Private equity and growth

Abstract

We study private equity in a dynamic general equilibrium model and ask two questions: (i) Why does the investment of venture funds respond more strongly to the business cycle than that of buyout funds? (ii) Why are venture fund returns higher than those of buyout? On (i), venture brings in new capital whereas buyout largely reorganizes existing capital; this can explain the stronger co-movement of venture with aggregate Tobin’s Q. On (ii), the cost of reorganized capital has been high compared to new capital. Our model embodies this logic and fits the data on investment and returns well. At the estimated parameters, the two PE sectors together contribute between 7 and 11% of observed growth relative to the extreme case where private equity is absent. Using an alternative plausible measure of PE excess returns in the literature, this contribution could be as low as 5.8–9.7%.

Appendices

Appendix A: Data and methods

In this appendix we document the data sources and methods used to construct the series depicted in our figures and included in the empirical analysis.

Figures 1 and 4. The “intakes” are the sum of investments made annually in U.S. venture capital and buyout funds, divided by annual estimates of gross private domestic investment from the BEA (2017, Table 5.2.5, line 4). Venture and buyout investments are from the April 2017 version of Thomson One’s VentureXpert database, and are the sum of all investments made in a given calendar year at any stage or round across funds of each type. The “intake ratio,” $n_{\text{v} }/n_{\text{b}}$, is the ratio of the respective investment sums in each year.

For aggregate $q_{t}$, we use fourth quarter observations underlying Hall (2001) for 1987–1999, and then join them with estimates underlying Abel and Eberly (2011) for 1999–2005. Abel and Eberly derive aggregate Tobin’s Q from the Federal Reserve Board’s Flow of Funds Accounts as the ratio of total market value of equity and bonds to private fixed assets in the non-financial corporate sector. We bring these estimates forward through 2016 using the same sources. Hall’s measure of Q in 1999 is higher than that of Abel and Eberly (3.376 vs. 1.819), so we use a ratio splicing factor of 1.856 to adjust the series from 2000 forward.

Figures 2, 3 and 5. Aggregate returns to U.S. venture capital funds for 1987–2016 are from Cambridge Associates (2016a), and are annualized returns constructed by compounding one quarter horizon pooled returns. Aggregate returns to U.S. buyout funds for 1987–2016 are from Cambridge Associates (2016b), and are also annualized returns constructed from single quarter horizon pooled returns. Both series are net of fees, expenses, and carried interest.¹⁷ The majority of the VC funds (1070 out of 1680) in our sample are in the early stage and earn large and positive returns. This is consistent with Korteweg and Nagel (2016), who document that VC start-up investments earn large positive abnormal returns whereas those in the later stage earn net returns close to zero. In addition, the positive VC returns could also reflect the possibility that general partners have considerable equity in the projects in addition to collecting fees. The average combined VC return ${\tilde{R}}_{VC}$ in our sample is 18.1%; this is in line with the estimates in Table IA.II in the Internet Appendix of Harris et al. (2014).

We convert each series into ex-post real returns using the annual growth of the consumer price index from the National Income and Product accounts. Annual returns to the S&P 500 are from Damodaran (2017),¹⁸ and deflated by the consumer price index. We then subtract 1.05% from both venture and buyout returns, which is the liquidity premium reported in Sorensen et al. (2014).

Tables 1 and 2. For $z_{t}$, we use private output, defined as GDP less government expenditures on consumption and investment from the BEA (2017) for 1987–2016. We then divide the result by $K_{t-1}$ after adjusting it for inflation during year $t-1$ by averaging the annual inflation factors across the 2 years that overlap $t-1$ and then using its square root as a deflator. The $K_{t}$ are end-year stocks of private fixed assets from BEA (2017, Table 6.1, line 1) for 1987–2016. The aggregate investment rate $i_{t}$ is constructed as annual gross private domestic investment from BEA (2017, Table 5.2.5, line 4) for 1987–2016 divided by $K_{t-1}$.

Figure 6. We obtained the data on venture and buyout funds from the Securities Data Company (SDC) Platinum and CRSP/Compustat Merged (CCM) Database.

For buyouts, the SDC lists the acquirer’s CUSIP and the market value before the merger. It also lists the year the merger occurred and the target’s assets. In the CCM, there are multiple CUSIPs per year in both the SDC and CRSP/Compustat data. The multiple CUSIPs per year in the SDC are due to an acquirer completing multiple mergers in a year. The multiple CUSIPs per year in the CRSP/Compustat are due to firms with different permanent numbers having the same CUSIPs. Unfortunately, the SDC does not contain permanent numbers (lpermno) so we can only match via CUSIPs. As a result, all observations in both datasets in which the same CUSIP appeared multiple times in a year were dropped from both datasets. We then merged the CCM and SDC based on the CUSIP and year of the merger. The combined value was then the shares outstanding at end of the year of the merger (CSHO from the CCM data) times the calendar year closing price ($\hbox {PRCC}_{\text{C}}$ from CCM).

For venture, we use the firms from CCM, only keeping data for the first year a firm appears in the CCM data, and defining IPO value as the shares outstanding at the end of that year (CSHO) times the closing price ($\hbox {PRCC}_{\text{C}}$).¹⁹

For both buyouts and venture, we further trim the sample based on firm’s value of common stock at 95% level (i.e. observations at the bottom 2.5% and top 2.5% of common stock values are dropped). Our final sample contains 8209 venture observations and 665 buyout observations. The annual data spans the periods from 1986 to 2017. For each venture and buyout funds in our sample, we compute the $\varepsilon _{\text{v}}$ and $\varepsilon _{ \text{b}}$ as defined in Eqs. (40) and (41), where IPO and combined value are defined as CSHO times the $\hbox {PRCC}_{\text{C}}$. We truncate the distribution of $\varepsilon _{\text{v}}$ and $\varepsilon _{\text{b}}$ by $\frac{1}{q_{t}}$ and $\frac{q_{t}+\tau }{q_{t}}$, respectively, for each time t.

Appendix B: Proofs

1.1 B1: Proof of Proposition 1

Let L satisfy the equation

$$\begin{aligned} L=\left( \beta \int \left( \frac{1+q^{1-1/\gamma }L}{z+\sum \pi _{j}+\theta q \left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q}\right) ^{\gamma }\left[ z+\left( 1-\delta \right) q \right] {\textit{dF}}\left( s\right) \right) ^{1/\gamma }. \end{aligned}$$

(64)

First, suppose that a solution for L exists (its existence will be shown at the end of this proof). Since $\frac{C^{\prime }}{C}=\frac{c^{\prime }}{c} \frac{k^{\prime }}{k}$, (4) implies

$$\begin{aligned} qc^{-\gamma }\left( \frac{k}{k^{\prime }}\right) ^{-\gamma }=\beta \int \left( c^{\prime }\right) ^{-\gamma }\left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\left( s^{\prime }\right), \end{aligned}$$

(65)

therefore

$$\begin{aligned} c\frac{k}{k^{\prime }}=\left( \frac{\beta }{q}\int \left( c^{\prime }\right) ^{-\gamma }\left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\left( s^{\prime }\right) \right) ^{-1/\gamma }, \end{aligned}$$

where $k^{\prime }/k$ is defined in (17).

