenders’ action is optimal given their beliefs. AII. If zsep ≥ zˇ, for actions that borrowers play in equilibrium, the
lenders’ beliefs are correct (since zˆL <(1−τw)w < zsep). AIII.i. Ifzsep ≥zˇ,
borrowers do not have a profitable deviation. Type H. Their equilibrium action, (zsep,0), gives payoff npvH(zsep). We want to show that (zsep,0) %H
(zH, bH) for any (zH, bH) 6= (zsep,0). We proceed in two steps. First, we
show that (zsep,0) %H (zH,0) for any zH 6= zsep. To see this, note that, if zH ∈ [0, zsep), (zH,0) gives payoff npvH(zH); and zH < zsep ≤ zˆH implies npvH(z
H) < npvH(zsep). If zH > zsep, (zH,0) gives payoff npvpoolH (zH) ≤ npvH
pool(ˆzpool), and zsep ≥ zˇ implies npvHpool(ˆzpool) ≤ npvH(zsep). Second, we
show that(zH,0)%H (zH, bH)for anyzH >0and feasiblebH >0. This follows
from the fact that the total cost of funding is higher in(zH, bH)than in(zH,0):
aH aL −1 bH higher if zH ∈ (0,(1−τw)w]; aH aL −1 (bH +zH −(1−τw)w)
higher ifzH ∈((1−τw)w, zsep]; and
aH aL −1 bH+ aH aL − aH a (zH−(1−τw)w)
higher ifzH > zsep. TypeL. Their equilibrium action,(ˆzL, b)(withb∈[0,zˆL]),
gives payoff npvL(ˆz
L). We want to show that (ˆzL, b) %H (zL, bL) for any
(zL, bL) 6= (ˆzL, b). We proceed in two steps. First, we show that (ˆzL, b) %H
(zL, bL)for any(zL, bL)such that eitherzL≤(1−τw)w, orbL >0. To see this,
note that any such (zL, bL) gives payoff npvL(zL), and npvL(zL) ≤ npvL(ˆzL).
Second, we show that (ˆzL, b) %H (zL,0) for any (zL,0) such that zL > (1− τw)w. To see this, note that, if zL ∈ ((1−τw)w, zsep], (zL,0) gives payoff
g
npvL(zL), and, by definition ofzsep,npvg
L
(zL)< npvL(ˆzL). If zL> zsep,(zL,0)
gives payoff npvL
pool(zL) ≤ npvLpool(ˆzpool). But zsep ≥zˇimplies npvLpool(ˆzpool) ≤ npvL(ˆz
L). AIII.ii. Ifzsep <zˇ, some borrowers have a profitable deviation. For
example, it is npvH
pool(ˆzpool) > npvH(zsep), implying that H have a profitable
deviation to (ˆzpool,0).
zsep ≤ zˇ. Fact BIII.ii. proves that it is not a PBE if zsep > zˇ. BI. For
every action that borrowers could play, the lenders’ action is optimal given their beliefs. BII. If zsep ≤ zˇ, for actions that borrowers play in equilibrium,
lenders’ beliefs are correct (sincezsep <zˆpool). BIII.i. Ifzsep ≤zˇ, borrowers do
not have a profitable deviation. Type H. Their equilibrium action, (ˆzpool,0),
gives payoffnpvH
pool(ˆzpool). We want to show that(ˆzpool,0)%L(zH, bH)for any
(zH, bH)6= (ˆzpool,0). We proceed in two steps. First, we show that(ˆzpool,0)%H
(zH,0) for any zH 6= ˆzpool. To see this, note that, if zH ∈ [0, zsep], (zH,0)
gives payoff npvH(zH), and zH ≤ zsep < zˆH implies npvH(zH) ≤ npvH(zsep);
furthermore, zsep ≤zˇimplies npvH(zsep)≤ npvpoolH (ˆzpool). IfzH > zsep,(zH,0)
gives npvH
pool(zH), and npvHpool(zH) ≤ npvHpool(ˆzpool). Second, by fact AIII.i.,
it is (zH,0) %H (zH, bH) for any zH > 0 and feasible bH > 0. Type L.
Their equilibrium action, (ˆzpool,0), gives payoff npvLpool(ˆzpool). We want to
show that (ˆzpool,0) %L (zL, bL) for any (zL, bL) 6= (ˆzpool,0). We show this
in two steps. First, we show that (ˆzpool,0) %L (zL, bL) for any (zL, bL) such
that either zL ≤ (1− τw)w, or bL > 0. To see this, note that any such
(zL, bL)gives payoffnpvL(zL). But npvL(zL)≤npvL(ˆzL), andzsep ≤zˇimplies npvL(ˆzL) ≤ npvpoolL (ˆzpool). Second, we show that (ˆzpool,0) %L (zL,0) for any
(zL,0)such thatzL>(1−τw)w. To see this, note that, ifzL∈((1−τw)w, zsep],
(zL,0) gives payoffnpvg
L
(zL), and, by definition of zsep, npvg
L
(zL)≤npvL(ˆzL);
furthermore, zsep ≤ zˇ implies npvL(ˆzL) ≤ npvpoolL (ˆzpool). If zL > zsep, (zL,0)
gives payoff npvL
pool(zL), and npvpoolL (zL)≤ npvLpool(ˆzpool). BIII.ii. If zsep >zˇ,
some borrowers have a profitable deviation. For example, it isnpvH
pool(ˆzpool)< npvH(z
sep), implying that H have a profitable deviation to (zsep,0).
B.2
Proofs
Lemma 1. There exists τw and τw, with −∞< τw < τw <1 such that:
• If τw ≤ τw, the high types are unconstrained: a separating equilibrium
realises, where zsep = ˆzH(τπ).
• If τw ∈(τw, τw], the high types are strategically constrained: a separating
• If τw > τw, the high types are constrained: a pooling equilibrium realises.
If τw ∈(τw, τw], zsep is decreasing in τw, increasing in τπ.
Proof. Points 2 and 3 in Theorem 2 imply that there exists ω < ω such that
zsep = ˇz iff (1−τw)w=ω. Letτw ≡argτ[(1−τw)w=ω] and τw ≡argτ[(1− τw)w=ω]. By Theorem 2, ifτw ≤τw, it iszsep = ˆzH >zˇ; ifτw ∈(τw, τw], it is zsep ∈[ˇz,zˆH); and if τw > τw it is zsep <zˇ. The first part of the Lemma then
follows from Theorem 3. Next, supposeτw ∈(τw, τw]. Theorem 2 implies that zsep is decreasing inτw.Differentiating both sides ofnpvg
L (zsep|τ) =npvL(ˆzL|τ) with respect to τπ: aLµ0(zsep) ∂zsep ∂τπ γπ(1−τπ)−aLµ(zsep)γπ− aL aH ∂zsep ∂τπ = aLµ0(ˆzL) ∂zˆL ∂τπ γπ(1−τπ)−aLµ(ˆzL)γπ− ∂zˆL ∂τπ ∂zsep ∂τπ = aaLµ(zsep)γπ−aLµ(ˆzL)γπ L aH [aHµ 0(z sep)γπ(1−τπ)−1] >0,
where we used aLµ0(ˆzL)γπ(1−τπ) = 1 to simplify.
Proposition 2. Lowering the tax on profit always boosts research effort by the low types, and by the high types at a separating equilibrium in which they are unconstrained (zsep = ˆzH(τπ)), or at a pooling equilibrium. At a
separating equilibrium in which the high types are strategically constrained (zsep < zˆH(τπ)), lowering the labor income tax boosts their research effort,