Online Appendix: The Aggregate Implications of Mergers and Acquisitions Joel M. David∗

∗

Email: [email protected].

A

Joint and Marginal Distributions - Sales

Figures A.1 and A.2 shows the joint and marginal distributions over the sales of acquirers and targets.

Figure A.1: Joint Distribution of Acquirers and Targets - Sales Panel A. Acquirers

Panel B. Targets

20

16 14 12

% of Targets

% of Acquirers

15

10

5

10 8 6 4 2

0

Low

0

High

Low

High

Decile

Decile

Figure A.2: Marginal Distributions of Transacting Firms - Sales

B

A detailed Q-Theory Model

In this appendix, I outline a detailed example of the q-theory that leads to a surplus function satisfying the conditions of Proposition 3. Assume that firms potentially operate multiple “segments,” j = 1, 2, 3, ...n, which can be interpreted, for example, as plants or products. Profits from segment j for firm i are equal to πij = Πzi , so that firm-level profits are linear in the number of segments, i.e., πi = ni Πzi .1 1

This easily obtains when (1) segments are plants, firms compete under perfect competition and production α is equal to qij = zu1−α lij , the production function I assume in Section 3.3, or (2) segments are products, firms 1

compete under monopolistic competition and production is qij = zij1−σ lij where σ is the elasticity of substitution between products.

2

For simplicity, I set the fixed cost of operation cf = 0. We have already seen that this feature alone is unable to generate matching patterns along the lines of those in the data.2 Each firm is indexed by the tuple (z, n) - its productivity and the number of segments it operates. The productivity of a firm is fixed through time but mergers enable firms to acquire additional segments and so expand their scale. Upon an acquisition, an acquirer with productivity za operating na segments purchases one of the target’s nt segments and continues to operate that segment with its own productivity za . This is the essence of the q-theory - the acquiring firm’s productivity extends to all the resources under its control. The target firm receives the payment from the acquirer and either exits the market (for a single segment firm) or continues to operate its nt − 1 remaining segments. The combined surplus from this transaction is equal to Σ ((za , na ) , (zt , nt )) = v (za , na + 1) + v (zt , nt − 1) − v (za , na ) − v (zt , nt ) where lower-case v denotes the firm-wide value (upper-case V will be used to denote the segment level value to match the notation in the main text). The purchase price satisfies P ((za , na ) , (zt , nt )) = v (zt , nt ) + (1 − β) Σ ((za , na ) , (zt , nt )) and the gains that accrue to an acquirer or target of type (z, n) partnering with a target of type (zt , nt ) or acquirer of type (za , na ) are, respectively, Σa ((z, n) , (zt , nt )) = βΣ ((z, n) , (zt , nt ))

Σt ((za , na ) , (z, n)) = (1 − β) Σ ((za , na ) , (z, n))

I assume the cost of search is a homogeneous of degree one function of the arrival rate of meetings and the number of segments the firm operates (the role of this assumption will become ¯ and µ apparent shortly). Denoting firm-level search intensities as λ ¯, we have  c (X, n) = nC

X n



 ¯ µ , X = λ, ¯

where C (x) = c (x, 1) is the intensive form of the cost function, which is increasing and strictly convex in x. The aggregate meeting rate is min

(Z ∞ X

¯ (z, n) dG (z, n) , λ

n=1 2

Z X ∞

) µ ¯ (z, n) dG (z, n)

n=1

All the results go through with the inclusion of a fixed cost paid at the segment level.

3

where dG (z, n) denotes the joint distribution of productivity and segment counts. We can express market tightness as  R P∞ ¯  λ (z, n) dG (z, n) n=1 θt = min R P∞ ,1 ¯ (z, n) dG (z, n) n=1 µ

  R P∞ µ ¯ (z, n) dG (z, n) n=1 ,1 , θa = min R P∞ ¯ n=1 λ (z, n) dG (z, n) and meeting rates µ ¯ (zt , nt ) dG (zt , nt ) ¯ (za , na ) θa R P λ , ∞ ¯ (z, n) dG (z, n) n=1 µ | {z }

µ ¯ (za , na ) dG (za , na ) µ ¯ (zt , nt ) θt R P∞ ¯ n=1 λ (z, n) dG (z, n) | {z }