To simplify notation, we now omit the input of the function and denote $G\equiv \ G^{\text{b}}\left( 1+\frac{\tau }{q}\right)$. From income identity (20)

$$\begin{aligned} i=\frac{z+\sum \pi _{j}+\theta q\left( 1-G\right) -c}{q} \end{aligned}$$

thus we have

$$\begin{aligned} c\frac{k}{k^{\prime }}&= \frac{c}{1-\delta +\frac{1}{q}\left[ z+\ \sum \pi _{j}+\theta q\left( 1-G\right) -c\right] } \\ &= \frac{1}{\frac{1}{c}\left( 1-\delta +\frac{1}{q}\left( z+\sum \pi _{j}+\theta q\left( 1-G\right) \right) \right) -\frac{1}{q}}, \end{aligned}$$

where the second line uses the identity (56) which implies $z-c=qx.$

Therefore

$$\begin{aligned} \frac{1}{\frac{1}{c}\left( 1-\delta +\frac{1}{q}\left( z+\sum \pi _{j}+\theta q\left( 1-G\right) \right) \right) -\frac{1}{q}}=\left( \frac{ \beta }{q}\int \left( c^{\prime }\right) ^{-\gamma }\left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\right) ^{-1/\gamma } \end{aligned}$$

i.e.,

$$\begin{aligned} \frac{1}{c}\left( 1-\delta +\frac{1}{q}\left( z+\sum \pi _{j}+\theta q\left( 1-G\right) \right) \right) =\frac{1}{q}+\left( \frac{\beta }{q}\int \left( c^{\prime }\right) ^{-\gamma }\left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\right) ^{1/\gamma } \end{aligned}$$

i.e.,

$$\begin{aligned} c&= {} \frac{1-\delta +\frac{1}{q}\left( z+\sum \pi _{j}+\theta q\left( 1-G\right) \right) }{\frac{1}{q}+\left( \frac{\beta }{q}\int \left( c^{\prime }\right) ^{-\gamma }\left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\left( s^{\prime }\right) \right) ^{1/\gamma }} \\ &= \frac{z+\sum \pi _{j}+\theta q\left( 1-G\right) +\left( 1-\delta \right) q }{1+q^{1-1/\gamma }\left( \beta \int \left( c^{\prime }\right) ^{-\gamma } \left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\left( s^{\prime }\right) \right) ^{1/\gamma }} \\ &= \frac{z+\sum \pi _{i}\left( q\right) +\theta q\left( 1-G\right) +\left( 1-\delta \right) q}{1+q^{1-1/\gamma }L} \end{aligned}$$

where L is defined in (64), and

$$\begin{aligned} i&= \frac{z+\sum \pi _{j}+\theta q\left( 1-G\right) -c}{q} \\ &= \frac{\left[ z+\sum \pi _{j}+\theta q\left( 1-G\right) \right] q^{-1/\gamma }L-\left( 1-\delta \right) }{1+q^{1-1/\gamma }L}. \end{aligned}$$

Existence of solution for L in Eq. (22). Next, we show that L exists when (23) holds. Divide both sides of (64) by L to get

$$\begin{aligned} 1&= \left( \beta \int L^{-\gamma }\left( \frac{1+q^{1-1/\gamma }L}{z+\sum \pi _{j}+\left( 1-\delta \right) q}\right) ^{\gamma }\left[ z+\left( 1-\delta \right) q\right] {\textit{dF}}\right) ^{1/\gamma } \\ &= \left( \beta \int \left( \frac{L^{-1}+q^{1-1/\gamma }}{z+\sum \pi _{j}+\left( 1-\delta \right) q}\right) ^{\gamma }\left[ z+\left( 1-\delta \right) q\right] {\textit{dF}}\right) ^{1/\gamma }. \end{aligned}$$

Since $L^{-1}$ ranges from zero to infinity as L ranges over the positive line, and since $\gamma >0$, a necessary and sufficient condition for a solution for L to exist is that

$$\begin{aligned} \left( \beta \int \left( \frac{q^{1-1/\gamma }}{z+\sum \pi _{j}+\left( 1-\delta \right) q}\right) ^{\gamma }\left[ z+\left( 1-\delta \right) q \right] {\textit{dF}}\right) ^{1/\gamma }\le 1. \end{aligned}$$

This is equivalent to

$$\begin{aligned} 1\ge \beta ^{1/\gamma }\int q^{\gamma -1}\frac{z+\left( 1-\delta \right) q}{ \left( z+\sum \pi _{j}\left( q\right) +\left( 1-\delta \right) q\right) ^{\gamma }}{\textit{dF}}. \end{aligned}$$

(66)

Since $\pi _{j}$ and $\gamma$ are positive, for (66) to hold it suffices that

$$\begin{aligned} 1>\beta ^{1/\gamma }\int q^{\gamma -1}\frac{z+\left( 1-\delta \right) q}{ \left( z+\left( 1-\delta \right) q\right) ^{\gamma }}{\textit{dF}}=\beta ^{1/\gamma }\int \left( 1-\delta +\frac{z}{q}\right) ^{1-\gamma }{\textit{dF}}, \end{aligned}$$

i.e., (23).

1.2 B2: Proof of Proposition 2

When $\gamma =1$, z and q are constant, Eq. (64) becomes

$$\begin{aligned} L\left( z+\sum \pi _{j}+\left( 1-\delta \right) q\right) =\beta \left( 1+L\right) \left[ z+\left( 1-\delta \right) q\right] . \end{aligned}$$

This implies that

$$\begin{aligned} L&= \frac{\beta \left[ z+\left( 1-\delta \right) q\right] }{\left( z+\sum \pi _{j}+\left( 1-\delta \right) q\right) -\beta \left[ z+\left( 1-\delta \right) q\right] } \\ &= \frac{\beta }{1-\beta +\left[ z+\left( 1-\delta \right) q\right] ^{-1}\sum \pi _{j}}. \end{aligned}$$

We denote

$$\begin{aligned} \mu =\left[ z+\left( 1-\delta \right) q\right] ^{-1}, \end{aligned}$$

then L can be expressed as,

$$\begin{aligned} L=\frac{\beta }{1-\beta +\mu \sum \pi _{j}}. \end{aligned}$$

We then have

$$\begin{aligned} i=\frac{\left( z+q\theta \int _{\varepsilon _{\text{b}}}^{\infty }\varepsilon dG^{\text{b}}\right) q^{-1}\beta -q\left( 1-\delta \right) \left( 1-\beta +\mu \sum \pi _{j}\right) }{1+\mu \sum \pi _{j}}. \end{aligned}$$

