Improving Outcomes in Child Care Subsidy Voucher Programs under Regional Asymmetries

Abstract

Related to several efforts by governments to attain United Nations Sustainable Development Goals, subsidy vouchers are used to provide monetary support to income‐eligible (IE) families who cannot afford child care. However, IE families are often unable to accept an offered voucher due to an inadequate supply of voucher‐accepting child care providers that best suit their needs. Child Care Resource and Referral agencies (CCR&Rs), which administer these programs, can influence the family's propensity to accept a voucher by investing funds both in outreach to increase the number of voucher‐accepting providers in each region, and in provider services to improve the service delivery to families. This study analyzes how a CCR&R should allocate its limited funds between these two types of activities to ensure equitable access to child care across the different regions of its service area. We show that it might be optimal for the CCR&R to invest more funds in outreach in the region with a lower proportion of IE families. This is especially true when: the external considerations (e.g., public transportation and infrastructure) in that region have a greater impact on a family's acceptance propensity; the marginal return of investment in outreach in that region is higher and abundant funds are available; the socioeconomic distress experienced by families in that region is significantly higher; or a large amount of funds is earmarked for outreach in that region. We contextualize our study for a CCR&R in Massachusetts and conclude that the proposed investment decisions can improve equity outcomes by 7.0%.

Keywords

child care subsidy vouchers socially responsible operations UN sustainable development goals resource allocation

Introduction

Child care is an important intervention for early childhood development, and provides valuable support to families with young children (Richter et al. 2017). Access to child care is considered critical to achieving a better and more sustainable future for all. As such, access to high‐quality child care directly affects the indicators of several United Nations Sustainable Development Goals (UN SDGs), which were adopted as “a universal call to action to end poverty, save the planet and improve the lives and prospects of everyone, everywhere." These goals include no poverty (SDG #1), quality education (SDG #4), gender equality (SDG #5), and reduced inequalities (SDG #10), among others. Past research has shown that providing access to high‐quality child care can help improve the long‐term outcomes of children from economically disadvantaged backgrounds. These outcomes include educational attainment, earning capacity, and reduced anxiety and depression (Barnett and Masse 2007, Herbst 2017). Furthermore, increasing the accessibility of child care helps parents in low‐income families maintain employment or attend school, which in turn helps them break the cycle of poverty and contribute to the economy (Hartmann et al. 2003). It also reduces barriers to work for mothers, who disproportionately leave the workforce when their families cannot find or afford child care (Parker 2015). The combination of these life‐altering benefits clearly indicates that access to affordable child care is a key enabler for the SDGs mentioned above.

While access to child care has these far‐reaching positive effects on children, families, and society, it is prohibitively expensive throughout the United States (U.S.). On average, families in the U.S. spend $10,000 per year on child care, which forms a substantial portion (e.g., as high as 60%) of their annual income (as reported by the Child Care Aware of America; CCAoA 2018). Also, on average, a family's child care expenses are 40% higher than the affordability threshold defined by the Department of Health and Human Services (CCAoA 2018). Among the member countries of the Organisation for Economic Cooperation and Development, the U.S. is the least affordable nation for child care for low‐income families, as measured by the percentage of family income (OECD 2018). Unable to access affordable child care, approximately two million parents in the U.S. are forced to make career sacrifices every year (Schochet and Malik 2017). Furthermore, CCAoA (2018) reports that the U.S. economy loses $28.9 billion in wages annually because low‐income families do not have access to affordable child care. Given these repercussions, unaffordable child care is considered not only a humanitarian crisis but also an economic crisis (Covert 2019, UNICEF 2020).

Child Care Subsidy Voucher Programs in the U.S.

To help people balance the demands of work, school, and parenting, the federal and state governments in the U.S. provide support to income‐eligible (IE) families through child care subsidy programs.¹ Moreover, child care subsidy voucher programs constitute the most significant proportion of the entire portfolio of the country's child care subsidy schemes (Schumacher et al. 2003). Overall, federal and state entities spend more than $20 billion annually on child care subsidy funding, providing 90% of it to voucher programs for IE families. This subsidization results in the provision of care to over 1.4 million children monthly (Rachidi 2017). Under these subsidy voucher programs, IE families have the freedom to choose any voucher‐accepting child care provider based on factors such as family structure, location, and other considerations. The voucher essentially assures a fixed amount of monetary support from the government to subsidize the fees at the voucher‐accepting child care provider.

For the administration and management of day‐to‐day operations of child care subsidy voucher programs, almost all the state governments in the U.S. (specifically, 47 states and the District of Columbia) partner with Child Care Resource and Referral agencies (CCR&Rs). These agencies are state‐funded not‐for‐profit organizations tasked with providing services to IE families and child care providers in their designated areas. A typical CCR&R provides the following services: adding families that are in need of subsidized child care to a centralized wait list; completing paperwork for families that are issued subsidy vouchers; increasing the supply of child care providers; supporting providers to ensure high quality of care; and collecting, analyzing, and disseminating child care data, among others. A CCR&R offers an available voucher to IE families on the wait list on a first‐come first‐served (FCFS) basis. Our study takes the perspective of a CCR&R and considers how the operational actions it takes (in the form of resource allocation decisions) affect societal outcomes of the child care subsidy voucher program in its service area.

Problem Description and Research Questions

Several practitioner reports emphasize the need to address the low quantity and low quality of voucher‐accepting providers in order to ensure that eligible families are able to access subsidized child care (Pilarz et al. 2016, Ullrich et al. 2019). Within the context of these supply‐related concerns, the fact that IE families are often unable to accept offered vouchers is noteworthy (CCAoA 2018). Two principal factors cause this inability to accept vouchers. First, a low number of voucher‐accepting providers that best suit IE families’ practical needs lead to a low propensity of IE families accepting the offered vouchers. For instance, some providers may not operate during nontraditional hours, despite the high level of need for child care during such hours within IE families (Anderson et al. 2005). Also, immigrant IE families without cars are less likely to enroll at providers outside their neighborhoods (Greenberg et al. 2016). Second, research has found that, since continuity and stability of child care are important factors for IE families, they often decline subsidy vouchers due to potential administrative hassles or errors (Isaacs et al. 2016, Speirs et al. 2015). For instance, Isaacs et al. (2016) state that a reduction of hassles faced by IE families at child care providers can significantly increase the chances that eligible families are able to accept offered vouchers.

Unable to accept the offered subsidy vouchers, such IE families continue to bear an avoidable financial burden. This burden could take the form of having to give up a job or drop out of school, using savings or retirement funds, or borrowing money at high‐interest rates to cover child care (Banerjee et al. 2017, Marshall et al. 2013). Furthermore, Malik et al. (2018) report a 3% lower maternal labor force participation in areas that have an insufficient number of voucher‐accepting providers, because mothers who would otherwise be able to accept and use vouchers cannot re‐enter the workforce. These studies underscore how IE families’ inability to accept available vouchers adversely impacts society by exacerbating economic and gender inequalities.

To reduce these negative impacts on IE families and society, CCR&Rs undertake two types of supply‐enhancing activities: outreach and provider services. As part of outreach, CCR&Rs invest funds in various activities to increase the number of voucher‐accepting providers across different regions of their service areas. The rationale of this approach is that the greater the number of voucher‐accepting child care providers in a specific region, the more likely an IE family in that region will have access to a provider that suits their needs. The outreach activities can include identifying and collaborating with local stakeholders, creating public–private partnerships that stimulate local provider participation in subsidy voucher programs, and organizing events to inspire corporate voices to speak up for promoting voucher acceptance among local providers.

As part of provider services, CCR&Rs dedicate funds to reduce several potential administrative and ancillary challenges that may prohibit parents from using a subsidy voucher at voucher‐accepting providers. CCR&Rs can signal and ensure better delivery of child care at voucher‐accepting providers in their areas by offering several provider services (Pilarz 2018). These services can include the following: assisting providers with accreditation and quality improvement programs; conducting centralized training sessions for providers on health and safety, child development, and effective business practices; creating professional development opportunities for providers’ workforce; and purchasing technological or software tools to streamline administrative processes such as reimbursements. Although these provider services require a significant upfront investment, they potentially benefit all the affiliated providers at a negligible additional investment. For instance, developing an online information repository, organizing webinars by industry experts on quality improvements, or purchasing a software package to automate exchange of information with providers will require a significant amount of investment by a CCR&R; however, this support will significantly improve service delivery at all providers in its area (see IDHS 2017). So, while the investment in provider services is typically made at a service area level (unlike the regional outreach investment), its positive benefits accrue across all regions in the area.