¯ t ,nt ) Γ(z

¯ a ,na ) Λ(z

Firm value is given by rv (z, n) =

max

¯ λ(z,n),¯ µ(z,n)


¯ (z, n) , n − c (¯ µ (z, n) , n) nΠz − c λ ¯ (z, n) θa E [Σa ((z, n) , (zt , nt ))] + µ + λ ¯ (z, n) θt E [Σt ((za , na ) , (z, n))]

where E [Σa ((z, n) , (zt , nt ))] =

Z X ∞

¯ (zt , nt ) max {Σa ((z, n) , (zt , nt )) , 0} Γ

nt =0

E [Σt ((za , na ) , (z, n))] =

Z X ∞

¯ (za , na ) max {Σt ((za , na ) , (z, n)) , 0} Λ

na =0

This is, in general, a complicated problem, but the assumptions on the cost function leads to a significant simplification. Specifically, I prove below that the value function is homogeneous of degree one in n, i.e., v (z, n) = nV (z) and search intensities satisfy ¯ (z, n) = nλ (z) , λ

µ ¯ (z, n) = nµ (z)

where V (z) solves rV (z) = Πz − C (λ (z)) − C (µ (z)) + λ (z) θa E [Σa (z, zt )] + µ (z) θt E [Σt (za , z)]

4

with expected gains Z max {Σa (z, zt ) , 0} Γ (zt )

E [Σa (z, zt )] = Z E [Σt (za , z)] =

max {Σt (za , z) , 0} Λ (za )

where Γ (zt ) and Λ (za ) denote the density of search intensities of segments of type zt and za , respectively. Match surplus is equal to Σ (za , zt ) = V (za ) − V (zt )

(1)

so that individual gains are Σa (z, zt ) = β (V (za ) − V (zt )) ,

Σt (za , z) = (1 − β) (V (za ) − V (z))

(2)

and λ (z) and µ (z) are characterized by C 0 (λ (z)) = θa E [Σa (z, zt )] ,

C 0 (µ (z)) = θt E [Σt (za , z)]

These equations are analogous to - and in the case of merger surplus, a special case of - the expressions in Section 3. The merger surplus in equation (1) clearly satisfies the condition in a ,zt ) > 0 as long proposition 3 (indeed, it is exactly the example given in that section), i.e., ∂Σ(z ∂(za −zt ) as V (z) is increasing in z, which I prove below is the case under appropriate conditions. Then the matching patterns characterized in that proposition hold: first, equation (1) shows that any meeting where za > zt has positive surplus and so will result in a transaction. The matching set is then simply the area above the 45 degree line. This implies that low z targets and high z acquirers are in a greater share of matching sets. Second, since gains are decreasing in zt and increasing in za , expected gains conditional on meeting a particular purchaser or target are also respectively decreasing and increasing in z. Finally, expected surplus as an acquirer is increasing in z, since the size of the matching set and the gains per match are increasing. The inverse is true for the expected surplus as target. The first order conditions then imply that high z firms search most intensively as acquirers, and low z firms as targets, so that they match with one another at the highest rate. The remainder of this appendix proves that the value function is homogeneous of degree one in n and increasing in z. Proof of homogeneity of v (z, n). Conjecture that v (z, n) = nV (z). The surplus from a match of types (za , na ) and (zt , nt ) is (na + 1) V (za ) + (nt − 1) V (zt ) − na V (za ) − nt V (zt ) = V (za ) − 5

V (zt ) and each party is entitled to their share, β and 1 −β. This  gives equations (1) and (2). ¯ = θa E [Σa (z, zt )]. The right The first order condition on acquirer search gives C 0 λ(z,n) n hand side is independent of n from which it follows that search intensities are linear in n, ¯ i.e., for each type z, λ(z,n) is equal to a constant, denote it λ (z), which is defined by λ (z) = n 0−1 C (θa E [Σa (z, zt )]). An analogous result holds for target search. Substituting into the value function: rnv (z) = nΠz − C (λ (z)) n − C (µ (z)) n + nλ (z) θa E [Σa (z, zt )] + nµ (z) θt E [Σt (za , z)] Dividing through by n completes the proof. Proof that V (z) is increasing in z. Looking at the terms involving acquisition and denoting by fa (z) the expected gains from acquisition, we have −C (λ (z)) + λ (z) θa fa (z). Since we have already shown that transactions are consummated if and only if za ≥ zt , we know z

