Furloughing Employees with Uncertain Return? Management of Labor Frictions and Inventory Under Demand Shock

Abstract

We assess how a retailer can manage inventories and labor under an extreme condition such as a temporary negative demand shock due to a pandemic or an economic turmoil. This analysis incorporates labor market frictions whereby firms incur deadweight costs associated with hiring and firing employees, as well as the option to furlough labor. We examine the impact of these frictions on a retail firm's optimal operating policies around inventory level, furlough, and layoffs. We find that labor market frictions condition the inventory and the level of employment in two ways. First, they lead to underinvestment in inventories, which limits recovery and employment in the postshock period. Second, high labor market frictions motivate the firm to conservatively downsize workforce during the negative demand period leading to higher employment, when compared to downsizing without friction. A significant contribution of our study lies in delineating the optimal furlough decisions and quantifying the impact of the furlough option on inventory and labor decisions. We demonstrate the conditions under which it is optimal for the retailer to either (i) fully downsize labor and leverage the furlough option in the labor market, or (ii) maintain excess labor while also opting to furlough a portion of the workforce. During an extreme event such as a temporary negative demand shock, our results highlight the need for a coordinated effort when implementing governmental subsidy policies on alleviating labor and inventory reductions by accounting for labor market frictions and furlough support.

Keywords

Labor Market Frictions Retail Inventory Furlough Negative Demand Shock

1. Introduction

Efficiently scaling down and then ramping up operations during and after an economic turmoil is vital for business resilience. Even temporary shocks may drive firms to bankruptcy if they are obliged to pay off their fixed costs in the short-term. Economic turbulences, such as the U.S. recession in 2001 and the global financial crisis in 2009, typically precipitate a downturn in consumer spending. This contraction in consumption predominantly affects discretionary sectors, leading to a decline in demand for nonessential items, particularly in the realms of entertainment and luxury goods. A recent driver of such demand shocks was the COVID-19 pandemic, during which retailers like car dealerships, beauty salons, and fashion stores struggled to survive as demand for their services and products vanished.

In this manuscript, we explore a retailer's optimal operations in the face of a negative demand shock, amidst labor market frictions and with the possibility of furloughing employees. We assume that the demand shock is temporary and the markets are expected to recover over time. If a demand shock results in a permanent change in supply and demand dynamics in an industry, such as changing consumption habits, then some firms may face scenarios such as voluntary liquidation, or alterations in business models, which are beyond the scope of this manuscript. That is, most retail firms hope to weather a finite and downward economic shock by temporarily downsizing and then returning to routine operations. This scenario is addressed in this paper.

Downsizing retail operations involves two crucial and interconnected decisions: (a) reducing labor through layoffs and/or furloughs, and (b) decreasing inventory levels. This dual decision strategy has been adopted during economic downturns and reported in popular press during the US recession of 2001, the global financial crisis of 2009, and more recently, the COVID-19 pandemic. Such periods often prompt retailers to simultaneously cut labor and adjust inventory. For instance, the COVID-19 pandemic led to widespread layoffs and furloughs, with a notable impact attributed to retailers (Reuters, 2020). High-profile retailers, including Ralph Lauren, Kohl's, Tailored Brands, JC Penney, PVH Corp, and Walgreens made significant inventory and labor reductions to preserve profitability (Business Insider, 2020).

While layoffs often dominate media headlines, reducing inventories is another critical aspect of retail downsizing during economic challenges. The downturns trigger a decline in consumer demand, compelling many retailers to shut stores and either postpone or completely cancel orders. This action, although less publicized, is essential for managing resources and adapting to reduced market demand.

As markets rebound from economic disruptions, restoring labor and inventory levels also pose operational challenges. Frictions in the labor market make it costly, and uncertain, to find and train new workers. The literature on labor economics has long argued that “Trade in the labor market is a nontrivial economic activity due to the existence of heterogeneities, frictions and information imperfections” (Pissarides, 2000). As explained by the Diamond-Mortensen-Pissarides (DMP) model of labor market frictions by Diamond (1982), Mortensen (1982) and Pissarides (1985), no retailer would want to lose experienced labor force, as replacing this labor later in the market will result in both indirect costs (associated with the loss of valuable tacit knowledge, reduced efficiency in meeting demand, and training costs) and direct costs of firing and hiring. Additionally, in retail, labor and inventory are complementary resources, i.e., the marginal value of each of these resources on the firms’ ability to meet customer demand increases with the availability of the other resource. Hence, scaling up and down inventory and labor are closely linked in practice.

Evidence from downturns suggests that furlough schemes play a pivotal role for firms in reorganizing operations during economic shocks, facilitating recovery without the destruction of capital and jobs (Görtz et al., 2022). This issue underscores the importance of furloughs in maintaining business resilience and aiding in postcrisis recovery. Typically, they are not part of standard business practices but are employed during significant market shocks that lead to a substantial reduction in demand. Resorting to furloughs during minor market fluctuations can negatively impact long-term labor recruitment and retention, eroding employee trust and loyalty. As such, businesses often view furloughs as a last-resort option, indicating a balance between immediate operational requirements and the long-term sustainability of the workforce (Sucher and Winterberg, 2014).¹ Thus, motivated by the observed operating decisions of retailers during the economic downturns driven by the recent pandemic or previous financial crises, we formulate the following research questions:

How do firms make integrated labor and inventory management decisions in the presence of labor market frictions? Labor management decisions include hiring, firing as well as furloughing labor in the face of an economic shock.

What is the value of the furlough option for firms, and how does it affect their decisions regarding inventory and labor scaling? Additionally, how is the value of the furlough option influenced by key market conditions such as labor market frictions, labor market uncertainties and retail margins?

We set up a stylized two-period model to examine our research questions. The first period pertains to the economic downturn and the second period pertains to economic recovery. We show how a retailer would manage its inventory and labor capacity decisions in an integrated manner in the face of a negative demand shock (driven by, e.g., a pandemic or an economic downturn) and labor market frictions. Labor market frictions capture various costs such as search, training, and hiring expenses for new employees, alongside opportunity costs from potential reductions in sales efficiency during transitions. They also account for increased hiring costs amidst potential future labor shortages. Our contributions to the literature are summarized as follows:

We begin with our analysis by showcasing the benchmark scenario when there are no labor market frictions. In this context, we establish that labor and inventory decisions are separable. We show that labor market frictions call for joint consideration of labor and inventory decisions.

When labor market frictions are introduced in a downsizing scenario, these frictions are expected to attenuate labor reductions during periods of low demand due to the costs associated with rehiring in subsequent periods — a standard inference derived from stochastic control literature. However, this intuition may not hold, as shown in our analysis, due to the interplay of labor, inventory, and furlough decisions.

We demonstrate that labor market frictions only mitigate labor reduction in the first period if the level of frictions exceeds a certain threshold identified in our model. If the level of frictions is below this threshold, the presence of frictions does not affect labor downsizing. However, regardless of the friction level, it invariably results in a reduction in new inventory orders. This has a nuanced impact on operations by limiting new hires in the second period, consequently restricting the firm's ability to recover or ramp up during the time of recovery.

Our analysis highlights a key mechanism for functioning of the labor market frictions on the operations of the firm. This functioning, in our model, is characterized by two separate and complementary impacts on labor level: (a) a positive main effect on the labor amount in the first period and (b) a negative secondary effect on the labor level in the second period induced through reduced inventory levels. The net employment effect of labor market frictions is the sum of these two affects, which can be positive depending on the level of labor market frictions, profit margin and demand uncertainty.

In addition, our findings on furloughs indicate that a furlough option benefits the firm by mitigating inventory downsizing and facilitating recovery after the demand shock. However, the impact of furloughs on labor downsizing varies with the level of labor friction. With low labor friction, furloughs do not influence the extent of labor downsizing but can convert some of the terminated positions into furloughed ones. Conversely, in environments of high labor friction, the furlough option may lead to increased labor downsizing by transitioning more employees to furlough status. We also identify specific thresholds related to demand, labor friction and furlough risk, determining when a firm might opt for furloughs, retain excess labor (minimize labor downsizing), or apply both strategies—using furloughs while also maintaining excess labor.

Through detailed numerical analysis, we also explore how the impact of labor frictions and furlough options is influenced by the retail margin. We show that under a low profit margin, it is typically optimal to fully downsize labor (i.e., reducing the labor to a level that is just sufficient to meet the demand); and the main positive impact of labor market frictions on employment disappears. Hence, low profit margin firms are more likely to experience an increase in employment due to a reduction in labor market frictions (possibly driven by a governmental policy action) as compared to high profit margin firms.

In terms of the value and impact of furloughs, we observe that firms derive greater benefits from furloughs when labor market frictions and retail profits are moderate—neither too low nor too high. Although firms with low profit margins tend to initiate furloughs earlier in response to labor market frictions, firms with intermediate profit margins tend to place more labor on furlough in total.

2. Literature Review

Our research intersects with the literature on the operations-finance interface, the management of workforce flexibility, and labor economics. Over the last ten years, researchers have shown increasing attention to problems at the interface of operations and financial management (Babich and Birge, 2022; Babich and Kouvelis, 2018; Mutha et al., 2021). This stream focuses exclusively on frictions (and their impact on operations) associated with the acquisition of a vital resource: capital. Besides capital, labor is another vital resource, and trading in the labor market is also subject to frictions (Pissarides, 2011). However, there is limited work on the interaction of operations and labor management decisions in the presence of labor market frictions. In this study, we provide an integrated approach to coordinating labor, inventory, and furlough decisions within a retail context, amid labor market frictions.

It is well-established in the finance literature that, within imperfect capital markets, financial decisions may impact other management decisions (such as inventory levels) within the firm, and ultimately also affect the firm's ability to create value (Froot et al., 1993; Mayers and Smith, 1982; Smith and Stulz, 1985; Stulz, 1996). For example, bankruptcy cost is a prevalent form of capital market friction, and it has been shown that this cost can lead to substantially conservative operating and investment decisions by firms compared to the perfect market case (Alan and Gaur, 2018; Kouvelis and Zhao, 2012, 2016; Tanrisever et al., 2021). Similarly, the operational role of other capital market frictions such as moral hazard (Yang et al., 2021), transaction costs (Goel and Tanrisever, 2017), informational asymmetry (Lee et al., 2022; Tang et al., 2018), and financial distress cost (Birge et al., 2017) has also been explored in the operations literature. This adjacent literature has informed our models and analysis.

