We consider an entrepreneurial firm which can be in one of the two macroeconomic conditions (or regimes), which we denote by $H$ (expansion) and $L$ (contraction). The firm faces different (stochastic) opportunities on investment, production, and financial trading in these two regimes, which furthermore affect the risk attitudes of the entrepreneur that runs the firm. To mimick the empirically observed alternations among different phases of the business cycle, we assume that the dynamics of $H$ and $L$ are governed by a two-state (continuous-time) Markov chain³. More specifically, the macroeconomic condition switches from $H$ to $L$ (or from $L$ to $H$) with a constant probability ${\lambda }^H\text{Δ}$ (or ${\lambda }^L\text{Δ}$) over a short time interval Δ. Thus, the expected duration of regime $i$ is ${\left({\lambda }^i\right)}^{-1}$ and the average fraction of time spent in that regime is ${\lambda }^{i^{\prime }}{\left({\lambda }^H+{\lambda }^L\right)}^{-1}$ for $i^{\prime }\ne i$; $i,i^{\prime }\in \left\{H,L\right\}$. In the following, we use $s\left(t\right)$ to denote the realized macroeconomic condition, which is observable to the entrepreneur.

An entrepreneurial firm obtains productivity from its capital stock $K_t$, which evolves according to

$d{K}_t=\left(I_t-{\delta }_K{K}_t\right)dt+{\sigma }_K{K}_{t}d{Z}_t+{ϵ}^{s\left(t\right)}{I}_{t}d{Z}_t^I,$(2.1)

where $I_t$ is investment; ${\delta }_K>0$ is the depreciation rate. Without loss of generality, we decompose the firm-level risk into two orthogonal components: the usual capital depreciation shock (e.g., Bolton, Wang, and Yang (BWY) [17] ) driven by $d{Z}_t$ with the implied volatility governed by ${\sigma }_K$, and an investment-specific shock driven by $Z_t^I$ where the volatility parameter ${ϵ}^{s\left(t\right)}$ loads on the macroeconomic conditions for $s\left(t\right)\in \left\{H,L\right\}$. The latter specification implies that output fluctuations arise from shocks to the marginal efficiency of investment (Keynes [18] ), and we use the regime-dependent ${ϵ}^{s\left(t\right)}$ to capture the differences in investment opportunities for the firm when it is subject to the macro-level shocks.

The gross output of the firm over the period $\left(t,t+dt\right)$ is proportional to its time- $t$ capital stock $K_t$ by $K_t{A}^{s\left(t\right)}$, where $A^{s\left(t\right)}\in \left\{A^H,A^L\right\}$ which captures the impact of macroeconomic conditions on the firm’s productivity. The firm’s operating profit $d{Y}_t$ over the same period is thus given by

$d{Y}_t=K_t{A}^{s\left(t\right)}dt-I_{t}dt-G\left(I_t,K_t\right)dt,$(2.2)

where $G\left(I,K\right)$ is the adjustment cost. Following Hayashi [19] , we assume that the adjustment cost $G\left(I,K,s\right)$ is convex in $I$ and homogeneous of degree one in $I$ and $K$ by

$G\left(I,K\right)=g\left(i\right)K=\frac{\theta i^2}{2}K,$(2.3)

where $i=I/K$ which denotes the investment-capital ratio; $\theta$ determines the degree of adjustment cost.

Besides capital investment and the resulting productivity, the firm can also invest in a risk-free asset which pays a constant rate of interest $r$ and the risky market portfolio. Assume that the incremental return $d{R}_t$ of the market portfolio over the time period $dt$ follows:

$d{R}_t={\mu }_R^{s\left(t\right)}dt+{\sigma }_{R}d{B}_t,$(2.4)

where $B_t$ is the standard Brownian representing the systematic (or market) risk which is correlated with the firm-level risks of $Z_t$ and $Z_t^I$ by and $\rho$ and ${\rho }_I$, respectively⁴. In (2.4), ${\sigma }_R$ denotes the constant volatility parameter of the market portfolio return process. Suited to our setup, we allow the average return of market portfolio, ${\mu }_R^{s\left(t\right)}$, to load on the macroeconomic conditions, which reflects the changable market conditions that the firm is facing.

Let $W$ and $X$ denote the firm’s liquid wealth and the amount invested in the risky asset, respectively. Their difference, $W-X$, is thus invested in the risk-free asset. Out of its liquid asset, the firm pays the investment cost and consumes. Thus, $W_t$ evolves according to

$d{W}_t=\left[r\left(W_t-X_t\right)+{\eta }^{s\left(t\right)}{\sigma }_R{X}_t+d{Y}_t-C_t\right]dt+{\sigma }_R{X}_{t}d{B}_t,$(2.5)

where $d{Y}_t$ is given by (2.2);

${\eta }^{s\left(t\right)}\equiv \frac{{\mu }_R^{s\left(t\right)}-r}{{\sigma }_R}$

which denotes the market Sharpe ratio⁵. Since ${\mu }_R^{s\left(t\right)}$ is regime-dependent, so is ${\eta }^{s\left(t\right)}$ which summarizes the impact of macroeconomic conditions on the firm’s stochastic opportunities on finanial trading.

