Fiscal stimulus as an optimal control problem

Abstract

During the Great Recession, Democrats in the United States argued that government spending could be utilized to “grease the wheels” of the economy in order to create wealth and to increase employment; Republicans, on the other hand, contended that government spending is wasteful and discourages investment, thereby increasing unemployment. This past year we have found ourselves in the midst of another crisis where government spending and fiscal stimulus is again being considered as a solution. In the present paper, we address this question by formulating an optimal control problem generalizing the model of Radner and Shepp (1996). The model allows for the company to borrow continuously from the government. We prove that there exists an optimal strategy; rigorous verification proofs for its optimality are provided. We proceed to prove that government loans increase the expected value of a company. We also examine the consequences of different profit-taking behaviors among firms who receive fiscal stimulus.

Introduction

The purpose of this paper is to mathematically model optimal fiscal policy with the hopes of contributing to the ongoing debate between United States’ Democrats and Republicans on what government measures should be taken to stimulate the economy. The model was originally developed by the second named and the third named authors in 2010, shortly after the Great Recession and the U.S. government’s subsequent bailout of large banks. In the midst of another major recession from the COVID-19 pandemic, this work offers a timely addition to the third named author’s seminal contributions to mathematical finance and optimal control.²

The key macroeconomic question considered in this paper is how much – if at all – should the government inject into the private sector to improve aggregate wealth of the economy? To this end, we assume that a company can be characterized by four parameters, $(x,\mu ,\sigma ,r)$: the present cash reserve, $x>0$, the profit rate, $\mu >0$, the riskiness, $\sigma >0$, and the prevailing interest rate, $r\ge 0$. Let $(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathsf{P})$ be a filtered probability space, where ${\mathcal{F}}_t$ represents the market information available at time $t$. Let $W\left(t\right)$ be a Wiener process w.r.t. ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$, representing uncertainty, and $Z\left(t\right)$ be the total dividends subtracted from the fortune up until time $t$. The cash reserve of the company at time $t$, denoted by $X\left(t\right)$, evolves according to the dynamics

$$X\left(t\right)=x+\mu t+\sigma W\left(t\right)-Z\left(t\right),t\ge 0,$$

where $W\left(0\right)=0$ and $Z\left(0\right)\ge 0$.

To model fiscal policies that can “grease the wheels” of the private sector, we assume that the government may choose to provide a loan to the firm in order to increase the firm’s expected profit rate from $\mu$ to ${\mu }^{\ast }$, where it is assumed that ${\mu }^{\ast }\ge \mu >0$. This loan is to be repaid at the interest rate $r$. It is assumed that the firm may borrow continuously at a limited rate from the government.³ The above considerations lead us to write the process $Z\left(t\right)$ as $Z\left(t\right)=Z_{+}\left(t\right)-Z_{-}\left(t\right)$ where $Z_{+}\left(t\right)$ and $Z_{-}\left(t\right)$, respectively, represent the total dividends (which may be taxed) paid up to time $t$ and the total loans taken up to time $t$. Hence, we require $Z_{\pm }\left(t\right)\ge 0$ for any $t$ and rewrite Eq. (1) as

$$X\left(t\right)=x+\mu t+\sigma W\left(t\right)+Z_{-}\left(t\right)-Z_{+}\left(t\right).$$

Let $\mathcal{A}$ be the collection of admissible controls $(Z_{+},Z_{-})$ which satisfy assumptions (A1)–(A3) below:

• $Z_{+}\left(t\right)$ is a nondecreasing and càdlàg process adapted to ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$.

• $Z_{+}\left(0\right)\in [0,x]$, and $\Delta Z_{+}\left(t\right)=Z_{+}\left(t\right)-Z_{+}(t-)\le X(t-)$ for every $t>0$.

• $Z_{-}\left(t\right)$ is a nondecreasing and differentiable process adapted to ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ with $Z_{-}\left(0\right)=0$ and $d{Z}_{-}\left(t\right)/dt\le {\mu }^{\ast }-\mu$ for some ${\mu }^{\ast }\ge \mu$.