Regardless of the particular form of capital market frictions, they induce potentially value-destroying operating behavior within firms. A number of studies have examined how the value-destroying impact of frictions can be mitigated by (a) using innovative financial technologies such as reverse factoring (Kouvelis and Xu, 2021; Tanrisever et al., 2015), trade credit (Devalkar and Krishnan, 2019; Yang and Birge, 2018), purchase order finance (Reindorp et al., 2018), and other forms of supply chain finance applications (Tunca and Zhu, 2018); (b) adopting operational flexibility (Boyabatli and Toktay, 2011; Chod and Zhou, 2014; Iancu et al., 2017); and (c) implementing financial hedging and risk management practices (Boyabatlı et al., 2011; Devalkar et al., 2018; Ding et al., 2007; Goel and Tanrisever, 2017; Kouvelis and Li, 2019).

Within this stream of operations and finance literature, our study is most closely related to the papers exploring the role of operational flexibility and hedging in the presence of capital market frictions (Boyabatlı et al., 2016; Boyabatlı and Toktay, 2011; Chod and Zhou, 2014; Iancu et al., 2017). Boyabatli and Toktay (2011) and Boyabatlı et al. (2016) are among the first studies exploring the interaction of capital market frictions and operational flexibility on the financial performance of the firm. These authors show how the presence of capital market frictions and budget limits may motivate the firm to adopt more flexible technology. In a similar spirit, Iancu et al. (2017) examine the impact of operating flexibility in the presence of capital market frictions and debt covenants. They find that operating flexibility may not always benefit the firm under imperfect capital markets.

In a manner that is analogous to the above-mentioned flexibility papers (Boyabatli and Toktay, 2011 and Iancu et al., 2017), the ability to furlough labor provides the firm with operational flexibility when trading in the labor market via mitigating labor market frictions. In the presence of labor market frictions, this flexibility directly interacts with the operating decisions. That is, we set up a broader context for examining the interaction of market frictions and operating flexibility, which leads to novel operational insights around downsizing and furlough options.

Labor economists have long argued that trading in the labor markets is subject to deadweight costs, i.e., frictions (Diamond, 1982; Mortensen, 1982; Pissarides, 1985). These frictions include direct and indirect costs such as lost tacit knowledge due to losing a trained employee, search and training costs for a new hire, etc. Diamond–Mortensen–Pissarides (DMP) model takes labor market frictions in its core and aims to provide an explanation for the unemployment phenomena in the markets. Following the DMP model, numerous works have incorporated labor market frictions into both micro- and macro-economic models to analyze labor markets. For instance, motivated by the COVID-19 pandemic, Bernstein et al. (2020) examine the U.S. labor markets under a significant job separation shock, and Birinci et al. (2021) study the implications of labor market policies that counteract the economic fallout from containment measures during the pandemic. Birinci et al. (2021) provide an equilibrium analysis of labor markets under a standard disease spread model to examine the impact of unemployment insurance programs and payroll subsidies. They find that payroll subsidies, which preserve match capital and enable a faster economic recovery, are preferred over unemployment insurance programs. In the retail context, in a make-to-stock business model, our work provides a detailed analysis of interaction of labor, furlough and inventory decisions in the presence of labor market frictions. We highlight the mechanism through which labor frictions interact with inventory and labor decisions possibly leading to nonmonotone impact on total employment.

Furlough dynamics in our model are also linked to the literature on workforce flexibility (Kesavan et al., 2014; Pinker et al., 2009; Pinker and Larson, 2003; Qin et al., 2015). These studies focus on trade-off between employing regular versus contingent workers, emphasizing the optimization of staffing decisions to minimize costs while enhancing operational responsiveness. Related findings suggest that while flexibility in labor can significantly benefit organizations by allowing them to adapt to demand fluctuations, there exists an optimal level of such flexibility beyond which the costs may outweigh the benefits.

3. Model Description

We develop a two-period model where the firm, in each period, decides the volume of inventory and labor to meet random demand. The inventory decisions follow a newsvendor setup such that the firm needs to order prior to observing the demand due to long lead times that are common in many retail businesses. The firm has some flexibility in accessing the labor market, and the labor capacity can be revised after observing the demand subject to market frictions. Retail products that typically involve contingent sourcing of labor but not inventory include fashion and apparel items, tech gadgets, and custom-designed goods. These products often require specialized skills for design, production, or customization, which can be sourced on a contingent basis. However, the inventory for these products, such as raw materials or premanufactured components, may require long-term planning and purchasing agreements, making contingent sourcing less feasible.

Matching supply and demand in retail requires the simultaneous presence of labor (such as sales assistants in physical stores or packaging workers in online retailers) and customer since labor facilitates the delivery of the goods and service to the customer (Boulding et al., 1994). For expositional simplicity, we will model this “facilitation function” of labor as a linear relationship between sales and labor capacity given that there is sufficient demand and inventory, i.e., availability of labor does not affect the demand, but labor enables the firm to generate sales by exploiting demand.² It is also possible to model labor as an aggregation of agents who can not only generate sales but also increase demand by exerting sales effort (see e.g., Dai and Jerath, 2013). We ignore this possibility and posit that our main findings will remain the same if this possibility is considered within our model. We also assume that there is no moral hazard, and labor exerts full effort. This situation can be achieved by properly designing labor compensation packages, which is beyond the scope of our work (Dai and Jerath, 2013).

The first stage of our model denotes the shock period, where the firm is subject to a significant negative shock; and the second stage corresponds to the recovery period where the demand restores back to preshock level. For ease of exposition, we explicitly assume that the shock and recovery periods are of the same length. By properly scaling problem parameters, it is possible to capture unequal period lengths. Timeline of events and decisions are detailed below and are depicted in Figure 1.

Figure 1.

Timeline of events and decisions.

Decisions and Events during the Shock Period: We assume that the firm embarks to the shock period with $L_{0}$ units of initial labor and $Q_{0}$ amount of inventory available to sell during that period. If there is enough demand, each labor may facilitate the sales of $β$ units of inventory, and the total labor capacity in terms of demand is given by $β L_{0} .$ In what follows, we will refer $β$ as labor productivity.

During the shock period, the firm observes a negative demand shock significantly reducing the demand (below its usual level) down to level $ξ_{0}$ . After observing the demand shock, the firm may revise its labor and inventory decisions to better match supply and demand. In particular, the firm aims to scale down operations to maintain profitability. As noted above, the firm has flexibility to quickly revise labor level, and it reduces its initial labor amount down to $L_{1} \leq L_{0}$ by laying off $\bar{L} = L_{0} - L_{1}$ units of labor.³ Hence, the available labor to meet the shock period demand becomes $L_{1}$ and the firm pays for the labor cost $c_{L} L_{1}$ where $c_{L}$ denotes the unit cost of labor per period including salaries and other payments. Reducing the labor level has a positive effect on immediate cash flows by cutting cost. However, scaling down labor may compromise firm's ability to exploit demand in the next period.

The firm may also choose to put some of the laid-off workers, l, on furlough by paying a unit cost of f. In the second period, furloughed workers may be called back to increase labor capacity without being subject to labor market frictions. After the first period, the potential return of furloughed labor to work remains uncertain. In practice, the duration of this shock period greatly influences whether furloughed employees will seek alternative employment or pursue new training opportunities, potentially leading to the discovery of different job prospects. To address this uncertainty, we assume that in the recovery period, the amount of furloughed labor available to call back to work is denoted by a random variable $χ$ which takes the value of l with probability $τ$ , and takes the value zero with probability ( $1 - τ$ ). That is, we posit that furloughed labor, denoted as l, will be available in the second period to call back to work with probability $τ$ . Conversely, there exists a probability of ( $1 - τ$ ) that they will secure new employment and consequently become unavailable for rehiring. Our examination extends to more general uniform and normal distributions, effectively validating our central findings. However, for the sake of analytical tractability, we have used a binary distribution.

During the shock period, the retailer sells its existing stock $Q_{0}$ with the help of the reduced labor $L_{1}$ . We assume that $Q_{0}$ is already paid for and received by the firm, and it is not possible to scale it down (or economically salvage) during the first period. Excess inventory has no salvage value at the end of the period.⁴ Nevertheless, the firm may select its inventory decision $Q_{1}$ for the next period after observing the demand during the shock period. Due to long manufacturing and shipping lead times, $Q_{1}$ needs to be ordered in the shock period so that it arrives at the beginning of the recovery period and becomes available to meet the demand in that period. (Since both $L_{1}$ and $Q_{1}$ are decided during the first period, we index them with subscript 1). We denote the unit inventory ordering cost as $c_{I}$ .

As noted above, labor is needed to exploit the observed demand and generate sales, and hence during the first period, the revenues of the firm can be stated as $R_{0} = p min {Q_{0}, min (β L_{1}, ξ_{0})}$ , where p denotes the unit sales price. Overall, during the first period, the firm collects revenues from sales $R_{0}$ , and pays for the labor cost $c_{L} L_{1}$ , furlough cost $f l$ , and the cost of new orders $c_{I} Q_{1}$ . Accordingly, the realized profits at the end of the first period is given by: $R_{0} - c_{L} L_{1} - f l - c_{I} Q_{1}$ .

Although, reduced inventory and labor levels ( $Q_{1}$ and $L_{1}$ ) can help preserve profitability during the first period, they also restrict the firm's ability to rebound after the negative demand shock. In particular, it is costly to hire new labor in the second period and it is not possible to revise the inventory level immediately.

Decisions and Events during the Recovery Period: During the recovery period, after observing the new inventory level $Q_{1}$ , the realization of recovery period demand $ξ_{1}$ , and the realization of the amount of available furloughed labor $χ$ , the firm decides the new level of labor $L_{2}$ to meet the realized demand.⁵ We denote the cumulative distribution function of the demand by $F_{1} (.)$ . In this period, $χ$ amount of furloughed labor (which depends on l) can be hired at no additional cost, while hiring beyond the furloughed amount is subject to labor market frictions. Hence, given $χ$ , increasing the labor amount to $L_{2}$ costs the firm $c_{L} min (L_{1} + χ, L_{2}) + c_{L} (1 + α) (L_{2} - L_{1} - χ)^{+}$ , i.e., only hiring beyond the furloughed amount is subject to labor market frictions, $α$ . Here $α$ captures the deadweight costs such as search, training, and hiring costs as well as opportunity costs due to possible reduction in sales efficiency. It also incorporates increased hiring costs due to possible labor shortages in the future.

Then, the firm meets the demand, collects revenues $R_{1} = p min {Q_{1}, min (β L_{2}, ξ_{1})}$ , and pays for the labor cost $c_{L} min (L_{1} + χ, L_{2}) + c_{L} (1 + α) (L_{2} - L_{1} - χ)^{+}$ . Accordingly, the realized profits at the end of the second period is given by: $R_{1} - c_{L} min (L_{1} + χ, L_{2}) - c_{L} (1 + α) (L_{2} - L_{1} - χ)^{+}$ .