The entrepreneur of the firm is equipped with a recursive preference (e.g., Epstein and Zin [20] ; Dufie and Epstein [21] ) by

$J_t=E_t\left[{\displaystyle {\int }_t^{\infty }f\left(C_s,J_s\right)\text{d}s}\right],$(2.6)

where $J$ denotes the entrepreneur’s utility; $f\left(C,J\right)$ is known as the normalized aggregator for consumption $C$ and it is regime dependent in our setup by

$f^{s\left(t\right)}\left(C_t,J_t^{s\left(t\right)}\right)=\frac{\zeta }{1-1/\psi }\frac{C_t^{1-1/\psi }-{\left(\left(1-{\gamma }^{s\left(t\right)}\right)J_t^{s\left(t\right)}\right)}^{{\chi }^{s\left(t\right)}}}{{\left(\left(1-\gamma \right)J_t^{s\left(t\right)}\right)}^{{\chi }^{s\left(t\right)}-1}},$(2.7)

where $\zeta$ denotes the usual subjective discount rate; $\psi >0$ measures the elasticity of substitution (EIS); $\gamma >0$ is the coefficient of relative risk aversion;

${\chi }^{s\left(t\right)}\equiv \frac{1-1/\psi }{1-{\gamma }^{s\left(t\right)}}.$⁶(2.8)

The regime-dependence of $f$ is attributed to the entrepreneur’s persistent preference shocks, which are induced by the macro-level fluctuations. Such shocks are captured by the entrepreneur’s regime-dependent risk aversion, indicated by ${\gamma }^{s\left(t\right)}$, which captures the usual intuition that she is more risk averse when the aggregate economy is in a bad condition.

In summary, our setup involves two different types of shocks: 1) a small diffusive shock as captured by $d{Z}_t$, $d{Z}_t^I$, and $d{B}_t$, and 2) a large shock when the macroeconomic condition changes. While potentially all model parameters are subject to the large shock, for parsimony we only allow four parameters, $ϵ$, $A$, $\eta$, and $\gamma$, to load on the macroeconomic conditions. Since these four parameters cover the firm’s investment ( $ϵ$) and production ( $A$), the market condition that it faces ( $\eta$), as well as the entrepreneur’s risk attitudes ( $\gamma$), our parametric choices for regime-dependences seem representative for understanding the impact of macro level shocks on an entrepreneurial firm’s operations.

Let $J^m\left(K,W\right)$ denote the value function for the entrepreneur when the macroeconomic condition is $m\in \left\{H,L\right\}$. By applying the principle of dynamic programming to $J^m$, we obtain the following Hamilton-Jacobi-Bellman (HJB) equation:

$\begin{array}{c}0=\underset{C,X,I}{\text{max}}{f}^m\left(C_t,J^m\right)+\left[r{W}_t+{\eta }^m{\sigma }_R{X}_t+A^m{K}_t-I_t-G\left(I_t,K_t\right)-C_t\right]J_W^m\\ +\left(I-{\delta }_{K}K\right)J_K^m+\frac{{\left({ϵ}^m\right)}^2{I}^2+{\sigma }_K^2{K}^2}{2}{J}_{KK}^m+\left({\rho }_I{ϵ}^{m}I+\rho {\sigma }_{K}K\right){\sigma }_{R}X{J}_{KW}^m\\ +\frac{{\sigma }_R^2{X}^2}{2}{J}_{WW}^m+{\lambda }^m\left[J^{m^{\prime }}\left(K,W\right)-J^m\left(K,W\right)\right],\end{array}$(3.1)

where $J_W^m$, $J_K^m$, $J_{KK}^m$, $J_{KW}^m$, and $J_{WW}^m$ denote the partial derivatives of $J^m$; $m^{\prime }\ne m$; we have used (2.1), (2.2), and (2.5) ⁷. As indicated by the last term of (3.1), $J^m$ transitions into $J^{m^{\prime }}$ with the intensity ${\lambda }^m$ when macroeconomic condition shifts from $m$ into $m^{\prime }$. Consequently, the two HJBs on $J^m$ and $J^{m^{\prime }}$ are intrinsically linked to one another. Intuitively, a rational entrepreneur adapts to the current macroeconomic condition while she anticipates the macro-level fluctuations would drive the corporate environment, together with her risk preferences, into a different regime at any time with certain probability.

By the first-order conditions (FOCs), the firm’s optimal policies on consumption $C^m$, investment $I^m$, and asset allocation $X^m$ during the macro regime $m$ are determined according to:

$\left[C^m\right]:f_C^m\left(C^m,J^m\right)=J_W^m\left(K,W\right),$(3.2)

$\left[I^m\right]:\left[1+G_I\left(I^m,K\right)\right]J_W^m=J_K^m+{\left({ϵ}^m\right)}^2{I}^m{J}_{KK}^m+{\rho }_I{ϵ}^m{\sigma }_R{X}^m{J}_{KW}^m,$(3.3)

$\left[X^m\right]:X^m=-\frac{{\mu }_R-r}{{\sigma }_R^2}\frac{J_W^m}{J_{WW}^m}-\frac{\left(\rho {\sigma }_{K}K+{\rho }_I{ϵ}^m{I}^m\right){\sigma }_R}{{\sigma }_R^2}\frac{J_{KW}^m}{J_{WW}^m},$(3.4)

when the aggregate economy is in state $m$, where $f^m$ is given by (2.7); $G$ is given by (2.3). Du [9] provides the financial interpretations of the above FOCs for the single-regime case which are largely applicable here except for the differences that $J^m$ for $m\in \left\{H,L\right\}$ have to be jointly determined in an internally compatible manner.