Z

(V (z) − V (zt )) Γ (zt ) Z z Z z V (zt ) Γ (zt ) Γ (zt ) − β = βV (z)

fa (z) = β

0

0

0

Then, ˆ (z) + βV (z) Γ (z) − βV (z) Γ (z) = βV 0 (z) Γ ˆ (z) fa0 (z) = βV 0 (z) Γ R ˆ (z) = z Γ (z) and I use the fundamental theorem of calculus and chain rule. We can where Γ 0 similarly write expected target gains as ∞

∞

 Λ (za ) V (za ) Λ (za ) − V (z) ft (z) = (1 − β) z  z Z z  Z z Λ (za ) − V (za ) Λ (za ) = (1 − β) V (z) Z

Z

∞

∞

  ˆ (z) − 1 . and similar steps as above give ft0 (z) = (1 − β) V 0 (z) Λ Taking the full derivative of the value function gives   0 ˆ ˆ rV (z) = Π + λ (z) θa βV (z) Γ (z) + µ (z) θt (1 − β) V (z) Λ (z) − 1 0

0

or V 0 (z) =

Π   ˆ (z) + µ (z) θt (1 − β) 1 − Λ ˆ (z) r − λ (z) θa β Γ

6

Because λ (z) is increasing in z and limz→∞ µ (z) = 0, limz→∞ λ (z) θa β < r is a sufficient condition for V 0 (z) to be everywhere positive. This is an analogous condition to expression (15) in Lucas and Moll (2014).

C

Knowledge Diffusion and Growth

In this appendix, I derive a mapping between my framework and the recent literature on growth through imitiation a la Lucas and Moll (2014). The derivations follow very closely the steps in that paper. The model of Perla and Tonetti (2014) is related, although there is not a direct mapping between the two. The notion of “imitation” in Lucas and Moll (2014) is closely related to the q-theory version of the model here - the idea of one firm imitating another’s productivity is similar to a process by which the more productive firm simply absorbs the resources of the least productive and applies its higher productivity to those inputs. In both cases, the low productivity firm has become like the high productivity one. To formally establish this connection, rewrite the firm’s value function from Appendix B in a potentially non-stationary environment as Π (t) z − C (λ (z, t)) − C (µ (z, t)) Z z [V (z, t) − V (zt , t)] Γ (zt , t) + λ (z, t) θa (t) β 0 Z ∞ ∂V (z, t) [V (za , t) − V (z, t)] Λ (zt , t) + + µ (z, t) θt (t) (1 − β) ∂t z

rV (z, t) =

max

λ(z,t),µ(z,t)

The firm’s value function and search choices are now potentially time-dependent, as are all aggregates (e.g., profit levels, market tightness and distributions). As proved in Online Appendix B, firms are willing to buy any targets with a lower z and sell themselves to any acquirer with a higher z. The law of motion for G (z, t), the (now time-dependent) distribution of firm types, in an interval (t, t + ∆) is given by G (z, t + ∆) = Prob (productivity below z at t and a merger does not occur with an acquirer of type za > z in (t, t + ∆))   Z z Z z = dG (y, t) 1 − µ (y, t) θt (t) ∆ + µ (y, t) θt (t) ∆ Λ (za , t) 0 0  Z z Z z = G (z, t) − ∆θt (t) 1 − Λ (za , t) µ (y, t) dG (y, t) 0

0

7

a ,t)dG(za ,t) is the conditional probability of meeting an where similar to the text, Λ (za , t) = Rλ(z ∞ 0 λ(z,t)dG(z,t) acquiring firm of type za . From here, we have

G (z, t + ∆) − G (z, t) ∂G (z, t) = lim ∆→0 ∂t Z z  Z z∆ Λ (za , t) µ (y, t) dG (y, t) = −θt (t) 1 − 0

(3)

0

  ˜ µ Define a balanced growth path as a growth rate κ and a set of functions dΦ, λ, ˜, v where dG (z, t) = e−κt dΦ ze−κt
V (z, t) = eκt v ze−κt
˜ ze−κt λ (z, t) = λ
µ (z, t) = µ ˜ ze−κt