However, as with most nonprofit organizations (NPOs), CCR&Rs have a limited amount of funds to allocate to these two types of activities. The levels of investments dedicated to regional outreach and provider services activities influence the supply (in terms of quantity and quality) of voucher‐accepting providers, which leads to disparities in access to subsidized child care for the IE families residing in different regions of the service areas (Chaudry et al. 2011, Pilarz et al. 2016). For example, within the Western Massachusetts service area, the ratio of the number of voucher‐accepting child care providers to the number of children on the wait list differs greatly across different regions, with Hampshire County having the highest ratio (0.28) and Western Worcester having the lowest ratio (0.08). Furthermore, research finds that residents and businesses in certain localities (e.g., ones with higher educational levels, stronger shared identities, or close‐knit communities) are more pro‐social and, thus, are more likely to help in community projects (Bierhoff 2005, Putnam 2001). As a result, the marginal benefit of funds invested in outreach in those localities may be higher than in other localities. Moreover, there are region‐based differences in the socioeconomic burden of distress faced by IE families when they are unable to accept offered subsidy vouchers. For example, Smith and Adams (2013) find that families in rural regions face more challenges and pay more when making child care arrangements than their urban counterparts.

Equity or fairness is a measure that reflects the notion that “rewards, punishments, and resources should be distributed according to a combination of different criteria: merit, need, equality, and procedural" (Leventhal 1980). Ideally, a CCR&R will split its budget to offer provider services in its service area and to conduct outreach in the regions within the service area, such that an IE family in any region can accept the offered voucher to access subsidized child care (see, e.g., the CDBG Act of 2014 and the report by National League of Cities 2017). However, as noted above, families are often unable to accept offered vouchers, and the probability of acceptance may differ from one region to the other. This leads to disparities in the distress experienced by IE families across different regions when they are unable to accept offered subsidy vouchers. Thus, from an equity perspective, the allocation of the CCR&R's funds for provider services and outreach should minimize this inequity in the distress experienced by IE families. The presence of various regional asymmetries—including transportation infrastructure and regulatory environment, marginal return of outreach investment, and distress experienced by families—complicates the allocation decisions, and it becomes important to answer the following research questions: (i) What are the optimal levels of investment by a CCR&R toward outreach and provider services activities that minimize the inequitable societal burden across different regions of its service area?; and (ii) How is this optimal allocation affected by activity‐ and region‐specific characteristics?

Summary of Main Findings

This study provides answers to these research questions by developing an analytical model that incorporates details of the subsidy voucher offer process, and that captures the challenges faced by a CCR&R when allocating funds for its outreach and provider services activities. To capture the influence of scarce funds and region‐specific differences, we consider that the CCR&R has a fixed amount of funds and its service area comprises two types of regions. We develop a measure of the inequity in distress between the two regions of its service area. To do so, we relate the propensity of acceptance/rejection of the available subsidy voucher by IE families residing in each of those regions with various region‐based asymmetries. With this equity‐based objective, we solve the CCR&R's optimization problem to obtain the optimal levels of investment that minimize inequity, and analyze the impact of activity‐ and region‐specific differences on the optimal investment decisions.

We show that, when the marginal return of provider services investment is higher (lower) than the marginal return of outreach investment, the CCR&R should invest more (less) in provider services than in outreach. While, in general, investing more funds in outreach in regions with more IE families is optimal, this may not be the case when regional asymmetries are incorporated in the fund allocation problem. Specifically, a CCR&R should invest more funds in outreach in a region with a relatively lower proportion of IE families in the following situations: (i) when the external considerations (those exogenous to the CCR&R) in that region have a greater impact on the IE family's propensity of acceptance, (ii) when the marginal return of outreach investment in that region is higher, and abundant funds are available, or (iii) when the distress faced by IE families in that region is significantly higher.

We also extend our model and analysis to understand the impact of the infusion of additional funds earmarked for outreach in one region. The related analysis shows that a substantially large amount of such earmarked funds may crowd out a CCR&R's investment in outreach in the non‐earmarked region, irrespective of the relative proportion of the IE families in each region. Furthermore, we contextualize these insights within a practical setting using a case study of a service area in Massachusetts. Through this real‐world application of our model, we conclude that the proposed investment decisions can improve equity outcomes by approximately 7.0%. Notably, this estimated value of the improvement in equity outcomes is directly related to the SDG Indicator 4.2, which includes reducing inequity in “access to quality early childhood development, care, and pre‐primary education" for rural versus urban recipients (SDG 2017).

The remainder of the paper is organized as follows: Section 2 presents a review of the related literature. Section 3 describes the model of a CCR&R's resource allocation problem. The results are discussed in Section 4. Section 5 presents a case study based on a real‐world example of the CCR&R that operates in Western Massachusetts. Finally, Section 6 concludes the paper by summarizing our study and providing directions for future research.

Literature Review

There is a growing interest within the field of operations management (OM) to study various aspects of socially responsible operations (Besiou and Van Wassenhove 2015, Lee and Tang 2017). Our paper contributes to this emerging line of inquiry along three dimensions: outcome improvement in education and child care programs; resource allocation by NPOs; and supply‐side decisions in socially responsible operations.

Outcome Improvement in Education and Child Care Programs. Several economists and sociologists, such as Barnett and Masse (2007) and Herbst (2017), have examined factors that can improve child‐related short‐term outcomes (e.g., performance on tests) and long‐term outcomes (e.g., educational attainment and earning capacity). However, few studies have considered operational challenges faced by organizations that provide education and child care (Kretschmer et al. 2014, Slaugh et al. 2016). In particular, Kretschmer et al. (2014) study food distribution across different regions through school feeding programs. Using a theoretical framework, they state that incorporating equity considerations is necessary in strategic resource planning, because remote regions are often the most impoverished and are logistically difficult to access. Also, a few recent papers analyzing the operational decisions of education systems have focused on issues of inequitable access (Bertsimas et al. 2019, Nguyen and Vohra 2019).

Similarly, within the context of child care subsidy programs in the U.S., several reports by government agencies and media organizations have emphasized the importance of increasing the number of voucher‐accepting providers, better allocating limited funds for more outreach across different regions, and enhancing the participation of low‐income families, all to reduce the societal burden resulting from inequitable access to subsidized child care (DHHS 2016, Keith 2019, Williams 2020). In their review of the gap between practitioner needs and academic publications, Besiou and Van Wassenhove (2020) find that OM researchers have not paid much attention to the operational aspects of voucher programs. To the best of our knowledge, ours is the first study to analyze the impact of operational decisions on equity outcomes within subsidy voucher programs. Given resource constraints and region‐specific differences in contextual parameters, such operational decisions constitute a resource allocation problem for CCR&Rs.

Resource Allocation by NPOs. OM scholars have given considerable attention to the challenges NPOs face when allocating limited resources across various value‐adding activities (Berenguer and Shen 2020). Considering a macro‐level view, several studies (Arya and Mittendorf 2016, Kotsi et al. 2019) have analyzed how NPOs should allocate their resources between fundraising, administration, and programs. While such studies offer interesting insights into when to prioritize program activities over fundraising and administrative tasks, they omit contextual details of how program activities create societal value. Considering the salient trade‐offs NPOs are forced to make, a few studies offer insights into NPOs that face the challenge of allocating limited resources across various program activities (Arora et al. 2020, McCoy and Lee 2014, de Véricourt and Lobo 2009). For instance, McCoy and Lee (2014) study the equitable allocation of limited resources for conducting outreach across different sites within the service area. Our study complements these previous studies by considering how an NPO should incorporate geographical differences and economic burdens on IE families and society into its resource allocation problem. Furthermore, by focusing on decisions that influence the supply side of child care subsidy programs, our study adds to the nascent literature on the supply‐side decisions in socially responsible operations, which are reviewed next.

Supply‐side Decisions in Socially Responsible Operations. Organizations that engage in socially responsible operations face unique challenges, owing to their distinct objectives and the significant gap between demand and supply of lifesaving products or services they offer (see Atasu et al. 2017, Lien et al. 2014, Martin et al. 2020). When the demand for these products or services is uncertain, a series of papers, including Urrea et al. (2019), Pedraza‐Martinez et al. (2020), and Zhang et al. (2020), have examined how mechanisms can be designed to ensure the right amount or the right type of products provided to beneficiaries. When demand is known, several studies have analyzed allocation policies that maximize social or individual welfare, with supply assumed to be exogenous. For instance, in the context of offering an available organ to patients on a wait list, Ata et al. (2016) consider geographic disparities, Bertsimas et al. (2013) consider fairness and flexibility, and Su and Zenios (2006) consider efficiency‐equity trade‐off to analyze health outcomes resulting from different policies for organ allocation.