Given the realized profits at the end of each period and the decisions in each stage, the retailer's two-stage optimization problem can be formalized as follows:

\begin{aligned} π_{0} (Q_{0}, L_{0}; ξ_{0}) & = max_{Q_{1} \geq 0, L_{1} \geq 0, \bar{L} \geq l \geq 0} R_{0} - c_{L} L_{1} - f l - c_{I} Q_{1} \\ + E_{ξ_{1}, χ} [π_{1} (Q_{1}, L_{1}, l; ξ_{1}, χ)], \end{aligned}

(1)

where,

\begin{aligned} π_{1} (Q_{1}, L_{1}, l; ξ_{1}, χ) & = max_{L_{2} \geq 0} {R_{1} - c_{L} min (L_{1} + χ, L_{2}) \\ - c_{L} (1 + α) (L_{2} - L_{1} - χ)^{+}}, \end{aligned}

(2)

and

R_{1} = p min {Q_{1}, min (β L_{2}, ξ_{1})}

and

R_{0} = p min {Q_{0}, min (β L_{1}, ξ_{0})}

To summarize, $π_{1} (Q_{1}, L_{1}, l; ξ_{1}, χ)$ in equation (2) denotes the recovery period optimal profits of the firm conditional on the level of inventory, labor and realized demand. In equation (1), $π_{0} (Q_{0}, L_{0}; ξ_{0})$ gives the optimal expected total profits at the beginning of the first period.

Since we model a negative demand shock, both $Q_{0}$ and $L_{0}$ are sufficiently large to meet the realized demand, $ξ_{0}$ . Substituting for the second period profits and rearranging the terms in equation (1), we obtain:

\begin{aligned} π_{0} (Q_{0}, L_{0}; ξ_{0}) & = max_{Q_{1} \geq 0, L_{1} \geq 0, \bar{L} \geq l \geq 0} R_{0} - c_{L} L_{1} - f l - c_{I} Q_{1} \\ + E_{ξ_{1}} [R_{1}] - E_{ξ_{1}, χ} [c_{L} L_{2} *] - L F C, \end{aligned}

(3)

where

L_{2} *

is the optimal recovery period labor and

L F C = E_{ξ_{1}, χ} [α c_{L} (L_{2} * - L_{1} - χ)^{+}]

denotes the expected cost of labor due to frictions.

4. Analysis

In this section, we begin with our analysis by showcasing the benchmark scenario when there are no labor market frictions. Next, we introduce labor market frictions and examine the interaction of labor and inventory decisions. Following this, we model the option for furloughing labor, examining both its value and its influence on firm decisions.

It is convenient to analyze the model backward starting with the last decision $L_{2}$ . At the beginning of the recovery period, the firm observes the demand $ξ_{1}$ , inventory level $Q_{1}$ and the labor position $(L_{1}, χ$ ), and determines the optimal second period labor level $L_{2} *$ , which is given in Lemma 1. All proofs are available in E-companion, Appendix A in the online supplemental materials.

Lemma 1 (Recovery stage)

The optimal second period labor level of the firm is given by

\begin{aligned} L_{2} * = {\begin{cases} min (Q_{1}, ξ_{1}) / β & i f p \geq (1 + α) c_{L} / β \\ min (L_{1} + χ, min (Q_{1}, ξ_{1}) / β) & i f c_{L} / β \leq p < (1 + α) c_{L} / β \end{cases} . \end{aligned}

The first scenario in Lemma 1 denotes the case when the unit sales price is sufficient to cover the marginal cost of hiring new labor, i.e., $p \geq (1 + α) c_{L} / β$ . This case implies that it is profitable to meet the demand with new labor such that $L_{2} * = min (Q_{1}, ξ_{1}) / β$ . The second condition denotes the case when the unit sales price is less than the marginal cost of hiring new labor such that $p < (1 + α) c_{L} / β$ . In this case, the firm does not hire new labor even when there is unmet demand. In what follows, we focus on the former case as it represents a more realistic scenario, i.e., it is profitable to meet the demand with new labor. This scenario is likely to hold in many retail sectors where value-added services or unique product offerings enable significant markups such as custom design services in the furniture industry, exclusive fashion lines in the apparel sector, and personalized health solutions in the wellness and healthcare market. The condition for the former case, can also be formulated as an upper bound on the labor market friction such that $α \leq \overset{˘}{α} = \frac{p - c_{L}}{c_{L}}$ . We retain this assumption throughout the paper. In Section 4.1, we discuss the impact of considering $α > \overset{˘}{α}$ on our results.

We begin our analysis with labor markets with no frictions in Proposition 1, i.e., $α = 0$ . We use this case as a benchmark for the follow-on analysis. From Section 4.1 and onwards, we include labor market frictions to our analysis illustrating how it conditions inventory, labor and furlough decisions.

Proposition 1 (Labor Market with no Frictions)

When there are no labor market frictions, i.e., $α = 0$ , the optimal labor and inventory decisions are given by $L_{1} * = ξ_{0} / β$ , $Q_{1} * = Q_{1}^{H} = F_{1}^{- 1} (\frac{p - c_{I} - c_{L} / β}{p - c_{L} / β})$ , $L_{2} * = min (Q_{1}, ξ_{1}) / β$ , and $l * = 0$ .

Table 1.
Optimal decisions without furlough.

Modiglianni-Miller (MM) results in the finance literature establish that financial and operational decisions are separable when there are no capital market frictions. Analogously, we also find that the first stage labor ( $L_{1} *$ ) and inventory decisions ( $Q_{1} *$ ) are also separable when there are no labor market frictions. In the first stage, the firm fully downsizes labor reducing it to $L_{1} * = ξ_{0} / β$ which is just sufficient to meet the first period demand $ξ_{0}$ . Inventory is not scaled down and is kept at its first-best level $Q_{1} * = Q_{1}^{H} = F_{1}^{- 1} (\frac{p - c_{I} - c_{L} / β}{p - c_{L} / β})$ (the superscript “H” denotes a high order quantity). Proposition 1 serves as benchmark for our follow-on analyses.

4.1 Model with Labor Market Frictions

In Theorem 1, we present the results when we introduce the labor market frictions to the case discussed in Proposition 1. Theorem 1 is instrumental in explaining the impact of labor market frictions.

Theorem 1 (Labor Market with Frictions and without Furlough)

Suppose there are labor market frictions ( $α > 0$ ), and there is no possibility of furlough $(l = 0)$ . Then, there are two cases:

(i)
If the level of labor market frictions is low, i.e., $α < 1$ , then
$\begin{aligned} (Q_{1} , L_{1} ) = {\begin{cases} Q_{1}^{L}, \frac{ξ_{0}}{β} & i f Q_{1}^{L} \geq ξ_{0} \\ ξ_{0}, \frac{ξ_{0}}{β} & i f Q_{1}^{H} > ξ_{0} > Q_{1}^{L} \\ Q_{1}^{H}, \frac{ξ_{0}}{β} & i f ξ_{0} \geq Q_{1}^{H} \end{cases} . \end{aligned}$

(ii)
If the level of labor market frictions is high, i.e., $α \geq 1$ , then
$\begin{aligned} (Q_{1} , L_{1} ) = {\begin{cases} Q_{1}^{L}, L_{1}^{H} & i f α \leq \bar{α} a n d β L_{1}^{H} \geq ξ_{0} \\ Q_{1}^{L}, \frac{ξ_{0}}{β} & i f α \leq \bar{α} a n d Q_{1}^{L} \geq ξ_{0} > β L_{1}^{H} \\ ξ_{0}, \frac{ξ_{0}}{β} & i f α \leq \bar{α} a n d Q_{1}^{H} > ξ_{0} > Q_{1}^{L} \\ Q_{1}^{B}, L_{1}^{B} & i f α > \bar{α} a n d Q_{1}^{B} \geq ξ_{0} \\ ξ_{0}, \frac{ξ_{0}}{β} & i f α > \bar{α} a n d Q_{1}^{H} > ξ_{0} > Q_{1}^{B} \\ Q_{1}^{H}, \frac{ξ_{0}}{β} & f o r a l l α a n d ξ_{0} \geq Q_{1}^{H} \end{cases}, \end{aligned}$

where $Q_{1}^{L} = F_{1}^{- 1} (\frac{p - c_{I} - (1 + α) c_{L} / β}{p - (1 + α) c_{L} / β})$ , $Q_{1}^{H} = F_{1}^{- 1} (\frac{p - c_{I} - c_{L} / β}{p - c_{L} / β})$ , $Q_{1}^{B} = F_{1}^{- 1} (\frac{p - c_{I} - 2 c_{L} / β}{p - c_{L} / β})$ , $L_{1}^{H} = F_{1}^{- 1} (\frac{α - 1}{α}) / β$ , $L_{1}^{B} = \frac{Q_{1}^{B}}{β}$ , $\bar{α} = \frac{p - c_{L} / β}{c_{I} + c_{L} / β}$ .

Under low labor market frictions ( $α < 1$ ), laying off a worker in the first period and hiring her back in the second period is less costly as compared to keeping her employed during both periods. For the high labor market frictions case ( $α \geq 1$ ), firing and hiring back a worker is more costly than keeping her employed during both periods.⁶ Below we explain the effect of labor market frictions under these two scenarios. Results are summarized in Table 1.

Impact of low labor market frictions ( $0 < α < 1$ ): In this case, the firm reduces the labor amount to a level that is just sufficient to meet the current demand, i.e., $L_{1} * = \frac{ξ_{0}}{β}$ , and it does not keep any idle labor in the first period. This nonidling policy is independent of the level of demand $ξ_{0}$ and the level of labor market frictions $α$ . The order quantity, however, depends on both $ξ_{0}$ and $α$ . If the demand is low, $ξ_{0} \leq Q_{1}^{L},$ the firm orders $Q_{1}^{L}$ which is calculated expecting that new labor will be hired in the next period (the superscript “L” denotes a lower order quantity relative to the first-best solution). If the demand is high, $ξ_{0} \geq Q_{1}^{H},$ the firm orders $Q_{1}^{H}$ which is now calculated expecting that no new labor will be hired in the next period. When the demand is moderate, $Q_{1}^{L} < ξ_{0} < Q_{1}^{H},$ order quantity is independent of the distribution of future demand and is equal to the shock period demand, $Q_{1} * = ξ_{0} .$

Recall that $Q_{1}^{H}$ represents the first-best order quantity in the absence of labor market frictions. Consequently, in the presence of labor market frictions, the firm underinvests in inventory, resulting in $Q_{1} * = Q_{1}^{L} < Q_{1}^{H}$ . This result, together with Proposition 1, implies that labor market frictions create a negative secondary lagged impact on future employment, which is bounded by the inventory orders $Q_{1} $ in the first stage.

Impact of high labor market frictions* ( $α \geq 1$ ): In this case, there are two distinct scenarios such that $1 \leq α \leq \bar{α}$ and $α > \bar{α}$ , which we discuss below.