Using (6) and (12), we have

$$\begin{aligned} \sum \pi _{j}&= q\theta \int _{\varepsilon _{\text{b}}}^{\infty }\varepsilon dG^{\text{b}}-\left( \tau +q\right) \theta \left[ 1-G^{\text{b}}\left( \varepsilon _{\text{b}}\right) \right] \\ &\quad+q\lambda \int _{\varepsilon _{\text{v}}}^{\infty }\varepsilon dG^{\text{v} }-\lambda \left[ 1-G^{\text{v}}\left( \varepsilon _{\text{v}}\right) \right] \\ &= q\left( \frac{\theta }{\rho _{\text{b}}-1}\left( \frac{\tau +q}{ q\varepsilon _{\text{b, }0}}\right) ^{1-\rho _{\text{b}}}+\frac{\lambda }{ \rho _{\text{v}}-1}\left( \frac{1}{q\varepsilon _{\text{v, }0}}\right) ^{1-\rho _{v}}\right) . \end{aligned}$$

This implies

$$\begin{aligned} i&= \frac{\left( z+q\theta \int _{\varepsilon _{\text{b}}}^{\infty }\varepsilon dG^{\text{b}}\right) q^{-1}\beta -q\left( 1-\delta \right) \left( 1-\beta +\mu \sum \pi _{j}\right) }{1+\mu \sum \pi _{j}} \\ &= \frac{\left( z+\sum \pi _{i}+\theta q\left( 1+\frac{\tau }{q\varepsilon _{ \text{b, }0}}\right) ^{-\rho _{b}}\right) q^{-1}\beta -q\left( 1-\delta \right) \left( 1-\beta +\mu \sum \pi _{j}\right) }{1+\mu \sum \pi _{j}}. \end{aligned}$$

Thus the growth $g=i-\delta$ becomes,

$$\begin{aligned} g=\frac{\left( z+\nu +\theta q\left( \frac{\tau +q}{q\varepsilon _{\text{b, } 0}}\right) ^{-\rho _{b}}\right) q^{-1}\beta -q\left( 1-\delta \right) \left( 1-\beta +\mu \nu \right) }{1+\mu \nu }-\delta , \end{aligned}$$

where

$$\begin{aligned} \nu =q\left( \frac{\theta }{\rho _{\text{b}}-1}\left( \frac{\tau +q}{ q\varepsilon _{\text{b, }0}}\right) ^{1-\rho _{\text{b}}}+\frac{\lambda }{ \rho _{\text{v}}-1}\left( \frac{1}{q\varepsilon _{\text{v, }0}}\right) ^{1-\rho _{v}}\right) . \end{aligned}$$

As for the comparative statics, first we have

$$\begin{aligned} \frac{\partial g}{\partial z}=q^{-1}\beta >0 .\end{aligned}$$

Next, we observe that,

$$\begin{aligned} \frac{\partial \nu }{\partial \rho _{b}}&= -q\theta \frac{\left[ \ln \left( \frac{\tau +q}{q\varepsilon _{\text{b, }0}}\right) \left( \rho _{b}-1\right) +1\right] }{\left( \rho _{\text{b}}-1\right) ^{2}}\left( \frac{\tau +q}{ q\varepsilon _{\text{b, }0}}\right) ^{1-\rho _{\text{b}}} \\ &< 0. \end{aligned}$$

The last line follows because $\varepsilon _{\text{b,}0}<\frac{\tau +q}{q}$ so that $\frac{\tau +q}{q\varepsilon _{\text{b, }0}}>1$ or $\ln \left( \frac{\tau +q}{q\varepsilon _{\text{b, }0}}\right) >0$. Therefore, we have,

$$\begin{aligned} \frac{\partial \nu }{\partial \rho _{b}}<0. \end{aligned}$$

Similarly,

$$\begin{aligned} \frac{\partial \nu }{\partial \rho _{v}}=-q\lambda \frac{\left[ \ln \left( \frac{1}{q\varepsilon _{\text{v, }0}}\right) \left( \rho _{v}-1\right) +1 \right] }{\left( \rho _{\text{v}}-1\right) ^{2}}\left( \frac{1}{q\varepsilon _{\text{v, }0}}\right) ^{1-\rho _{\text{b}}}<0, \end{aligned}$$

where the inequality follows because $\frac{1}{q\varepsilon _{\text{v, }0}} >1$ or $\ln \left( \frac{1}{q\varepsilon _{\text{v, }0}}\right) >0$. This implies that

$$\begin{aligned} \frac{\partial g}{\partial \nu }&= \left( q^{-1}\beta -q\left( 1-\delta \right) \mu \right) \left( 1+\mu \nu \right) \\ &\quad-\mu \left[ \left( z+\nu +\theta q\left( \frac{\tau +q}{q\varepsilon _{ \text{b, }0}}\right) ^{-\rho _{b}}\right) q^{-1}\beta -q\left( 1-\delta \right) \left( 1-\beta +\mu \nu \right) \right] \\ &= q^{-1}\beta \left[ 1+\mu \nu -\mu \left( z+\nu +\theta q\left( \frac{\tau +q}{q\varepsilon _{\text{b, }0}}\right) ^{-\rho _{b}}\right) \right] +q\left( 1-\delta \right) \beta \mu \\ &= q^{-1}\beta \left[ \underset{>0}{\underbrace{\frac{\left( 1-\delta \right) q}{z+\left( 1-\delta \right) q}+\theta \mu q\left( \frac{\tau +q}{ q\varepsilon _{\text{b, }0}}\right) ^{-\rho _{b}}}}\right] +q\left( 1-\delta \right) \beta \left[ z+\left( 1-\delta \right) q\right] ^{-1} \\ &> 0, \end{aligned}$$

which implies that

$$\begin{aligned} \frac{\partial g}{\partial \rho _{v}}&= \frac{\partial g}{\partial \nu } \frac{\partial \nu }{\partial \rho _{v}}<0 \\ \frac{\partial g}{\partial \rho _{b}}&= \frac{\partial g}{\partial \nu } \frac{\partial \nu }{\partial \rho _{b}}<0. \end{aligned}$$

Similarly,

$$\begin{aligned} \frac{\partial \nu }{\partial \tau }=-\theta \left( \frac{\tau +q}{ q\varepsilon _{\text{b, }0}}\right) ^{-\rho _{\text{b}}}<0, \end{aligned}$$

as stated in the proposition.