In the context of child care subsidy voucher programs, while the demand for vouchers is known in the form of data collected by the government agencies, an IE family's acceptance of the available voucher can be influenced by investments of resources in supply‐side activities. Thus, we focus on the following distinguishing element of child care subsidy voucher programs: since some low‐income families cannot accept the vouchers offered to them (due to inadequate quantity and quality of the supply base), CCR&Rs can significantly reduce socioeconomic burdens through the investments in provider services and outreach activities.

Most NPOs depend on altruistic donations for their supplies of products or services, which leads to several limitations and challenges for NPOs when attempting to enhance their levels of supply (e.g., as studied in Arora and Subramanian 2019, Mehrotra and Natarajan 2020, Solak et al. 2018). With the exception of a study by Mehrotra and Natarajan (2020), most related studies in this stream do not consider other socially beneficial activities undertaken by NPOs, such as community outreach and partner development, to enlarge their supply of critical products or services. Mehrotra and Natarajan (2020) study the monetary incentive design problem of a healthcare provider in a developing country that has the goals of improving its delivery of services and encouraging more patients to seek care. Whereas, in the supply‐side decision problem considered in our setting, CCR&Rs need to allocate their limited financial funds between outreach and provider services activities (instead of offering monetary incentives) to increase the quantity and quality of service delivered to IE families in their service areas.

Overall, the review of the literature above underscores the importance of studying the supply‐enhancing investment decisions of CCR&Rs to improve outcomes for low‐income families and to benefit society. The scarcity of resources and the increased pressure to reach SDGs make a focus on examining the operations of humanitarian and nonprofit organizations necessary and timely (Besiou and Van Wassenhove 2020). In recent years, OM scholars have studied the equity outcomes of humanitarian aid delivery models (Ferrer et al. 2018, Gralla et al. 2014). While these studies focus on resource allocation problems within natural disaster settings, our study considers the resource allocation problem (and the salient trade‐offs) in child care subsidy voucher programs. Moreover, even though our study is motivated by a problem in a developed country, the regional asymmetries considered play similar roles in other voucher programs in developed and developing countries alike (Hatt et al. 2010, Ir et al. 2010). Therefore, this study sheds light on how the optimal allocation of funds across supply‐side activities can reduce inequitable access to subsidized services across different regions in a wide variety of pro‐social settings (see discussion in Section 6).

Model

In this section, we present our model of a CCR&R's allocation of funds between provider services and outreach activities in its designated service area. For these supply‐enhancing activities, the CCR&R has limited funds, denoted by F > 0. Next, we describe the key elements of the model in detail. (Table A1 in the appendix summarizes the key notation of this study.)

Region‐ and Activity‐specific Elements of Model

Types of Regions in a Service Area: Differences in geography, income levels, and racial/ethnic compositions lead to differences in demand for vouchers across the various regions within a CCR&R's service area (Swenson 2008). Furthermore, since the outreach activity is conducted at a local level, the extent of its impact is dependent on these region‐specific differences. In order to capture the heterogeneity of regions within the CCR&R's service area, we consider that the service area comprises two types of regions, namely, region 1 and region 2. Let γ ∈ (0, 1) denote the proportion of IE families in region 1 and 1−γ denote the proportion of IE families in region 2 (for simplicity, we assume one child per eligible family). In Section 5, we present a case study to generalize our model and results for a service area with more than two types of regions.

Types of Supply‐Side Activities: We consider that the CCR&R invests x amount of funds in provider services‐related activities in its service area. As noted in Section 1.2, these activities are undertaken at the service area level (for all voucher‐accepting providers in the service area) because of the large amount of funds required to, say, hire quality management consultants, and purchase attendance and workflow management software tools (Isaacs et al. 2016). Also, given that the investments in provider services are aimed at improving administrative and business practices at voucher‐accepting providers, they are made at an area level to ensure a standardized quality of child care. However, since investments in outreach‐related activities are targeted at promoting voucher‐acceptance among local providers at a regional level, we consider that the CCR&R invests

y_{1}

amount of funds for outreach in region 1 and

y_{2}

in region 2.

Voucher Offer to IE Families: When the state agency makes a voucher available for an IE family in the CCR&R's service area,² the CCR&R goes down the wait list of the IE families until a family accepts the voucher and enrolls for child care at the voucher‐accepting provider that best suits family's needs. Although the demand for vouchers in the service area is known in the form of a wait list, the wait list data are typically inaccurate, imperfect, and not up‐to‐date. Isaacs et al. (2015) emphasize that the poor quality of wait list data on IE families makes it difficult to estimate the caseload of CCR&Rs (as the basis for making operational decisions) and suggest using the U.S. Census Bureau data instead. Accordingly, in our model, the proportions of IE families in regions 1 and 2 (γ and 1−γ, respectively) also represent the probabilities that IE families that are offered the voucher reside in one region or the other. The probability that an IE family residing in region i accepts the offered voucher is denoted by

p (x, y_{i})

, where i ∈ {1, 2}. Accordingly,

1 - p (x, y_{i})

denotes the probability that the family is unable to accept the voucher, under which the CCR&R offers the voucher to the next eligible family in its service area. Figure 1 illustrates the progression of a voucher offered by the CCR&R to the IE families.

Figure 1

Progression of a Voucher Offered by the CCR&R to IE Families in Different Regions

Acceptance Probability: Consistent with the discussion above, we consider that the probability

p (x, y_{i})

increases in the levels of investment made by the CCR&R in provider services as well as outreach activities, that is,

\frac{\partial p (x, y_{i})}{\partial x} > 0

and

\frac{\partial p (x, y_{i})}{\partial y_{i}} > 0

for i ∈ {1, 2}. Furthermore, investing more in provider services yields a large benefit when the investment in outreach is high, and vice versa (CCAoA 2018, p. 21–23). Based on our conversations with officials at different CCR&Rs, it is observed in practice that any increase in the likelihood of IE family's voucher acceptance that results from an increased provider services investment is greatly enhanced by an increased outreach investment. This is because improvements in administrative and business practices at voucher‐accepting providers will be beneficial for the IE families (in the form of ease of doing business) when they are more likely to find the local voucher‐accepting provider that best suits their needs. Mathematically, this implies that the probability

p (x, y_{i})

exhibits the supermodularity property:

\frac{\partial^{2} p (x, y_{i})}{\partial x \partial y_{i}} > 0

. Finally, it is reasonable to assume that the probability exhibits diminishing returns with respect to the CCR&R's investments in each of the two types of activities, that is,

\frac{\partial^{2} p (x, y_{i})}{\partial x^{2}} < 0

and

\frac{\partial^{2} p (x, y_{i})}{\partial y_{i}^{2}} < 0

A function of the form of the Cobb–Douglas function,

x^{α} y_{i}^{β_{i}}

with 0 < α < 1 and

0 < β_{i} < 1

, satisfies these mathematical properties. Within OM, this specific functional form is commonly used in situations in which the output is co‐produced by two inputs (e.g., Andritsos and Tang 2018, Roels et al. 2010), which is true for this context.³ Also, it allows for the analytical characterization of the effect of different marginal returns of investments from different activities, which is an important consideration for NPOs. Specifically, we consider

p (x, y_{i}) ≐ min {{\underset{̲}{p}}_{i} + p_{oi} x^{α} y_{i}^{β_{i}}, 1}

, in which

0 < {\underset{̲}{p}}_{i} < 1

and

0 < p_{oi} < 1

for i ∈ {1,2}. The parameter

{\underset{̲}{p}}_{i}

denotes the baseline acceptance probability in region i, that is, the probability that an IE family accepts the offered voucher when the CCR&R invests no amount of funds toward either provider services or outreach in that region. In Section 3, we explore how the magnitudes of

{\underset{̲}{p}}_{1}

and

{\underset{̲}{p}}_{2}

affect the optimal investments.

External Environmental Factors: The parameters

p_{o 1}

and

p_{o 2}

capture the impact of external environmental factors on the probability that an offered voucher will be accepted by an IE family in regions 1 and 2, respectively. The term “external environmental factors" is commonly used in marketing parlance to refer to external legal, regulatory, and technical considerations that have an impact on the entity's decisions (Kotler and Armstrong 2017). In this context, in addition to the supply‐enhancing activities of the CCR&R, several reports underscore the role of external environmental factors, such as transportation infrastructure and regulatory environment, in determining an IE family's ability to accept an available subsidy voucher. For instance, Chaudry et al. (2011) explain that a wider coverage of the local public transportation network, higher service frequency of buses, and greater lobbying efforts by social activists to reduce paperwork requirements of immigrant IE families can increase the IE family's propensity to accept the offered voucher. In essence, the parameter

p_{oi}

captures those considerations that are external to the CCR&R's two activities, yet they have an influence on the probability of an IE family's acceptance of an offered voucher in region i. Furthermore, to analyze the effect of region‐based asymmetry in these external environmental factors on the optimal investments, we allow for

p_{o 1}

to be different from

p_{o 2}

. We consider

p_{o} ≐ \frac{p_{o 2}}{p_{o 1}}

and refer to

p_{o}

as the relative external environmental factor.