Case (i): If $1 \leq α \leq \bar{α}$ , the optimal solution is identical to the low friction case except for the scenario $ξ_{0} \leq β L_{1}^{H}$ . In this situation, where demand during the shock period is sufficiently low, it is optimal to keep some idle workers in the first period ( $L_{1} * = L_{1}^{H} > \frac{ξ_{0}}{β}$ ), in anticipation of market recovery and the costs of rehiring in the subsequent period. Specifically, when the labor market friction is such that $1 \leq α \leq \bar{α}$ , then a “diverging strategy” in terms of labor and inventory is optimal, $(Q_{1} , L_{1} ) =$ $(Q_{1}^{L}, L_{1}^{H})$ . In this case, the firm aggressively scales down the inventory level to $Q_{1}^{L} < Q_{1}^{H}$ , but acts conservatively when scaling down the labor, i.e., $L_{1}^{H} > \frac{ξ_{0}}{β}$ . This implies that (i) the firm keeps excess labor during the first period ( $L_{1}^{H} - \frac{ξ_{0}}{β} > 0$ ), and (ii) this labor amount is less than the labor amount needed to clear the ordered volume ( $L_{1}^{H} < \frac{Q_{1}^{L}}{β}$ ) in the second period.

Case (ii): If $\bar{α} < α < \overset{˘}{α}$ , the firm becomes even less aggressive in downscaling the labor when compared with the previous case (i). In this case, the firm partially downscales labor even under relatively higher demand realizations, i.e., when $ξ_{0} < Q_{1}^{B}$ where $Q_{1}^{B} > β L_{1}^{H}$ (the superscript “B” denotes a boundary solution). The optimal labor level is also higher as compared to the previous case, i.e., $L_{1} * = L_{1}^{B} > L_{1}^{H}$ . Overall, (i) the firm keeps more excess labor during the first period ( $L_{1}^{B} - \frac{ξ_{0}}{β} > 0$ ) and (ii) this labor amount is exactly equals to the labor needed to clear the ordered volume ( $L_{1}^{B} = \frac{Q_{1}^{B}}{β}$ ) in the second period. (The case when $α \geq \overset{˘}{α}$ yields qualitatively similar results to case (ii), except that the firm refrains from reducing labor below new inventory orders. This is due to Lemma 1, which deems new labor hiring suboptimal in this case).

Theorem 1 highlights the role of labor market frictions in shaping operating decisions. This role, in our model, is characterized by two separate and complementary impacts on labor level: (i) a positive main effect on the labor amount in the first period and (ii) a negative secondary effect on the labor level in the second period induced through reduced inventory levels. On the one hand, increased labor market frictions may lead to less aggressive downsizing of labor, i.e., partial downsizing may be possible. On the other hand, reduced orders for the second period limits firm's ability to meet the demand and hence leads to reduced hiring in the second period. The net employment effect of labor market frictions is the sum of these two affects in our model. In Section 5, we numerically illustrate the economic conditions under which the net effect of labor market frictions is positive on total employment.
4.2 Model with Labor Market Frictions and Furlough

So far, we have conducted our analysis in isolation of furlough possibility. In this section, we expand our examination to include furlough, assuming that furloughed employees may return to work with a probability $τ \leq 1$ . For expositional ease, we will present our findings in two parts: initially, we explore the low labor friction scenario in Theorem 2, followed by an analysis of the high friction scenario in Theorem 3.⁷

Theorem 2 (Labor Market with Low Labor Frictions and with Furlough)

Assuming labor market frictions such that $0 < α < 1$ , and considering the option to furlough labor where furloughed employees return to work with a probability $τ \leq 1$ , it follows that:

\begin{aligned} (Q_{1} *, L_{1} *, l *) \\ = {\begin{cases} Q_{1}^{L}, \frac{ξ_{0}}{β}, \hat{l} - \frac{ξ_{0}}{β} & i f α \leq \hat{α} a n d β \hat{l} \geq ξ_{0} \\ Q_{1}^{L}, \frac{ξ_{0}}{β}, 0 & i f α \leq \hat{α} a n d Q_{1}^{L} \geq ξ_{0} > β \hat{l} \\ ξ_{0}, \frac{ξ_{0}}{β}, 0 & i f α \leq \hat{α} a n d Q_{1}^{H} > ξ_{0} > Q_{1}^{L} \\ {\hat{Q}}_{1}^{B}, \frac{ξ_{0}}{β}, \frac{{\hat{Q}}_{1}^{B}}{β} - \frac{ξ_{0}}{β} & i f α > \hat{α} a n d {\hat{Q}}_{1}^{B} \geq ξ_{0} \\ ξ_{0}, \frac{ξ_{0}}{β}, 0 & i f α > \hat{α} a n d Q_{1}^{H} > ξ_{0} > {\hat{Q}}_{1}^{B} \\ Q_{1}^{H}, \frac{ξ_{0}}{β}, 0 & f o r a l l α a n d ξ_{0} \geq Q_{1}^{H} \end{cases}, \end{aligned}

where

Q_{1}^{L} = F_{1}^{- 1} (\frac{p - c_{I} - (1 + α) c_{L} / β}{p - (1 + α) c_{L} / β})

\hat{l} = F_{1}^{- 1} (\frac{α c_{L} τ - f}{α c_{L} τ}) / β

{\hat{Q}}_{1}^{B} = F_{1}^{- 1} (\frac{p - c_{I} - f / β - (1 + α - α τ) c_{L} / β}{p - (1 + α - α τ) c_{L} / β})

, and

\hat{α} = \frac{(p - c_{L} / β) f}{(c_{I} τ + f / β) c_{L}}

The labor decisions in Theorem 2 align with those presented in Theorem 1-(i). That is, under low labor friction, where $α < 1$ , the presence of a furlough option does not alter the optimal labor level, $L_{1} *$ , which stays at $ξ_{0} / β$ , resulting in a complete downsizing of labor. However, depending on the cost of furlough, level of labor frictions and demand shock, the firm may furlough some labor. There are two scenarios under which furloughing becomes a preferred strategy:

When labor market frictions are relatively low ( $α \leq \hat{α}$ ) and demand is sufficiently weak ( $ξ_{0} \leq β \hat{l}$ ), the quantity of furloughed labor, $\hat{l} - ξ_{0} / β$ , increases with labor market friction $α$ . In this case, $\hat{l}$ represents the unconstrained labor amount in the second period calculated by assuming the firm recalls furloughed labor. If this figure falls below the labor remaining at the end of the first period, $ξ_{0} / β$ , then there is no need to furlough. Otherwise, furloughing the difference, $\hat{l} - ξ_{0} / β$ , is optimal. From the definition of $\hat{l}$ , the decision to furlough hinges on three factors: $f / τ,$ $α c_{L}$ and $ξ_{0}$ . Intuitively, furloughing is only favored when $f / τ < α c_{L}$ , indicating that the effective cost of furloughing is less than that of laying off and rehiring. In this case, we also note that the inventory decision, $Q_{1} * = Q_{1}^{L}$ is not affected by the furlough option.

When the labor market frictions exceed the threshold $\hat{α}$ , i.e., $α > \hat{α}$ , the furloughed labor quantity is similarly determined by $({\hat{Q}}_{1}^{B} - ξ_{0}) / β$ and also increases with $α$ . For $α > \hat{α}$ , the inventory decision becomes dependent on both $α$ and f, suggesting that furloughing may influence both inventory levels and future employment.

Overall, Theorem 2 reveals that in low friction markets ( $α < 1$ ), firms opt to fully downsize labor ( $L_{1} * = ξ_{0} / β$ ) but consider furloughing for future labor planning in response to low demand, as it offers a cost-effective alternative. The question then arises: in high-friction scenarios ( $α \geq 1$ ), will firms leverage both excess labor retention and furlough? This query is addressed in Theorem 3.

Theorem 3 (Labor Market with High Labor Frictions and with Furlough)

Assuming labor market frictions such that $α \geq 1$ , and considering the option to furlough labor whereby furloughed employees return to work with a probability $τ \leq 1$ , it follows that:

(i)
Suppose $α \leq \bar{α} a n d τ < τ_{0}$ (low friction $α$ and low return probability $τ$ ), then
$\begin{aligned} (Q_{1} , L_{1} , l ) = {\begin{cases} Q_{1}^{L}, L_{1}^{H}, 0 & i f β L_{1}^{H} \geq ξ_{0} \\ Q_{1}^{L}, \frac{ξ_{0}}{β}, 0 & i f Q_{1}^{L} \geq ξ_{0} > β L_{1}^{H} \\ ξ_{0}, \frac{ξ_{0}}{β}, 0 & i f Q_{1}^{H} > ξ_{0} > Q_{1}^{L} \\ Q_{1}^{H}, \frac{ξ_{0}}{β}, 0 & i f ξ_{0} \geq Q_{1}^{H} \end{cases} \end{aligned}$

(ii)
Suppose* $α \leq \bar{α} a n d τ_{0} \leq τ \leq τ_{1}$ (low friction $α$ and medium return probability $τ$ ), then
$\begin{aligned} (Q_{1} , L_{1} , l ) = {\begin{cases} Q_{1}^{L}, {\hat{L}}_{1}^{H}, \hat{l} - {\hat{L}}_{1}^{H} & i f β {\hat{L}}_{1}^{H} \geq ξ_{0} \\ Q_{1}^{L}, \frac{ξ_{0}}{β}, \hat{l} - \frac{ξ_{0}}{β} & i f β \hat{l} \geq ξ_{0} > β {\hat{L}}_{1}^{H} \\ Q_{1}^{L}, \frac{ξ_{0}}{β}, 0 & i f Q_{1}^{L} \geq ξ_{0} > β \hat{l} \\ ξ_{0}, \frac{ξ_{0}}{β}, 0 & i f Q_{1}^{H} > ξ_{0} > Q_{1}^{L} \\ Q_{1}^{H}, \frac{ξ_{0}}{β}, 0 & i f ξ_{0} \geq Q_{1}^{H} \end{cases} \end{aligned}$

(iii)
Suppose* $α \leq \bar{α} a n d τ > τ_{1}$ (low friction $α$ and high return probability $τ$ ), then
$\begin{aligned} (Q_{1} , L_{1} , l ) = {\begin{cases} {\hat{Q}}_{1}^{B}, {\hat{L}}_{1}^{H}, \frac{{\hat{Q}}_{1}^{B}}{β} - {\hat{L}}_{1}^{H} & i f β {\hat{L}}_{1}^{H} \geq ξ_{0} \\ {\hat{Q}}_{1}^{B}, \frac{ξ_{0}}{β}, \frac{{\hat{Q}}_{1}^{B}}{β} - \frac{ξ_{0}}{β} & i f {\hat{Q}}_{1}^{B} \geq ξ_{0} > β {\hat{L}}_{1}^{H} \\ ξ_{0}, \frac{ξ_{0}}{β}, 0 & i f Q_{1}^{H} > ξ_{0} > {\hat{Q}}_{1}^{B} \\ Q_{1}^{H}, \frac{ξ_{0}}{β}, 0 & i f ξ_{0} \geq Q_{1}^{H} \end{cases} \end{aligned}$

