Daycare quality and regulation: A queuing-theoretic approach

Economics of Education Review, Vol. 17, No. 1, pp. 1–13, 1998  1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0272-7757/98 $19.00+0.00

Pergamon

PII: S0272-7757(97)00005-8

Daycare Quality and Regulation: A QueuingTheoretic Approach James G. Mulligan and Saul D. Hoffman Department of Economics, University of Delaware, Newark, DE 19716, U.S.A.

Abstract — Establishing national quality standards has become a central issue in the national debate about child care. Advocates of expanded regulation argue in favor of lower child–staff ratios, higher educational standards for caregivers, and smaller group sizes, but these tighter standards do not come without costs. We develop a formal model of the child care environment, concentrating on the impact that child–staff ratios, group size, and caregiver ability have on the amount of time children spend on-task and the intensity of caregiver–child interactions. The model clarifies the role of these variables and provides guidance concerning the potential costs and benefits of regulation. A major policy implication of this research is that regulation of child–staff ratios, group sizes and caregiver qualifications is too blunt an instrument for improving the overall quality level of child care. The model presented in this paper has wide application to other educational and related human services, since new technology makes it more economical to provide these services with individualized components. [JEL J13, J18, I21]  1998 Elsevier Science Ltd. All rights reserved higher thresholds than at lower thresholds and more important for infants and toddlers than for older children. Nor are low ratios the only way to ensure certain quality improvements. Although low ratios increase interaction between the individual child and the provider, improved training of providers accomplishes the same goal (Gormley, 1990, p. 25).

1. INTRODUCTION With the dramatic and continued increase in the use of daycare by preschool children in the U.S.,1 the proper role of government oversight and regulation has emerged as an important issue in the national debate about child care. On the one hand are national child care groups who have called attention to the low quality of some daycare environments. They have advocated an expanded role for regulation with the goal of assuring high quality daycare. One such highly publicized blueprint was featured in Who Cares for America’s Children?, a report by the National Research Council (1991) (NRC) that called for expanded regulations of child–staff ratios, group sizes, and caregiver training. On the other hand are economists and other public policy makers who have voiced the well-known concerns about ill-conceived regulations—that they may lead to inefficiencies and increased costs, thus exacerbating in this instance the already serious problems of daycare availability and affordability. Gormley (1990), Fuchs (1990), Fuchs and Coleman (1991), and Lowenberg and Tinnin (1992), among others, question the value of these regulations on cost–benefit grounds. For example, Gormley suggests that the biggest financial problem for child care centers is the cost of compliance with the child–staff ratio requirements:

With the daycare industry virtually certain to continue to expand, these issues are more important than ever. Conspicuously absent from the debate about child care and regulation is any solid evidence about or careful analysis of the likely effects of regulation on the quality of daycare. Child–staff ratios, group sizes, and caregiver training are the key regulated variables, but the research linking these factors to child outcomes is not particularly strong. There is little research, for example, that provides evidence on the functional relationship between marginal changes in child–staff ratios or group size and some measure of child care quality. The NRC report acknowledged that the research literature “has not addressed the question of acceptable or unacceptable ranges” of the regulated aspects of child care (Hayes et al., 1990, p. 86). On the basis of the empirical literature, it is difficult to defend any particular set of standards. In this paper, we construct a queuing model of the child care environment and then use it to simulate the effects of regulations on the quality of child care. Queuing models are widely used to describe and determine the optimal staffing for a wide range of

I do not dispute the connection between low [child–staff] ratios and quality. But that connection is stronger at

[Manuscript received 24 March 1995; revision accepted for publication 20 November 1996]

1

2

Economics of Education Review

services. Examples include fast food restaurants, hot lines, reservation systems, machine repair and checkout counters. The point in common with these examples is the presence of randomness in the timing of the request or need for service and in the duration of time needed to perform the service. Our model captures the randomness in both the timing of a child’s need for direct attention by a caregiver and in the time that the caregiver needs to attend to the child. By accounting for the inherent randomness that occurs in child care environments, the model provides a direct link between the three major regulated variables—child–staff ratios, group size, and caregiver qualifications—and measures of quality, such as the time that a child is engaged “on-task” in the activities designated by the child care staff and the extent of interactions between caregivers and children. The queuing model approach imposes a formal structure on the benefits side of the benefit–cost debate concerning the appropriate scope and focus of regulation.2 After laying out our application of the queuing model to the daycare environment, we use data from a national child care study to simulate the effects of variation in child–staff ratios, group size, and the ability level of caregivers on time on-task. This enables us to evaluate how changes in regulations would likely affect the quality of daycare provided. We also suggest how child care centers can collect and evaluate their own data to understand better the marginal impact of altering the child–staff ratio for different activities and children populations at their centers. The main policy implication of the queuing model is that regulations that limit child care centers’ flexibility in choosing the appropriate child–staff ratio and group size for the specific activity engaged in by the children and type of child can adversely affect the overall level of quality. In addition, maintaining excessively low child–staff ratios may limit a center’s capacity to hire and retain at higher wages those caregivers who, regardless of their academic qualifications, demonstrate an ability to work with children in effective ways. The outline of the paper is as follows. The next section presents a brief description of the U.S. child care market and the nature of regulation. The rationale for taking a queuing-theoretic approach to child care is presented in section 3. Section 4 contains an illustrative simulation based on data from the National Day Care Study. We present our results in section 5. Section 6 discusses additional applications of the modeling approach taken in this article. 2. BACKGROUND In this section we briefly describe the child care market and the nature and extent of child care regulation in the United States. A more detailed discussion of the regulatory environment appears in Gormley