As described by Lucas and Moll (2014), this is a path on which the quantiles of the productivity distribution all grow at a constant rate κ. Since G (z, t) = Φ (ze−κt ), equation (3) implies Z x  Z x ˜ ˜ µ ˜ (y) dΦ (y) Φ (x) κx = θt 1 − Λ (xa ) 0 0 R∞ ˜ (y) dΦ (y) Z x λ x ˜ = θt R ∞ µ ˜ (y) dΦ (y) ˜ (y) dΦ (y) 0 λ 0 0

where x = ze−κt . Assuming that Φ has a Pareto tail, i.e., there are k, ξ > 0 such that limx→∞ 1−Φ(x) = k, the xξ −ξ left-hand side is equal to κkξx for large x and we can rewrite the expression as "

κkξx−ξ

R∞

# ˜ (y) dΦ (y) Z ∞ λ = θ˜t lim Rx∞ µ ˜ (y) dΦ (y) ˜ (y) dΦ (y) 0 x→∞ λ 0

To maintain constant growth, the expression in brackets on the right-hand side, i.e., the conditional probability of meeting an acquirer of a type greater than x must be proportional to x−ξ as x grows large. When will this condition hold? First, notice that the setup in Lucas R∞ ˜ λ(y)dΦ(y) x and Moll (2014) has only “targets” searching (in the language of M&A), so that R ∞ λ(y)dΦ(y) = ˜ 0 R∞ dΦ (y) = κx−ξ . In this case, the probability of meeting a particular za is proportional to the x share of that type in the population of firms, exactly the necessary condition. More generally, search intensities must remain constant as x approaches infinity. With constant arrival rates

8

for acquirers, for example, the same equation would hold. With one-sided search, it is straightforward to derive the growth rate on the BGP: 1 κ= ξ

Z

∞

µ ˜ (y) dΦ (y) 0

The growth rate depends on the dispersion in firm types, ξ, and the average search intensity of targets. This is the same expression as in Lucas and Moll (2014) (although, in general, there is no reason to expect the search intensities to be the same). Indeed, setting β = 0, so that targets capture all the gains from merger (so essentially eliminating the presence of a “market”), makes the mapping between the models exact. In the case of two-sided search, and assuming the condition on acquirer search holds, the growth rate is given by: Z 1˜ ∞ µ ˜ (y) dΦ (y) κ = θt ξ 0 Notice the effects of two-sided search: if θ˜t = 1, so that average search is greater on the acquirer side of the market than the target (or the two are equal), the growth rate is constrained by the average search of targets (or equivalently, acquirers, if theyR are equal). On the other hand, if ∞˜ λ(x)dΦ(x) targets search more intensively than acquirers so that θ˜t = R0∞ µ˜(x)dΦ(x) , then the growth rate is 0 constrained by the search rate of acquirers. Formally,  R R∞ R∞ 1 ∞ µ ˜ (x) dΦ (x) ˜ (y) dΦ (y) , if 0 µ ˜ (x) dΦ (x) < 0 λ ξ 0 κ= R R R ∞ ∞˜ 1 ˜ λ ˜ (y) dΦ (y) , if ∞ µ ˜ (x) dΦ (x) > 0 λ (x) dΦ (x) ξ 0 0 R∞ R∞ ˜ (x) dΦ (x). where the two expressions are equivalent if 0 µ ˜ (x) dΦ (x) = 0 λ Thus, the Lucas and Moll (2014) is one variant of the q-theory model, where search is onesided on the part of targets and targets capture the entire match surplus. In my setting, search is two-sided (so that the short side of the market constrains the growth rate) and the parties bargain over the surplus, which will, in part, determine their search intensities.

D

Non-Parametric Approach

Hagedorn et al. (2017) propose a related approach to non-parametrically identify the match production function using data on pre-match values and wages in a labor market setting in which agent types are not observable. Their analysis rests on two key building blocks - first, they demonstrate how to identify firm and worker types and construct values. Second, they derive an expression relating the wage to the match production function and pre-match values only, which can be inverted to find the production function. Because I observe pre-match types, applying 9