(iv)
Suppose* $α > \bar{α} a n d τ < τ_{2}$ (high friction $α$ and low return probability $τ$ ), then
$\begin{aligned} (Q_{1} , L_{1} , l ) = {\begin{cases} Q_{1}^{B}, \frac{Q_{1}^{B}}{β}, 0 & i f Q_{1}^{B} \geq ξ_{0} \\ ξ_{0}, \frac{ξ_{0}}{β}, 0 & i f Q_{1}^{H} > ξ_{0} > Q_{1}^{B} \\ Q_{1}^{H}, \frac{ξ_{0}}{β}, 0 & i f ξ_{0} \geq Q_{1}^{H} \end{cases} \end{aligned}$

(v)
Suppose* $α > \bar{α} a n d τ \geq τ_{2}$ (high friction $α$ and high return probability $τ$ ), then
$\begin{aligned} (Q_{1} , L_{1} , l ) = {\begin{cases} {\hat{Q}}_{1}^{B}, {\hat{L}}_{1}^{H}, \frac{{\hat{Q}}_{1}^{B}}{β} - {\hat{L}}_{1}^{H} & i f β {\hat{L}}_{1}^{H} \geq ξ_{0} \\ {\hat{Q}}_{1}^{B}, \frac{ξ_{0}}{β}, \frac{{\hat{Q}}_{1}^{B}}{β} - \frac{ξ_{0}}{β} & i f {\hat{Q}}_{1}^{B} \geq ξ_{0} > β {\hat{L}}_{1}^{H} \\ ξ_{0}, \frac{ξ_{0}}{β}, 0 & i f Q_{1}^{H} > ξ_{0} > {\hat{Q}}_{1}^{B} \\ Q_{1}^{H}, \frac{ξ_{0}}{β}, 0 & i f ξ_{0} \geq Q_{1}^{H} \end{cases}, \end{aligned}$

where*
$\begin{aligned} \bar{α} & = \frac{p - c_{L} / β}{c_{I} + c_{L} / β}, τ_{0} = \frac{f}{c_{L}}, τ_{1} = \frac{(p - (1 + α) c_{L} / β)}{α c_{I}} \frac{f}{c_{L}}, \\ τ_{2} & = \frac{f p - (f / β + p - α c_{I}) c_{L} + (1 + α) c_{L}^{2} / β}{α c_{L} (c_{I} + c_{L} / β)}, \\ Q_{1}^{H} & = F_{1}^{- 1} (\frac{p - c_{I} - c_{L} / β}{p - c_{L} / β}), \\ Q_{1}^{L} & = F_{1}^{- 1} (\frac{p - c_{I} - (1 + α) c_{L} / β}{p - (1 + α) c_{L} / β}), \\ \hat{l} & = F_{1}^{- 1} (\frac{α c_{L} τ - f}{α c_{L} τ}) / β, L_{1}^{H} = F_{1}^{- 1} (\frac{α - 1}{α}) / β, \\ Q_{1}^{B} & = F_{1}^{- 1} (\frac{p - c_{I} - 2 c_{L} / β}{p - c_{L} / β}), \\ {\hat{Q}}_{1}^{B} & = F_{1}^{- 1} (\frac{p - c_{I} - f / β - (1 + α - α τ) c_{L} / β}{p - (1 + α - α τ) c_{L} / β}), and \\ {\hat{L}}_{1}^{H} & = F_{1}^{- 1} (\frac{α c_{L} (1 - τ) + (f - c_{L})}{α c_{L} (1 - τ)}) / β . \end{aligned}$

Theorem 3 identifies five scenarios depending on the level of furlough risk $τ$ and labor market friction $α$ :

Table 2.
Impact of furlough.

$0 < α < 1$ $1 \leq α \leq \bar{α}$ $\bar{α} < α < \overset{˘}{α}$

Labor market with no frictions (Proposition 1)
Fully downsize labor: $L_{1} * = ξ_{0} / β$

Order the first best level: $Q_{1} * = Q_{1}^{H}$

No need for Furlough: $l * = 0$

Labor market with frictions and without furlough (Theorem 1) -
Fully downsize labor:

$L_{1} * = ξ_{0} / β$
-
Under-invest in inventory:

$Q_{1} * = Q_{1}^{L} < Q_{1}^{H}$
-
Carry excess labor:

$L_{1} * = L_{1}^{H} > ξ_{0} / β$
-
Under-invest in inventory:

$Q_{1} * = Q_{1}^{L} < Q_{1}^{H}$
-
Carry excess labor:

$L_{1} * = L_{1}^{B} > L_{1}^{H}$
-
Under-invest in inventory:

$Q_{1} * = Q_{1}^{B} < Q_{1}^{H}$

Labor market with frictions and uncertain furlough (Theorems 2 and 3) -
Fully downsize labor: $L_{1} * = ξ_{0} / β$
-
Under-invest in inventory:

$Q_{1} * = Q_{1}^{L} < Q_{1}^{H}$
-
Furlough labor
If $τ < τ_{0}$ : -
Results are identical to “Labor market with frictions and without furlough” case
If $τ \geq τ_{0}$ : -
Carry less excess labor: $L_{1} * = {\hat{L}}_{1}^{H} < L_{1}^{H}$
-
Order less conservatively:

$Q_{1} * = min {{\hat{Q}}_{1}^{B}, Q_{1}^{L}} \geq Q_{1}^{L}$
-
Furlough labor
If $τ < τ_{2}$ : -
Results are identical to “Labor market with frictions and without furlough” case
If $τ \geq τ_{2}$ : -
Carry less excess labor

$L_{1} * = {\hat{L}}_{1}^{H} < L_{1}^{H}$
-
Order less conservatively:

$Q_{1} * = {\hat{Q}}_{1}^{B} > Q_{1}^{B}$
-
Furlough labor

Note. In Table 2, the x-axis presents thresholds for level of labor market frictions. The first row of the table summarizes the benchmark case with no labor market frictions, and the second row provides the case with labor market friction but without furlough. The table's third row contrasts the outcomes of the case with both furlough and labor market frictions against those without the furlough. The results in the table refer to the scenarios in respective theorems when the shock period demand $ξ_{0}$ is sufficiently low that the firm may furlough or partially downsize labor.

Case (i): With low labor market friction (1 $\leq α \leq \bar{α}$ ) and high furlough risk ( $τ < τ_{0}$ ), furloughing labor is not economically advantageous. In this case, if the demand is low ( $β L_{1}^{H} \geq ξ_{0}$ ), the firm manages its labor by partially downsizing its workforce, $L_{1} * = L_{1}^{H} > \frac{ξ_{0}}{β}$ , i.e., the firm prefers to retain excess labor over furloughing.

Case (ii): For low friction levels (1 $\leq α \leq \bar{α}$ ) coupled with moderate furlough risk ( $τ_{0} \leq τ \leq τ_{1}$ ), the firm employs both furloughing and partial downsizing under low demand $(β {\hat{L}}_{1}^{H} \geq ξ_{0})$ . That is, the firm uses both of its levers in the labor market to manage its workforce ( $L_{1} * = {\hat{L}}_{1}^{H} > \frac{ξ_{0}}{β}$ and $l * = \hat{l} - {\hat{L}}_{1}^{H} > 0$ ). As the demand increases such that $β \hat{l} \geq ξ_{0} > β {\hat{L}}_{1}^{H}$ , the strategy shifts to solely furloughing, sequentially adjusting to demand levels by initially relying on furloughs before resorting to carrying excess labor.

Case (iii): This case mirrors Case (ii) in strategy for low friction and high return probability of furloughed labor (low risk), differing in the specific quantities of labor adjustment.

Case (iv): When friction level is high ( $α > \bar{α}$ ) and the furloughed labor is less likely to return ( $τ < τ_{2}$ ), then it is optimal to partially downsize without furloughing when demand is low ( $Q_{1}^{B} \geq ξ_{0})$ , i.e., $L_{1} * = \frac{Q_{1}^{B}}{β} > \frac{ξ_{0}}{β}$ . That is, the firm relies on carrying excess labor only as in Case (i).

Case (v): This case, with high friction and low furlough risk, also mimics Case (ii).

In Table 2, similar to Table 1, we provide a summary of our results illustrating the impact of market frictions and furlough option on labor and inventory decisions. The first row of the table (from Proposition 1) shows the results in the absence of labor market frictions, i.e., the firm fully downsizes labor, places the first best order quantity, and there is no need for furlough. The second row (from Theorem 1) summarizes the impact of introducing labor market frictions. It reveals that with low friction ( $α < 1$ ), only the ordering decisions are impacted ( $Q_{1}^{L} < Q_{1}^{H}$ ), while the downsizing decision remains unaffected ( $L_{1} * = ξ_{0} / β$ ). In contrast, when the friction level is high $α \geq 1$ , friction affects both the order quantity and the labor downsizing decisions, i.e., the firm also carries excess labor.

Finally, the last row of the table summarizes the impact of introducing the furlough option (from Theorems 2 and 3) in the presence of labor market frictions. In this case, if the friction level is low $α < 1$ , the firm still fully downsizes labor but now relies on furloughed labor for the next period. However, with high friction ( $α \geq 1$ ) and a low likelihood of furloughed labor returning ( $τ < τ_{0}$ for $1 \leq α \leq \bar{α}$ or $τ < τ_{2}$ for $\bar{α} < α < \overset{˘}{α}$ ), the firm opts to retain excess labor without furloughing. Conversely, when the probability of furloughed labor's return is high ( $τ \geq τ_{0}$ for $1 \leq α \leq \bar{α}$ or $τ \geq τ_{2}$ for $\bar{α} < α < \overset{˘}{α}$ ), the firm maintains excess labor and also utilizes furloughs. That is, the firm uses both its levers (carrying excess labor and furlough option) to manage its labor force.

