Do Managers Do Good With Other People s Money? Online Appendix

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1 Do Manages Do Good With Othe People s Money? Online Appendix Ing-Haw Cheng Haison Hong Kelly Shue Abstact This is the Online Appendix fo Cheng, Hong and Shue 2013) containing details of the model. Datmouth College, Tuck School of usiness, ing-haw.cheng@tuck.datmouth.edu. Pinceton Univesity Depatment of Economics and endheim Cente fo Finance, and NER, hhong@pinceton.edu. Univesity of Chicago, ooth School of usiness, kelly.shue@chicagobooth.edu.

2 We stat with the manage s poblem is as in Chetty and Saez 2010): [ max α 1 τ) 1 + η) D + f K) + Γ D ] g Γ K D) +, K,D whee η is the monitoing paamete, τ is the dividend tax ate, α is the owneship of the manage, Γ is total cash in the fim, K is poductive capital spending, D is the dividend paid, and is the discount ate thee is no uncetainty in this model). Fo simplicity, assume η 0 fo now. The function f epesents the net pofits of the fim; goss poduction may be thought of as F K) fk) + K. Suppose that both f and g ae stictly inceasing and concave, with f0) g0) 0. Following Chetty and Saez 2010), the capital G used fo investment in goodness at peiod 0 is etuned to shaeholdes at peiod 1. One may assume that G is buned up without changing any pedictions by witing the model in tems of goss poduction and simply eplacing the second tem in backets with F K) and assuming that 1+ F is stictly inceasing and concave, with F 0) 0. The geneic fist-ode condition fo this poblem is: α 1 τ) f K) g Γ K D), α 1 τ) g Γ K D) with stict equality if and only if D > 0. Define ᾱ to be the citical owneship level at which manages stat paying dividends: ᾱ g Γ K ) 1 τ), whee K is the fist-best investment, detemined by f K ). Let α 1 be the owneship level of any high owneship manage α 1 > ᾱ) and let α 2 be the owneship level of any low owneship manage α 2 < ᾱ). Using a subscipt of 1 to denote the investment of the high owneship manage and 2 fo the low owneship manage, we can e-wite the optimal investment and goodness spending fo the high owneship manage as: and: f K 1 ), α 1 τ) g G 1 ), D > 0, Γ D + K 1 + G 1, α 1 τ) f K 2 ) g G 2 ), g G 2 ) > α 1 τ), D 0, G 2 Γ K 2, fo the low owneship manage. At a basic level, we ae inteested in how goodness spending G esponds to changes in 1

3 the tax ate, τ. Applying the implicit function theoem eveals that: G 1 α g G 1 ) > 0, αf K 2 ) g G 2 ) + α 1 τ) f K 2 ) > 0, 0 fo α 2 0, which demonstates the most basic implication of the model: G/ ω > 0. Since G η G < 0, this shows Pediction 2 as well. Fo Pediction 1, we ae inteested in the compaative esponse of goodness spending to the tax cut acoss high and low owneship manages, G i acoss i {1, 2}. We ae also inteested in whethe thee is heteogeneity within each goup, 2 G i within i {1, 2}. We α show that the pediction holds fo a boad class of smooth concave poduction and goodness functions f and g. Fo intuition, we begin with the simple case whee g is linea. Linea g case. Fist conside the simple case whee gg) G, maintaining the assumption that f is stictly inceasing and concave. Suppose < 1 τ) so that ᾱ < 1. Fo the moment, ename fims with α 0, ᾱ) medium owneship fims subscipted with 2), while still calling fims with α ᾱ, 1] high owneship fims subscipted with 1). The case whee α 0 gives ise to a cone solution; call this fim the zeo-owneship fim, subscipted with 0. Fo high owneship fims, poductive capital K is the fist best given by the solution to f K), and anything left ove is paid out as dividends since α 1 τ) >. Nothing is invested in goodness, and the fim s goodness spending does not espond to the tax cut at all. The zeo owneship fim is a cone solution who invests nothing in poductive capital K 0 0), pays no dividends and invests eveything in goodness G 0 Γ). This fim s goodness spending does not espond to the tax cut at all eithe. Fo medium owneship fims, no dividends ae paid, and thee is an inteio solution fo capital and goodness spending given by α 1 τ) f K 2 ) and G 2 Γ K 2 > 0. Applying the implicit function theoem eveals that: This implies: K 2 α f > 0, αf K 2 f < 0. 1 τ) f Lemma 1. [Pediction 1, Zeo Owneship.] Suppose < 1 τ). Then manages with medium owneship cut goodness moe than manages with zeo owneship and also moe than high owneship in esponse to the dividend tax cut. Poof. This immediately follows fom the fact that zeo and high owneship manages spend G 0 Γ and G 1 0, that this does not espond to the tax cut, and that K 2 > 0. If the tax cut is discete and lage such that a medium owneship fim becomes a high 2