their methodology entails the second step only - finding an analogous expression relating the production function to observables. It turns out that deriving this link is challenging in my framework with repeat matching and endogenous search intensity. Consider the following system of equations for pre and post match values rV (za ) = Πza − C (λ (za )) − C (µ (za )) + λ (za ) θa E [Σa (za , z)] + µ (za ) θt E [Σt (z, za )] rV (zt ) = Πzt − C (λ (zt )) − C (µ (zt )) + λ (zt ) θa E [Σa (zt , z)] + µ (zt ) θt E [Σt (z, zt )] rV (zm ) = Πzm − C (λ (zm )) − C (µ (zm )) + λ (zm ) θa E [Σa (zm , z)] + µ (zm ) θt E [Σt (z, zm )] zm = s (za , zt ) where search intensities satisfy the firms’ optimality conditions (I abstract from the fixed cost for simplicity, although all results go through with it). Surplus sharing gives βΣ (za , zt ) = V (zm ) − V (za ) − P (za , zt ) (1 − β) Σ (za , zt ) = P (za , zt ) − V (zt ) Σ (za , zt ) = V (zm ) − V (za ) − V (zt ) In their baseline setup, Hagedorn et al. (2017) assume exogenous search intensities and no on-the-job-search. The analogous setup in my framework implies no repeat matching. Under this assumption, the matched value is simply V (zm ) = Πzrm (recall that the discount rate, r, includes both the interest rate and the death shock). From here, use the sharing rule to derive  P (za , zt ) = V (zt ) + (1 − β) Σ (za , zt ) = βV (zt ) + (1 − β)

 Πzm − V (za ) r

which can be inverted to find s (za , zt ) =

r P (za , zt ) + (1 − β) V (za ) − βV (zt ) Π 1−β

(4)

so that, for a given value of β, the production function is non-parametrically identified (up to a constant). This is the analogous expression to that in Section 3.4.4 in their paper. Hagedorn et al. (2017) propose a strategy to identify β using the response of wages to either idiosyncratic or aggregate shocks to match output. This approach is challenging in the context of M&A since, in general, there is no ongoing relationship between the acquirer and target, i.e., there is typically a one-time payment rather than a continual flow of wages as in the labor context. But a more fundamental challenge is that the derivation of (4) depends on the assumption of one-time matching. In the case with repeat matching (and endogenous search 10

intensities), similar steps give P (za , zt ) = βV (zt ) − (1 − β) V (za ) Πzm − C (λ (zm )) − C (µ (zm )) + (1 − β) r λ (zm ) θa E [Σa (zm , z)] + µ (zm ) θt E [Σt (z, zm )] + (1 − β) r which shows that the costs of search and the expected future gains of the firm from continued matching all enter the equation and confound the mapping from values and prices to the technology. Hagedorn et al. (2017) extend their framework to incorporate on-the-job-search and derive an expression analogous to (4). However, this derivation rests on a particular set of assumptions on bargaining - take-it-or-leave-it offers on the part of unemployed workers and Bertrand competition for employed workers such that moving workers obtain the full surplus generated from their current match. This is challenging to port into my framework, in which there is no distinction between “merged” and “un-merged” firms. Further, allowing any party to make take-it-or-leave-it offers in a setting with two-sided endogenous search is problematic, since there is no incentive for search.

E

Detailed Kolmogorov-Smirnov Test Results

Table E.1 reports the results of the 45 individual KS tests from Appendix C.4.

11

Table E.1: Kolmogorov-Smirnov Test Results Target Profitability Interval

Acquirer Comparison Sets

Min

Max

Low Decile

High Decile

1.78 2.34 3.31 5.82 13.41 3.70 4.90 7.40 10.45 17.33 6.47 8.70 11.91 19.67 36.10 10.85 14.90 20.20 29.57 49.77 14.67 20.19 28.66 40.31 71.07 20.29 29.64 44.73 77.11 132.09 35.88 52.88 80.67 129.73 254.89 44.47 73.09 116.40 191.02 349.52 67.50 110.70 194.02 349.17 711.65

2.32 3.31 5.69 13.37 164.07 4.83 7.39 10.24 17.32 2539.24 8.67 11.89 18.95 35.21 1226.39 14.86 20.18 28.98 49.73 1003.66 19.92 28.63 40.22 70.16 405.91 29.03 44.16 76.87 129.39 781.31 51.69 80.13 128.08 254.55 1385.98 73.02 115.25 190.44 348.23 1465.75 110.32 193.01 340.99 702.29 12313.05