Provider Services and Outreach Elasticities: The parameters α and

β_{i}

capture the elasticity of an IE family's probability of accepting a voucher with respect to levels of investment in provider services and regional outreach activities, respectively. To relate with practice the relative magnitudes of α and

β_{i}

, consider a service area in which parents in most IE families work on a daily‐wage basis (e.g., a metro service area). In such an area, ease of doing business with child care providers (say, in terms of billing processes and limited unplanned shut‐down of centers) is a more prominent factor when parents decide whether to accept a voucher for child care (i.e.,

α > β_{i}

) (Adams et al. 2003). In contrast, consider a service area with a significant lack of infrastructure. In such an area, the inconvenience of lengthy commute times plays a significantly greater role in determining a family's decision to accept or decline a voucher, as compared to the factors related to the ease of doing business with a child care provider (i.e.,

β_{i} > α

) (Walker and Reschke 2004).

Socioeconomic Burden of Distress: In practice, a family may decline an offered voucher due to an inadequate supply of voucher‐accepting providers that best suit the family's child care needs. In such a scenario, the family faces a financial burden because a family member may have to give up a job, drop out of school, or use savings and retirement funds in order to take care of the child (Banerjee et al. 2017, Marshall et al. 2013). Meanwhile, society suffers from loss in productivity (e.g., absenteeism from work due to unavailability of reliable child care) and increased unemployment rates among parents in IE families (CCAoA 2018). Let

ξ_{i} > 0

denote the socioeconomic burden of distress borne by an IE family in region i that is unable to accept an offered voucher. Furthermore, to capture region‐based differences in distress, we allow

ξ_{1}

and

ξ_{2}

to be different.

The CCR&R's Resource Allocation Problem

The CCR&R operates under the state mandate that allows it to go down the wait list on a FCFS basis until an IE family accepts an available subsidy voucher (MEEC 2019). This implies that one of the IE families on the list is able to access child care using the available voucher. In that regard, any feasible distribution of funds among its supply‐enhancing activities will generate a positive societal outcome (since the acceptance probability increases in x and

y_{i}

). With no constraint on its resources, the CCR&R should be able to build the supply base to the extent that each IE family on the top of the wait list can accept the subsidy voucher when it becomes available (i.e.,

p (x, y_{i}) = 1

). However, due to the scarcity of resources and various regional asymmetries, the CCR&R's investment decisions lead to regional differences in the probability of accepting the offered voucher. Thus, in terms of equity, each feasible distribution of funds leads to varying levels of disparities in the distress experienced by IE families in different regions of the service area.

According to the World Health Organization, the term inequity refers to “avoidable or remediable differences among groups of people, whether those groups are defined socially, economically, demographically, or geographically" (WHO 2020). In our context, these differences arise when, because of CCR&R's levels of investments in each activity, an IE family in a given region is unable to accept an offered subsidy voucher. As per government regulations (the CDBG Act of 2014 and the CCDF Ruling 45 CFR 98.46(b) on September 30, 2016) and based on our interviews with experts in the field of child care subsidy voucher programs, the CCR&R must make operational decisions that ensure equitable access to affordable child care in different regions of its service areas. Moreover, several reports have underscored the need to address inequitable access to affordable child care across different parts of the nation (e.g., Malik et al. 2018).

Thus, the CCR&R should allocate its limited funds between x,

y_{1}

, and

y_{2}

such that the resulting inequity in distress experienced by IE families (that are unable to accept the offered voucher) across different regions is minimized. We formulate an expectation‐based measure of inequity in distress to ensure that the allocation of funds is based on an assessment of disparities across IE families in different regions of its service area. Next, we describe the formulation of this measure, which is in line with the relative equity principle that views the FCFS policy as the most equitable policy.

Objective Function: We consider that the CCR&R decides on investment levels x,

y_{1}

, and

y_{2}

such that when a voucher becomes available within the planning horizon, the expected measure of inequity in distress, denoted by

M I (x, y_{1}, y_{2})

, is minimized. Recall that with probability γ, the CCR&R offers the available voucher to an IE family residing in region 1. With probability

1 - p (x, y_{1})

, this family declines the voucher. In this scenario, since an IE family is unable to accept the subsidy voucher, the CCR&R increments the inequity measure by the magnitude of distress

ξ_{1}

. Next, the voucher is offered to an IE family in either region 1 (with probability γ) or region 2 (with probability 1‐γ). This voucher offer process continues till an IE family in the service area accepts the available voucher (see Figure 1). Using the probabilities γ and

p (x, y_{i})

, and the magnitude of distress

ξ_{i}

, the expected value of the measure of inequity in distress—as a function of the CCR&R's investment levels—can be expressed as follows:

\begin{matrix} M I (x, y_{1}, y_{2}) & ≐ γ (1 - p (x, y_{1})) [ξ_{1} + \underset{Per progression of voucher offer after the first rejection in region 1}{\underset{⏟}{γ (1 - p (x, y_{1})) (ξ_{1} + \dots) + (1 - γ) (1 - p (x, y_{2})) (ξ_{2} + \dots)}}] + \\ (1 - γ) (1 - p (x, y_{2})) [ξ_{2} + \underset{Per progression of voucher offer after the first rejection in region 2}{\underset{⏟}{γ (1 - p (x, y_{1})) (ξ_{1} + \dots) + (1 - γ) (1 - p (x, y_{2})) (ξ_{2} + \dots)}}], \end{matrix}

which can be re‐written in a recursive form as

M I (x, y_{1}, y_{2}) = γ (1 - p (x, y_{1})) [ξ_{1} + M I (x, y_{1}, y_{2})] + (1 - γ) (1 - p (x, y_{2})) [ξ_{2} + M I (x, y_{1}, y_{2})] .

And, on simplification, we obtain the following:

M I (x, y_{1}, y_{2}) = \underset{Conditional rejection probability in region 1}{\underset{⏟}{(\frac{γ (1 - p (x, y_{1}))}{γ p (x, y_{1}) + (1 - γ) p (x, y_{2})})}} ξ_{1} + \underset{Conditional rejection probability in region 2}{\underset{⏟}{(\frac{(1 - γ) (1 - p (x, y_{2}))}{γ p (x, y_{1}) + (1 - γ) p (x, y_{2})})}} ξ_{2} .

The terms within parentheses in Equation (1) capture the probability that an IE family in region i is unable to accept the offered voucher conditioned on the voucher being eventually accepted by an IE family in one of the regions. This implies that the expression for

M I (x, y_{1}, y_{2})

yields the expected distress value in the CCR&R's service area when the IE families in its different regions are unable to accept the offered voucher. In essence, the objective function ensures that the limited funds for the regional outreach activities are allocated proportionally between the two regions, wherein these proportions are governed by the contextual parameters. Furthermore, it turns out that the formulated measure

M I (x, y_{1}, y_{2})

captures the notion of proportional fairness. One of the first definitions of a proportional fairness objective was presented by Mo and Walrand (2000) in their model of equitable congestion controls. Specifically, they establish that a proportionally fair vector maximizes the (weighted) sum of logarithmic utility functions. In a lemma in the online companion, we illustrate the equivalence of the vector of optimal levels of investments resulting from our objective function with their definition.⁴

Optimization Problem: The measure

M I (x, y_{1}, y_{2})

obtains the highest value of

\frac{γ (1 - {\underset{̲}{p}}_{1}) ξ_{1} + (1 - γ) (1 - {\underset{̲}{p}}_{2}) ξ_{2}}{γ ({\underset{̲}{p}}_{1}) + (1 - γ) ({\underset{̲}{p}}_{2})} > 0

when x=0 or

y_{1} = y_{2} = 0

. For a given amount of funds F, the CCR&R can minimize

M I (x, y_{1}, y_{2})

by solving the following optimization problem:

min_{{x, y_{1}, y_{2}}} M I (x, y_{1}, y_{2}) = \frac{γ ξ_{1} + (1 - γ) ξ_{2} - γ ξ_{1} p (x, y_{1}) - (1 - γ) ξ_{2} p (x, y_{2})}{γ p (x, y_{1}) + (1 - γ) p (x, y_{2})}

s.t., x + y_{1} + y_{2} \leq F,

x \geq 0, y_{1} \geq 0, y_{2} \geq 0,

where the expression for the objective function is obtained by re‐arranging terms in Equation (1). The optimal investments in provider services and outreach activities are denoted by

(x^{*}, y_{1}^{*}, y_{2}^{*})

In the next section, we analyze the CCR&R's optimal levels of investments in each activity, and investigate the effect of contextual parameters on these optimal decisions. Proofs of the main analytical results are presented in Appendix B. Also, additional technical details are available in the online companion. Throughout the rest of the paper, the analysis is restricted to those combinations of parameters that ensure the probability

p (x, y_{i})

is less than unity, for i ∈ {1,2}. This allows for a focus on non‐trivial optimal decisions and on scenarios that are interesting as well as practically relevant. For instance, proof of Proposition 4 shows that

p (x^{*}, y_{i}^{*}) < 1

when the amount of funds available to the CCR&R for supply‐enhancing activities is low, which is true for most, if not all, CCR&Rs (CCAoA 2018).