Table 2 highlights key insights into the impact of furlough on operating decisions. When labor market frictions are low, furlough merely converts some laid-off workers to furloughed status without affecting inventory decisions, resulting in a limited effect on recovery in the second period. However, at higher levels of labor frictions, the furlough option also increases stocking levels, leading to faster recovery. Additionally, the furlough option is not effective when the probability of furloughed workers returning falls below a certain threshold, which depends on the level of labor market frictions.
5. Numerical Analysis

	$0 < α < 1$	$1 \leq α \leq \bar{α}$	$\bar{α} < α < \overset{˘}{α}$
Labor market with no frictions (Proposition 1)	Fully downsize labor: $L_{1} * = ξ_{0} / β$ Order the first best level: $Q_{1} * = Q_{1}^{H}$ No need for Furlough: $l * = 0$
Labor market with frictions and without furlough (Theorem 1)	- Fully downsize labor: $L_{1} * = ξ_{0} / β$ - Under-invest in inventory: $Q_{1} * = Q_{1}^{L} < Q_{1}^{H}$	- Carry excess labor: $L_{1} * = L_{1}^{H} > ξ_{0} / β$ - Under-invest in inventory: $Q_{1} * = Q_{1}^{L} < Q_{1}^{H}$	- Carry excess labor: $L_{1} * = L_{1}^{B} > L_{1}^{H}$ - Under-invest in inventory: $Q_{1} * = Q_{1}^{B} < Q_{1}^{H}$
Labor market with frictions and uncertain furlough (Theorems 2 and 3)	- Fully downsize labor: $L_{1} * = ξ_{0} / β$ - Under-invest in inventory: $Q_{1} * = Q_{1}^{L} < Q_{1}^{H}$ - Furlough labor	If $τ < τ_{0}$ : - Results are identical to “Labor market with frictions and without furlough” case If $τ \geq τ_{0}$ : - Carry less excess labor: $L_{1} * = {\hat{L}}_{1}^{H} < L_{1}^{H}$ - Order less conservatively: $Q_{1} * = min {{\hat{Q}}_{1}^{B}, Q_{1}^{L}} \geq Q_{1}^{L}$ - Furlough labor	If $τ < τ_{2}$ : - Results are identical to “Labor market with frictions and without furlough” case If $τ \geq τ_{2}$ : - Carry less excess labor $L_{1} * = {\hat{L}}_{1}^{H} < L_{1}^{H}$ - Order less conservatively: $Q_{1} * = {\hat{Q}}_{1}^{B} > Q_{1}^{B}$ - Furlough labor

In this section, we setup a numerical experiment to demonstrate the implications of the theorems in Section 4 under a wide range of conditions. This experiment is designed to not only confirm the theoretical findings but also to deepen our understanding of the interplay among critical model parameters, including profit margin, furlough risk, and labor friction. That is, through this experiment, we aim to provide a more comprehensive perspective on how these factors collectively influence the profit, labor, and inventory decisions.

5.1 Impact of Labor Frictions on Profit, Labor, and Inventory

First, we numerically illustrate the impact of frictions on optimal labor, inventory, and firm profit over the two periods. Table 3 summarizes our base-case parameters and experimental setup for our numerical analysis, which cover the entire relevant range of important model parameters.

Table 3.
Setup for numerical analysis.

Profit margin ratio ( $m = 1 - \frac{c_{L} + c_{I}}{p}$ ) We set the unit sales price to $p = 20$ , and cover the entire range of profit margin ratio from 0 to 1, by varying the total unit cost of inventory and labor ( $c_{L} + c_{I}$ ) from 0 to $p = 20$ . For the base-case analysis, we let $\; c_{L}$ = 5 and $c_{I}$ = 3. Whenever it is necessary to illustrate the impact of other parameters in their full ranges, we sample three values of total unit cost from the interval $[0, \; 20]$ corresponding to low, medium and high levels of profit margin, i.e., $c_{L} + c_{I} = 12, 8$ and $4$ , respectively, and conduct our analysis. We keep the labor content of unit cost, $c_{L} / (c_{L} + c_{L})$ , fixed to 5/8 unless otherwise stated.

Labor market friction ( $α$ ) We consider the entire relevant range of labor market frictions $α \in [0, p / c_{L} - 1]$ . Note that Lemma 1, in our case, implies an upper bound on labor market frictions such that $α \leq p / c_{L} - 1$ . Whenever it is necessary to illustrate the impact of other parameters in their full ranges, we sample three values from the interval $[0, p / c_{L} - 1]$ corresponding to low, medium and high levels of labor market frictions, i.e., $\; α \in {0.5, \; 1.5, \; 2.5}$ and conduct our analysis.

Furlough cost and furlough risk We set the furlough cost $f = 1$ and vary the furlough risk such that $τ \in {0.2, 0.5, 0.8}$

Demand variability We consider three levels of coefficient of variation (CV) for the demand corresponding to low, medium and high levels of demand variability, i.e., $\; C V \in {0.19, \; 0.38, \; 0.57}$ . Note that 0.57 is the highest CV under uniform demand distribution.

Other parameters We let demand shock $ξ_{0} = 5$ , regular demand $ξ_{1} \sim U [0, 100]$ , initial inventory $Q_{0} = 90$ , initial labor $L_{0} = 90$ and labor productivity $β = 1$ . Note that we set the demand shock significantly below the expected regular demand to reflect the nature of the shrinkage of the business during the first period. Accordingly, we also set initial inventory and labor to significantly large levels as compared to the shock period demand.

In Figure 2, we present the key performance measures—inventory, labor, and total profit—as a function of the levels of demand variability and labor market frictions. The x-axis in all figures is set to the full range of labor market frictions ( $α$ ).

Figure 2.

Impact of labor friction.

As we have noted in Section 4, we observe that orders for the second period ( $Q_{1}$ ) are decreasing with labor market friction (Figure 2(a)). Regarding the labor decision (Figure 2(b)), as predicted by Theorem 1, we observe that for low levels of labor market frictions the firm fully downscales initial labor ( $L_{0}$ is reduced to $L_{1} = ξ_{0} / β = 5$ ). For intermediate level of labor market frictions, however, labor is not fully downsized and it increases with labor market frictions ( $α)$ . For sufficiently high levels of labor market frictions, the firm conservatively downscales labor, as hiring labor back in the second period is very costly.

The expected second period labor ( $E [L_{2}]$ ) follows the pattern of new inventory orders $Q_{1}$ (Figure 2(c)). In the second period, firm's ability to exploit demand is limited by new orders $Q_{1}$ . In other words, declining new orders in the retail business is a signal of declining employment (or lagged recovery) in the following period. Note that this observation is consistent with our analytical results in Section 4.

Figure 2(d) presents the total expected employment over the two periods (this figure can also be interpreted as the total wages or labor surplus). The total expected employment is nonmonotone in the level of labor market frictions. As explained in the previous section, labor market frictions have two distinct effects on employment: a main direct effect and a secondary lagged effect through inventory. Firstly, the presence of labor market frictions serves as a deterrence against laying off labor as it makes it costly to hire and fire labor. This effect is observed in the first period when the firm downscales labor (see Figure 2(b)), i.e., as the labor market frictions increase, firm tends to downscale less aggressively and keeps more labor during the first stage. Hence, the main effect of labor market frictions on total employment is positive and nondecreasing with friction level. The indirect secondary effect of labor market frictions on employment is generated through under-investment in inventories. Labor market frictions motivate downscaling of inventories ( $Q_{1}$ decreases with $α$ ), which eventually limits firm's need for labor and the ability to capture demand in the next period. So, the expected employment level in the second period ( $E [L_{2}]$ ) decreases with labor market frictions, i.e., the secondary effect of labor market frictions is negative on total employment (Figure 2(c)). The sum of these two effects (shown in Figure 2(d)) on employment first decreases and then increases with labor friction.

Intuitively, total profit decreases with increasing labor market frictions (Figure 2(e)). Additionally, we find that the inventory decisions of firms with volatile demand are more sensitive to labor market frictions, and these firms are also more aggressive in downsizing labor in the first period (Figure 2(a) and (b)).

The impact of labor market frictions on the retailer depends on the firm's ability to generate profits and the composition of this profit margin, i.e., the relative dependence of the profits on unit labor and inventory costs ( $c_{L}$ and $c_{I}$ ). In Figure 3, we examine how the profit margin of the firm conditions the impact of frictions on key performance measures. In this figure, we set $c_{L}$ = $c_{I}$ , and vary the total unit cost $c_{L} + c_{I}$ such that the profit margin ratio of the firm, $1 - \frac{c_{L} + c_{I}}{p}$ , ranges from 0 to 1. A detailed discussion on the impact of composition of profit margin (i.e., the relative size of $c_{L}$ and $c_{I}$ ) is available in E-companion, Appendix B in the online supplemental materials.

Figure 3.

Interaction of labor frictions and profit margin.

Intuitively, profits decrease when we increase labor market frictions (Figure 3(a)). However, friction's impact on profit depends on the profit margin. When the profit margin is low, labor market frictions have little impact on profits. In this case, the firm carries a low level of inventory, and it also needs a low amount of labor to sell it. Hence, the firm's expected trade volume in the labor market (firing and hiring labor) would be low limiting firm's exposure to labor market frictions.

As the profit margin increases, firm's order quantity quickly goes up in the absence of labor market frictions (Figure 3(b)). In this case, introducing labor market frictions also quickly erodes the critical fractile for inventory, making firm's total profit highly sensitive to labor market frictions, particularly when the profit margin ratio is neither too high nor too low (Figure 3(a)). If the profit margin ratio becomes very high, then the impact of labor market frictions is absorbed by the operating margin, i.e., operating margin is so high that the order quantity does not change too much when labor market frictions are introduced (Figure 3(b)).

The interaction of profit margin ratio and labor market frictions on total employment is less intuitive (Figure 3(c)–(e)). When the profit margin ratio is low, introducing labor market frictions reduces total employment. When low profit margin firms face labor market frictions, they are more likely to fully downscale labor as in the no friction case (Figure 3(c)). Hence, changes in the level of labor market frictions are less likely to impact employment in the first period for these firms. However, since the inventories are also scaled down in response to labor market frictions, the second period labor goes down with frictions (Figure 3(d)). Overall, the introduction of labor market frictions results in a reduction in total employment for low profit margin firms (Figure 3(e)). At high profit firms, however, labor market frictions motivate managers to keep excess labor in the first period via conservatively scaling down labor. These firms also act more conservatively when scaling down inventories leading to higher total employment in the presence of labor market frictions (Figure 3(e)).

5.2 Policy Implications - Furlough

Furloughing labor provides the firm with flexibility when trading in the labor market, as furloughed employees can be called back to work – after observing the demand- without being subject to labor market frictions. That is, the firm avoids incurring search and training expenses, along with other opportunity costs, when recalling furloughed employees. However, furloughing carries a risk: employees might find alternative employment and not be available when needed later. Our model accounts for this risk through a parameter, $τ$ , representing the likelihood that furloughed employees will be available for rehire.

In Figure 4, we illustrate optimal decisions in the presence of furlough to highlight the impact of the furlough option on operations as a function of furlough risk and level of labor market friction. Figure 4(a)–(c) should be interpreted together. First consider the case when $τ = 0.5$ . When the labor friction is low ( $α < 1$ ), the firm fully downsizes labor (as in Theorem 1), but now it may use furlough to manage its labor force. When the friction level exceeds $α = 1.7$ , then simply furloughing labor is not enough to manage the labor and the firm also partially downsizes labor ( $L_{1} * > ξ_{0} / β$ ) to hedge against the cost of hiring in the second period. So, the firm uses both levers (furloughing and carrying excess labor) to manage its labor force. As the friction level further increases, the amount of furloughed labor decreases, and the firm's reliance on excess labor increases. This is because under very high labor friction, the risk associated with furloughing dominates outcomes, leading the firm to increase its reliance on excess labor. As discussed in the previous section, inventory ( $Q_{1} *$ ) is decreasing with friction, but using furlough reduces the negative impact of labor friction on inventory.