To actually solve the above HJBs with the stochastic controls, we conjecture (and verify later) that $J^m$ can be written as

$J^m\left(K,W\right)=\frac{{\left(b^m{P}^m\left(K,W\right)\right)}^{1-{\gamma }^m}}{1-{\gamma }^m}=\frac{{\left(b^{m}K{p}^m\left(w\right)\right)}^{1-{\gamma }^m}}{1-{\gamma }^m},$(3.5)

where $b^m$ is a constant for the given $m\in \left\{H,L\right\}$; $w\equiv W/K$ which denotes the firm’s financial slack; $P^m\left(K,W\right)\equiv K{p}^m\left(w\right)$ is interpreted as the certainty-equivalent (CE) valuation of the firm by the entrepreneur for the given macroeconomic condition⁸. Under (3.5), we have the following expressions for $J^m$-derivatives:

$J_W^m={\left(b^m\right)}^{1-{\gamma }^m}{\left(p^m\left(w\right)K\right)}^{-{\gamma }^m}{p}^m\left(w\right),$(3.6)

$J_K^m={\left(b^m\right)}^{1-{\gamma }^m}{\left(p^m\left(w\right)K\right)}^{-{\gamma }^m}\left(p^m\left(w\right)-w\cdot {\left(p^m\right)}^{\prime }\left(w\right)\right),$(3.7)

$J_{WW}^m={\left(b^m\right)}^{1-{\gamma }^m}{\left(p^m\left(w\right)K\right)}^{-{\gamma }^m-1}\left[p^m\left(w\right)\cdot {\left(p^m\right)}^{\prime \text{}\prime }\left(w\right)-{\gamma }^m{\left({\left(p^m\right)}^{\prime }\left(w\right)\right)}^2\right],$(3.8)

$J_{KW}^m={\left(b^m\right)}^{1-{\gamma }^m}{\left(p^m\cdot K\right)}^{-{\gamma }^m-1}\left[-w{p}^m\cdot {\left(p^m\right)}^{\prime \text{}\prime }-{\gamma }^m{\left(p^m\right)}^{\prime }\left(p^m-w\cdot {\left(p^m\right)}^{\prime }\right)\right],$(3.9)

$J_{KK}^m={\left(b^m\right)}^{1-{\gamma }^m}{\left(p^m\cdot K\right)}^{-{\gamma }^m-1}\left[w^2{p}^m\cdot {\left(p^m\right)}^{\prime \text{}\prime }-{\gamma }^m{\left(p^m-w\cdot {\left(p^m\right)}^{\prime }\right)}^2\right].$(3.10)

Consistent with the formulation of (3.5), we treat the firm’s capital $K_t$ as the scaling factor and use lower case letters to denote the following variables: firm’s liquid wealth $w_t=W_t/K_t$, the agent’s CE valuation of the firm $p_t^m=P_t^m/K_t$, consumption $c_t^m=C_t^m/K_t$, investment $i_t^m=I_t^m/K_t$, and risky asset al. location $x_t^m=X_t^m/K_t$. Using (3.5) to simplify (2.7), which are then substituted into (3.2), we obtain the following consumption rule after making use of (3.6):

$c^m\left(w\right)={\zeta }^{\psi }{\left(b^m\right)}^{1-\psi }{\left[{\left(p^m\right)}^{\prime }\right]}^{-\psi }{p}^m.$(3.11)

By substituting (3.6)-(3.10) into (3.4)-(3.3) and performing necessary manipulations, the optimal investment and asset allocation policies as functions of $w$ are given by

$i^m\left(w\right)=\frac{p^m-\left(w+1\right){\left(p^m\right)}^{\prime }-{\rho }_I{ϵ}^m{\left(p^m\right)}^{\prime }\frac{{\gamma }^m{p}^m-w{h}^m}{p^m}\left[{\eta }^m\frac{p^m}{h^m}-\rho {\sigma }_K\left(\frac{{\gamma }^m{p}^m}{h^m}-w\right)\right]}{\theta {\left(p^m\right)}^{\prime }+\left(1-{\rho }_I^2\right){\left({ϵ}^m\right)}^2\frac{{\left(p^m\right)}^{\prime }}{p^m}\frac{{\left({\gamma }^m{p}^m-w{h}^m\right)}^2}{h^m}-{\gamma }^m{\left({ϵ}^m\right)}^2\frac{{\left(p^m\right)}^2{\left(p^m\right)}^{\prime \text{}\prime }}{h^m{\left(p^m\right)}^{\prime }}}$(3.12)

$x^m\left(w;i^m\right)=\frac{{\eta }^m}{{\sigma }_R}\frac{p^m}{h^m}-\frac{{\rho }_Iϵi^m+\rho {\sigma }_K}{{\sigma }_R}\left(\frac{{\gamma }^m{p}^m}{h^m}-w\right),$(3.13)

where $m\in \left\{H,L\right\}$;

$h^m\equiv {\gamma }^m{p}^m-\frac{p^m{\left(p^m\right)}^{\prime \text{}\prime }}{{\left(p^m\right)}^{\prime }}$(3.14)