Results

Proposition 1 shows the effects of activity‐specific characteristics on the CCR&R's optimal levels of investment in each activity. To highlight the relative role of provider services and outreach elasticities in the allocation decision, we initially consider situations with no regional differences (specifically,

p_{o} = 1

ξ_{1} = ξ_{2}

, and

β_{1} = β_{2} ≐ β

Proposition 1

The CCR&R's optimal levels of investment for 0 < α < 1 and 0 < β < 1 are:

x^{*} = \frac{F α}{α + β} > 0

y_{1}^{*} = \frac{F β γ^{\frac{1}{1 - β}}}{(α + β) (γ^{\frac{1}{1 - β}} + {(1 - γ)}^{\frac{1}{1 - β}})} > 0

, and

y_{2}^{*} = \frac{F β {(1 - γ)}^{\frac{1}{1 - β}}}{(α + β) (γ^{\frac{1}{1 - β}} + {(1 - γ)}^{\frac{1}{1 - β}})} > 0 .

In addition, (i)

x^{*} > y_{1}^{*} + y_{2}^{*}

if α > β; and

x^{*} \leq y_{1}^{*} + y_{2}^{*}

otherwise; (ii) There exists a threshold

\hat{α} ≐ β {(γ^{\frac{1}{1 - β}} + {(1 - γ)}^{\frac{1}{1 - β}})}^{- 1} min \{γ^{\frac{1}{1 - β}}, {(1 - γ)}^{\frac{1}{1 - β}}\}

such that,

x^{*} < min \{y_{1}^{*}, y_{2}^{*}\}

if and only if

α < \hat{α}

The analytical finding in Proposition 1 that the inequity‐minimizing split of funds depends on the relative magnitude of marginal returns is in line with the general view of the extant OM literature. As such, in practical terms, this finding is useful for NPOs that aim toward generating societal impact in resource‐constrained environments, a conclusion also noted by McCoy and Lee (2014) and Kotsi et al. (2019) in their baseline analyses of optimal investment decisions by NPOs. In our context, it is particularly important because most CCR&Rs neglect outreach activities in the presence of scarcity of resources (CCAoA 2018, p. 22). Proposition 1 shows that when α < β, it is optimal for the CCR&R to invest less in provider services than the total investment in outreach in regions 1 and 2. Moreover, when the provider services elasticity is below a threshold (

\hat{α}

), it is optimal for the CCR&R to spend more on outreach in each region. This is because of the relatively higher impact of outreach on a family's propensity to accept an offered voucher.⁵

We next analyze the effects of various region‐based asymmetries on the CCR&R's optimal levels of investments—specifically, we consider region‐based asymmetries in: external environmental factors (Section 4.1); outreach elasticity (Section 4.2); IE family's distress (Section 4.3); and earmarking of funds for outreach in a region (Section 4.4). To simplify exposition and understand the pure effect of each asymmetry, we present our analytical findings by introducing one asymmetry in each subsection (with other factors considered to be symmetric). We later show in Section 5 that similar insights hold for a general setting.

Effect of Region‐based Asymmetry in External Environmental Factors

In this subsection, we study how differences in the external environmental factors in regions 1 and 2 (captured by the relative external environmental factor

p_{o} = p_{o 2} / p_{o 1}

) affect the optimal levels of investments in outreach activities in each region. Our analysis shows that the effect of

p_{o}

on the optimal outreach investments (

y_{1}^{*}

and

y_{2}^{*}

) depends on the mix of families in each region and outreach elasticity, as presented in the following proposition.

Proposition 2

Define

{\hat{p}}_{o} ≐ \frac{γ}{1 - γ}

. Then, (i)

y_{1}^{*}

is decreasing in

p_{o}

, and

y_{2}^{*}

is increasing in

p_{o}

; (ii)

y_{1}^{*} < y_{2}^{*}

if and only if

p_{o} > {\hat{p}}_{o}

; (iii)

y_{1}^{*}

is unimodal with respect to β, and

y_{2}^{*}

increases in β when

p_{o} > {\hat{p}}_{o}

, and

y_{1}^{*}

increases in β, and

y_{2}^{*}

is unimodal with respect to β when

p_{o} \leq {\hat{p}}_{o}

Proposition 2(i) shows that as

p_{o}

gets larger, that is, when the external environmental factors in region 2 play an increasingly positive role as compared with those in region 1, the CCR&R should increase its outreach investment in region 2, at the expense of outreach investment in region 1. Next, when the effect of the external environmental factors in region 2 is significantly higher than in region 1 (i.e., when

p_{o} > {\hat{p}}_{o}

), Proposition 2(ii) shows that the CCR&R should invest more funds in conducting outreach in region 2 than that in region 1. By doing so, it can further reduce the chances that the IE families in region 2 will reject offered vouchers, and as a result, improve the overall societal outcomes in the service area. This analytical finding implies that the optimal outreach investment in region 2 (

y_{2}^{*}

) can be greater than the optimal outreach investment in region 1 (

y_{1}^{*}

), even when the majority of IE families reside in region 1.

Also, in situations when

p_{o} > {\hat{p}}_{o}

, Proposition 2(iii) shows that the CCR&R should respond to an increase in the outreach elasticity β by increasing

y_{2}^{*}

. Whereas, an increase in β has a more nuanced effect on

y_{1}^{*}

. When β increases, but remains below a threshold, the optimal investment

y_{1}^{*}

increases. However, when β increases beyond a threshold,

y_{1}^{*}

undergoes a decrease. This happens because, for a significantly high marginal benefit of outreach investment, the CCR&R should optimally channel its funds toward outreach in region 2 instead of region 1 (since

p_{o} > {\hat{p}}_{o}

Furthermore, based on expressions for optimal outreach investments, we obtain that the optimal balancing ratio of the CCR&R's investments in outreach activities, that is,

y_{1}^{*} / y_{2}^{*}

, is equal to

{(\frac{γ}{p_{o} (1 - γ)})}^{\frac{1}{1 - β}}

. This expression shows that the optimal outreach investments in both region are more balanced when the outreach elasticity is low. Whereas, when the outreach elasticity is high (β is large), the ratio of optimal investments becomes more skewed. Eventually, when β → 1, the CCR&R should invest in outreach activity in only one region. The linear returns to scale reduce the proportional fairness objective to a utilitarian objective, and it becomes optimal for the CCR&R to conduct outreach in only the more “efficient" region (based on

γ ≶ p_{o} (1 - γ)

).⁶

Effect of Region‐based Asymmetry in Outreach Elasticity

In this subsection, we analyze the effect of asymmetry in outreach elasticities on the CCR&R's optimal investments. Our analysis reveals that, when

β_{1} \neq β_{2}

, the CCR&R should determine the optimal balancing ratio of investments in outreach activities in its two regions, not only by considering the proportion of IE families and the outreach elasticity in each region, but also by taking into account the available funds and provider services elasticity. This finding is characterized in the following proposition.

Proposition 3

When the outreach elasticity is different for different regions, there exists a unique allocation strategy

(x^{*}, y_{1}^{*}, y_{2}^{*}) > 0

, such that

x^{*} = F - y_{1}^{*} - y_{2}^{*}

y_{2}^{*} = {(\frac{(1 - γ) β_{2}}{γ β_{1}})}^{\frac{1}{1 - β_{2}}} {(y_{1}^{*})}^{\frac{1 - β_{1}}{1 - β_{2}}}

, and

y_{1}^{*}

is the unique solution to the following equation:

F - (1 + \frac{α}{β_{1}}) y_{1} - (1 + \frac{α}{β_{2}}) {(\frac{(1 - γ) β_{2}}{γ β_{1}})}^{\frac{1}{1 - β_{2}}} {(y_{1})}^{\frac{1 - β_{1}}{1 - β_{2}}} = 0

. Furthermore, (i)

\frac{\partial}{\partial F} (\frac{y_{1}^{*}}{y_{2}^{*}}) > 0

if and only if

β_{1} > β_{2}

, and (ii)

\frac{\partial}{\partial α} (\frac{y_{1}^{*}}{y_{2}^{*}}) < 0

if and only if

β_{1} > β_{2}

Proposition 3 shows that, when the total amount of available funds increases (i.e., F increases), the CCR&R should increase its investment in outreach in the region with the higher outreach elasticity relative to the outreach investment in the other region. In contrast, when the provider services elasticity is high (i.e., α is large), it becomes optimal for the CCR&R to raise its investment in outreach in the region with the lower outreach elasticity relative to the region with the higher outreach elasticity. This is because, for a large α, the joint impact of investments in provider services and outreach activities on a family's acceptance probability would be higher when the levels of each investment are higher. This implies that the CCR&R would be able to overcome the adverse impact of smaller outreach elasticity in the region by investing more funds in outreach (in relative terms) in that region, when its provider services elasticity is high.