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9

2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 10 10 10 10 10

D-statistic

P-value

0.15 0.24 0.13 0.22 0.10 0.15 0.30 0.17 0.12 0.17 0.18 0.15 0.20 0.16 0.26 0.14 0.18 0.13 0.19 0.18 0.17 0.12 0.14 0.17 0.15 0.20 0.17 0.18 0.13 0.17 0.15 0.17 0.12 0.19 0.21 0.21 0.14 0.20 0.21 0.12 0.20 0.24 0.11 0.23 0.19

0.71 0.08 0.54 0.07 0.88 0.59 0.01 0.39 0.77 0.23 0.37 0.50 0.17 0.39 0.03 0.68 0.28 0.73 0.25 0.25 0.48 0.85 0.73 0.36 0.37 0.25 0.37 0.32 0.70 0.28 0.61 0.40 0.81 0.25 0.14 0.17 0.69 0.24 0.19 0.75 0.27 0.08 0.82 0.16 0.16

Notes: Table reports results of Kolmogorov-Smirnov tests for overlapping intervals of targets across different intervals of acquirers as described in the text. The first two columns report the interval of targets. The second two columns report the intervals of acquirers, which are adjacent deciles. The fifth column reports the KS test statistic and the sixth column the associated p-value.

12

F

Details of the Planner’s Problem

The steady state costate equation is given by rW (z) =

∂H ∂M dG (z)

= Πz − C (λ (z)) − C (µ (z)) − cf − δW (z) Z µ (zt ) dG (zt ) 1 + θa λ (z) W (z, zt ) Φ (Σ (z, zt )) 2 µ ¯ Z 1 λ (za ) dG (za ) + θa µ (z) W (s (za , z)) Φ (Σ (za , z)) 2 µ ¯ Z Z 1 µ (z) − θa 2 v (s (za , zt )) Φ (Σ (za , zt )) λ (za ) dG (za ) µ (zt ) dG (zt ) 2 µ ¯ Z Z 1 µ (z) − θa λ (z) W (s (za , zt )) Φ (Σ (za , zt )) λ (za ) dG (za ) µ (zt t) dG (zt ) + ¯µ 2 λ¯ µ ¯ if θa = ¯ , o/w equals 0 λZ λ (za ) dG (zt ) 1 + θt µ (z) W (za , z) Φ (Σ (za , z)) ¯ 2 λ Z 1 µ (zt ) dG (zt ) + θt λ (z) W (s (z, zt )) Φ (Σ (z, zt )) ¯ 2 λ Z Z 1 λ (z) − θt ¯ 2 v (s (za , zt )) Φ (Σ (za , zt )) λ (za ) dG (za ) µ (zt ) dG (zt ) 2 λ Z Z 1 λ (z) − θt µ (z) + W (s (za , zt )) Φ (Σ (za , zt )) λ (za ) dG (za ) µ (zt t) dG (zt ) ¯ 2 µ ¯λ ¯ λ if θt = , o/w equals 0 µ ¯ Z µ (zt ) dG (zt ) Φ (Σ (z, zt )) − θa W (z) λ (z) µ ¯ Z λ (za ) dG (za ) − θa µ (z) W (za ) Φ (Σ (za , z)) µ ¯ Z Z µ (z) + θa 2 W (za ) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) µ ¯ Z Z µ (z) − θa λ (z) − W (za ) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) ¯µ λ¯ µ ¯ if θa = ¯ , o/w equals 0 λ Z λ (za ) dG (za ) − θt W (z) µ (z) Φ (Σ (za , z)) ¯ λ Z µ (zt ) dG (zt ) − θt λ (z) W (zt ) Φ (Σ (z, zt )) ¯ λ