By substituting (2.7), optimal policies of (3.11)-(3.12), and the expressions of (3.6)-(3.10) into (3.1), making use of the conjectured (3.5), which applies to both $J^H$ and $J^L$, and performing necessary simplifications, tedious algebra gives:

$\begin{array}{c}0=\frac{{\zeta }^{\psi }{\left(b^m\right)}^{1-\psi }{\left({\left(p^m\right)}^{\prime }\right)}^{1-\psi }-\zeta \psi }{\psi -1}{p}^m+\left(rw+A^m\right){\left(p^m\right)}^{\prime }+\frac{1}{2}\frac{{\left({\eta }^m\right)}^2{p}^m{\left(p^m\right)}^{\prime }}{h^m}\\ -{\delta }_K\left(p^m-w\cdot {\left(p^m\right)}^{\prime }\right)-\frac{{\sigma }_K^2\left(1-{\rho }^2\right)}{2}\frac{p^m{\left(p^m\right)}^{\prime }}{h^m}{\left({\gamma }^m-\frac{w{h}^m}{p^m}\right)}^2\\ -\rho {\sigma }_K{\eta }^m{\left(p^m\right)}^{\prime }\left(\frac{{\gamma }^m{p}^m}{h^m}-w\right)+\frac{{\gamma }^m{\sigma }_K^2}{2}\frac{{\left(p^m\right)}^2{\left(p^m\right)}^{\prime \text{}\prime }}{h^m{\left(p^m\right)}^{\prime }}\\ +\frac{1}{2}\frac{{\left[p^m-\left(w+1\right){\left(p^m\right)}^{\prime }-{\rho }_I{ϵ}^m{\left(p^m\right)}^{\prime }\frac{{\gamma }^m{p}^m-w{h}^m}{p^m}\left({\eta }^m\frac{p^m}{h^m}-\rho {\sigma }_K\left(\frac{{\gamma }^m{p}^m}{h^m}-w\right)\right)\right]}^2}{\theta {\left(p^m\right)}^{\prime }+\left(1-{\rho }_I^2\right){\left({ϵ}^m\right)}^2\frac{{\left(p^m\right)}^{\prime }}{p^m}\frac{{\left(\gamma p^m-w{h}^m\right)}^2}{h^m}-{\gamma }^m{\left({ϵ}^m\right)}^2\frac{{\left(p^m\right)}^2{\left(p^m\right)}^{\prime \text{}\prime }}{h^m{\left(p^m\right)}^{\prime }}}\\ +\frac{{\lambda }^m{p}^m\left(w\right)}{1-{\gamma }^m}\left[{\left(\frac{b^{m^{\prime }}\cdot p^{m^{\prime }}\left(w\right)}{b^m\cdot p^m\left(w\right)}\right)}^{1-{\gamma }^m}-1\right]\end{array}$(3.15)

where $b^{m^{\prime }}$ and $p^{m^{\prime }}\left(w\right)$ in the last term come from the expression of $J^{m^{\prime }}\left(K,W\right)$ after the switch of the macroeconomic condition; $m,m^{\prime }\in \left\{H,L\right\}$ with $m\ne m^{\prime }$. A simplified version of (3.15) is provided in Equation (3.17) of Du [9] for the single-regime case where macro-level fluctuations are shut down such that 1) the transition density $\lambda$ is set at zero; and 2) the regime-dependences of $b$, $p$, $h$ and the four parameters of $ϵ$, $A$, $\eta$, and $\gamma$ are all shut down.

At the upper end when $w\to \infty$, the firm achieves the first-best (FB) for the given macroeconomic condition $m$ as follows:

$\underset{w\to \infty }{\text{lim}}{p}^m\left(w\right)=p^{FB,m}=w+q^{FB,m},$(3.16)

where $p\left(w\right)$ satisfies (3.15); $q^{FB,m}$ denotes the constant valuation of the firm-held capital under FB. Intuitively, $p\left(w\right)$ can be decomposed into $w+q\left(w\right)$ where $w$ denotes the liquid wealth per unit of capital and $q\left(w\right)$ denotes the liquid-wealth valuation of capital. The $w$-dependence of $q$ is attributed to the time-varying liquidation risk since a higher $w$ provides the better buffer against the potential liquidation. At $w\to \infty$ that shuts down the liquidation risk, $q$ becomes a constant within the given macroeconomic condition.

To identify $q^{FB,m}$ for $m\in \left\{H,L\right\}$, substituting (3.16) into (3.12) and (3.15), taking the limit $w\to \infty$, and simplying⁹, we obtain