Another implication of asymmetry in outreach elasticities is that, under certain conditions, it may be optimal for the CCR&R to invest more funds in outreach activity in the region with a lower proportion of IE families. Specifically, when the CCR&R has a sufficiently high amount of funds (F), it should invest more funds in outreach in the region with a higher outreach elasticity, even when a higher proportion of IE families resides in the other region.⁷ This finding can be explained as follows: When the CCR&R has an abundance of resources, the effect of higher marginal returns of investment in outreach is more dominant in reducing the overall inequity as compared to the effect of having a higher proportion of IE families (since F is large). As a result, in such scenarios, it is beneficial to invest more funds toward outreach in a region with a higher marginal return than in a region with a higher proportion of IE families.

Effect of Region‐based Asymmetry in the IE Family's Distress

In this section, we study the impact of asymmetry on the socioeconomic burden of the distress experienced by IE families in regions 1 and 2. Without loss of generality, we consider the situations in which

ξ_{2} > ξ_{1}

. To parse out the pure effect of asymmetry in distress, we conduct analysis by considering the absence of other activity‐ and region‐specific differences. We later show in Section 5 that the generated insights are robust to situations in which other parameters are asymmetric.

Proposition 4

Consider α = β = 1/2, γ = 1/2,

p_{o 1} = p_{o 2} ≐ p^{o}

, and

{\underset{̲}{p}}_{1} = {\underset{̲}{p}}_{2} ≐ \underset{̲}{p}

. Then,

(x^{*}, y_{1}^{*}, y_{2}^{*})

(\frac{F}{2}, \frac{F}{2} (\frac{k_{1}^{2}}{k_{1}^{2} + k_{2}^{2}}) - \sqrt{\frac{F}{2} (\frac{k_{1}^{2}}{k_{1}^{2} + k_{2}^{2}}) - \frac{{(p^{o})}^{2} F^{3} {(1 - ξ)}^{2}}{2 (k_{1}^{2} + k_{2}^{2})}}, \frac{F}{2} (\frac{k_{1}^{2}}{k_{1}^{2} + k_{2}^{2}}) + \sqrt{\frac{F}{2} (\frac{k_{1}^{2}}{k_{1}^{2} + k_{2}^{2}}) - \frac{{(p^{o})}^{2} F^{3} {(1 - ξ)}^{2}}{2 (k_{1}^{2} + k_{2}^{2})}})

, where

k_{1} ≐ 2 ((\underset{̲}{p} + 1) ξ - \underset{̲}{p} + 1) > 0

k_{2} ≐ 2 ((1 - \underset{̲}{p}) ξ + 1 + \underset{̲}{p}) > 0

, and

ξ ≐ \frac{ξ_{2}}{ξ_{1}}

. Also,

y_{1}^{*}

is decreasing in ξ, and

y_{2}^{*}

is increasing in ξ.

Proposition 4 shows that a relative increase in the distress experienced by families in a particular region makes it optimal for the CCR&R to increase its investment in outreach in that region. This allows the CCR&R to increase the propensity of an IE family in such a region to accept an offered voucher, and, as a result, to minimize the adverse impact on the families in that region and on society. This finding also implies that, when the regional disparity in distress is large, it may be optimal for the CCR&R to allocate more funds to outreach in the region with the relatively higher socioeconomic burden of distress, even when the proportion of IE families in that region is lower. Consider that the majority of IE families reside in region 1. Then, per Proposition 4, it may be optimal for the CCR&R to invest a relatively higher amount of funds in outreach in region 2 as compared to region 1 when

ξ_{2}

is substantially larger than

ξ_{1}

Furthermore, we find that such a scenario (i.e.,

y_{1}^{*} < y_{2}^{*}

, even when γ>1/2) is more likely to arise in the following two situations. First, when the provider services elasticity α increases: This is because, when α is high, it is optimal for the CCR&R to increase investment in provider services by reducing investment in outreach in region 1 relatively more than the outreach investment in region 2 (since

ξ_{2} > ξ_{1}

). Second, when the baseline acceptance probabilities increase: Consider that the baseline acceptance probability in region 1

({\underset{̲}{p}}_{1})

increases, that is, the chance that an IE family in region 1 will reject an offered voucher reduces. As a result, the CCR&R should decrease outreach investment in region 1, and utilize those funds to increase its outreach investment in region 2 (since

ξ_{2} > ξ_{1}

). The explanation for the effect of the baseline probability

{\underset{̲}{p}}_{2}

is analogous.

Effect of Region‐based Earmarking of Funds for Outreach Activity

In most humanitarian and nonprofit settings, additional funds that are made available by a donor agency may be earmarked for investment in a specific activity only (Pedraza‐Martinez et al. 2020). In our context, the CCR&R may receive funds that can only be invested in outreach in a certain region (Keith 2019). Without loss of generality, we consider the case in which the CCR&R receives δ>0 additional funds to invest exclusively in outreach in region 1. Accordingly, the CCR&R's resource allocation problem described in Section 3 can be modified by replacing the previous resource constraint (in Equation (3)) with the following two constraints:

x + y_{1} + y_{2} \leq F + δ

and

y_{1} \geq δ

. The latter constraint implies that the funds earmarked for outreach in region 1 cannot be invested in provider services or outreach in region 2. The analysis of this modified optimization problem shows that the earmarked funds have a non‐trivial impact on the CCR&R's optimal decisions (as stated in the proposition below).

Proposition 5

Suppose δ > 0 units of additional funds are earmarked for outreach in region 1. Then, there exists a unique threshold

\bar{δ} > 0

, such that: (i) When

δ \leq \bar{δ}

(x^{*}, y_{1}^{*}, y_{2}^{*})

(\frac{(F + δ) α}{α + β}, \frac{(F + δ) β γ^{\frac{1}{1 - β}}}{(α + β) (γ^{\frac{1}{1 - β}} + {(1 - γ)}^{\frac{1}{1 - β}})}, \frac{(F + δ) β {(1 - γ)}^{\frac{1}{1 - β}}}{(α + β) (γ^{\frac{1}{1 - β}} + {(1 - γ)}^{\frac{1}{1 - β}})})

and; (ii) When

δ > \bar{δ}

y_{1}^{*} = δ

x^{*} = F - y_{2}^{*}

, and

y_{2}^{*}

is the unique solution to the equation

(1 - γ) (α + β) y_{2} + γ α δ^{β} y_{2}^{1 - β} - (1 - γ) β F = 0

. Furthermore,

y_{2}^{*}

is increasing in δ if

δ \leq \bar{δ}

, and decreasing otherwise.

When the amount of earmarked funds is below the threshold

\bar{δ}

, it remains optimal for the CCR&R to allocate the total funds (F+δ) between provider services and outreach activities in the same fashion as shown in Proposition 4. In contrast, when the amount of earmarked funds is greater than

\bar{δ}

, it is optimal for the CCR&R not to invest anything more than the amount earmarked for investing in outreach in region 1. This is because, when the amount of earmarked funds for outreach in region 1 is sufficiently high, the CCR&R should utilize the remaining funds (F) to reduce the overall inequity outcomes through outreach in region 2 and improvement of service experiences delivered by the providers in its service area.

Another implication of the availability of the earmarked funds is that, when the amount of earmarked funds is substantially large, the CCR&R's optimal investment in outreach in the non‐earmarked region reduces. The intuition behind this implication is as follows: Recall that large amounts of investment in provider services and outreach in a region will lead to a significant improvement in the propensity of IE families to accept offered vouchers. As a result, with large amounts of earmarked funds for one region, the CCR&R can reduce the inequity outcomes in its area by increasing investment in provider services, even at the expense of investment in outreach in the other region. Thus, while additional funds always help reduce inequity outcomes, the funds earmarked for outreach in one region may crowd out the investment in outreach in the non‐earmarked region. This finding on the unintended consequences of earmarked funds is consistent with findings in the existing literature on counter‐productive effects of earmarked funds (Arora et al. 2020, Besiou et al. 2014). For instance, Besiou et al. (2014) show that earmarked funds for use in a specific zone can decrease the fleet size available for disaster response in the other zones.