13

Z Z λ (z) W (zt ) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) + θt ¯ 2 λ Z Z λ (z) − θt µ (z) − W (zt ) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) ¯ µ ¯λ ¯ λ if θt = , o/w equals 0 µ ¯ R ¯ = λ (z) dG (z) and analogously for µ where λ ¯. Simplifying and rearranging gives (28). The FOC on λ is Z 1 µ (zt ) dG (zt ) 0 Φ (Σ (z, zt )) C (λ (z)) = θa W (s (z, zt )) R 2 µ (z) dG (z) Z Z 1 1 − W (s (za , zt )) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) ¯2 2λ µ ¯ if θa = ¯ , o/w equals 0 λ Z λ (za ) dG (za ) 1 Φ (Σ (za , z)) θt W (za , z) + ¯ 2 λ Z Z 1 θt W (za , zt ) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) − ¯2 2λ Z Z 1 1 + W (za , zt ) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) ¯ 2µ ¯λ ¯ λ if θt = , o/w equals 0 µ ¯ Z µ (zt ) dG (zt ) − θa W (z) Φ (Σ (z, zt )) µ ¯ Z Z 1 W (za ) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) + ¯2 λ µ ¯ if θa = ¯ , o/w equals 0 λ Z µ (zt ) dG (zt ) Φ (Σ (z, zt )) − θt W (zt ) ¯ λ Z Z θt − ¯2 W (zt ) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) λ Z Z 1 + W (zt ) λ (za ) dG (za ) µ (zt ) dG (zt ) Φ (Σ (za , zt )) ¯ µ ¯λ ¯ λ if θt = , o/w equals 0 µ ¯ and simplifying and rearranging gives (29). Similar steps for µ give (30).

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G

Computational Algorithms

Estimation. To estimate the model, I use a simulated method of moments estimator with a minimum distance criterion to find the parameter values. There are six parameters to pin down in this way, which I collect in the vector Θ = {γ, ν, A, β, η, B}. Formally, the parameter vector Θ∗ is chosen to solve
0
Θ∗ = arg min Ψs (Θ) − Ψd I Ψs (Θ) − Ψd

(5)

where I is the identity matrix, Ψs (Θ) the simulated moments for a candidate value of Θ and Ψd the empirical moments. I outline the computational algorithm in Table G.1. As mentioned in footnote 26 in the text, rather than iterate on the entry distribution dF (z), I directly impose the target distribution dG (z). This entails directly constructing the density at each z and using the stationary conditions in (16) to infer the primitive distribution dF (z). Table G.1: Estimation Algorithm 1. Construct z, dG (z) and set direct parameters. 2. Guess candidate vector Θc = {γ, ν, A, β, η, B}. 3. Construct merger matrix s (z, z0 ). 4. Guess candidate Π. Compute π (z). 5. Guess candidate V (z). Evaluate merger matrix. 6. Guess candidate µR(z) and θa . 7. Solve for λ (z), λ (z) dG (z) and θt . Iterate on θt and µ (z). 8. Solve for cf s.t. V (ˆ z ) = 0 and construct new V (z). Iterate on V (z). 9. Construct dF (z) using (16) and check FE condition. Update Π until FE satisfied. 10. Simulate data and construct target moments. 11. Compute objective function in (5) and iterate on Θc until minimized. I now describe the mechanics of the numerical algorithm in more detail. I discretize the productivity distribution over z into 500 points from a z of 1, which corresponds to the normalization of zˆ described above, up to a z of 10,000. I follow Restuccia and Rogerson (2008) in constructing the grid such that the largest operating firm will be 10,000 times the size of the smallest, and additionally in log-spacing the grid to ensure greater accuracy over the lower tail of the distribution, where most firms reside. I then construct the endogenous distribution dG (z) over this grid such that dG (z) takes on a Pareto with shape parameter ξ. Next, I guess a candidate value of Θc = {γ, ν, A, β, η, B}. With the candidate values of A, γ, and ν, I can construct a “merger matrix” which represents the zm resulting from each combination of za and zt , where the two pre-merger firms are drawn from the entire set of z’s. That is, the 15

merger matrix contains the productivity of a merged entity formed by the merger of all possible combinations of z’s. Computation of the equilibrium begins by guessing the industry aggregate Π. I perform value function iteration to find V (z) , λ (z) , µ (z) , Υa (z) , Υt (z). For a candidate V (z), I use the merger matrix to compute the value of each potential transaction on the merger market and in particular, to find those generating positive gains. I then use an iterative procedure to construct optimal search intensities, by which I guess a candidate vector µ (z) and value of θa , solve for λ (z) and θt and recompute µ (z) and θa . I iterate on this process until convergence. It is now straightforward to construct new values of V (z) in accordance with (8). In doing so, I compute the fixed cost cf that is consistent with this equilibrium by solving V (ˆ z ) = 0. Next, I use firm search and matching decisions to construct the flows in (16) and in conjunction with the distribution dG (z), I infer the entry distribution dF (z). Here, I must make a normalization of the minimum possible draw of z, zmin , which I set to 0.3. Figure G.1 displays the CDFs of the endogenous distribution dG (z) and the estimated entry distribution dF (z). Finally, I use V (z) and dF (z) to construct the free entry condition (13) and iterate on the candidate value of Π until the free entry condition is satisfied. 1 0.9 0.8