(1990). In the United States, child care for preschoolers outside the home is provided in centerbased programs, which include both large franchise operations and non-profits, and in family daycare, which are much smaller and typically operate out of the provider’s home. More than five million preschool children with employed mothers now attend some form of child care provided by a non-relative outside the home. Of these, slightly more than half are in center-based programs, with the remainder cared for in family daycare. These figures represent almost a threefold increase in the number of preschool children receiving child care outside the home and a fourfold increase in the number attending center-based programs in a decade. Currently, center-based child care is regulated in virtually all states, and family daycare is regulated in about half of the states.3 The child–staff ratio is the most frequently regulated aspect of child care. Because each state establishes its own regulations, there is considerable variation in the allowable ratios. For example, 28 states require child–staff ratios of 5:1 or lower for children up to age 1.5 years, while 12 states permit ratios greater than 7:1. For 3-yearolds, the overall range is from 7:1 in North Dakota to 15:1 in Arizona, North Carolina, and Texas (National Research Council, 1991). Expert panels advocate continued regulation of center-based programs and the adoption of national standards. For example, the National Research Council’s report suggests that “In order to ensure the best possible experiences for our children, child care policies must address: group size, child/staff ratios, caregiver qualifications, stability and continuity of caregivers, structure and content of daily activities, and organization of space” (Hayes et al., 1990, p. 38). Its report recommends maximum child–staff of ratios 4:1 for infants and 1-year-olds, 4 to 6:1 for 2-year-olds, 5 to 10:1 for 3-year-olds, and 7 to 10:1 for 4- or 5year-olds.4 In general, these ratios are well below those currently permitted in many states. For example, Gormley (1991) notes that for 4-year-olds, the average child–staff ratio ranges from 10.6:1 in the Northeast to 15.8:1 in the South. Regulation of group size and child–staff ratios does not, of course, come without problems. Even the NRC report noted that “tough regulations may lead to higher costs that price many people out of the market or into illegal ’black market’ arrangements,” which are typically unregulated. As a result, it recognized that “State regulators are thus left with an ongoing dilemma of how to balance the competing goals of quality and financial access.” Fuchs and Coleman (1991) go further, arguing that child–staff ratios ought to be increased not decreased. Since labor costs represent between 70 and 80% of the cost of child care, they argue that increasing the child–staff ratio appears to be the sole possibility for increasing the quantity of child care services given current budgetary constraints. They write that “there is simply no way sim-

Daycare Quality and Regulation ultaneously to hold down the cost of care, reduce the child–staff ratio, and increase the pay and qualifications of child care workers” (p. 79). In addition, regulation may reduce the needed flexibility of daycare providers. Child–staff ratios in the U.S. are in marked contrast to those in other countries. One notable example is France, where the highly praised nationally run “e´cole maternelle” program provides universal coverage to preschool age children using highly educated caregivers but with much higher child–staff ratios (approximately 22:1) than in the U.S. (Richardson and Marx, 1990). Gormley indicates that ratios are nearly in this range in Germany as well. This suggests the point, obvious to economists but not to many policy makers, that there may well be alternative ways of producing high quality daycare using differing combinations of moreskilled and less-skilled labor. Restrictions on maximum child–staff ratios, which do not recognize this distinction, may inadvertently affect the incentives of providers to hire more qualified caregivers.5 A simple example of this is shown in Figure 1. Suppose a firm can produce daycare services using various combinations of less-skilled and more-skilled workers, with the amount of capital held constant. Below, we address what we mean by more skilled in the context of a queuing model, but for the moment simply assume that more-skilled workers have a higher marginal product. The firm’s choice between more and less-skilled workers is represented by a standard production isoquant. With more-skilled workers represented on the vertical axis, the isoquants would be relatively flat, reflecting the higher marginal product of these workers. The isocost line has a slope with absolute value less than 1, since more-skilled workers will also have a higher wage. In the absence of regulation, this leads to cost-minimizing choices Ls*(ws,wu,C) and Lu*(ws,wu,C), shown by point A. If regulations establish a maximum ratio of children per staff member (R), without reference to skill levels, then the firm faces the additional constraint Ls

Number of skilled workers

C/R

A Isocost line

Regulatory constraint

B C

C/R

Number of unskilled workers

Figure 1. The effect of regulation on input choices.