1.3 B3: Taylor approximation for growth and proof of Corollary 1

We consider a first-order Taylor expansion of the growth $g\left( \lambda ,\theta \right)$ at $\lambda =\theta =0$. First we notice that

$$\begin{aligned} v\left( 0,0\right) =0. \end{aligned}$$

Therefore we have,

$$\begin{aligned} g\left( 0,0\right) =zq^{-1}\beta -q\left( 1-\delta \right) \left( 1-\beta \right) -\delta . \end{aligned}$$

The first-order Taylor approximation reads,

$$\begin{aligned} g=g\left( 0,0\right) +g_{\lambda }\left( 0,0\right) \lambda +g_{\theta }\left( 0,0\right) \theta . \end{aligned}$$

Note that,

$$\begin{aligned} v_{\lambda }= & {} \frac{q}{\rho _{\text{v}}-1}\left( \frac{1}{q\varepsilon _{ \text{v, }0}}\right) ^{1-\rho _{v}} \end{aligned}$$

(67)

$$\begin{aligned} v_{\theta }= & {} \frac{q}{\rho _{\text{b}}-1}\left( \frac{\tau +q}{ q\varepsilon _{\text{b, }0}}\right) ^{1-\rho _{\text{b}}}. \end{aligned}$$

(68)

Therefore, we have

$$\begin{aligned} g_{\lambda }\left( 0,0\right)&= \frac{\left\{ \begin{array}{c} \left[ 1+\left[ z+\left( 1-\delta \right) q\right] ^{-1}\nu \right] \left( q^{-1}\beta \nu _{\lambda }-q\left( 1-\delta \right) \mu v_{\lambda }\right) \\ -\left\{ \begin{array}{c} \left( z+\nu +\theta q\left( \frac{\tau +q}{q\varepsilon _{\text{b, }0}} \right) ^{-\rho _{b}}\right) q^{-1}\beta \\ -q\left( 1-\delta \right) \left( 1-\beta +\mu \nu \right) \end{array} \right\} \left[ \left[ z+\left( 1-\delta \right) q\right] ^{-1}\nu _{\lambda }\right] \end{array} \right\} }{\left[ 1+\left[ z+\left( 1-\delta \right) q\right] ^{-1}\nu \right] ^{2}} \\ &= \left( q^{-1}\beta \nu _{\lambda }-q\left( 1-\delta \right) \mu v_{\lambda }\right) -\left\{ zq^{-1}\beta -q\left( 1-\delta \right) \left( 1-\beta \right) \right\} \left[ \mu \nu _{\lambda }\right] \\ &= \frac{\left( 1-\delta \right) q}{z+\left( 1-\delta \right) q}v_{\lambda }q^{-1}\beta -q\left( 1-\delta \right) \beta \left( \mu \nu _{\lambda }\right) . \end{aligned}$$

Similarly,

$$\begin{aligned} g_{\theta }\left( 0,0\right)&= \left\{ \begin{array}{c} \left( \left( v_{\theta }+q\left( \frac{\tau +q}{q\varepsilon _{\text{b, }0}} \right) ^{-\rho _{b}}\right) q^{-1}\beta -q\left( 1-\delta \right) \left( \left[ z+\left( 1-\delta \right) q\right] ^{-1}\nu _{\theta }\right) \right) \\ -\left[ zq^{-1}\beta -q\left( 1-\delta \right) \left( 1-\beta \right) \right] \left[ z+\left( 1-\delta \right) q\right] ^{-1}\nu _{\theta } \end{array} \right\} \\ &= \frac{\left( 1-\delta \right) q}{z+\left( 1-\delta \right) q}\left[ \left( v_{\theta }+q\left( \frac{\tau +q}{q\varepsilon _{\text{b, }0}} \right) ^{-\rho _{b}}\right) \right] q^{-1}\beta -q\left( 1-\delta \right) \beta \left( \mu \nu _{\theta }\right) , \end{aligned}$$

so we have

$$\begin{aligned} g&\approx \left( z+\frac{\lambda \left( 1-\delta \right) q}{z+\left( 1-\delta \right) q}v_{\lambda }+\frac{\theta \left( 1-\delta \right) q}{ z+\left( 1-\delta \right) q}\left( v_{\theta }+q\left( \frac{\tau +q}{ q\varepsilon _{\text{b, }0}}\right) ^{-\rho _{b}}\right) \right) q^{-1}\beta \\ &\quad-q\left( 1-\delta \right) \left( 1-\beta -\beta \left( \mu \lambda \nu _{\lambda }\right) -\beta \left( \mu \theta \nu _{\theta }\right) \right) -\delta \\&= \left( z+\frac{\lambda \left( 1-\delta \right) q}{z+\left( 1-\delta \right) q}v_{\lambda }+\frac{\theta \left( 1-\delta \right) \rho _{b}q}{ z+\left( 1-\delta \right) q}v_{\theta }\right) q^{-1}\beta -q\beta \left( 1-\delta \right) \left[ \frac{1-\beta }{\beta }-\mu v\right] -\delta , \end{aligned}$$

where the last line follows

$$\begin{aligned} \lambda \nu _{\lambda }+\theta \nu _{\lambda }=v. \end{aligned}$$

Now we plug in (67) and (68), and get

$$\begin{aligned} g&\approx q^{-1}\beta \left( z+\frac{\lambda \left( 1-\delta \right) q}{ z+\left( 1-\delta \right) q}\frac{q}{\rho _{\text{v}}-1}\left( \frac{1}{ q\varepsilon _{\text{v, }0}}\right) ^{1-\rho _{v}}+\frac{\theta \left( 1-\delta \right) q}{z+\left( 1-\delta \right) q}\frac{q\rho _{b}}{\rho _{ \text{b}}-1}\left( \frac{\tau +q}{q\varepsilon _{\text{b, }0}}\right) ^{1-\rho _{\text{b}}}\right) \\ &\quad-q\beta \left( 1-\delta \right) \left[ \frac{1-\beta }{\beta }-\mu v\right] -\delta . \end{aligned}$$

We then get Eq. (29) using the following notation

$$\begin{aligned} {\tilde{A}}&= q^{-1}\beta \left( z+\frac{\left( 1-\delta \right) q}{z+\left( 1-\delta \right) q}\left[ \frac{\lambda q}{\rho _{\text{v}}-1}\left( \frac{1 }{q\varepsilon _{\text{v, }0}}\right) ^{1-\rho _{v}}\right] \right. \\ &\quad \left. +\frac{\left( 1-\delta \right) q}{z+\left( 1-\delta \right) q}\left[ \frac{\theta q\rho _{b}}{\rho _{\text{b}}-1}\left( \frac{\tau +q}{q\varepsilon _{\text{b, }0}} \right) ^{1-\rho _{\text{b}}}\right] \right) \\ a&= q\beta \left( 1-\delta \right) \left( \frac{1-\beta }{\beta }-\left[ z+\left( 1-\delta \right) q\right] ^{-1}v\right) . \end{aligned}$$