Figure 4.

Optimal decisions with furlough.

When the furlough risk is high $τ = 0.2$ , firm tends to rely more on excess labor and does not use furlough option. Also in this case, the decline in inventory due to labor friction is more pronounced. For low furlough risk $τ = 0.8$ , the firm instead relies on furlough rather than carrying excess labor to manage its labor force.

In Figure 5, we illustrate the optimal furlough amount and its impact on profits, inventory and labor decisions as we vary profit margin, $m \in {0.4, 0.6, 0.8}$ and furlough risk, $τ \in {0.2, 0.5, 0.8}$ . The first row of Figure 5 shows the value of furlough defined as

\begin{aligned} V a l u e o f F u r l o u g h (Δ π) \\ = (O p t i m a l P r o f i t s w i t h F u r l o u g h \\ - O p t i m a l P r o f i t s w i t h o u t F u r l o u g h) \end{aligned}

Figure 5.

Impact of furlough.

The second row of the figure shows the optimal amount of furloughed labor. The third row shows the impact of furlough on labor downsizing decision $L_{1}$ defined as:

\begin{aligned} I m p a c t o f F u r l o u g h o n L a b o r D o w n s i z i n g (Δ L_{1}) \\ = (L_{1} w i t h F u r l o u g h - L_{1} w i t h o u t F u r l o u g h) . \end{aligned}

The fourth row shows the impact of furlough on optimal inventory decision

Q_{1}

defined as:

\begin{aligned} I m p a c t o f F u r l o u g h o n I n v e n t o r y (Δ Q_{1}) \\ = (Q_{1} w i t h F u r l o u g h - Q_{1} w i t h o u t F u r l o u g h) . \end{aligned}

Each cell in Figure 5 illustrates the dynamics in one of the situations mentioned above with respect to changing levels of labor market friction

α

Impact of labor friction: One would expect that as the labor market friction increases (plotted in the x-axis in Figure 5), trading in the labor market becomes more costly increasing the value of furlough flexibility. An analysis of the first and second rows of Figure 5 reveals that this intuition, however, is not fully correct. Value of furlough, across all profit margin levels and furlough risk scenarios, initially increases and then decreases in response to labor friction. When labor friction is low, it is more cost-effective to avoid furloughs, as firing and rehiring workers is cheaper. Conversely, at high friction levels, firms tend to eschew furloughs due to the uncertainty of furloughed labor returning, opting instead to retain excess labor. Therefore, in these scenarios, furloughs have a minimal impact on operations and profits.

However, at intermediate friction levels, furloughs are employed more strategically. Here, the furlough serves as an effective tool for balancing the costs associated with market friction and labor downsizing. This leads to significant improvements in both operating decisions and overall profits.

Impact of profit margin:

For high profit margin firms (for $m = 0.8$ in Figure 5), the availability of a furlough option does not impact inventory levels ( $Δ Q_{1} = 0$ ). However, these firms do exhibit an increase in labor downsizing ( $Δ L_{1} \leq 0$ ), driven by furloughed labor, when faced with sufficiently high levels of labor friction. For such firms, the use of furloughs has a limited effect on the recovery phase in the subsequent period and also on total profits. This is because the unchanged inventory levels continue to constrain hiring decisions and profits in the second period.

For intermediate profit margin firms (for $m = 0.6$ in Figure 5), the furlough option influences both labor and inventory decisions. Specifically, when labor market friction reaches sufficiently high levels (e.g., $α = 1.5$ for $τ = 0.5$ ), not only does furlough lead to increased labor downsizing, but it also prompts an increase in inventory orders. This demonstrates how furlough becomes a more integrated part of operational strategy under certain market conditions. Consequently, we observe that firms with intermediate profit margin benefit the most from the furlough option.

For low profit margin firms (for $m = 0.4$ in Figure 5), the threshold for initiating furlough is lower, meaning these firms resort to furloughs even with relatively lower levels of labor friction. Furloughing in these firms has a more pronounced positive impact on inventory levels. Unlike in medium margin firms, however, labor downsizing remains unaffected ( $Δ L_{1} = 0$ ) by the furlough strategy in low margin firms. This indicates that while furloughs do not impact layoffs in firms with lower profit margins, they do contribute to an increase in new orders.

Impact of furlough risk: Impact of furlough risk is intuitive. Higher furlough risk results in lower furloughed labor, lower impact on operations and lower value for furlough option.

6. Discussion

We examine the interaction of inventory, labor, and furlough decisions in the presence of labor market frictions under a demand shock driven by a pandemic or an economic turmoil. A policy debate in such extreme settings is often linked with the impact of labor market frictions and furlough option on employment and firm profits (Bruhn, 2020; Eichhorst et al., 2010). We find that the net impact of labor market frictions on retail profits and employment is sizable, and it must be assessed together with the inventory and furlough decisions.

6.1 Impact of Labor Frictions on Operations

Our analysis highlights a key operational mechanism for the functioning of labor market frictions in the context of retail operations management during an economic shock (Theorem 1). It is well established in OM-finance literature that firms under-produce/order in the presence of costly external funds (a form of capital market frictions). Our findings extend this key result to the labor market friction context. That is, firms under-invest in inventories in the presence of labor market fictions since labor and inventories present complementarity in retail markets. However, the impact of labor frictions on employment is not obvious.

We show that the labor market frictions have two separate effects on labor level: (i) a positive main effect on the labor amount during the period of economic shock and (ii) a negative secondary effect on the labor level during the economic recovery period induced through reduced inventory levels. The net employment effect of labor market frictions is the sum of these two effects which can be either positive or negative depending on the severity of the economic shock, level of labor market frictions and profit margin of the firm. In particular, labor market frictions are likely to increase total employment if labor market frictions and the retail margin are high.

6.2 The Role of Profit Margin

We find that the retailer's profit margin is a key factor conditioning the impact of frictions on operating decisions and profits. In particular, when the profit margin is neither too low nor too high, labor market frictions result in a sharper reduction in critical fractile (Figure 3(b)). That is, for medium levels of profit margin, firm's total profit becomes highly sensitive to labor market frictions.

Labor market frictions are also more likely to result in a reduction in total employment for low profit margin firms (Figure 3(e)) as these firms cannot afford to keep excess labor in the first period. For high profit margin firms, however, labor market frictions motivate them to keep excess labor in the first period via conservatively scaling down labor. So, these firms also act more conservatively when scaling down inventories leading to higher total employment in the presence of labor market frictions.

6.3 Impact of Furlough Option

Our analysis shows how the strategic option to furlough, contingent upon the likelihood of furloughed employees’ return ( $τ$ ), interacts with labor market frictions ( $α$ ) to shape inventory decisions, labor utilization, and overall firm profitability. We find that under low labor friction scenarios ( $α < 1$ ), firms are inclined to fully downsize their workforce in the face of reduced demand, yet the option to furlough offers a cost-effective alternative to navigate through periods of uncertainty without fully severing ties with the workforce.

As labor market frictions intensify ( $α \geq 1$ ), the dynamics of furloughing versus downsizing shifts, with firms increasingly leveraging furlough as a means to retain a ready-to-mobilize workforce, albeit at the risk of employees not returning postfurlough. This strategy, while mitigating some of the search and training costs associated with rehiring in tight labor markets, introduces a layer of uncertainty in workforce planning. Our model captures this complexity by integrating the probability of furloughed labor returning to work ( $τ$ ), revealing that as $τ$ decreases, firms are more likely to resort to carrying excess labor, thus underscoring the critical balance between maintaining operational flexibility and managing labor costs.

Furthermore, our results illustrate a significant interaction between furlough decisions, labor market frictions, and retail margins, indicating that the strategic value of furloughing is magnified under certain economic conditions. Specifically, we observe that in environments with low furlough risk and intermediate levels of labor market frictions and profit margins, firms adjust both their labor and inventory strategies to optimize profitability and operational resilience. This adjustment manifests in a more nuanced approach to furloughing—weighing the cost of potential workforce unavailability against the benefits of having a flexible, scalable labor force ready to respond to market recoveries. Through our comprehensive analysis, we shed light on the pivotal role of furloughing in retail operations management, offering insights into how firms can manage the complexities of labor dynamics to sustain profitability in the face of fluctuating market conditions.

7. Limitations and Future Research

In this study, we focus on understanding the impact of labor market frictions on labor, inventory, and furlough decisions of a retailer during an economic shock. Our researches can be extended in multiple ways. Firstly, our analysis does not explicitly incorporate risk aversion in the decision-making process of the retailer. Future studies could enhance our model by incorporating risk aversion, providing a more nuanced understanding of how retailer's risk aversion condition the impact of labor frictions on the operations of the firm.

Secondly, the development of a general equilibrium model represents a promising avenue for extending our findings. Such a model would allow for a comprehensive analysis of the interactions between multiple markets and the broader economic environment, capturing the dynamic effects of supply and demand across sectors. Building on this research by incorporating general equilibrium considerations could significantly deepen our understanding of the systemic impacts of retail inventory, labor, and furlough decisions on economic stability and growth. In a related manner, explicitly modelling heterogeneity among labor productivity may also lead to interesting managerial insights. Also, treating inventory and labor as substitutes would significantly alter the outcomes and insights of our model.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478241273876 - Supplemental material for Furloughing Employees with Uncertain Return? Management of Labor Frictions and Inventory Under Demand Shock

Supplemental material, sj-pdf-1-pao-10.1177_10591478241273876 for Furloughing Employees with Uncertain Return? Management of Labor Frictions and Inventory Under Demand Shock by Fehmi Tanrisever and Nitindra Joglekar in Production and Operations Management

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Fehmi Tanrisever

Supplemental Material

Supplemental material for this article is available online ().

Notes

How to cite this article

Tanrisever F and Joglekar N (2024) Furloughing Employees with Uncertain Return? Management of Labor Frictions and Inventory Under Demand Shock. Production and Operations Management 33(11): 2139–2157.

References

Alan

Gaur

(2018) Operational investment and capital structure under asset-based lending. Manufacturing & Service Operations Management 20(4): 637–654.

Babich

Birge

(2022) Interface of operations and finance: a tutorial. In: Innovative Technology at the Interface of Finance and Operations. NY: Springer International Publishing, 43–99.

Babich

Kouvelis

(2018) Introduction to the special issue on research at the interface of finance, operations, and risk management (iFORM): recent contributions and future directions. Manufacturing & Service Operations Management 20(1): 1–18.