4 owneship fim, goodness spending falls to zeo fom a positive numbe and the poposition is also tue. Suppose the poduction function f satisfies the egulaity condition that f /f is diffeentiable and a monotone deceasing function. As an example, any f K) AK γ with γ 0, 1) and A > 0 satisfies this popety. Lemma 2. Suppose f /f is diffeentiable and a monotone deceasing function. Then < 0 fo α 0, ᾱ). K 2 α Poof. Applying the implicit function theoem to K 2 eveals that: [ ] K 2 α f f f α 1 τ) f f ) 2 1, which is negative, since K [ f f ] 1 f f f ) 2 < 0. Now conside any α L 0, ᾱ) and e-define medium owneship manages to be those with α α L, ᾱ). Define low manages to be those with α 0, α L ). Poposition 1. [Pediction 1, Linea g.] Suppose < 1 τ) and that f /f is diffeentiable and a monotone deceasing function. In esponse to a tax cut, medium owneship manages cut moe than low owneship manages, high owneship manages, and zeo owneship manages. Poof. Follows diectly fom the pevious two lemmas and the obsevation that K 2 > 0 and K 2 > 0. α α oade Poduction Functions. Retuning to the geneal case, the tade-off between spending on K and G depends on the elative concavity of f and g. Fo tactability, we impose moe stuctue within the class of inceasing and concave functions. Let: f K) AK γ, g G) G γ, whee γ < 1. The paametes A and contol the elative concavity of the two functions. 1 The fist-best investment and citical owneship level ᾱ in this poblem ae: K ᾱ ) 1 Aγ, ) γ 1 Γ 1. A 1 τ) whee fo convenience we denote Γ Γ/K. To make the poblem non-tivial, assume Γ > 1, ) γ 1 and that is small enough such that Γ 1 0, 1). This ensues the fim will A1 τ) 1 If one wee to assume that G is buned up in the pocess of investing in goodness, eplacing the assumption of f K) AK γ with F K) AK γ yields identical pedictions. One would eplace in all subsequent calculations with

5 have enough cash on hand to make the fist-best investment if it chooses to do so, and fo thee to be both dividend and non-dividend-paying manages. The high owneship α > ᾱ) fim s optimal investment, goodness spending, and dividends ae given by: Diect computation yields: ) 1 Aγ K 1 K, ) 1 γ 1 G 1, 1 τ) α 1 D 1 Γ G 1 K 1. G 1 α 1 τ γ γ 1 1 τ) 1 γ G 1 G 1 α 1 α 1 1 γ 1 1 τ) α1 1 τ) γ α 1 ) 2 γ γ 1 ) 2 γ ) 2 γ γ < 0, α1 γ > 0, ) ] 2 γ [ 1 1 γ < 0. Fo the low owneship α < ᾱ) fim, the capital-to-goodness spending atio is fixed: K 2 G 2 [ ] 1 A α 2 1 τ). }{{} C This implies: G 2 Γ 1 + C, K 2 CΓ 1 + C. 4

6 Diect computation yields: α α C Γ α C) 2 1 τ A Γ α2 1 τ)a 1 + C) 2 < 0 fo α 2 > 0, 0 fo α 2 0, [ 1 Aα C Γ 2 Γ 1 + C) C) 2 > 0 fo α 2 > 0, 0 fo α 2 0, ) [ 1 + C) 2 Γ 1 1 τ) γ +Γ 1 τ A < 0 if and only if α > A α1 τ)a ) γ ) α2 1 τ)a ) γ 1 τ) A. ) γ ]) α1 τ)a ) C) 1 + C) 4 ) 2γ 1 Aα 1 Aα ) α1 τ)a ) γ α1 τ)a ) ) γ ] Poposition 2. [Within High and Within Low Owneship Fims.] Within high-owneship G fims, 1 is positive and monotonically deceasing in α. Within low owneship fims, > 0 fo α 2 > 0 and 0 fo α 2 0. If Γ 2, then is monotone inceasing in α fo α 2 0, ᾱ); if Γ ) < 2, then is non-monotone in α: inceases fo α 2 0,, 1 τ)a achieves a maximum at, and deceases fo α 1 τ)a 2 )., ᾱ 1 τ)a Poof. Diect algebaic manipulation yields the esult. Note that if Γ 1, 2) then 1 τ)a < ᾱ. Compaing acoss the two diffeent egions, a geneal necessay and sufficient condition fo α 2 ) / > G 1 α 1 ) / is then [ ) γ ]) 1 Aα Γ 2 α2 1 τ)a 1 + [ A α 2 1 τ) ] ) 1 2 > 1 ) 2 γ α1 1 τ) γ 1 α1 γ 1 γ γ. 1) Lemma 3. Let α 1 > ᾱ and α 2 < ᾱ be given. only if We have α 2 ) / > G 1 α 1 ) / if and α α 1 2 [ A 1 τ) ] 2 ) 1 α1 A < γ Γ 1 τ ) 2 2 [ A 1 τ) ] 1, 2) 5