Last, we notice that

$$\begin{aligned} \frac{\partial {\tilde{A}}}{\partial q}&= \left[ \begin{array}{c} -\frac{z\beta }{q^{2}}-\frac{\left( 1-\delta \right) q}{z+\left( 1-\delta \right) q}\left[ \underset{>0}{\underbrace{\lambda \left( \frac{1}{ q\varepsilon _{\text{v, }0}}\right) ^{-\rho _{v}}\frac{1}{q^{2}\varepsilon _{ \text{v, }0}}+\theta \rho _{b}\left( \frac{\tau }{q\varepsilon _{\text{b, }0} }\right) ^{-\rho _{b}}\frac{\tau +q}{q^{2}\varepsilon _{\text{v, }0}}}} \right] \\ -\frac{q^{-2}}{\left[ z\left( 1-\delta \right) ^{-1}q^{-1}+1\right] ^{2}} \left[ \underset{>0\text{ because }\rho _{v}>1\text{ and }\rho _{b}>1}{ \underbrace{\frac{\lambda q}{\rho _{\text{v}}-1}\left( \frac{1}{q\varepsilon _{\text{v, }0}}\right) ^{1-\rho _{v}}+\frac{\theta q\rho _{b}}{\rho _{\text{b }}-1}\left( \frac{\tau +q}{q\varepsilon _{\text{b, }0}}\right) ^{1-\rho _{ \text{b}}}}}\right] \end{array} \right] \\ &< 0 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial a}{\partial q}&= \left( 1-\beta \right) \beta \left( 1-\delta \right) +\frac{\beta \left( 1-\delta \right) q^{-2}}{\left( zq^{-1}+\left( 1-\delta \right) \right) ^{2}}v \\ &> 0\text{ because }v>0. \end{aligned}$$

Therefore we have

$$\begin{aligned} \frac{\partial g}{\partial q}=\left( \underset{<0}{\underbrace{\frac{ \partial {\tilde{A}}}{\partial q}}}-\underset{>0}{\underbrace{\frac{\partial a }{\partial q}}}\right) <0 \end{aligned}$$

as stated in Proposition 2 and corollary 1.

1.4 B4: Derivations of Eqs. (35) and (37)

Derivation of (35). We first show the derivation for venture. Since $1-G^{\text{v}}\left( \varepsilon _{\text{v}}\right) =e^{-\lambda _{\text{v}}\varepsilon _{\text{v}}}$, the first equality in ( 33) reads

$$\begin{aligned} \frac{R_{\text{v}}}{R_{\text{E}}}=q\lambda _{\text{v}}e^{\lambda _{\text{v} }\varepsilon _{\text{v}}}\int _{\varepsilon _{\text{v}}}^{\infty }\varepsilon e^{-\lambda _{\text{v}}\varepsilon }d\varepsilon . \end{aligned}$$

(69)

By a change of variable from $\varepsilon$ to $u=\varepsilon -\varepsilon _{ \text{v}}$, the RHS of (69) reads

$$\begin{aligned} \lambda _{\text{v}}e^{\lambda _{\text{v}}\varepsilon _{\text{v} }}\int _{0}^{\infty }\left( \varepsilon _{\text{v}}+u\right) e^{-\lambda _{ \text{v}}\left( \varepsilon _{\text{v}}+u\right) }du=\varepsilon _{\text{v}}+ \frac{1}{\lambda _{\text{v}}}. \end{aligned}$$

(70)

Substitution for $\varepsilon _{\text{v}}$ from (11) into (70) and using (69) yields the first equality in Eq. (35).

Next, the derivation for buyout. Equation (34) reads

$$\begin{aligned} \frac{q}{\tau +q}e^{\lambda _{\text{b}}\varepsilon _{\text{b}}}\lambda _{ \text{v}}\int _{\varepsilon _{\text{b}}}^{\infty }\varepsilon e^{-\lambda _{ \text{b}}\varepsilon }d\varepsilon =\frac{q}{\tau +q}\left( \varepsilon _{ \text{b}}+\frac{1}{\lambda _{\text{b}}}\right) =\frac{q}{\tau +q}\left( \frac{\tau +q}{q}+\frac{1}{\lambda _{\text{b}}}\right) , \end{aligned}$$

i.e., the second equality in (35).

Derivation of (37). When $\varepsilon _{i,0}$ is sufficiently small, Since the parameter $\varepsilon _{\text{i, }0}$ cancels from the ratio $\left( 1-G\left( \varepsilon _{\text{i}}\right) \right) dG^{ \text{i}}=\rho _{\text{i}}\varepsilon ^{-1-\rho _{\text{i}}}$ for $i\in \left\{ \text{v, b}\right\}$, the first equality in Eq. (33) reads

$$\begin{aligned} \frac{\text{R}_{\text{v}}}{R_{\text{E}}}=q\left( \frac{1}{q}\right) ^{\rho _{ \text{v}}}\int _{1/q}^{\infty }\rho _{\text{v}}\varepsilon ^{-\rho _{\text{v} }}d\varepsilon =\left( \frac{1}{q}\right) ^{\rho _{\text{v}}-1}\frac{\rho _{ \text{v}}}{\rho _{\text{v}}-1}\left( \frac{1}{q}\right) ^{1-\rho _{\text{v} }}=\frac{\rho _{\text{v}}}{\rho _{\text{v}}-1}. \end{aligned}$$

(71)

The second equality in Eq. (33) reads

$$\begin{aligned} \frac{\text{R}_{\text{b}}}{R_{\text{E}}}&= q\left( \frac{\tau +q}{q}\right) ^{\rho _{\text{b}}}\frac{1}{\tau +q}\int _{\left( \tau +q\right) /q}^{\infty }\rho _{\text{b}}\varepsilon ^{-\rho _{\text{b}}}d\varepsilon =\frac{\rho _{ \text{b}}}{\rho _{\text{b}}-1}\left( \frac{\tau +q}{q}\right) ^{\rho _{\text{ b}}-1}\left( \frac{\tau +q}{q}\right) ^{1-\rho _{\text{b}}} \\ &= \frac{\rho _{ \text{b}}}{\rho _{\text{b}}-1}. \end{aligned}$$

(72)

Thus we have the two equalities stated in Eq. (37).

1.5 B5: Value function under EZ preferences

Nothing in Proposition 1 changes with EZ preferences, so the proof of Proposition 1 holds.