3

+ Lu ⱖ C/R. This is shown in Figure 1 by the line with slope equal to ⫺ 1 and intercepts C/R. All points to the northeast of this line satisfy the regulatory constraint. Since the slope of this constraint does not equal the slope of the isocost line, then, as long as R is low enough to be binding, it necessarily alters the firm’s choices and raises its costs. Specifically, the firm now chooses the input combination at B, substituting less-skilled for more-skilled workers. Additionally, the firm’s costs of serving C children and quality level Q rises, since the chosen combination is not cost-minimizing.6 To date, the benefits of regulating child–staff ratios, group sizes and caregiver qualifications have been evaluated in an ad hoc manner. The widely divergent views on the merits of the regulation of these staffing variables in the first place along with the lack of consensus among the 50 states concerning appropriate ratios calls for a more theoretically consistent approach to the impact of these factors on the quality of child care. In the next section, we use a queuing-theoretic approach to child care to examine these factors. 3. A QUEUING-THEORETIC APPROACH TO CHILD CARE 3.1. Overview In this section we present a queuing-theoretic approach to child care. A queuing model can provide a consistent quantitative framework for examining the relationship between the key regulatory variables (inputs) and important aspects of a child care environment, such as the expected proportion of time that a child is engaged “on-task” in some productive activity and not either waiting for the assistance of a caregiver who is attending to another child’s needs or engaged in a non-productive activity. We also explain how the technique may be used by child care providers to determine the effects of altering the child– staff ratio to best fit the activity assigned. Child care providers have the potential to determine the cost and benefits of altering the child–staff ratios for their own mix of activities, children and staff members. While providers may be inclined to form their own analysis in an intuitive informal way, the queuing approach can serve as a formal means of analyzing the stochastic nature of the child care environment. In our formal model, we assume that children in a child care environment alternate between two behavioral “states.” In one, they are engaged in an activity and function either independently or in a group in a productive manner without need of individual corrective attention. We refer to this as “ontask” or “engaged” time. In the other state, they are “off-task” and require the direct intervention of a caregiver. “Off-task” events would include a child’s needing to be fed, changed, comforted, directed to another activity, and so on depending on the age of

4

Economics of Education Review

the child and the nature of the activity engaged in by the child. In this framework, the child care staff designs an appropriate set of activities and then responds to the needs of the children and interacts with them when they need assistance. The remainder of the caregiver’s time is spent observing the children and monitoring the activities plus offering positive encouragement when appropriate. Since interactions due to off-task events are generally random in nature, a queuing model is the most appropriate modeling approach for assessing staffing needs. While we are hesitant to describe the amount of on-task time as “the output” of a daycare center, we think it is likely to be highly correlated with the outputs of child care services, however measured. In addition, it is probably useful as a measure of child care quality in and of itself. We think that most parties involved in assessing the quality of child care would agree that time spent engaged in an assigned activity is good and time spent either waiting for assistance or involved in an unproductive activity is generally bad. For example, Howes (1990) argues that “Children in low quality child care may be more likely to be hostile and aggressive because they have spent their days either aimlessly wandering within a large peer group or competing for adult attention” (Howes, 1990, p. 293). In our simulations below, we use aimless wandering and inappropriate competition for adult attention as examples of off-task time.7 Alternatively, a queuing model provides a direct measure of another outcome: the degree and intensity of the individual interactions between a caregiver and child. The intensity of caregiver–child interactions has become an important measure of quality to researchers in recent years as center-based programs have moved to more individually paced activities for children. For example, Caring for America’s Children suggested that “Long-term studies found that children in a teacher-directed preschool program demonstrated less adequate social adaptation than children assigned to preschool programs in which children initiated and paced their own learning activities in environments prepared by teachers” (Hayes et al., 1990, p. 30). The study suggests further that “Children’s active initiation and pacing of their learning activities appears to have positive implications for their social development. Research on learning processes also points to the need for curricula to allow for individual differences in learning styles and to the importance of learning through interactions [among caregivers and children]” (Hayes et al., 1990, p. 17). 3.1.1. Model.8 The child care environment can be formally modeled as a queuing model characterized by a finite source of “arrivals”—the number of children needing assistance—and a service process—the time taken by the provider to interact with each child who needs assistance—both of which follow some probability distribution.9 In addition, in a learning

environment, the nature of the activity assigned the children and the abilities and motivation of the child are important factors that influence the arrival and service rates. For example, more challenging activities will increase the children’s reliance on the provider for direct, personal interaction and may require longer interaction times. In the remainder of this section we consider each of these elements of the queuing process and outline a method for incorporating this approach at the child care center. 3.1.2. The arrival process. In a queuing model, the arrival process characterizes the occurrence of some stochastic event. For example, arrivals at banks, car washes, and gas stations follow a random process that can be analyzed with a queuing model. When applied to the child care environment, an arrival is the movement from the state of being on-task to that of being off-task, which we refer to as a “breakdown.” While each child may go off-task (breakdown) after a regular, predictable length of time, it is likely that the timing of the transition from on-task to off-task states is stochastic. A formal queuing model is based on an assumption about the nature of this stochastic process. While a priori any stochastic process could be used to characterize the arrival process, it is likely that the breakdown process will follow one of a general class of distributions called Erlang distributions. Erlang distributions are characterized by two parameters, r and ␭, where r is an integer taking values from 1 to infinity and ␭ is the mean arrival rate. The Erlang distribution has probability density function, fT(t;r,␭) = [(␭r)r/(r ⫺ 1)!] tr ⫺ 1 e ⫺ r␭t for t ⱖ 0 with a mean, E[T], of 1/␭ and a variance, Var[T], of 1/r␭2. As r increases to infinity, the variance decreases and the density function approaches in the limit that of a deterministic arrival process at constant interarrival time of 1/␭. At the other extreme (r = 1) is the exponential distribution, which has probability density function f(t) = ␭e ⫺ ␭t for t ⱖ 0 with an expected value of 1/␭, a variance of 1/␭2 and a coefficient of variation of 1.10 3.1.3. The service process. The service process (here, helping children in need of assistance) can also be characterized as an Erlang distribution with parameters r and ␮, where r is, again, an integer ranging from 1 to infinity and ␮ is the average service rate (the number of children who can be assisted per unit of time). 1/␮ is the average length of time for each interaction. The notation used above for the arrival process is the same for the service process with ␮ replacing ␭. As a result, as r goes to infinity, the variance goes to zero and interaction (that is, service) time approaches 1/␮ with certainty. The exponential assumption for interarrival times and service times is equivalent to assuming that the arrival rate and the service rate are random variables that follow a Poisson distribution. The assumption of