F(z),G(z)

0.7 0.6 0.5 0.4 0.3 0.2 F(z) G(z)

0.1 −2

−1

0

1

2

3

4

5

6

log(z)

Figure G.1: Entry and Endogenous Productivity Distributions To simulate the economy, I draw 1 million firms from the stationary distribution dG (z). Standard arguments show that each acquirer has a probability of meeting a target in a single
period equal to 1 − e−λ(z) θa . Using these probabilities, I calculate the set of potential acquirers and match them to a set of potential targets who are drawn randomly according to their meeting probabilities 1 − e−µ(z) . Elimination of matches that generate negative gains gives a simulated merger dataset with matched acquirers and targets analogous to the actual data. It is then straightforward to calculate the target moments and compute the value of the objective function in (5). I iterate on the guess of Θc until this function is minimized.

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Equilibria in counterfactual economies. Computation of the equilibrium in the counterfactual economies follows a fairly similar process to that just described. I again begin by guessing the aggregate level of profits Π. I then guess a candidate threshold entrant zb, value function V (z) for z ≥ zb, productivity distribution dG (z), target search intensities µ (z) and acquirer market tightness θa . For this guess of V (z), I evaluate the merger matrix and compute the gains from all potential transactions. Together, the guess of µ (z), dG (z), θa and the expected gains from merger imply values of acquirer search intensities λ (z) using (9). In turn, λ (z) implies a value for θt and new values of µ (z). I iterate on the guess of µ (z) and θa until convergence, delivering the equilibrium values of µ (z), λ (z), θa and θt for this particular candidate V (z) and dG (z). Next, I use the stationary conditions in (16) to construct the flows into and out of each firm type, from which we can compute a new distribution dG (z). I iterate on dG (z) until convergence, yielding the counterfactual productivity distribution. The solved values of µ (z), λ (z), θa , θt and dG (z) imply a new value function V (z) and I iterate on the value function until convergence. I then check the guess of the the threshold entrant by checking if a unilateral deviation by the next best firm would be optimal, i.e., I compute the value of entry by the next best firm, and if it is positive, I set this firm to zb and reperform all the calculations just described. I continue in this way until I have found the true marginal entrant, that is, the firm with positive value from entry where the next best firm would have negative value. Finally, it remains to check the free entry condition (13) and iterate on the initial guess of Π until it is satisfied. Aggregate outcomes. The computed equilibrium gives the level of profits Π, the marginal entrant zb, the average productivity level Z, the aggregate exit rate (inclusive of merger) and the average costs of search Ys . With these objects in hand, it is straightforward to back out the value of the aggregate variables in the economy. From equation (12), we can compute the wage, w, from the equilibrium value of Π. From (15), total revenue is equal to α1 Lα and dividing by the inverse of the wage, which is the real price level, gives output, Y . Labor market α Π clearing implies 1−α M Z = L, which gives the mass of firms, M . Using the fact that in the w  R  R steady state we have [1 − F (ˆ z )] Me = δ + µ (z) θt Φ (Σt (za , z)) Λ (za ) dG (z) M where the expression in braces on the right-hand side is the aggregate exit rate, i.e., the aggregate flow of firms into and out of the economy must balance, along with the value of zb gives Me . Finally, we can use Me and M along with the value of Ys to obtain consumption, C.

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References Hagedorn, M., T. H. Law, and I. Manovskii (2017): “Identifying Equilibrium Models of Labor Market Sorting,” Econometrica, 85, 29–65. Lucas, R. E. and B. Moll (2014): “Knowledge Growth and the Allocation of Time,” Journal of Political Economy, 122, 1–51. Perla, J. and C. Tonetti (2014): “Equilibrium Imitation and Growth,” Journal of Political Economy, 122, 52–76. Restuccia, D. and R. Rogerson (2008): “Policy Distortions and Aggregate Productivity with Heterogeneous Establishments,” Review of Economic Dynamics, 11, 707–720.

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