$i^m=i^{FB,m}=\frac{q^{FB,m}-1-{\rho }_I{ϵ}^m\eta q^{FB,m}}{\theta }$(3.17)

and

$\begin{array}{c}0=\underset{i}{\text{max}}\left(\frac{{\zeta }^{\psi }{\left(b^m\right)}^{1-\psi }-\psi \zeta }{\psi -1}+r+\frac{{\left({\eta }^m\right)}^2}{2{\gamma }^m}\right)\left(w+q^{FB,m}\right)\\ +\frac{{\lambda }^m\cdot \left(w+q^{FB,m}\right)}{1-{\gamma }^m}\left[{\left(\frac{b^{m^{\prime }}}{b^m}\right)}^{1-{\gamma }^m}{\left(\frac{\underset{w\to \infty }{\text{lim}}{p}^{m^{\prime }}\left(w\right)}{\underset{w\to \infty }{\text{lim}}{p}^m\left(w\right)}\right)}^{1-{\gamma }^m}-1\right]\\ +A^m-i-\frac{1}{2}\theta i^2-\left(r+{\delta }_K-i\right)q^{FB,m}-\left({\rho }_I{ϵ}^{m}i+\rho {\sigma }_K\right){\eta }^m{q}^{FB,m},\end{array}$(3.18)

respectively. Since (3.18) has to hold for all $w$, $b^m$ has to satisfy

$0=\frac{{\zeta }^{\psi }{\left(b^m\right)}^{1-\psi }-\psi \zeta }{\psi -1}+r+\frac{{\left({\eta }^m\right)}^2}{2{\gamma }^m}+\frac{{\lambda }^m}{1-{\gamma }^m}\left[{\left(\frac{b^{m^{\prime }}}{b^m}\right)}^{1-{\gamma }^m}-1\right],$(3.19)

for $m,m^{\prime }\in \left\{H,L\right\}$ with $m\ne m^{\prime }$, where we’ve used (3.16) for both $H$ and $L$ so that the limiting ratio in (3.18), $\underset{w\to \infty }{\text{lim}}{p}^{m^{\prime }}\left(w\right)/\underset{w\to \infty }{\text{lim}}{p}^m\left(w\right)$, is simply one. (3.18) now degenerates to

$0=\underset{i}{\text{max}}{A}^m-i-\frac{1}{2}\theta i^2-\left(r+{\delta }_K-i\right)q^{FB}-\left({\rho }_I{ϵ}^{m}i+\rho {\sigma }_K\right){\eta }^m{q}^{FB,m},$(3.20)

and a substitution of (3.17) into (3.20) gives the valuation of capital under FB as follows:

$q^{FB,m}=\frac{1+\theta i^{FB,m}}{1-{\rho }_I{ϵ}^m{\eta }^m},$(3.21)

where¹⁰

$i^{FB,m}=\frac{r+\rho {\sigma }_K{\eta }^m+{\delta }_K}{1-{\rho }_I{ϵ}^m{\eta }^m}-\sqrt{{\left[\frac{r+\rho {\sigma }_K{\eta }^m+{\delta }_K}{1-{\rho }_I{ϵ}^m{\eta }^m}\right]}^2-\frac{2}{\theta }\left[A^m-\frac{r+\rho {\sigma }_K{\eta }^m+{\delta }_K}{1-{\rho }_I{ϵ}^m{\eta }^m}\right]}.$(3.22)

Turning to boundary conditions at the lower end, the firm gets liquidated when $w$ becomes sufficiently negative upon which the firm-held capital stock yields a residual value of $l{K}_t$ for $l\in \left(0,1\right)$. Upon liquidation during the macro regime $m\in \left\{H,L\right\}$, the entrepreneur sells the firm for $W+lK$ and becomes a Merton consumer (Merton [22] ), where her value function takes the form of

$J^{M,m}\left(W+lK\right)=\frac{{\left[b^m\left(W+lK\right)\right]}^{1-{\gamma }^m}}{1-{\gamma }^m}.$(3.23)

In (3.23), $b^m$ is solved by (3.19)¹¹ and our quantitative analyses show that $W_t+l{K}_t>0$ always holds so that the entrepreneur as a Merton consumer always starts with a positive wealth.

Let $W_d^m$ denote the firm’s liquidation boundary, quoted in terms of the firm’s liquid wealth, when the macroeconomic condition is $m$. Since $W_d$ is optimally chosen, we have the following value matching and smooth-pasting conditions:

$J^m\left(K,W_d^m\right)=J^{M,m}\left(W_d^m+lK\right),$(3.24)

${\frac{\partial J^m\left(K,W\right)}{\partial W}|}_{W=W_d^m}={\frac{\partial J^{M,m}\left(W+lK\right)}{\partial W}|}_{W=W_d^m}.$(3.25)

where $J^m\left(K,W\right)$ and $J^{M,m}\left(W\right)$ are defined by (3.5) and (3.23), respectively. Simplifying (3.24)-(3.25) by making use of (3.5), (3.23), and the scaled variables, we obtain

$p^m\left(w_d^m\right)=w_d^m+l,$(3.26)

${\left(p^m\right)}^{\prime }\left(w_d^m\right)=1.$(3.27)

where $w_d^m\equiv W_d^m/K$.

In our model, $J^H$ and $J^L$ denote the entrepreneur’s value function during the expansion and the contraction regime, respectively. Correspondingly, $J^{FB,H}$ and $J^{FB,L}$ denote the entrepreneur’s value function under first-best which characterize the boundary conditions at the upper end for $J^H$ and $J^L$, respectively, while $J^{M,H}$ and $J^{M,L}$ denote a Merton consumer’s value function during the expansions and contractions which characterize the boundary conditions at the lower end for $J^H$ and $J^L$, respectively. For a direct comparison, let $J$, $J^{FB}$, and $J^M$ denote the entrepreneur’s value function, her value function under first-best, and a Merton consumer’s value function, respectively, in the single-regime case when macro-level fluctuations are shut down¹². Table 1 summarizes the connections among $J$, $J^H$, $J^L$, $J^{FB}$, $J^{FB,H}$, $J^{FB,L}$, $J^M$, $J^{M,H}$, $J^{M,L}$.

As illustrated in Table 1 , $J^H$ and $J^L$ serve as the continuation value for each other. Apart from such linkages, $J^m$ for $m\in \left\{H,L\right\}$ is similarly characterized as its single-regime counterpart $J$ in terms of the two boundary conditions when $w\to \infty$ and when $w\to w_d^m$. While Table 1 focuses on the interconnections among the different value functions, the involved mechanics within a given value function are provided in Du [9] which remain largely unaffected by the macro-level fluctuations.

When the aggregate economy is in recession and there is no hope to get out of it, Figure 2 shows (lines in circle) that the firm’s investment and asset allocation, as well as its capital’s valuation, would be severely depressed, which is accompanied with the enhancement of the entrepreneur’s consumption. Once we account for the empirically observe business cycle alternations by setting a positive ${\lambda }^L$, however, the entrepreneur, by anticipating the recovery of the economy from contraction into expansion, would adjust the firm-level policies and assign a higher value to capital during the bad times. We refer to such changes as the recovery effect, which is officially plotted in Panel A-D of Figure 3 where the firm’s financial status, as measured by $w$, is set at 0.5 which ensures an alive firm at all ${\lambda }^{L}s$.

As ${\lambda }^L$ rises from 0 to its baseline level of 0.5, the entrepreneur in bad times becomes more optimistic in that she anticipates a sooner recovery of the aggregate economy. In response, she re-allocates more resources from consumption to investment as indicated by Panel A&B of Figure 3 , so that she can benefit from the firm’s enhanced capital accumulation and the resulting higher outputs to be realized in the future. The rising $i^L$ as the function of ${\lambda }^L$ (Panel B) is consistent with Rafferty’s [33] finding that firm’s R&D expenditures, as a type of investment, drops during initial recessionary periods but rise later periods when entrepreneurs are expecting a high probability of recovery. Simultaneously and as plotted in Panel C, the higher ${\lambda }^L$ prompts the entrepreneur to keep adjusting $w_d^L$ downwards so as to maintain the control of the firm for a longer term. Specifically, the firm can easily get liquidated at a very high $w_d^L$ of 0.413 when there is no hope to leave the contraction regime at ${\lambda }^L=0$, while firm’s liquidation is substantially delayed at $w_d^L=-0.633$ when the contraction regime is expected to only last for two years at ${\lambda }^L=0.5$.

¹⁸Mechancially, $b^L$ decreases from 0.151 to 0.0473 as $1/{\lambda }^L$ rises from 2 to $\infty$.

Given that the baseline value of ${\gamma }^L$ is greater than one, (3.5) implies that the entrepreneur’s utility $J^L$, which measures her welfare with the firm during the bad times, is negative as indicated by plot in Panel D of Figure 3 . As the duration of the contraction regime $1/{\lambda }^L$ rises from 2 to $\infty$, $J^L$ decreases dramatically from −23.5 to −1144 which implies a huge welfare loss borne by the entrepreneur when bad times are expected to last longer. Intuitively, as ${\lambda }^L$ decreases so that $1/{\lambda }^L$ rises, the entrepreneur gradually loses the hope for recovery which depresses her utility in a substantial way. Our numerical solutions show that this impact on welfare is mainly driven by 1) a smaller $b^L$ at a lower ${\lambda }^L$¹⁸ which implies a less efficient transfer of firm valuations into the entrepreneur’s utility; 2) a higher degree of risk-adjustment in bad times when the entrepreneur is more risk averse with a higher ${\gamma }^L$. Indeed, in an unreported exercise we find that $p^L$, the entrepreneur’s CE-valuation of the firm per unit of capital, only decreases from 1.60 to 1.40 when $1/{\lambda }^L$ rises from its baseline level to $\infty$. We conclude that the recovery effect has its largest impact on the entrepreneur’s welfare and the implied liquidation policy¹⁹.

While varying ${\lambda }^L$ has the direct impact on variables during the contraction regime through the recovery effect, it also affects the entrepreneur’s behaviors and the resulting welfare during the expansion regime. We refer to such influences as the spillover effect, which is plotted in Panel E-H of Figure 3 (solid lines) where $w$ is once again set at 0.5. When compared to the recovery effect shown in Panel A-D, the spillover effect is qualitatively similar but quantitatively weaker. Take the implied $w_d^H$ (Panel G) as the example. As ${\lambda }^L$ rises, an entrepreneur in good times also delays the firm’s liquidation because the potential recession, which occurs at the rate of ${\lambda }^H=0.1$, feels less costly when the subsequent recovery comes sooner at the higher ${\lambda }^L$. While this impact of ${\lambda }^L$ is foreseen by the entrepreneur during the expansion regime, its actual effect on $w_d^H$ is indirect and quantitatively much weaker than that on $w_d^L$. Indeed, while raising ${\lambda }^L$ from 0 to 0.5 substantially reduces $w_d^L$ (from 0.413 to −0.633), it reduces $w_d^H$ only marginally (from −0.835 to −0.838).

To further gauge the spillover effect, we also plot in Panel E-H the expansion-regime implications when the macroeconomic dynamics are shut down (dashed lines). Without the interactions between the two regimes, the entrepreneur’s behaviors and her welfare in good times no longer react to variations of ${\lambda }^L$. The interesting observation is that the discrepancies between the dashed and the solid lines are the largest at ${\lambda }^L=0$. Financially, setting ${\lambda }^L=0$ but maintaining ${\lambda }^H$ at its baseline level, as plotted in solid lines at their leftmost points, implies that contraction becomes an absorbing regime for firms in good times. With the anticipating of the permanent recession, the entrepreneur in good times mimicks her bad-time behaviors the most, which gives rise to the largest discrepancy mentioned above. As ${\lambda }^L$ rises, however, the economy recovers from contraction more easily so that the entrepreneur’s behaviors are more influenced by the current macroeconomic conditions. Consequently, the implied discrepancies between the solid and the dashed lines gradually shrink. As ${\lambda }^L\to \infty$, the spillover effect and hence the implied discrepancy vanishes because the contraction regime now exists for only an instant of time, which is no longer a concern for entrepreneur during the good times.

Our analyses on the impact of ${\lambda }^L$ provide a new insight about the impact of the macroeconomic environment on firm operations: It will be wrong to conclude that recessions have small effect on firm’s policies just because the ex-post responses to the large shock, i.e., the regime switch from $L$ to $H$, are small. Indeed, Panel A-C of Figure 3 shows that the entrepreneur takes actions ahead of the realization of the shock. For example, she substantially scales back her consumption in regime $L$ when ${\lambda }^L$ rises from 0 to 0.5 which is a main contributor to the overall reduction in $c$ from $c^{L,0}$ ( = 0.0593) at ${\lambda }^L=0$ as plotted in Panel A, to $c^H$ (= 0.170) at ${\lambda }^L=0.5$ as plotted in Panel E. The same logic also applies to the policy makings with $i$, $x$ (not plotted) and $w_d$. In other words, the ex-ante responses of the firm in regime $L$ already contribute to the actual behaviors observed in $H$ so a small observed policy responses to macro-level shocks from $H$ to $L$ does not imply that such shocks are unimportant to the firm’s optimal operations.

Figure 4 plots the comparatics analysis with respect to $A^L$ which measures the capital’s productivity during bad times. A higher $A^L$ raises the valuation of capital $q$ (Panel A&E) and capital investment $i$ (Panel C&F) in both regime $L$ and regime $H$ irrespective of the firm’s financial status $w$. This implication helps explain the findings by Zona [34] that firms respond to economic downturn of 2008-2009 with innovation investment: Such investment enhances $A^L$ which not only benefits the firm in the short-term but also delivers the longer-term beneficial effects after the aggregate economy has recovered. While a higher $A^L$ also encourages consumption $c^L$ (Panel B), the implied magnitudes are much smaller which is attributed to the entrepreneur’s intertemporal smoothing motives²⁰.

²⁰While not plotted, we find that raising $A^L$ leaves $c^H$ largely unaffected.

Since holding the risky market portfolio on average yields negative returns during the contraction regime, the entrepreneur chooses to take a short position in the stock market as indicated by a negative $x^L$ (Panel D). Raising $A^L$ from 0.15 (solid line) to 0.2 (dashed line) in general prompts the entrepreneur to take a more aggressive short position since the implied higher $i^L$, as discussed in last paragraph, induces a larger capital stock that is used as the collateral for taking the short positions. When $w$ is sufficiently negative, however, a lower (solid line in Panel D) instead of a higher $A^L$ induces the more aggressive position in the stock market. Intuitively, the entrepreneur has the incentive to gamble for resurrection when the firm is close to its liquidation and this effect is particularly strong when $A^L$ is small. While not plotted, we find the same gambling motive also leads to a more aggressive (long) position in the stock market during the expansion regime, albeit by a much smaller magnitude.

The top two panels of Figure 5 plot the impact of investment risk as measured by its volatility parameter $ϵ$. Once again we focus on parametric variations during the bad times which seems a bigger concern to the entrepreneur, so that $ϵ={ϵ}^L$. Panel A of Figure 5 confirms the intuition that lowering investment risk by reducing ${ϵ}^L$ encourages investment when $i^L>0$. When the firm is in a relatively bad financial status (as measured by $w$), it disinvests and a lower ${ϵ}^L$ (dashed line) facilitates the disinvestment as well which makes the implied $i^L$ even more negative²¹. In sum, a lower ${ϵ}^L$ raises $\left|i^L\right|$. In other words, a higher investment risk not only depresses investment but also depresses the disinvestment.

Consistent with the plot in Panel D of Figure 4 , the implied $x^L$ plotted in Panel B of Figure 5 is negative, whose absolute value rises with $w$, indicating a more aggressive stock market position when the firm has acheived a higher degree of financial slack. Varying ${ϵ}^L$ also affects $x^L$ because it is intrinsically linked to

$i^L$ through the intertemporal hedging component $-\frac{{\rho }_I{ϵ}^L{i}^L}{{\sigma }_R}\left(\frac{\gamma p^L}{h^L}-w\right)$, where $\frac{\gamma p^L}{h^L}-w$ can be roughly interpreted as a risk-adjusted version of the capital’s

valuation during bad times. Given a positive ${\rho }_I$ (see Table 2 ), this component and hence $x^L$ itself is turned more negative when we raise ${ϵ}^L$ from 0.1 (dashed line) to 1 (solid line) at $i^L>0$. Naturally, the opposite holds when the firm starts to disinvest at relatively low ws. In that case, raising ${ϵ}^L$ may actually raise $x^L$

by making it less negative as also plotted in Panel B. Financially speaking, our model predicts that a higher investment risk induces a more (less) aggressive asset allocation policy when the firm’s financial slack is high (low), and this prediction is clearly testable. Our numerical results further show that ${ϵ}^L$ has little impact on other variables and we omit their plots for brevity.