Daycare Quality and Regulation exponentially distributed arrival or service time is directly related to the corresponding assumption of a Poisson arrival or service rate. The mean time between arrivals (the “interarrival” time) is the reciprocal of the mean breakdown rate, while the mean service time is the reciprocal of the mean service rate. For example, if, on average, there are 10 breakdowns per hour, the mean interarrival time is 6 min. Similarly, if the mean service time is 10 min, then the mean service rate is six. The assumption of Poisson arrival and service rates has numerous convenient properties for analysis and is widely used in queuing applications, but it also is likely to be a reasonable, if not conservative assumption for most applications characterized by random behavior. 4. EXPERIMENTAL IMPLEMENTATION OF THE QUEUING-THEORETIC APPROACH Determination of which distributional assumption is appropriate for a specific daycare activity requires site data on interarrival and service times. The information can be expressed as cumulative frequency plots and compared to the cumulative frequency plots for different values of r to assess the appropriateness of the Erlang distribution assumption. Since Erlang distributions are completely characterized by the mean and variance of the sample data, the method of moments can be used to estimate the parameters of the distribution. Since E[T] = 1/␭ and Var[T] = 1/(r␭2), the sample variance and mean can be used to approximate ␭ and r for the arrival process. Equivalently, the sample mean and variance of the service process can be used to determine ␮ and r. Since the Erlang distribution is in terms of integer values of r, r must be adjusted to the nearest integer value. While it is likely that the Erlang distribution with r greater than 1 will fit the data better than the exponential distribution, assuming that the distribution is exponential increases the mathematical tractability of the analysis. In addition, the exponential distribution depends only on the sample mean, since the mean equals the standard deviation. Once having determined the nature of the processes governing the breakdown and repair processes and estimates of the relevant parameters (␭, ␮, and r) for the arrival and service processes for specific daycare activities, an analyst can simulate the effects of altering the child–staff ratio, group size, the type of children in the group and the ability of the caregivers. In the remainder of this section we illustrate the usefulness of the queuing approach with an extended example. The data serving as a basis for the simulations come from the National Day care Study (NDCS), a study of 3- and 4-year-olds in 57 child care centers in Atlanta, Detroit and Seattle in 1976 and 1977.11 The sample used in the National Day Care Study is not representative of the national population. There is

5

a disproportionately large number of minority, low income, and urban families in the sample. The study includes observational data on approximately 1800 children, each of whom was observed for four separate 20 min minute periods. During each observation period, each child’s activity was recorded every 12 sec, and was coded in terms of a set of 37 activities and three activity continuation codes that indicate whether the child’s behavior was a new activity, an old activity, or no identifiable activity.12 The NDCS researchers collected the type of site data needed for a direct determination of the the stochastic arrival and service processes in these child care centers. Unfortunately, since the data is not publicly available in disaggregated form, we are unable to implement the procedures discussed in section 3 to determine exactly which distributions best fit the underlying data for each of the daycare centers in the study. With direct access to these data, we would have been able to do so. As a result, in order to illustrate the usefulness of the queuing approach, we must make an assumption about the nature of the arrival and service processes. As indicated in the previous section, the assumption of an exponential distribution is often invoked even when site data suggests that an Erlang distribution with r greater than but close to 1 is more appropriate. Here, we assume that both the arrival time and the service time distributions are exponentially distributed. If the distribution of these variables were less random than implied by the exponential assumption (that is, more deterministic with r greater than 1), the effects of increasing the child–staff ratio would be less harmful to on-task time than with the exponential case. If child arrivals or the helping services performed by child care workers is more deterministic, child care workers are more able to manage their schedules so as to avoid having several children left unattended in the “queue”. It is the randomness of the breakdown and service rates that creates the unexpected bunching of children in the queue that reduces on-task time and increases the marginal value of lower child–staff ratios. As a result, the exponential/Poisson assumption leads to predictions that are at worst biased toward smaller child–staff ratios. As will be shown later in the simulations, even with this bias the data used in this illustration suggest that child–staff ratios at these centers may be unjustifiably small on a cost–benefit basis. We now develop in greater detail the queuing model to be used in the simulations.13 For an exponentially distributed interarrival time with mean 1/␭, the probability that a child will need attention within some time period t is P(t) = 1 ⫺ exp( ⫺ ␭t); the probability that exactly c children will need attention in a given time interval is P(c) = ␭ce ⫺ ␭/c!. The probability that a child who needs attention will have to wait in a queue for assistance depends on how often children are likely to need help (i.e. ␭), how quickly breakdowns can be fixed (␮), on the number of chil-