As for the value function, we first notice that the resource constraint is

$$\begin{aligned} z+\pi _{v}+\pi _{b}+\frac{q}{\tau +q}n_{b}=c+qi. \end{aligned}$$

(73)

The law of motion is

$$\begin{aligned} \frac{k^{\prime }}{k}=1-\delta +i. \end{aligned}$$

(74)

Combining (73) and (74), we get

$$\begin{aligned} c=z-q(\varGamma -1+\delta )+\pi _{v}+\pi _{b}+\frac{q}{\tau +q}n_{b}. \end{aligned}$$

(75)

We will verify that the value function takes the conjectured form:

$$\begin{aligned} v(z,q)k&= \max _{C,I}\left[ (1-\beta )(C)^{1-\psi }+\beta \left( E[v(z^{\prime },q^{\prime })k^{\prime }]^{\prime 1-\gamma }\right) ^{\frac{1-\psi }{ 1-\gamma }}\right] ^{\frac{1}{1-\psi }} \\ &= \max _{C,I}\left[ (1-\beta )(ck)^{1-\psi }+\beta \left( E[v(z^{\prime },q^{\prime })\varGamma k]^{1-\gamma }\right) ^{\frac{1-\psi }{1-\gamma }} \right] ^{\frac{1}{1-\psi }}. \end{aligned}$$

Using Eq. (75), then

$$\begin{aligned} v(z,q)k&\equiv \max _{\varGamma }\left[ \begin{array}{c} (1-\beta )\left( \left( z-q(\varGamma -1+\delta )+\pi _{v}+\pi _{b}+\frac{q}{ \tau +q}n_{b}\right) k\right) ^{1-\psi } \\ +\beta \left( E(v(z^{\prime },q^{\prime })\varGamma k)^{1-\gamma }\right) ^{ \frac{1-\psi }{1-\gamma }} \end{array} \right] ^{\frac{1}{1-\psi }} \\ &= \max _{\varGamma }\left[ \begin{array}{c} (1-\beta )\left( z-q(\varGamma -1+\delta )+\pi _{v}+\pi _{b}+\frac{q}{\tau +q} n_{b}\right) ^{1-\psi }k^{1-\psi } \\ +\beta \left( E(v(z^{\prime },q)\varGamma )^{\prime 1-\gamma }\right) ^{\frac{ 1-\psi }{1-\gamma }}k^{1-\psi } \end{array} \right] ^{\frac{1}{1-\psi }} \\ &= \max _{\varGamma }\left[ \begin{array}{c} (1-\beta )\left( z-q(\varGamma -1+\delta )+\pi _{v}+\pi _{b}+\frac{q}{\tau +q} n_{b}\right) ^{1-\psi } \\ +\beta \left( E(v(z^{\prime },q)\varGamma )^{\prime 1-\gamma }\right) ^{\frac{ 1-\psi }{1-\gamma }} \end{array} \right] ^{\frac{1}{1-\psi }}k. \end{aligned}$$

Therefore, we define

$$\begin{aligned} v(z,q)=\max _{\varGamma }\left[ \begin{array}{c} (1-\beta )\left( z-q(\varGamma -1+\delta )+\pi _{v}+\pi _{b}+\frac{q}{\tau +q} n_{b}\right) ^{1-\psi } \\ +\beta \left( E(v(z^{\prime },q^{\prime })\varGamma )^{1-\gamma }\right) ^{ \frac{1-\psi }{1-\gamma }} \end{array} \right] ^{\frac{1}{1-\psi }} \end{aligned}$$

(76)

so that $v\left( z,q,k\right) =v(z,q)k$ as conjectured.

1.6 B6: Proof of Proposition 4

Taking first order conditions from 76:

$$\begin{aligned}&\left[ (1-\beta )(1-\psi )\left( z-q(\varGamma -1+\delta )+\pi _{v}+\pi _{b}+ \frac{q}{\tau +q}n_{b}\right) ^{-\psi }q\right] \\ &\quad =\beta (1-\psi )\varGamma ^{-\psi }\left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{ 1-\psi }{1-\gamma }}, \end{aligned}$$

which implies that

$$\begin{aligned} \Longrightarrow (1-\beta )qc^{-\psi }=\beta \varGamma ^{-\psi }\left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{1-\psi }{1-\gamma }}. \end{aligned}$$

Substituting our conjectures:

$$\begin{aligned}&(1-\beta )q\left( \frac{z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q}{1+q^{1-1/\psi }L}\right) ^{-\psi } \\ &\quad =\beta \left( 1-\delta +\frac{\left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] \right) q^{-1/\psi }L-\left( 1-\delta \right) }{1+q^{1-1/\psi }L}\right) ^{-\psi } \\ &\qquad \times \left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{ 1-\psi }{1-\gamma }} \end{aligned}$$

and multiplying both sides by $(1+q^{1-\frac{1}{\psi }}L)^{-\psi }$

$$\begin{aligned}&(1-\beta )q\left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{ \text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q\right) ^{-\psi } \\ &\quad =\beta \left( \left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{ \text{b}}\left( 1+\frac{\tau }{q}\right) \right] \right) q^{-1/\psi }L-\left( 1-\delta \right) +(1-\delta )(1+q^{1-\frac{1}{\psi }}L)\right) ^{-\psi } \\ &\qquad \times \left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{ 1-\psi }{1-\gamma }}, \end{aligned}$$

which implies that,

$$\begin{aligned}&\Longrightarrow (1-\beta )q\left( z+\sum \pi _{i}\left( q\right) +\theta q \left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q\right) ^{-\psi } \\ &\quad =\beta \left( \left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{ \text{b}}\left( 1+\frac{\tau }{q}\right) \right] \right) q^{-1/\psi }L+(1-\delta )q^{1-\frac{1}{\psi }}L)\right) ^{-\psi } \\ &\qquad \times \left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{ 1-\psi }{1-\gamma }}. \end{aligned}$$

Multiplying both sides by $L^{\psi }$, we have

$$\begin{aligned}&(1-\beta )q\left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{ \text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q\right) ^{-\psi }L^{\psi } \\ &\quad =\beta \left( \left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{ \text{b}}\left( 1+\frac{\tau }{q}\right) \right] \right) q^{-1/\psi }+(1-\delta )q^{1-\frac{1}{\psi }})\right) ^{-\psi } \\ &\qquad \times \left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{ 1-\psi }{1-\gamma }}, \end{aligned}$$

or equivalently,

$$\begin{aligned} L^{\psi }=\frac{\beta }{1-\beta }\left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{1-\psi }{1-\gamma }}. \end{aligned}$$

This delivers a recursion for L that satisfies the FOC.