Key Takeaways for CCR&Rs: Figure 2 summarizes the results discussed in Sections 4.1–4.4. In general, investing more funds in outreach in a region with more IE families is optimal (as depicted by portions I and III in Figure 2(a)). However, as depicted by portion II, it might be optimal for the CCR&R to invest relatively more funds in outreach in, say, region 2, even when the majority of IE families reside in region 1. This is especially true in the following situations, which are also presented in Figure 2(b):

Figure 2

Key Takeaways for CCR&Rs

The external considerations (e.g., public transportation and infrastructure) in region 2 have a greater impact on a family's acceptance propensity;

The rate of return of investment in outreach in region 2 is higher and funds are in abundance;

The socioeconomic distress experienced by families in region 2 is significantly higher; or

A large amount of funds is earmarked for outreach in region 2.

Additionally, portion II expands (and portion III shrinks) when the marginal return of investment in provider services is high, or when the baseline probability that an IE family accepts the offered voucher is high.

Practical Implementation and Case Study Based on a CCR&R in Massachusetts

In this section, we illustrate how the results discussed in Section 4 can apply in practice. We present a case study based on the real‐world example of the CCR&R operating in the Western Massachusetts service area. We use data collected from the U.S. Census Bureau, the child care literature, and our interviews with multiple officials (director and front‐line staff) at the CCR&R under study, to estimate values of the model parameters. Over a period of 2 years, we conducted 10 semi‐structured interviews with multiple field experts to elicit the CCR&R‐specific parameters.

Based on the U.S. Census Bureau's categorization of the U.S. landscape, we categorize the CCR&R's service area into three types of regions: urbanized (denoted by 1), urban clusters (denoted by 2), and rural (denoted by 3). Table 1 lists the considered values of parameters and the sources used to estimate them. The online companion presents a generalized optimization model (for n≥2 different types of regions) and describes the steps taken to estimate the model parameters.

Table 1

Estimated Values of the Model Parameters for the Case Study

Parameter Value	Sources/Methods (see details in the online companion)
$γ_{1} = 0.27$ , $γ_{2} = 0.05$ , $γ_{3} = 0.68$	The U.S. Census Bureau data on number of households and household income, and the state's income eligibility criteria
α=0.55	Expert elicitation approach
$β_{1} = 0.26$ , $β_{2} = 0.22$ , $β_{3} = 0.23$	Expert elicitation approach, child care literature, the local wage rate, and the U.S. general election data on voting
${\underset{̲}{p}}_{1} = 0.3$ , ${\underset{̲}{p}}_{2} = 0.3$ , ${\underset{̲}{p}}_{3} = 0.2$	Field interviews
$p_{o 1} = p_{o 2} = 6 x 10^{- 7}$ , $p_{o 3} = 4 x 10^{- 7}$	Field interviews, and child care literature
$ξ_{1} = 1$ , $ξ_{2} = 1.41$ , $ξ_{3} = 1.87$	The U.S. Census Bureau data on household income
F = $260,000	The Massachusetts Department of Early Education and Care's data on the CCR&R's budget

Notes

We also conduct sensitivity analysis by varying values of α from 0.25 to 0.85,

β_{3}

from 0.20 to 0.30,

ξ_{2}

from 1.41 to 7.05,

ξ_{3}

from 1.87 to 9.35, and F from $260,000 to $780,000. See the summary description of model parameters in Table A1 in the appendix.

Optimal Investment Decisions for the CCR&R in Western Massachusetts

We solve the optimization problem by using the estimated parameters to compute the optimal levels of investments in provider services and outreach activities. We find that the optimal amounts of investment are as follows: $180,681 for provider services

(x^{*})

, $33,362 for outreach in region 1

(y_{1}^{*})

, $1,972 for outreach in region 2

(y_{2}^{*})

, and $43,985 for outreach in region 3

(y_{3}^{*})

. For different values of the available funds F, we find that the optimal amounts of investments in provider services and outreach activities as a percentage of F remain almost identical: approximately,

x^{*} / F = 70 %

and

(y_{1}^{*} + y_{2}^{*} + y_{3}^{*}) / F = 30 %

Notably, while the ratio of the distress faced by IE families scaled by the proportion of IE families and the external environmental factors in regions 3 and 1

(\frac{γ_{3} p_{o 3} ξ_{3}}{γ_{1} p_{o 1} ξ_{1}})

is equal to 3.14, the ratio of optimal investments in outreach activities in regions 3 and 1

(\frac{y_{3}^{*}}{y_{1}^{*}})

to minimize overall inequity outcomes is around 1.32. The difference in this magnitude arises because the outreach elasticity in region 3 is lower than the outreach elasticity in region 1. Furthermore, even though the proportion of IE families residing in region 2 is quite low and the outreach elasticity in that region is the lowest, the optimal outreach investment in region 2 is a non‐trivial percentage value of the CCR&R's total investment in all outreach activities in its service area (this is due to the fact that

ξ_{2} = 1.41 ξ_{1}

). This optimal investment

y_{2}^{*}

, although relatively smaller in magnitude, in practice can cover the expenses of a few visits to the region to identify and reach out to influential voices within the local community. The next subsection discusses how the optimal investments change as estimates of the model parameters vary.

Sensitivity Analysis: We perform sensitivity analysis by varying estimates of provider services elasticity, outreach elasticities, and socioeconomic burdens of distress. As a summary measure of the CCR&R's allocation of funds, we consider the ratio of optimal investment in provider services to available funds, as well as the ratio of optimal investment in outreach in region i to total investment in outreach activities in all three regions, that is,

x^{*} / F

and

y_{i}^{*} / (y_{1}^{*} + y_{2}^{*} + y_{3}^{*})

, respectively. Figure 3 illustrates the findings, which are discussed in items (a)–(c) below.

Figure 3

Sensitivity Analysis of the CCR&R's Optimal Investments with Respect to Model Parameters

The estimate of the provider services elasticity α is based on the relationship between increments in funding provided to the team focusing on provider services and the increase in the chance of an IE family accepting an offered voucher. The plots in Figure 3(a) depict changes in the CCR&R's optimal investments with respect to the required increment in investment of funds toward the service team to improve the acceptance chances from 1/5 to 1/4, which we refer to as the provider services return factor. Naturally, the optimal investment in provider services is reduced when the provider services return factor increases (because α decreases as a function of this factor). Interestingly, we find that a higher portion of freed up resources (due to reduction in

x^{*}

) should be directed to outreach in the urbanized region rather than in the urban clusters and the rural regions (because

β_{1}

is greater than both

β_{2}

and

β_{3}

The estimate of outreach elasticity

β_{3}

is based on the information that the hourly wage rate of the personnel specializing in outreach‐related events is $30.2 per hour. The plots in Figure 3(b) depict changes in the CCR&R's optimal investments as the hourly wage rate changes. As expected, the optimal investment in provider services increases when the hourly wage rate increases (because the outreach elasticities decrease with this hourly wage rate). However, the decrease in the total optimal outreach investments (to free up resources to increases

x^{*}

) is such that the CCR&R takes away funds from its outreach investments in the urbanized and rural regions only. This can be explained as follows: Since the optimal amount of outreach investment in the urbanized clusters is already significantly smaller (due to low

γ_{2}

), the effect of diminishing returns of the outreach investment is not as pronounced in the urbanized clusters as it is in the urbanized and rural regions. As a result, the CCR&R will find it optimal to use a portion of freed‐up resources from

y_{1}^{*}

and

y_{3}^{*}

toward outreach investment in urbanized cluster

(y_{2}^{*})

The estimate of distress

ξ_{i}

is based on the U.S. Census Bureau data on household income in region i. Additional reasons (other than the cost of living) can amplify these quantified values of an IE family's distress. The plots in Figure 3(c) depict changes in the CCR&R's optimal investments for larger values of the relative distress factors in the urban clusters and the rural region (specifically, when both

ξ_{2}

and

ξ_{3}

are amplified by a factor of 2, 3, and so on). We find that it becomes optimal for the CCR&R to increase its investment in outreach in the rural region, even at the expense of reducing its investment in outreach in the urbanized region despite the latter having both a larger outreach elasticity (

β_{1} > β_{3}

) and a greater impact of external environmental factors (

p_{o 1} > p_{o 3}

We also perform sensitivity analysis by varying the amount of additional funds (δ) that are earmarked for conducting outreach in the rural region. We find that, when δ is greater than $52,000, the optimal outreach investment in the rural region

(y_{3}^{*})

is equal to the amount of the earmarked funds. Moreover, at such large values of δ, it becomes optimal to reduce the outreach investments in both the urbanized region and the urban clusters in order to invest more funds in the provider services. This specific finding with regard to the reduction in both

y_{1}^{*}

and

y_{2}^{*}

stands in contrast to the finding discussed in (b) above, wherein it is optimal for the CCR&R to use a portion of the freed‐up resources (from the reduction in

y_{1}^{*}

) to increase

y_{2}^{*}

. This result is due to the complementary nature of the provider services and outreach activities. These findings can help donors make informed decisions on donating earmarked funds when earmarking might be unavoidable due to regulatory, accounting, or monitoring considerations of the donor entity.

