Entrepreneurs face high uncertainty and liquidity constraints, both of which have significant influences on the business decision making process. These uncertainty and constraints are important determinants for capital ac- cumulation, exit decision and asset allocation. Since [1] , the real option approach has become an essential part of optimization problem for the entrepreneurs. [2] develop an incomplete-market q-theoretic model to study entrepreneurship dynamics and find that the illiquidity, idiosyncratic risks and borrowing constrains result in business decisions, consumption and asset allocation decisions different with the standard complete markets profit-maximizing analysis for entrepreneurial firms.

In the standard optimization model framework, it is assumed that agents have a constant rate of time pre- ference. However, virtually every experimental research on time preference indicates that this assumption is unrealistic. The most relevant effect of time inconsistent is the preference change with time. Usually, an agent has different preferences at different stages. In this article, we focus on the entrepreneurs’ decision making under non-constant preference.

In our paper, we try to focus on the following interesting questions: What is the impact of time inconsistent preference on the entrepreneurial valuation? How could the time varying discount rate affects the liquidation choices? How would the entrepreneur allocate the wealth between investment, consumption and public equity? We extend the entrepreneurial optimization model in [2] by incorporating the time inconsistence preferences.

Our model gives three main contributions. First, the time inconsistent agent will under invest when his liquid wealth is close to liquidation boundary and over invest when the agent is far away from the exit threshold. Second, time inconsistent preferences weaken the effects of risk aversion on the liquidation decisions. Third, increasing the possibility of the birth of the future stage with utility discount has ambiguous influences on the exit decision and investment strategy and finally the decisions approach the steady result in the future stage.

Our research relates to the literature about time inconsistency model and its application. [3] models time varying impatience with quasi hyperbolic discount functions and explains why consumers have asset-specific marginal propensities. [4] describe the equilibrium of a discrete-time exchange economy in which consumers with arbitrary subjective discount factors and homothetic period utility functions follow linear Markov consumption and portfolio strategies. [5] consider two types of goods: goods with immediate costs and delayed benefits, and goods with immediate benefits and delayed costs. With time inconsistency model, they explain how to design optimal contract respond to consumer biases. [6] extend the real option framwork to model the investment timing decisions of entrepreneurs with time inconsistent preferences.

The remainder of the paper proceeds as follows: Section 2 presents the model; Section 3 derives the solutions; Section 4 provides the quantitative results and Section 5 concludes.

In this section, we set out the framework for the basic model of entrepreneurial optimization problem. The liqui- dation option is described and followed by a discussion of the nature of time inconsistent preferences.

Time is continuous and horizon is infinite. There is a single perishable consumption good. The agent derives utility from a consumption process according to

where is a concave function. For tractability, we choose, where is the coefficient

of relative risk aversion. denotes the agent’s intertemporal discount function: the agent’s value at time of $1 received at the future time s. We thus have

for. As in [7] , the present stage could last for a random duration of time. For simplicity, we assume that the lifespan of present agent is exponentially distributed with parameter. Stated in another way, the birth of future agent is modeled as a Poisson process with intensity. We define where measures the degree of the agent’s utility discount in future stage. We assume the agent is in the present stage, and thus the value function is:

Consider the setting for a standard entrepreneurial problem as in [2] . The agent possesses a firm and our pro- duction specification features the widely used “AK” technology augmented with capital adjustment costs. Let I denote the gross investment. The change of capital stock is given by:, where is the rate of depreciation. The firm’s productivity shock over the period is independently and identically distributed (iid), and is given by:, where is a standard Brownian motion, and are the mean and volatility of the productivity shock respectively. The firm’s operating revenue over period is proportional to and is given by. The firm’s operating profit over the same period is given by:

where is the adjustment cost. We assume that the firm’s adjustment cost is homogeneous of degree one in and, and write in the homogeneous form as:, where

is the firm’s investment-capital ratio. For simplicity, we assume, where the parameter

measures the degree of the adjustment cost.

The entrepreneur has an option to liquidate capital at any moment. Liquidation is irreversible and gives a terminal value, where is a constant. Let denote the optimal liquidation time. When the entre- preneur is not well-diversified, liquidation provides an important channel to manage the downside risk exposure.

The agent can invest in risk free asset and public equity. These two financial asset represent the standard investment opportunities in the classical [8] model. The risk free asset accumulates with a constant interest rate. Incremental return of public equity, , over time period is iid: , where is a standard Brownian motion, and are the constant expected mean and volatility, respectively. The sharp

ratio for the public equity is:. Let denote the correlation coefficient between the public equity

and the entrepreneurial business. The non diversifiable risks play a role in the decision making process.

Let be the amount allocated to the risky public equity at time, and denote the liquid financial wealth process. Before, the agent holds the firm and acts as an entrepreneur. The liquid financial wealth process evolves as follows:

After exiting from the business, the agent retires and accumulates in the following form:

The agent is allowed to borrow against capital at all times in our model. To make sure the debt is risk free, we set the liquidation value of the capital greater than outstanding debt:

The optimization problem of the agent involves the maximization of the utility defined as (3). First, before liquidation, the entrepreneurial objective is to choose a consumption process, a portfolio allocation rule, the investment process and an optimal liquidation timing strategy to maximize the utility subject to the wealth dynamics (5) and borrowing constrain (7). After the liquidation option has been exercised, the entrepreneur collects the liquidation proceeds and retires. And then the agent chooses optimal allocation between the risk free asset, public equity and consumption.

Parameter choices. Where possible, we borrow the parameters from [2] . We set. For public equity, and. Adjusted cost and depreciation. For production shock, and. Capital liquidation price.

This paper extends the entrepreneurial dynamics model to account for time inconsistent preferences. Entrepreneurs need to formulate the investment decisions taking into account the possibility of future stage with utility discount. This sets up a conflict between two opposing forces. First, the agent desires to take advantage of the option to exit, and also has an incentive to invest more and longer to offset the utility discount in future stage. Second, the time inconsistent preference lead to motivation of precautionary saving and thus reducing the investment. We extend the model of [2] to consider the decision making process for an industry made up of time inconsistent entrepreneurs.

We find that time inconsistency leads to under investment when the entrepreneurial liquid wealth is close to the liquidation boundary and over investment when the liquid wealth is far away from the exit threshold. For further analysis, we study the effects of some key factors in our model. Inconsistency weakens the effects of risk aversion which accelerates the liquidation. The effects of the correlation between the firm and the public equity are ambiguous and non monotonic in both cases, but the inconsistent setting delays the exit decision compared to consistent model. The magnitude of Poisson process intensity and the utility discount determine the degree of the entrepreneurial time inconsistency. Increasing promotes the possibility for the birth of the future stage in which there exists utility discount. We find that the entrepreneurial CE wealth is negatively correlated with and will approach the future stage steady value with increasing. The effects of on the liquidation decisions and the investment strategy are ambiguous. At a low level, increasing delays the liquidation and promotes the investment. But a higher accelerates the exit and reduces the investment. measures the utility discount in future stage and thus the entrepreneurial CE wealth decreases sharply when reducing. The investment is negatively correlated to. The liquidation boundary becomes stable when is close to 1 but decreases significantly when is lower.

Jinqiang Yang acknowledges support by Natural Science Foundation of China (#71202007 #71472117), New Century Excellent Talents in University (#NCET-13-0895) and Chen Guang Project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation (#12CG44).