Now we need to verify that (42) holds. Notice that

$$\begin{aligned} v(z,q)&= \left[ (1-\beta )\left( z+\sum \pi _{i}\left( q\right) +\theta q \left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] \right. \right. \\ &\quad\left. \left. +\left( 1-\delta \right) q\right) ^{1-\psi }(1+q^{1-1/\psi }L)^{\psi }\right] ^{ \frac{1}{1-\psi }} \\ &= \left[ (1-\beta )\left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q\right) \right] ^{\frac{1}{1-\psi }} \\ &\quad\times \left[ \left( \frac{z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q}{(1+q^{1-1/\psi }L)}\right) ^{-\psi }\right] ^{\frac{1}{1-\psi }} \\ &= \left[ (1-\beta )\left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q\right) c^{-\psi }\right] ^{\frac{1}{1-\psi }}. \end{aligned}$$

If our value function guess holds, then (24) implies

$$\begin{aligned}&\left[ (1-\beta )\left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q\right) c^{-\psi }\right] ^{\frac{1}{1-\psi }} \\ &\quad =\left[ (1-\beta )c^{1-\psi }+\beta \varGamma ^{1-\psi }\left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{1-\psi }{1-\gamma }}\right] ^{ \frac{1}{1-\psi }}. \end{aligned}$$

This implies that,

$$\begin{aligned}&\Longrightarrow (1-\beta )\left( z+\sum \pi _{i}\left( q\right) +\theta q \left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q\right) c^{-\psi } \\ &\quad =(1-\beta )c^{1-\psi }+\beta \varGamma ^{1-\psi }\left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{1-\psi }{1-\gamma }}. \end{aligned}$$

Or equivalently,

$$\begin{aligned}&\Longrightarrow \frac{z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{ \text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q}{ c}\\ &=1 +\left( \frac{\varGamma }{c}\right) ^{1-\psi }\frac{\beta }{1-\beta } \left( Ev(z^{\prime },q^{\prime })^{1-\gamma }\right) ^{\frac{1-\psi }{ 1-\gamma }} \\ &\Longrightarrow \frac{z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{ \text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q}{ c}=1+\left( \frac{\varGamma }{c}\right) ^{1-\psi }L^{\psi }. \end{aligned}$$

Plugging the conjecture for consumption on the LHS:

$$\begin{aligned} 1+q^{1-\frac{1}{\psi }}L=1+\left( \frac{\varGamma }{c}\right) ^{1-\psi }L^{\psi } \end{aligned}$$

which implies

$$\begin{aligned} q^{-\frac{1}{\psi }}L=\frac{\varGamma }{c}. \end{aligned}$$

(77)

Using the conjectures for c and $\varGamma$, we can write the ratio $\frac{ \varGamma }{c}$ as

$$\begin{aligned} \frac{g}{c}&=\left( \frac{z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q}{1+q^{1-\frac{1}{\psi }}L}\right) ^{-1} \\ &\quad\times \left( 1-\delta +\frac{\left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] \right) q^{-\frac{1}{\psi }}L-(1-\delta )}{1+q^{1-\frac{1}{\psi }}L}\right) \\ &=\frac{(1-\delta )(1+q^{1-1/\psi }L)+\left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] \right) q^{-\frac{1}{\psi }}L-(1-\delta )}{z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q} \\ &=\frac{(1-\delta )q^{1-1/\psi }L+\left( z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] \right) q^{-\frac{1}{\psi }}L}{z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q} \\ &=q^{-\frac{1}{\psi }}L .\end{aligned}$$

From Eqs. (77) and (23) it is clear that the conjectured decision rules satisfy the problem.

Last, from the budget constraint, we have

$$\begin{aligned} i=\frac{z+\sum \pi _{i}\left( q\right) +\theta q\left( 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right) -c}{q}, \end{aligned}$$

and we plug in $c=\frac{z+\sum \pi _{i}\left( q\right) +\theta q\left( 1-G\right) +\left( 1-\delta \right) q}{1+q^{1-1/\psi }L}$, we have

$$\begin{aligned} i&= \frac{q^{1-1/\psi }L\left[ z+\sum \pi _{i}\left( q\right) +\theta q\left( 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right) \right] -\left( 1-\delta \right) q}{q\left( 1+q^{1-1/\psi }L\right) } \\ &= \frac{\left[ z+\sum \pi _{i}\left( q\right) +\theta q\left( 1-G^{\text{b} }\left( 1+\frac{\tau }{q}\right) \right) \right] q^{-1/\psi }L-\left( 1-\delta \right) }{1+q^{1-1/\psi }L} \end{aligned}$$

as stated in the proposition.

Proof that $L_{\text{EZ}}=L$ when $\psi =\gamma$. To show that $L_{\text{EZ}}=L,$ we set $\psi =\gamma$ in Eq. (44), which then reads

$$\begin{aligned} L_{\text{EZ}}= & {} \left[ \beta \int \frac{v(z,q)^{1-\gamma }}{1-\beta }{\textit{dF}} \right] ^{1/\gamma }, \end{aligned}$$

(78)

$$\begin{aligned} \text{where}\,v(z,q)^{1-\gamma }= & {} \frac{\left( 1-\beta \right) \left( z+\left( 1-\delta \right) q\right) \left( 1+q^{1-1/\gamma }L_{\text{EZ} }\right) ^{\gamma }}{\left[ z+\sum \pi _{i}\left( q\right) +\theta q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q\right] ^{\gamma }}. \end{aligned}$$

(79)

Substituting from (79) into (78) and simplifying makes the resulting equation the same as Eq. (22) in Proposition 2.

1.7 B7: Proof of Propositions 6 and 7

We first give the proof for venture: Using (53) we find that relative to k the number of ideas is $\lambda h/k=\lambda \kappa$, and the intake of VC funds (6) becomes

$$\begin{aligned} n_{\text{v}}\equiv \#\,\text{VC-backed projects} = \lambda \left[ 1-G^{\text{v}}\left( \varepsilon _{\text{v}}\right) \right] . \end{aligned}$$

(80)

Their payouts now are

$$\begin{aligned} D_{\text{v}}\left( s^{\prime }\right) =\left( \alpha z^{\prime }+\left( 1-\delta \right) q^{\prime }\right) x_{\text{v}}, \end{aligned}$$

and $x_{\text{v}}$ in (8) is normalized by $\kappa$:

$$\begin{aligned} x_{\text{v}}=\lambda \int _{\varepsilon _{\text{v}}}^{\infty }\varepsilon dG^{ \text{v}}. \end{aligned}$$

(81)