6

Economics of Education Review

dren in the group, and on the number of caregivers. Clearly, if the ratio of children to caregivers is 1:1, then all breakdowns are immediately attended to and no queue develops. But for all other ratios, there is some probability that a child will remain in a queue for attention for some period of time while the caregiver(s) attends to the needs of other children. For a group consisting of C children and a single caregiver, it can be shown that the steadystate expected number of children in the queue needing attention (i.e. the queue length) at any point in time is: L = C ⫺ (1 ⫺ P0)(1 + (␮/␭))

(1)

where L is the queue length and P0, the probability that no child is either being helped or is in need of attention, is:

冘 C

P0 = 1/

[(C!/(C ⫺ c)!)(␭/␮)c].

(2)

C=0

Alternatively, for the single caregiver case, (1 ⫺ P0) is the percentage of time that the caregivers are interacting in a meaningful way with some of the children. The nature of these interactions depends on the activity and the objectives of the caregiver. For example, when children are involved in free time or recreation, a caregiver’s interactions may be limited to disciplinary actions or resolving crises. Depending on the age of the children involved these interventions may be of a limited nature and require low caregiver– child ratios. Relatively better trained and skilled caregivers may be able to manage large numbers of children in these environments. On the other hand, in learning environments with relatively greater complexity interventions may occur on a more frequent basis in order to help children having difficulties or engaged in aimless wandering. Regardless of the activity chosen, it is unlikely that 1 ⫺ P0 will approach 1, since this would imply that there are time periods when children not receiving the attention of the caregiver will be forced to wait much longer periods of time. For the case of more than one caregiver, the expected number of children in need of individual attention is:

冘 C

L(S) =

(c ⫺ S)Pc

(3)

c=S

where S is the number of caregivers and Pc is the probability that exactly c children are either being attended to or in need of attention. The full derivation of Pc is presented in Appendix A. The proportion of time that a child would spend engaged or “on-task” (OTT) is the proportion of time not spent either waiting for assistance or engaged in non-productive activities:

OTT = 1 ⫺ Lq/C.

(4)

Substituting from Equations (1)–(3) and noting that ␭ and ␮ appear throughout exclusively in ratio form, this measure can be expressed as: OTT = f(C/S,C,␭/␮).

(5)

Equation (5) can be regarded as a production function for on-task time, with C/S, C, and ␮ the inputs; ␭ is best regarded as a characteristic of the children and is not a choice variable for the daycare provider. Equivalently, for the case when there is only one caregiver, 1 ⫺ P0 is the proportion of time spent by the caregiver actively intervening. P0 is therefore the time that the caregiver can use to observe and encourage children. The queuing model imposes the necessary structure to estimate the relationship between the on-task time and the three inputs. With information on ␭/␮, we can use Equations (1)–(4) to compute the expected child care on-task time for various child–staff ratios and groups sizes. Since the disaggregated Abt data is not publicly available, we inferred the value of ␭/␮ from published information on OTT, C, and S. Tabulations of the proportion of time spent in each activity are reproduced here as Table 1. We used this information to construct two independent estimates of the average on-task time for the entire sample. As Table 1 shows, 5.3% of the time was spent in “aimless wandering,”16 which yields an on-task time rate of 0.947. Alternatively, 7.3% of the time was spent “not involved in an activity,” which yields an on-task time rate of 0.927. These values describe an average child care environment in which children spend between 5 and 7% of the day—3–4 min/h—either waiting for attention or engaged in activity considered unacceptable by the child care staff. Since there is likely to be variation in these on-task rates from center to center in the sample and across type of activity assigned, these estimates provide an approximate average across the entire sample. With an average of 2.4 caregivers per group and 6.8 children per caregiver in the NDCS, an expected on-task rate of 0.947 implies that ␭/␮ equals 0.117 while an expected on-task rate of 0.927 implies ␭/␮ equals 0.132.17 Equivalently, they imply that the service rate is between 7.5 and 8.5 times the mean breakdown rate. There is a second completely independent method for identifying these parameters. In addition to observing specific child activities, observers independently recorded the children’s longest sustained time interval of engaged activity during a 20 min observation period. Eleven minutes was the mean duration of each child’s longest activity during the 20 min interval. Since the observation period was only 20 min, the actual longest activity per day was most likely much longer than 11 min. This suggests that

7

Daycare Quality and Regulation

Activity codes

Frequency

Group closed, structured activity Group open, expressive activity Monitors environment (looks, watches) Gives opinions Wanders aimlessly, does nothing Group passive behavior Moves with purpose Individual open, expressive activity Adds prop or idea Considers, contemplates, tinkers Individual closed, structured activity Gives orders, directs others Intrudes playfully Asks for attention Selects activity (with others) Shares, helps Asks for information Asks for turn Selects activity (alone) Isolates self Asserts rights Cries Sees pattern, solves problem Intrudes hostilely, bullies Hostilely asserts rights, anger Hostile exchange Avoids, withdraws Individual passive activity Asks for assistance, help Offers sympathy, comfort Asks for comfort Intrudes unintentionally Experiences rejection Quits activity after frustration Angry reaction to frustration Experiences accident Temper tantrum Activity continuity codes Longest activity Not involved in activity