Condition (A2) implies that the company cannot make a lump-sum dividend payment greater than its current fortune. Condition (A3) means that the company may be supplied any amount less than or equal to $({\mu }^{\ast }-\mu )dt$ in each interval $dt$. It is worth noting that the restriction we put on $Z_{-}\left(t\right)$ is different from the usual setup of dividend problems where singular controls are used to represent instantaneous large-scale capital injections. We require that the government loan is paid back at interest rate $r$, and the objective of the firm is to maximize its expected net value (to be defined) by choosing the policy $(Z_{+},Z_{-})$ optimally. Mathematically, this means we want to compute the value function defined as⁴

$$V\left(x\right)=V(x;\mu ,{\mu }^{\ast })=\underset{(Z_{+},Z_{-})\in \mathcal{A}}{sup}{\mathsf{E}}_x{\int }_{0-}^{{\tau }_0}{e}^{-rt}[d{Z}_{+}\left(t\right)-d{Z}_{-}\left(t\right)],$$

where ${\tau }_0=inf\{t:X\left(t\right)\le 0\}$ is the time that the company goes bankrupt and the notation ${\int }_{0-}^{\tau }dZ\left(t\right)$ should be understood as $Z\left(0\right)+{\int }_0^{\tau }dZ\left(t\right)$. The definition implies that $V\left(0\right)=0$. Further, $V\left(x\right)\ge x$ for any $x$ since one strategy for taking profit is to take the fortune immediately; bankruptcy occurs at time ${\tau }_0=0$.

If we replace $\mathcal{A}$ with a smaller admissible class ${\mathcal{A}}^{\prime }=\{(Z_{+},Z_{-})\in \mathcal{A}:Z_{-}\left(t\right)\equiv 0\}$ and define the corresponding value function by

$$\bar{V}\left(x\right)=\underset{(Z_{+},Z_{-})\in {\mathcal{A}}^{\prime }}{sup}{\mathsf{E}}_x{\int }_{0-}^{{\tau }_0}{e}^{-rt}d{Z}_{+}\left(t\right),$$

the problem then reduces to that considered in the seminal paper of Radner and Shepp [24]. But this is equivalent to considering the function $V(x;\mu ,\mu )$, which corresponds to the case where the maximum loan rate is zero. Since ${\mathcal{A}}^{\prime }\subset \mathcal{A}$, we have

$$V(x;\mu ,{\mu }^{\ast })\ge V(x;\mu ,\mu )=\bar{V}\left(x\right).$$

The question is as follows: if ${\mu }^{\ast }>\mu$, whether $Z_{-}\left(t\right)=0$ is strictly suboptimal, i.e., whether we have $V\left(x\right)>\bar{V}\left(x\right)$? It turns out that we do indeed have strict inequality, which implies that the expected additional dividend payouts from having the government funds are strictly greater than the loan cost (until bankruptcy), provided that the company takes profits in an optimal way to maximize its presumed objective. More generally, we have $V(x;\mu ,{\mu }^{\ast })>V(x;\mu ,{\mu }^{\prime })$ for any ${\mu }^{\ast }>{\mu }^{\prime }\ge \mu$; that is, the more the fiscal stimulus offered by the government, the larger net value of the company (see Section 3.)

Now, what if the company, after borrowing from the government, chooses a “greedy” policy that maximizes its own dividend payouts without caring to repay the loans? In Section 5.2, we will show that such a strategy is socially undesirable in that the expected net value of the company could be smaller than that with no government loans and, moreover, the expected dividend payouts may not even cover the loan cost. This represents an interesting caveat to the results of our model: in order to ensure that the mathematically optimal and socially optimal solution is achieved, the government must play some role in enforcing how the firms who take government money operate.

The paper is organized as follows. In Section 2 we review the related literature and, in particular, the results of the seminal work of Radner and Shepp [24], which can be viewed as a baseline model where the government does not offer any loans to companies. In Section 3 we derive the corresponding free-boundary problem for the value function given in Eq. (3) and prove the existence of the solution. Further optimal control results for our problem are provided in Section 4, including how the optimal dividend payout policy changes with the model parameters. In Section 5, we discuss the economic implications of different dividend payout policies. Section 6 concludes the paper with the requisite technical proofs.