Bernstein

Richter

Throckmorton

(2020) COVID-19: a view from the labor market. FRB of Dallas Working Paper No. 2010 . Available at SSRN: https://ssrn.com/abstract=3587636.

Birge

Parker

, et al. (2017) When customers anticipate liquidation sales: managing operations under financial distress. Manufacturing & Service Operations Management 19(4): 657–673.

Birinci

Karahan

Mercan

, et al. (2021) Labor market policies during an epidemic. Journal of Public Economics 194: 104348.

Boulding

Lee

Staelin

(1994) Mastering the mix: do advertising, promotion, and sales force activities lead to differentiation? Journal of Marketing Research 31(2): 159–172.

Boyabatli

Kleindorfer

Koontz

(2011) Integrating long-term and short-term contracting in beef supply chains. Management Science 57(10): 1771–1787.

Boyabatli

Leng

Toktay

(2016) The impact of budget constraints on flexible vs. dedicated technology choice. Management Science 62(1): 225–244.

10.

Boyabatli

Toktay

(2011) Stochastic capacity investment and flexible vs. dedicated technology choice in imperfect capital markets. Management Science 57(12): 2163–2179.

11.

Bruhn

(2020) Can wage subsidies boost employment in the wake of an economic crisis? Evidence from Mexico. The Journal of Development Studies 56(8): 1558–1577.

12.

Business Insider (2020) The coronavirus outbreak has triggered unprecedented mass layoffs and furloughs. Available at: https://www.businessinsider.com/coronavirus-layoffs-furloughs-hospitality-service-travel-unemployment-2020#kohls-is-cutting-15-of-its-corporate-workforce-the-unspecified-cuts-will-save-the-company-65-million-annually-according-to-a-september-15-securities-filing-17 (accessed 5 May 2024).

13.

Chod

Zhou

(2014) Resource flexibility and capital structure. Management Science 60(3): 708–729.

14.

Dai

Jerath

(2013) Salesforce compensation with inventory considerations. Management Science 59(11): 2490–2501.

15.

Devalkar

Anupindi

Sinha

(2018) Dynamic risk management of commodity operations: model and analysis. Manufacturing & Service Operations Management 20(2): 317–332.

16.

Devalkar

Krishnan

(2019) The impact of working capital financing costs on the efficiency of trade credit. Production and Operations Management 28(4): 878–889.

17.

Diamond

(1982) Aggregate demand management in search equilibrium. Journal of Political Economy 90(5): 881–894.

18.

Ding

Dong

Kouvelis

(2007) On the integration of production and financial hedging decisions in global markets. Operations Research 55(3): 470–489.

19.

Eichhorst

Escudero

Marx

, et al. (2010) The impact of the crisis on employment and the role of labour market institutions. IZA Discussion Paper No. 5320 .Available at SSRN: https://ssrn.com/abstract=1712627.

20.

Froot

Scharfstein

Stein

(1993) Risk management: coordinating corporate investment and financing policies. The Journal of Finance 48(5): 1629–1658.

21.

Goel

Tanrisever

(2017) Financial hedging and optimal procurement policies under correlated price and demand. Production and Operations Management 26(10): 1924–1945.

22.

Görtz

McGowan

Mortimer-Lee

(2022) Box B: a targeted Furlough Scheme to help the economy in downturns. National Institute UK Economic Outlook – Winter 2022.

23.

Iancu

Trichakis

Tsoukalas

(2017) Is operating flexibility harmful under debt? Management Science 63(6): 1730–1761.

24.

Kesavan

Staats

Gilland

(2014) Volume flexibility in services: the costs and benefits of flexible labor resources. Management Science 60(8): 1884–1906.

25.

Kouvelis

(2019) Integrated risk management for newsvendors with value-at-risk constraints. Manufacturing & Service Operations Management 21(4): 816–832.

26.

Kouvelis

(2021) A supply chain theory of factoring and reverse factoring. Management Science 67(10): 6071–6088.

27.

Kouvelis

Zhao

(2012) Financing the newsvendor: supplier vs. bank, and the structure of optimal trade credit contracts. Operations Research 60(3): 566–580.

28.

Kouvelis

Zhao

(2016) Supply chain contract design under financial constraints and bankruptcy costs. Management Science 62(8): 2341–2357.

29.

Lee

Tang

Yang

, et al. (2022) Dynamic trade finance in the presence of information frictions and fintech. Manufacturing & Service Operations Management 25(6): 2038–2055.

30.

Mayers

Smith

(1982) On the corporate demand for insurance. In: Dionne

Harrington

(eds) Foundations of Insurance Economics. Dordrecht: Springer, 190–205.

31.

Mortensen

(1982) Property rights and efficiency in mating, racing, and related games. The American Economic Review 72(5): 968–979.

32.

Mutha

Bansal

Guide

VDR

(2021) Managing the inter-functional tension between accounting-and financial-profits in remanufacturing multiple-use cycle products. Production and Operations Management 30(9): 2993–3014.

33.

Pinker

Lee

Berman

(2009) Can flexibility be constraining? IIE Transactions 42(1): 45–59.

34.

Pinker

Larson

(2003) Optimizing the use of contingent labor when demand is uncertain. European Journal of Operational Research 144(1): 39–55.

35.

Pissarides

(1985) Short-run equilibrium dynamics of unemployment, vacancies, and real wages. The American Economic Review 75(4): 676–690.

36.

Pissarides

(2000) Equilibrium Unemployment Theory. MA: MIT press.

37.

Pissarides

(2011) Equilibrium in the labor market with search frictions. American Economic Review 101(4): 1092–1105.

38.

Qin

Nembhard

Barnes

II (2015) Workforce flexibility in operations management. Surveys in Operations Research and Management Science 20(1): 19–33.

39.

Reindorp

Tanrisever

Lange

(2018) Purchase order financing: Credit, commitment, and supply chain consequences. Operations Research, 66(5), 1287–1303.

40.

Reuters (2020) U.S. companies criticized for cutting jobs rather than investor payouts. Available at: https://www.reuters.com/article/business/us-companies-criticized-for-cutting-jobs-ratherthan-investor-payouts-idUSKBN21Q24X/ (accessed 7 June 2024).

41.

Smith

Stulz

(1985) The determinants of firms’ hedging policies. The Journal of Financial and Quantitative Analysis, 20(4), 391–405.

42.

Stulz

(1996) Rethinking risk management. Journal of Applied Corporate Finance 9(3): 8–25.

43.

Sucher

Winterberg

(2014) Furloughs: an alternative to layoffs for economic downturns. Harvard Business School Background Note, pp.314–097.

44.

Tang

Yang

(2018) Sourcing from suppliers with financial constraints and performance risk. Manufacturing & Service Operations Management 20(1): 70–84.

45.

Tanrisever

Cetinay

Reindorp

, et al. (2015) Value of reverse factoring in multi-stage supply chains. Available at SSRN: https://ssrn.com/abstract=2183991.

46.

Tanrisever

Joglekar

Erzurumlu

, et al. (2021) Managing capital market frictions via cost-reduction investments. Manufacturing & Service Operations Management 23(1): 88–105.

47.

Tunca

Zhu

(2018) Buyer intermediation in supplier finance. Management Science 64(12): 5631–5650.

48.

Yang

Bakshi

Chen

(2021) Trade credit insurance: operational value and contract choice. Management Science 67(2): 875–891.

49.

Yang

Birge

(2018) Trade credit, risk sharing, and inventory financing portfolios. Management Science 64(8): 3667–3689.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.73 MB

Profit margin ratio ( $m = 1 - \frac{c_{L} + c_{I}}{p}$ )	We set the unit sales price to $p = 20$ , and cover the entire range of profit margin ratio from 0 to 1, by varying the total unit cost of inventory and labor ( $c_{L} + c_{I}$ ) from 0 to $p = 20$ . For the base-case analysis, we let $\; c_{L}$ = 5 and $c_{I}$ = 3. Whenever it is necessary to illustrate the impact of other parameters in their full ranges, we sample three values of total unit cost from the interval $[0, \; 20]$ corresponding to low, medium and high levels of profit margin, i.e., $c_{L} + c_{I} = 12, 8$ and $4$ , respectively, and conduct our analysis. We keep the labor content of unit cost, $c_{L} / (c_{L} + c_{L})$ , fixed to 5/8 unless otherwise stated.
Labor market friction ( $α$ )	We consider the entire relevant range of labor market frictions $α \in [0, p / c_{L} - 1]$ . Note that Lemma 1, in our case, implies an upper bound on labor market frictions such that $α \leq p / c_{L} - 1$ . Whenever it is necessary to illustrate the impact of other parameters in their full ranges, we sample three values from the interval $[0, p / c_{L} - 1]$ corresponding to low, medium and high levels of labor market frictions, i.e., $\; α \in {0.5, \; 1.5, \; 2.5}$ and conduct our analysis.
Furlough cost and furlough risk	We set the furlough cost $f = 1$ and vary the furlough risk such that $τ \in {0.2, 0.5, 0.8}$
Demand variability	We consider three levels of coefficient of variation (CV) for the demand corresponding to low, medium and high levels of demand variability, i.e., $\; C V \in {0.19, \; 0.38, \; 0.57}$ . Note that 0.57 is the highest CV under uniform demand distribution.
Other parameters	We let demand shock $ξ_{0} = 5$ , regular demand $ξ_{1} \sim U [0, 100]$ , initial inventory $Q_{0} = 90$ , initial labor $L_{0} = 90$ and labor productivity $β = 1$ . Note that we set the demand shock significantly below the expected regular demand to reflect the nature of the shrinkage of the business during the first period. Accordingly, we also set initial inventory and labor to significantly large levels as compared to the shock period demand.

Furloughing Employees with Uncertain Return? Management of Labor Frictions and Inventory Under Demand Shock

Abstract

Keywords

1. Introduction

2. Literature Review

3. Model Description

Lemma 1 (Recovery stage)

Proposition 1 (Labor Market with no Frictions)

Table 1. Optimal decisions without furlough.

Theorem 1 (Labor Market with Frictions and without Furlough)

Theorem 2 (Labor Market with Low Labor Frictions and with Furlough)

Theorem 3 (Labor Market with High Labor Frictions and with Furlough)

5.1 Impact of Labor Frictions on Profit, Labor, and Inventory

6.1 Impact of Labor Frictions on Operations

6.2 The Role of Profit Margin

6.3 Impact of Furlough Option

7. Limitations and Future Research

Supplemental Material

sj-pdf-1-pao-10.1177_10591478241273876 - Supplemental material for Furloughing Employees with Uncertain Return? Management of Labor Frictions and Inventory Under Demand Shock

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iD

Supplemental Material

Notes

How to cite this article

References

Supplementary Material

Table 1.
Optimal decisions without furlough.