21.1 13.2 11.9 8.0 5.3 4.8 3.1 2.9 2.8 1.7 1.5 1.0 0.9 0.9 0.6 0.6 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 ⬍ 0.1 ⬍ 0.1 ⬍ 0.1 ⬍ 0.1 54.8 7.3

the number of breakdowns per child was no less than one and most likely no more than three per 20 min time period. This implies an average number of six breakdowns per hour (assuming an average of two breakdowns per 20 min). With ␭/␮ of 0.117, a typical caregiver would need approximately 1 min 10 sec to respond to the average need for intervention; for ␭/␮ equal to 0.132, the average service time would be about 1 min 20 sec.18 5. RESULTS In the simulations to follow we first show how deviations from the sample average child–staff ratio are likely to affect on-task time and the intensity of caregiver–child interactions. We then examine how increasing the group size and using higher ability caregivers would affect those outcomes. Finally, we use the queuing model to examine substitution between

inputs in producing a specified level of child care output. 5.1. Child–Staff Ratios What would happen to on-task time of children in child care if child–staff ratios were increased? In 1988, the average staff/child ratio for 3- and 4-yearolds was 8.4:1. State regulations ranged from 7:1 to 15:1, while the National Research Council recommended ratios between 5:1 and 10:1. Our findings of the likely effects of these ratios are presented in Figure 2, which shows on-task time for the two estimates of ␭/␮ derived above and also for two values (0.100, 0.140) that bracket those values. We focus on the modal case of two caregivers and on group sizes from 10 to 30 (i.e. ratios from 5:1 to 15:1). For child–staff ratios of 7:1 or less, on-task time is well over 90% for all of the ␭/␮ values. For ␭/␮ = 0.117, on-task time is 0.965 at a 6:1 ratio and 0.924 at 8:1, and then falls more sharply as the ratio rises, dropping to 0.859 at 10:1 and 0.635 at 15:1. For ␭/␮ equal to 0.132, on-task times are, of course, lower and they also fall more rapidly, from 0.895 at a ratio of 8:1 to 0.573 at 15:1. More importantly, though, Figure 2 indicates that marginal increments in the child–staff ratio would be expected to cause relatively small decreases in ontask time. For example, increasing the average ratio from 8:1 to 9:1—which would increase the number of available child care spaces by 12.5%—causes ontask time to fall by between 3.0 and 4.5% depending on the value of ␭/␮. Even an increase to 10:1, which would increase child care availability by 25%, would cause on-task time to fall by 9.5% for ␭/␮ = 0.132 and about 7% for ␭/␮ = 0.117. In contrast, in the vicinity of current ratios, further decreases in child– staff ratios have very small estimated improvements in on-task time. The relationship represented in Figure 2 reflects the outcome for the average child receiving child care services at the specified child–staff ratios. While ontask time per child falls as the child–staff ratio increases, it is not immediately obvious what happens to total on-task time, measured over all children, since

1

On-task time

Table 1. Activity and activity continuity codes, National Daycare Staffing Study

0.9 0.8 0.7 0.6 0.5 4

6

8

10

12

14

16

Child-staff ratio Lambda/Mu = 0.100 Lambda/Mu = 0.132

Figure 2. On-task

Lambda/Mu = 0.117 Lambda/Mu = 0.140

time and child–staff caregivers).

ratios

(two

8

Economics of Education Review service rates ␮) caring for physically separated but otherwise identical groups of children (that is, the same number of children each with the same ␭). Due to the stochastic nature of breakdowns, there will be some occasions when no children in one group require individual attention, while two or more children require direct intervention in the other. If the two groups are combined, the caregiver having no children in need of individual attention would be free to assist the already engaged caregiver by helping one of his or her children in need of personal attention. By combining efforts in this manner the caregivers reduce the expected amount of time that children will spend being non-productive. The extent of this group size effect depends on the mutual occurrence of an available caregiver and an overextended caregiver.21 Figure 3 shows these effects for ratios from 5:1 to 10:1 for one, two, and three caregivers, using ␭/␮ = 0.117.22 Group size effects are uniformly positive, but are modest in size. We find that in moving from one caregiver to two, the improvement in on-task time (holding the child–staff ratio constant), averages about 4–5%. In moving from two to three caregivers, the effects are smaller, averaging about 1.5 to 2.5%. These estimates imply that constant on-task time can be achieved at higher child–staff ratios as the number of caregivers increases. Thus, for example, a 5:1 ratio with one caregiver yields the same expected on-task rate (0.945, for ␭/␮ = 0.117) as a 7:1 ratio with two caregivers and an 8:1 ratio with three caregivers. As

5.1.1. Group size. Child care advocates argue that smaller group sizes (holding the child–staff ratio constant) are better, a view that has received some statistical support from regression analyses of child outcomes (Abt Associates, 1980). In contrast, queuing models suggest that on-task time may rise with group size, holding the child–staff ratio and ␭/␮ constant. To see why this occurs, assume that there are two caregivers of identical ability (that is, they have equal