7 and ᾱ) / > G 1 α 1 ) / if and only if α 1 > A 1 τ) ) ) Γ ˆα. 3) Γ 1 Poof. The fist pat follows diectly fom manipulation of the necessay and sufficient condition, equation 1). Substituting Γ ΓK, eplacing K Aγ eveals that ᾱ) / > G 1 α 1 ) / if and only if ) 1 Γ 1 [ Γ ] A 1 τ) Γ 1 ) 1 < α 1 1, and the definition of ᾱ fom which the conclusion follows. Poposition 3. [Pediction 1.] Suppose < 1 1 Γ). A1 τ) Thee exist cut-offs αl and α H with 0 < α L < ᾱ < α H < 1 such that medium owneship fims with α α L, ᾱ) have α) / > G α) / fo any α 0, α L ) α H, 1) and those with α ᾱ, α H ) have G 1 α) / > G α) / fo any α similaly given, whee G α) / is defined as: The cut-off α H ight-side. G α) / α) / if α < ᾱ, G 1 α) / if α > ᾱ. may be chosen to be abitaily close to ˆα A1 τ) ) Γ 1) on the Poof. Suppose < 1 1 Γ). A1 τ) Fist note that ᾱ < ˆα < 1. The fist inequality follows since, by the definition of ᾱ, ) γ 1 ᾱ Γ 1 A 1 τ) ) 1 A 1 τ) Γ 1 < ˆα, since Γ > 1. The second inequality follows by supposition. Take any α H ˆα, 1). y Lemma 3 note the stict inequalities): ᾱ) > G 1α H ) > 0. Recall that 0) 0. Conside the case whee Γ 2. y the Intemediate Value Theoem, thee exists an α L 0, ᾱ) such that α L ) G 1α H ) D. Futhemoe, this α L must be unique in 0, ᾱ) given any choice of α H > ˆα) since / is monotonically inceasing between 0 and ᾱ, by Poposition 2. Note that thee is no α ᾱ, α H ) such that G 1 α) / D since G 1 / 6

8 is monotonic, so α L is unique in 0, α H ). y constuction, α) > D fo α α L, ᾱ), D > α) fo α 0, α L ), G 1 α) > D fo α ᾱ, α H ), D > G 1 α) fo α α H, 1), whee the inequalities in the fist ow follow since / is monotonically inceasing on 0, ᾱ) and the inequalities in the second ow follow since G 1 / is monotonically deceasing, fom Poposition 2. Theefoe, α) G 1 α) > D > G α) > D > G 1 α) fo α, α) α L, ᾱ) [0, α L ) α H, 1)], fo α, α) ᾱ, α H ) [0, α L ) α H, 1)]. Fo Γ ) < 2, note that fo α 0, ᾱ), / inceases fo α 2 0,, achieves a 1 τ)a maximum at, and deceases fo α 1 τ)a 2 )., ᾱ Since G 1 τ)a 2 / has one maximum ) in the inteval 0, ᾱ) at < ᾱ with G 1 τ)a 2 / > G 1 τ)a 1 α H ) /, it must be by the ) Intemediate Value Theoem that thee is a α L 0, such that α L ) G 1α H ) 1 τ)a D. Futhemoe, we must have α) > D fo α α L, ᾱ) and α) ) < D fo α 0, α L ), by constuction. Finally, since / > G 1 τ)a 2 ᾱ) / > G 1 α H ) /, and / ) is monotonic ove 0, and ),, ᾱ this α 1 τ)a 1 τ)a L is unique given any choice of α H > ˆα. The est of the poof follows similaly. Refeences Chetty, R., and E. Saez, 2010, Dividend and Copoate Taxation in an Agency Model of the Fim, Ameican Economic Jounal: Economic Policy, 23),

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics