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We propose a tractable model of entrepreneur dynamics where the investment conditions are stochastic. Applying the approach of stochastic control and optimization, we solve the dynamics of the entrepreneur’s optimal investment, consumption and portfolio allocation under regime switching. We find that the interactions of precautionary savings and liquidation boundary advance/postpone motives generate rich implications for entrepreneur dynamics. Facing the threat of financial crisis, entrepreneurs build cash reserves and bring forward liquidation option exercise to mitigate downside risk. During the bad times, entrepreneurs value financial slack and postpone liquidation boundary to maintain the business and wait for the good state to come.

The ability of regime-switching models to capture the cyclical features of real macroeconomic variables as proposed by Hamilton [

To address these questions, we propose a quantitative model to study the dynamics of entrepreneur’s optimal consumption, investment and portfolio allocation when the dynamics of decision variables are subject to discrete regime shifts at random times. Our model builds on the dynamic framework of entrepreneur’s optimal policies (see Wang, Wang and Yang [

The analysis in the present paper relates to two different strands of literature. First, it relates to the entrepreneurship literature. Hurst, Lusardi [

Second, the present paper relates to a series of recent papers on regime switching. Driffill, Kenc, Sola [

The paper that is most closely related to the present analysis is Bolton, Chen and Wang [

The remainder of the paper is organized as follows. Section 2 presents the model of entrepreneur dynamics under regime switching. Section 3 derives the model solution. Section 4 presents parameter choices and quantitative results. Section 5 concludes.

In this section, we propose our quantitative model to study the dynamics of entrepreneurship under regime switching. Our model is built on WWY [

Stochastic investment conditions and production technology. The entrepreneur employs only capital and cash as the factor of production. We denote as the gross investment and as capital stock. As is standard in capital accumulation models, the change of capital stock evolves according to

where is the rate of depreciation.

We assume the cumulative productivity shock follows arithmetic Brownian motion process. Thus, the firm’s productivity shock over the period is given by

where is a standard Brownian motion, is the mean of the productivity shock in state, and is the volatility of the productivity shock in both states.

The firm’s operating revenue over period is given by. After investment and adjustment cost, the firm’s operating profit over the same period is given by

where the price of the investment good is set to unity and is the adjustment cost.

Following Hayashi [

where is the firm’s investment-capital ratio and is an increasing and convex function. To make the analysis simple and tractable ( see WWY [

where the parameter measures the degree of the adjustment cost. The higher, the more adjustment cost occurs.

Finally, the entrepreneur can liquidate its assets at any moment. The flexibility of liquidation option makes the entrepreneur optimally manage his downside business risk. Liquidation gives a terminal value, where depends on the state. A higher implies a higher liquidation value. Obviously, we have, which makes liquidation in the good state much more attractive than in the bad state. Let denote the entrepreneur’s optimally chosen liquidation time.

Financial investment opportunities. In our model, the entrepreneur has financial investment opportunities to partially hedge his business risk. The entrepreneur allocates his liquid financial wealth between a risk-free asset which pays a constant rate of interest and the risky market portfolio (Merton [

where is the mean of the market portfolio return in state and is the volatility of the market portfolio return in both states, and B is a standard Brownian motion. The correlation coefficient between and is less than 1, which implies there exists nondiversifiable risk and the entrepreneur can’t completely hedge his business risk. Let denote the Sharpe ratio of the market portfolio in state, which is given by

We denote and as the agent’s financial wealth and the amount invested in the risky asset respectively. Given the entrepreneur’s operating profit, investment, consumption and portfolio allocation, we can write the dynamics of liquid financial wealth as follows:

where the firm term is the return on risk-free asset, the second and third terms are the return on market portfolio, the fourth term is the entrepreneur’s consumption, and the last term is the firm’s cash flows from operations.

We assume the entrepreneur can use capital as collateral to borrow and the borrowing is risk-free. Thus, the liquidation value of capital must be greater than outstanding liability. So the following Equation holds,

The Entrepreneur’s Objective. The entrepreneur maximizes his utility defined as,

where for Epstein-Zin non-expected homothetic recursive utility (Duffie and Epstein [

where

In the above utility function, the parameter is the elasticity of intertemporal substitution (EIS), and the parameter is the coefficient of relative risk aversion. The parameter is the agent’s subjective discount rate. To maximize his utility, the entrepreneur chooses investment, consumption and portfolio allocation subject to the collateralized borrowing limit (1.9). And the agent optimally chooses the liquidation time.

The entrepreneurial value depends on both its capital stock and its cash holdings. Thus, let denote the value function in state. It satisfies the following system of Hamilton-Jacobi-Bellman (HJB) Equations:

The first term on the right side of the HJB Equation (1.13) represents the one-period utility. The second term represents the effect of capital stock changes on entrepreneurial value. The third and fourth terms represent the effect of the expected change in cashing holdings and volatility of on entrepreneurial value and the last term is the expected change of entrepreneurial value when the state changes form to.

Next, we solve the first-order conditions (FOC) with respect to consumption, investment and portfolio choice respectively. The FOC with respect to consumption is

which shows the marginal utility of consumption is equal to the marginal utility of wealth at optimality.

The FOC with respect to investment is given by

which implies the entrepreneur’s marginal cost of investing is equal to the marginal benefit of adding a unit of capital in state.

The FOC with respect to portfolio choice is given by

which states that the optimal amount investing in market portfolio is equal to the mean-variance demand (the first term on the right side) plus the hedging demand (the second term).

Then, we conjecture that the value function is given by

where is given in appendix B and is the entrepreneur’s certainty equaivalent (CE) wealth.

Using the homogeneity property, we can write scaled CE wealth as in each state. We substitute the FOCs with respect to, , , the value function form and the scaled CE wealth into the HJB Equation (1.13), and then get the systems of ordinary differential Equations (ODE) for scaled CE wealth. The results are summarized in the following theorem.

Theorem 1 The scaled CE wealth solves the following system of ODEs,

where is given by

and is given by

Intuitively, can be referred to as the idiosyncratic component of the total volatility of the productivity shock and is the effective risk aversion (see WWY [

Next, we specify the boundary conditions. When approaches infinity, approaches the first-best solution given by

where is calculated in the appendix B.

At the endogenous liquidation boundary, we have the following value matching and smooth pasting conditions for,

Finally, the optimal scaled consumption, investment, and market portfolio allocationcapital ratio are given by

where is the marginal value of cash in state.

In this section, we illustrate the quantitative results for given parameter choices of the model. First, we specify our choice of parameters. In state G, the expected productivity is set to be (Eberly, Rebelo, and Vincent [

The other parameters remain the same in the two states, and we set the parameters by widely-used numbers (see WWY [

In state G, the optimal liquidation boundary . At this point, the firm hasn’t reached the boundary compared with the benchmark. Further running the business would help the entrepreneur earn more profit. However, doing so would mean taking the risk that the state of nature switches to the bad state when the investment condition is much worse. The entrepreneur makes the trades off and optimally exercises the liquidation option by liquidating when hits the lower barrier. However, the optimal liquidation boundary in state B is lower than the benchmark, which implies that the firm struggles to maintain the business and waits for the possible arrival of good state.

Panel C of

Comparing the transitory case with the benchmark case, displays different dynamics. In good state, for an entrepreneur with sufficient financial slack, is higher than the benchmark, since wealth can mitigate the additional risk associated with regime switching. For an entrepreneur with low financial slack, is lower than the benchmark.

However, in the bad state, the entrepreneur values his cash more than in the benchmark, no matter his financial slack is sufficient or not. Intuitively, the entrepreneur may value wealth less, aware of the possibility of the state of nature switching to the good state. It is incorrect, because the entrepreneur needs more cash to maintain the business and postpone the liquidation option exercise.

Panel D plots marginal value of capital, which is also referred to as the marginal. The entrepreneur with medium cash holdings values capital more in the bad state and the one with extreme low (near the liquidation boundary) and high cash holdings values capital more in the good state. When, the marginal approaches to average shown in Panel B.

Compared to the benchmark, investment in the good state is lower for the entrepreneur with sufficient financial slack. Aware of the threat of switching to the bad state, the entrepreneur cuts investment and hoards cash for precautionary purpose. However, the entrepreneur

reduces disinvestment and even boosts investment to the level higher than the benchmark near the liquidation boundary. When is near the liquidation boundary, the cash hoarding motive is dominated by the early liquidation option exercise motive, thus leading to the investment boost. In the bad state, the investment is higher than the benchmark since the possible arrival of the good state. Similarly, the investment in state B in lower than the benchmark when is sufficiently low. Having the possibility of the state of nature switching to the good state in mind, the entrepreneur cuts investment to maintain the business and wait for the good state to come.

Panel B of

Panel A of

As expected, the portfolio allocation in the good state is higher than the portfolio allocation in the bad state and so does consumption. Similar to the investment, the entrepreneur chooses to accelerate consumption and financial investment when the firm approaches the liquidation boundary. Strikingly, and decrease as w increases near the liquidation boundary, which is counterintuitive.

In fact, the entrepreneur needs to burn cash to bring forward the liquidation option exercise. In the bad state, the difference between and the benchmark when w is large is significantly larger than that when w is near the liquidation boundary. Intuitively, the entrepreneur needs to hoard cash to postpone the liquidation option exercise.

Apparently, the impact of regime switching on portfolio allocation and consumption is different. Comparing Panel A and Panel B in

We propose a quantitative model to study the dynamics of entrepreneur’s optimal consumption, investment and portfolio allocation when the dynamics of decision variables are subject to discrete regime shifts at random times. Our model builds on the dynamic framework of entrepreneur’s optimal policies by adding stochastic investment conditions.

We have shown that when the entrepreneur faces uncertain macro conditions, it is optimal for them to hoard cash for precautionary reasons. In addition, the entrepreneur may choose the optimal liquidation boundary to eliminate the risk associated with regime switching. The analysis shows that precautionary savings and liquidation boundary advance/postpone can have significant value and generate rich implications for entrepreneur dynamics.

During favorable macroeconomic conditions, the cash hoarding motive is reflected in the lower investment,

consumption and portfolio allocation for the entrepreneur with sufficient cash holdings. Interestingly, we find that the need to burn cash to bring forward liquidation option exercise dominates the cash hoarding motive, thus making the boost of investment, consumption and portfolio allocation when the firm is near the liquidation boundary.

During a financial crisis, the entrepreneur cuts investment, delays consumption, lowers portfolio allocations and sometimes engages in asset sales. This is especially true when the entrepreneur enters the crisis with low cash reserves. These predictions are consistent with the stylized facts about firm behaviour during the recent financial crisis. In prospect of the possible arrival of the good state, the investment, consumption and portfolio allocation is higher than the benchmark for the entrepreneur with sufficient wealth. As the firm is near the liquidation boundary, the entrepreneur brings investment, consumption and portfolio allocation back to the benchmark to maintain the business and wait for the good state to come.

The authors acknowledge financial support from Natural Science Foundation of China (# 71202007), Innovation Program of Shanghai Municipal Education Commission (# 13ZS050), “Chen Guang” Project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation (# 12CG44) and Innovation Research Fund of Shanghai University of Finance and Economics (# CXJJ-2013-317 and # CXJJ-2013-312)

We conjecture that the value function is given by Equation (1.17). Using homogeneity property of, we can obtain Equation(1.25), (1.26) and (1.27) for, and respectively. Substituting these results into Equation (1.13), we obtain the system of ODEs, i.e. Equations (1.18) and (1.19).

We consider the lower liquidation boundary. When, the entrepreneur liquidates the firm. Using the value-matching condition at, we have

where is given by

is the agent’s value function after liquidation and with no business. The entrepreneur’s optimal liquidation strategy implies the following smooth-pasting condition at the endogenously determined liquidation boundary:

Using, Equations (1.1.1)-(1.1.3), and simplifying, we obtain the scaled value-matching and smooth pasting conditions given in Equations (1.23) and (1.24), respectively.

As approaches infinity, firm value approaches the first-best value and

which implies Equation (1.22).

The CE wealth, where is given by

Let, , and replace them in Equations (1.18) and (1.19), we have

When w converges to infinity, approaches to, approaches to and the coefficient before w is zero:

Solving Equations (1.2.3) and (1.2.4), we obtain that and satisfy

Consider the remaining part, and we find that and satisfy