1

On-task time

the number of children in the system is obviously higher at the higher ratios. Whether total on-task time rises or falls depends on the rate at which average on-task time falls and on the net gain in on-task time to the marginal children added to the system. This latter measure depends on the new on-task time in the child care system and the on-task time of the marginal children when they are outside of the regulated child care system.19 Table 2 provides information on this question. For the two main values of ␭/␮ and the modal case of two caregivers, we use the OTT values from Figure 1 to compute total on-task time (C*OTT) and the critical value of OTT for the marginal children. If OTT time for the marginal children, when they are outside the regulated child care system, is below the critical value shown, then total on-task time rises at the higher ratios. Table 2 indicates that total on-task time rises as the child–staff ratio rises, but at a decreasing rate. This is true for both ␭/␮ values, reflecting the eventually falling profile of average on-task time in Figure 2. Consistent with this, the critical values also fall. As the child–staff ratio is increased from 5:1 to 6:1, the increase in total system on-task time is substantial enough that total on-task time will rise unless the marginal children had on-task times in excess of 0.900 (for ␭/␮ = 0.117) and 0.856 (for ␭/␮ = 0.132). But our simulations suggest that for child–staff ratios beyond 10:1 and above—and for these values of ␭/␮—the critical value falls sharply so that total ontask time will rise only if marginal children have relatively low on-task times.20

0.95 0.9 1 Caregiver 2 Caregivers 3 Caregivers

0.85 0.8 5

6

7

8

9

Child–staff ratio

Figure 3. On-task time and group size.

Table 2. Total child on-task time and critical values for marginal children (two servers)

␭/␮ = 0.117

Number of children per group*

10 12 14 16 18 20 25 30

␭/␮ = 0.138

Total on-task time

Critical value

Total on-task time

Critical value

9.78 11.58 13.26 14.78 16.18 17.18 18.70 19.05

0.900 0.839 0.763 0.699 0.499 0.304 0.070

9.70 11.41 12.98 14.32 15.41 16.20 17.08 17.19

0.856 0.783 0.671 0.544 0.396 0.175 0.023

*A group consists of the specified number of children and two caregivers.

10

9

Daycare Quality and Regulation far as we know, current regulations do not adjust the child–staff ratio for the number of caregivers, although our analysis suggests that this might be a sensible approach. 5.1.2. Relative ability of caregivers. Figure 4 shows the relationship between caregiver ability relative to the “type” of children cared for, measured by ␮/␭, and on-task time for four different child–staff ratios ranging from 6:1 to 15:1.23 For our purposes here, we think of ␭ as fixed, so that the ␮/␭ ratio reflects caregiver ability “normalized” by the underlying value of ␭ and increases in ␮/␭ reflect the use of more skilled caregivers; it equals the expected maximum number of children that a server could assist in some time period divided by the expected number of arrivals. The figure illustrates a number of important points. First, there are substantial increases in on-task time as ␮/␭ increases from 5 to 12.5. This is especially true at higher child–staff ratios. For example, increasing ␮/␭ from 8.33 (a number within the range of our estimated value of ␮/␭) to 10 increases on-task time per child by approximately 3% if the child–staff ratio is 5:1, but increases it by approximately 17% if the child–staff ratio is 15:1. Intuitively, at higher child– staff ratios, there are more opportunities for intervention, and thus more able caregivers are more valuable. In contrast, where breakdowns are less frequent, as at lower child–staff ratios, the greater ability of more skilled caregivers is unneeded, since even a less skilled caregiver may be adequate to prevent a queue from developing. Indeed, our estimates suggest that when the child–staff ratio is as low as 5:1 on-task time changes relatively little for ␮/␭ between 5 and 12.5. This suggests that regulations that limit child– staff ratios may impose de facto limits on the ability of caregivers employed in the child care system. Second, for the child–staff ratios considered here, there appears to be an effective upper limit on the value of more skilled caregivers. For example, when ␮/␭ exceeds 12.5, on-task time is essentially independent of caregiver ability, regardless of the child– staff ratio. That is, marginal increments in ability do not appreciably increase on-task time, since the

ability level is already sufficient to prevent a queue from developing. Empirical evidence from the child care literature is quite consistent with these findings. Econometric results suggest that higher educational levels and specialized training in child care tend to increase the productivity of caregivers (Hayes et al., 1990), although the magnitudes have typically been quite small. One possible explanation for the statistical results concerns the usefulness of formal education as a proxy for ␮. Another possible explanation is that most of the empirical estimates have been based on child– staff ratios that are relatively low—typically, 7 or 8:1. Since on-task rates at these ratios are above 92%, Figure 4 indicates that increasing ␮ has much smaller effects on on-task time and caregiver–child interactions than it would at much higher child–staff ratios. Since the OLS regression results reported in earlier statistical studies were based on a narrow range of the child–staff and a linear specification of the relationship between output and caregiver qualifications, the importance of higher ability caregivers at higher child–staff ratios may have been significantly underestimated. Our results suggest that the marginal impact of more able caregivers is greatest at higher child–staff ratios. 5.1.3. Substitution possibilities. While the previous discussions have focused on the effects of changing one of the three regulated variables while holding the others constant, child care centers also face a possible tradeoff between the number of caregivers per child and the ability level of caregivers. As indicated earlier, there is variation in child–staff ratios and the training of caregivers within the United States and especially between the United States and other countries, such as France. This underlying relationship between the ability of caregivers and the child– staff ratio is at the heart of the national debate on regulatory standards. Regulations that constrain the child care center to low child–staff ratios also limit its incentive to hire higher ability caregivers. Figure 5 shows alternative combinations of ␭/␮ and C/S that yield the same on-task time (in this case, an on-task time equal to 0.947). As the diagram 0.175