A VC fund chooses $\varepsilon _{\text{v}}$ to maximize the expected utility of its investors exactly as in (9). Since $\int \left( \frac{C}{C^{\prime }}\right) ^{\gamma }D_{\text{v}}{\textit{dF}}=x_{\text{v}}\int \left( \frac{C}{C^{\prime }}\right) ^{\gamma }\left( \alpha \frac{y^{\prime } }{k^{\prime }}+\left( 1-\delta \right) q^{\prime }\right) {\textit{dF}}=\frac{q}{\beta } \lambda \kappa \int _{\varepsilon _{\text{v}}}^{\infty }\varepsilon dG^{\text{ v}}$, the problem in (9) reduces to

$$\begin{aligned} \pi _{\text{v}}\left( q\right) \equiv \max _{\varepsilon _{\text{v}}}\left( q\lambda \int _{\varepsilon _{\text{v}}}^{\infty }\varepsilon dG^{\text{v} }-n_{\text{v}}\right) . \end{aligned}$$

(82)

Since $n_{\text{v}}$ in (80) is now also normalized by $\kappa$, we have

$$\begin{aligned} \pi _{\text{v}}\left( q\right) =\pi _{\text{v}}\left( q\right) , \end{aligned}$$

(83)

and the minimal accepted quality of VC-backed projects is still $\varepsilon _{\text{v}}=\frac{1}{q}$ as stated in (11)

Buyout Funds Instead of (12), total buyout fund investment is

$$\begin{aligned} n_{\text{b}}\equiv \left( \tau +q\right) \theta \left[ 1-G^{\text{b}}\left( \varepsilon _{\text{b}}\right) \right] . \end{aligned}$$

(84)

and buyout payouts become

$$\begin{aligned} D_{\text{b}}\left( s^{\prime }\right) =\left( \alpha \frac{y^{\prime }}{ k^{\prime }}+\left( 1-\delta \right) q^{\prime }\right) x_{\text{b}}, \end{aligned}$$

where instead of (14), we have

$$\begin{aligned} x_{\text{b}}=\theta \int _{\varepsilon _{\text{b}}}^{\infty }\varepsilon dG^{ \text{b}}. \end{aligned}$$

(85)

The buyout fund chooses $\varepsilon _{\text{b}}$ to maximize

$$\begin{aligned} \beta \int \left( \frac{C}{C^{\prime }}\right) ^{\gamma }D_{\text{b}}\left( s^{\prime }\right) {\textit{dF}}\left( s^{\prime },s\right) -n_{\text{b}}, \end{aligned}$$

and using the same logic as that behind the proof of (11), we get the buyout fund’s decision problem

$$\begin{aligned} {\hat{\pi }}_{\text{b}}\left( q\right) \equiv \max _{\varepsilon _{\text{b} }}\left( q\theta \int _{\varepsilon _{\text{b}}}^{\infty }\varepsilon dG^{ \text{b}}-n_{\text{b}}\right) , \quad \text{subject to }(\mathrm{A21}), \end{aligned}$$

(86)

with its optimal cutoff rule $\varepsilon _{\text{b}}$ also the same as in (16). Now let L satisfy the equation

$$\begin{aligned} L=\left( \beta \int \left( \frac{1+q^{1-1/\gamma }L}{z+\sum \pi _{j}+\theta \kappa q\left[ 1-G^{\text{b}}\left( 1+\frac{\tau }{q}\right) \right] +\left( 1-\delta \right) q}\right) ^{\gamma }\left[ z+\left( 1-\delta \right) q \right] {\textit{dF}}\left( s\right) \right) ^{1/\gamma }. \end{aligned}$$

(87)

Since $\frac{C^{\prime }}{C}=\frac{c^{\prime }}{c}\frac{k^{\prime }}{k}$, ( 4) implies

$$\begin{aligned} qc^{-\gamma }\left( \frac{k}{k^{\prime }}\right) ^{-\gamma }=\beta \int \left( c^{\prime }\right) ^{-\gamma }\left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\left( s^{\prime }\right), \end{aligned}$$

(88)

therefore

$$\begin{aligned} c\frac{k}{k^{\prime }}=\left( \frac{\beta }{q}\int \left( c^{\prime }\right) ^{-\gamma }\left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\left( s^{\prime }\right) \right) ^{-1/\gamma }, \end{aligned}$$

where $k^{\prime }/k$ is defined in (17). To simplify notation, we now omit the input of the function and denote $G\equiv \ G^{ \text{b}}\left( 1+\frac{\tau }{q}\right)$. From income identity (20)

$$\begin{aligned} i=\alpha \frac{z+\sum \pi _{j}+\theta \kappa q\left( 1-G\right) -c}{q} \end{aligned}$$

thus we have

$$\begin{aligned} c\frac{k}{k^{\prime }}&= \frac{c}{1-\delta +\frac{\alpha }{q}\left[ z+\sum \pi _{j}+\theta \kappa q\left( 1-G\right) -c\right] } \\ &= \frac{1}{\frac{1}{c}\left( 1-\delta +\frac{\alpha }{q}\left( z+\sum \pi _{j}+\theta \kappa q\left( 1-G\right) \right) \right) -\frac{1}{q}}, \end{aligned}$$

where the second line uses the identity (56) which implies $z-c= \frac{q}{\alpha }x.$ Therefore

$$\begin{aligned} \frac{1}{\frac{1}{c}\left( 1-\delta +\frac{\alpha }{q}\left( z+\sum \pi _{j}+\theta \kappa q\left( 1-G\right) \right) \right) -\frac{1}{q}}=\left( \frac{\beta }{q}\int \left( c^{\prime }\right) ^{-\gamma }\left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\right) ^{-1/\gamma } \end{aligned}$$

i.e.,

$$\begin{aligned} c&= \frac{1-\delta +\frac{\alpha }{q}\left( z+\sum \pi _{j}+\theta \kappa q\left( 1-G\right) \right) }{\frac{1}{q}+\left( \frac{\beta }{q}\int \left( c^{\prime }\right) ^{-\gamma }\left[ z^{\prime }+\left( 1-\delta \right) q^{\prime }\right] {\textit{dF}}\left( s^{\prime }\right) \right) ^{1/\gamma }} \\ &= \frac{z+\sum \pi _{i}\left( q\right) +\theta \kappa q\left( 1-G\right) +\left( 1-\delta \right) q}{1+q^{1-1/\gamma }L} \end{aligned}$$

where L is defined in (64), and

$$\begin{aligned} i&= {} \alpha \frac{z+\sum \pi _{j}+\theta \kappa q\left( 1-G\right) -c}{q} \\ &= \alpha \frac{\left[ z+\sum \pi _{j}+\theta \kappa q\left( 1-G\right) \right] q^{-1/\gamma }L-\left( 1-\delta \right) }{1+q^{1-1/\gamma }L}. \end{aligned}$$

as stated in Proposition 7.

Appendix C: Additional tables and figures

See the Table 10 and Fig. 7.

Table 10 Parameters for the estimation under EZ preferences, Freely Estimate $\left( \gamma ,\psi \right)$

Fig. 7

Growth and PE: a numerical example for the Pareto distribution in Eq. (26)