Panel C-F of Figure 5 plot the impact of ${\eta }^L$ which denotes the market Sharpe ratio during the bad times. When ${\eta }^L$ is reduced from 0 (dashed lines) to −0.2 (solid lines), the firm’s position in the stock market (Panel E) changes from near 0 ²² to negative and this effect is very pronounced when the firm is in a healthy financial status as indicated by relatively large $w$s. Taking the proceeds from its short position, the firm simultaneoulsy raises its investment at ${\eta }^L=-0.2$ (Panel D). The same logic seems to suggest that the entrepreneur’s consumption will also get raised. However, our numerical solution shows that the opposite holds as plotted in Panel C, which is consistent with the usual intuition that a worse market condition as measured by a lower ${\eta }^L$ tends to depress consumption. Since the entrepreneur is forward-looking, a lower ${\eta }^L$ simulatenously depresses her consumption in the expansion regime as plotted in Panel F.

While not plotted, we find that varying ${\eta }^L$ has little effects on other variables. In particular, it has little impact on $x^H$ which is worth some more comments. Financially, the firm’s asset allocation policy is mainly driven by its myopic component (see discussions in footnote 12). For $x^H$, this component is determined by ${\eta }^H$ which apparently cannot exert its influence until the occurrence the macro-level shock from $L$ to $H$. In other words, the impact of ${\eta }^m$ on firm’s asset allocation policy is myopic and contingent on regime $m\in \left\{H,L\right\}$ which is in sharp contrast to the persistent impact of $A^m$ (see plots in Figure 4 with $A^L$ as the example).

We now examine the impact of the entrepreneur’s persistent preference shocks which are captured by her regime-dependent risk aversion ${\gamma }^m$ suited to the macroeconomic condition $m$ for $m\in \left\{H,L\right\}$. Panel A-D of Figure 6 plot the implied $q^H$, $c^H$, $i^H$, and $x^H$ as the function of $w$ under two scenarios of ${\gamma }^H$: ${\gamma }^H=2$ (solid lines) and ${\gamma }^H=4$ (dashed lines). Reducing ${\gamma }^H$ from 4 to 2 implies less degree of discounting which naturally raises the capital’s valuation $q^H$ (Panel A). In terms of the firm’s optimal policies, the lower degree of risk aversion enables the entrepreneur to bear more of the investment and the market risks. Consequently, she transfers resources from consumption (Panel B) to investment (Panel C) and asset allocation (Panel D) for obtaining a higher productivity and a higher expected return to be realized in the future²³. In unreported exercises, we find that reducing ${\gamma }^H$ has a similar impact on $q^L$, $c^L$, and $i^L$ with comparable magnitudes, which tells the persistent influences of the entrepreneur’s preference shocks²⁴.

Turning to the impact of ${\gamma }^L$, Panel E-H of Figure 6 plot the implied $q^L$, $c^L$, $i^L$, and $x^L$ for ${\gamma }^L=3$ and ${\gamma }^L=5$. As illustrated in Panel E, reducing ${\gamma }^L$ from 5 (dashed lines) to 3 (solid lines) raises the capital’s valuation $q^L$ which simultaneously delays the firm’s liquidation. Empirically, small firm are subject to higher risks (Beaver and Ross [35] ), so it seems reasonable that entrepreneurs of smaller business have higher appetites for risks, i.e., a lower $\gamma$. Interpreted this way, the implication from Panel E helps explain the findings by Latham [36] that smaller firms tend to be more resistant to recessionary pressures.

In general, a lower ${\gamma }^L$ prompts the entrepreneur to invest more (or divest less; Panel G) and take a more aggressive (short) position in the stock market (Panel H). The differences is that when $w$ is sufficiently close to $w_d^L$, it is a higher (dashed lines) instead of a lower (solid lines) degree of risk aversion that encourages investment which is financed by proceeds from a more aggressive short position in the stock market. This is because the entrepreneur is subject to the unhedgeable business risk (see footnote 4) and abandoning the firm through liquidation provides the benefit of risk diversification. Realizing this benefit, the entrepreneur has weaker incentives to delay liquidation by cutting investment when she is more risk averse. Since the unhedgeable business risk is more costly during bad times as indicated by a higher $w_d^L$ than $w_d^H$, the implied diversification effect is stronger which may convert a negative $i^L$ serving to delay liquidation, into a positive $i^L$ so as to obtain a larger liquidation payment²⁵. These mixed results on $i^L$ support Ghemawat [37] who makes a strong case that managers during economic downturns face a heightened tension on the financial risk of investing against the competitive risk of not investing. Our numerical solution further suggests that such a positive $i^L$ at very negative ws is financially supported by proceeds of a rather aggressive short position in the stock market²⁶.