Value of Optimal Allocation of Funds

To quantify the value of the proposed optimal allocation of funds, we calculate the percent value reduction in inequity outcomes by comparing values of the inequity measure under both the optimal and the current fund allocation strategies. The CCR&R currently allocates only 10% of the total working hours of an existing employee (working at a wage rate of $15.44 per hour). Considering 2080 working hours in a year, the CCR&R's investment in outreach in its rural region (denoted by, say,

{\hat{y}}_{3}

) is $3,211.52. In the current scenario, the interviews with the CCR&R officials indicate that the outreach investments in different regions are only proportional to the number of IE families residing in each region. Thus, the CCR&R's current investments in outreach in the urbanized (

{\hat{y}}_{1}

) and urban cluster (

{\hat{y}}_{2}

) regions are $1275.16 and $236.14, respectively. Next, to calculate the improvement in

M I (x^{*}, y_{1}^{*}, y_{2}^{*}, y_{3}^{*})

relative to

M I (F - {\hat{y}}_{1} - {\hat{y}}_{2} - {\hat{y}}_{3}, {\hat{y}}_{1}, {\hat{y}}_{2}, {\hat{y}}_{3})

, we run a simulation experiment using ranges of practical values for the activity‐ and region‐specific parameters (formulated around the specific values estimated above). We find the CCR&R can achieve an average improvement of 7.0% in the equity outcomes, with a 95% confidence interval of [6.0%, 8.0%], by using the optimal allocation strategy instead of the current allocation strategy.⁸

As emphasized in SDGs, such a reduction in inequity in access to child care, especially for children in low‐income families, plays a pivotal role in determining the future performance of the children and the lowering of the socioeconomic burden (EAC 2016). In particular, SDG #4 emphasizes the need to address equity issues related to opportunities provided to rural children. In that regard, the estimated value of the improvement in equity outcomes in the CCR&R's service area is related to the SDG Indicator 4.2, which includes reducing inequity in “access to quality early childhood development, care, and pre‐primary education" for rural versus urban recipients (SDG 2017). Furthermore, the magnitude of benefit of the proposed optimal allocation of funds should motivate all CCR&Rs (nearly 700 of them in the U.S.) to utilize this study to estimate the model parameters and calculate their optimal levels of investment in provider services and outreach in their service areas.

Conclusion and Future Research

In this study, using an equity‐based objective, we analyze how a CCR&R should allocate its limited funds between its regional outreach activities and the provider services it offers in its service area. Our model‐based analysis shows that it is important for the CCR&R to take into account regional asymmetries—including differences in transportation infrastructure and regulatory environment, marginal return of outreach investment, and distress experienced by families—when balancing its investments in outreach across different regions. In summary, we offer the following key takeaway for managers at CCR&Rs: In the presence of regional differences, it can be beneficial to invest more funds in outreach in a region even when that region has a relatively lower proportion of IE families. We also contextualize our study for a CCR&R in Massachusetts and demonstrate that the proposed investment decisions can improve equity outcomes by approximately 7.0%.

Most recently, the COVID‐19 pandemic has underscored the need to ensure equitable access to affordable and high‐quality child care in order to facilitate a safe and robust reopening of the economy. As a result, CCR&Rs should appropriately allocate their funds to offering provider services, such as cleaning and sanitation and other safety‐related programs, and to conduct outreach, especially in rural regions (Johnson‐Staub 2020, Williams 2020). We view this study as the first step in exploring the operational complexities on the supply side of child care subsidy voucher programs. At a broader level, this study can also help any organization administering and operating subsidy voucher programs in other pro‐social settings to allocate its limited funds across different supply‐enhancing activities optimally.

Across the globe, as a means of attaining SDGs, subsidy vouchers have become a widely accepted form of providing support to beneficiaries who do not have access to affordable essential services (e.g., child care, housing, and maternity care). It is worth noting that several other voucher programs exhibit similar supply‐side concerns. Hatt et al. (2010) and Ir et al. (2010) provide empirical evidence on families’ inability to accept maternity care vouchers offered to low‐income households in Bangladesh and Cambodia, respectively. The commonly stated reasons for such low acceptance probabilities are the inadequate number of service providers in different regions and the low quality of service delivery by the voucher‐accepting providers. Furthermore, given that region‐based socioeconomic and demographic differences play a similar role in these related contexts, the insights generated here pertaining to the effects of regional asymmetries will be particularly useful to balancing investments in outreach activities across different regions.

Our study can be extended for future research in the following directions. First, while we consider an equity‐based objective (with efficiency only captured implicitly through marginal returns), future research can consider a multi‐objective structure that explicitly combines equity and efficiency. For example, a multi‐objective problem setup that considers social benefits and operational costs of various supply‐enhancing interventions can help understand the effect of other efficiency related factors on the investment decisions. Also, we consider the mandated FCFS allocation policy and, accordingly, our equity‐based measure captures the resulting disparities between the families in different regions in terms of the likelihood of acceptance of an offered voucher. Future research can consider alternative allocation policies, wherein it might also be meaningful to capture disparities within the families in each region by using, say, a combination of mean waiting time until acceptance and the likelihood of acceptance. Finally, we believe capturing the interplay between demand‐ and supply‐side factors will be valuable to further improving societal outcomes.

Footnotes

A. Notation

B. Proofs of Propositions

Acknowledgments

This work has been supported by the University of Massachusetts Amherst Public Service Endowment Grant Award Number P1PSEG000000033. The authors appreciate Sara Garvin of New England Farm Workers’ Council for her time and insights on the practical aspects of the context under study. The authors also thank the special issue editors, a senior editor, and three anonymous reviewers for their valuable suggestions, and Sripad Devalkar and Beril Toktay for their helpful feedback on earlier drafts.

Each state or territory in the U.S. sets an eligibility criterion for families to qualify for obtaining support through child care subsidy programs. This criterion is typically income based; for instance, in Massachusetts, the eligibility criteria state that the family's household income should be below 50% of the state median income (MEEC ).

Due to budgetary constraints, state agencies typically fund a fixed number of vouchers. A voucher is made available for offering to an IE family only when an existing voucher‐utilizing family's placement is terminated for failure to meet any eligible requirements (MEEC 2019, Pilarz et al. ). From the CCR&R's perspective, availability of a voucher is unknown, whereas the number of vouchers being utilized in its area remains at a fixed level.

Our numerical studies show that insights obtained by using other functional forms that capture diminishing returns (e.g., exponential functions) are qualitatively similar to the ones obtained by using the Cobb–Douglas function.

This equivalence also implies that the vector of optimal investments satisfies the four axioms pertaining to proportional fairness—namely, Pareto Optimality, Symmetry, Affine Invariance, and Independence of Irrelevant Alternatives (Bertsimas et al. ).

It is possible that the CCR&R may possess knowledge only on the probability distributions of the elasticities α and

β_{i}

. In such a scenario, our extensive numerical studies show that: (i) the first‐order insights (based on point estimates) generated in this study remain unchanged, and (ii) the CCR&R should hedge against a higher uncertainty in the estimate of elasticity of its investment in an activity by increasing its investment in that activity.

In an extension in the online companion, we analyze the CCR&R's fund allocation problem using a max–min fairness objective. Based on the preliminary analysis, we find that effects of the region‐specific characteristics on the optimal investments are qualitatively similar under both proportional and max–min notions of fairness.

We characterize these conditions for a special case when either

β_{1}

β_{2}

is equal to 1/2 (Lemma 9 in the online companion). However, our numerical studies show that similar insights hold for any general case where

β_{1} \neq β_{2}

We obtain these values of the percentage improvement in inequity outcomes by using uniform distributions for the model parameters. As a robustness check, we instead use censored normal distributions for these model parameters, and find that the CCR&R can improve the inequity outcomes, on an average, by 5.6%, with a 95% confidence interval of [4.7%, 6.5%]. Furthermore, the magnitude of improvement (7.0% or 5.6%) is consistent with the improvement estimate of yearly societal benefits (in the range of 6.0‐10.0%) in the study by Heckman (), which focuses on the effectiveness of child development programs, such as birth‐to‐five and preschool programs.

ORCID

Priyank Arora

Wei Wei

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