1

0.15

Lamda/Mu

On-task time

0.9 0.8 C/S C/S C/S C/S

0.7 0.6 0.5

= = = =

5 7.5 10 15

0.125 0.1 0.075

On-task time = 0.947

0.05

0.4

0.025 5

0.3 4

6

8

10

12

14

16

18

20

6

7

8

9

10 11 12

13 14 15

Child–staff ratio

Caregiver ability (Mu/Lambda)

Figure 4. Caregiver ability and on-task time.

Figure 5. Constant quality tradeoffs between caregiver ability and child–staff ratio.

10

Economics of Education Review

shows, there is a familiar-looking tradeoff: when ␭/␮ is large, the designated on-task time can be achieved only at a low C/S ratio and as ␭/␮ falls, C/S rises.24 The French and German child care situation would be represented by points at the bottom right of the curve and U.S. points further toward the top. Suppose that ␭ is constant and that variation in ␮ is the source of the variation in ␭/␮. Then the diagram shows that more skilled caregivers are able to produce the same on-task time at higher child–staff ratios. The curve in Figure 5 exhibits the property that an increase in the child–staff ratio (holding on-task time per child constant) requires a somewhat smaller percentage increase in ␮ (holding ␭ constant).25 For example, an increase in the child–staff ratio from 8:1 to 9:1 (that is, an increase of 12.5%) requires an increase in ␮/␭ from 9.681 to 10.787 (that is, an increase of 11.4%). This general result holds regardless of the child–staff ratio. The tradeoff in Figure 5 also suggests that regulation that sets maximum child–staff ratios, without regard for caregiver skill, may lead to inefficiency. The cost-minimizing child care center would choose the combination of average caregiver ability and child–staff ratio that minimizes its costs at each output level for each type of activity. Given constraints on child–staff ratios, the center may be unable to choose the cost-minimizing combination. For example, limiting the child–staff ratio explicitly rules out the French solution of higher child–staff ratios, more skilled caregivers, and universal coverage. 6. ADDITIONAL APPLICATIONS While the focus of this article is the efficient organization of the daycare environment, the modeling approach presented here is directly applicable to a wide range of educational and related human services. Rapid advances in technology are greatly reducing computing costs making instruction more personalized to the needs of the students than ever before. As a result, the staffing issues associated with providing instruction through hot lines and other online help services are direct applications of this queuing methodology. The computerized classroom environment is an even closer application. In a computer-assisted environment students work at their own pace, but, as in the case of the daycare setting, may break down and need direct assistance of the instructor. In addition, building criterion-referenced tests into computerized learning software provides feedback not only to the student but to the instructor, as well. For example, the instructor can determine the number of times a specific criterionreferenced question is attempted without success by each student. Given this information, the instructor may intervene effectively even when the student does not specifically request assistance. The instructor has a number of options for responding to the information about the success that

students are having answering specific questions. For example, the instructor can interact in person with the student to correct the problem or make suggestions. Alternatively, the instructor can alter the nature of the feedback that the computer provides students who are unable to answer a question after a specified number of tries or alter the degree of difficulty of the material being studied. With sufficient experience using specific software, one should be able to determine the expected values of the parameters identified in this article in order to assist educators in determining the most efficient staffing of their classrooms.26 7. CONCLUSION Child care in the United States is currently regulated by the individual states. There have been recent efforts to nationalize standards that at present vary considerably from one state to the next. Advocates of high quality care argue in favor of lower child–staff ratios, higher educational standards for child caregivers, and smaller group sizes. These tighter standards do not come without higher costs. Unless accompanied by government subsidies, child care centers would most likely have to raise their prices to meet the new standards and would, as a result, force more children outside of the formal child care system. In addition, stricter limits on child–staff ratios may force providers to employ workers in inefficient ways, since the optimal child–staff ratio is likely to vary depending on the activity assigned and the ability levels of the children. In this paper we have developed a formal model of one aspect of the child care environment, concentrating on the impact that child–staff ratios, group size, and caregiver ability have on the amount of time children spend on-task and the intensity of caregiver– child interactions. Assuming a positive correlation between on-task time, caregiver–child interactions and longer term output measures, the three regulable variables have a direct impact on the output achieved in the child care environment. The model is a means of clarifying the role of these three variables in order to understand better the potential costs and benefits of regulating them. The empirical evidence based on the theoretical model suggests, that in the absence of significant increases in government support, the only feasible method for accommodating more children in formal child care rests with higher child–staff ratios. At current median child–staff ratios, average on-task rates appear to be well above 90%, while children outside the system are likely to be experiencing much lower on-task rates and meaningful interactions with caregivers. While the model also indicates that there are advantages to grouping caregivers together, these gains in on-task time are small beyond the grouping of two or three caregivers. These gains would have to be weighed against the added time cost of covering

Daycare Quality and Regulation larger distances to help children that would otherwise be in a smaller group with a separate caregiver and the potential negative health consequences of encouraging the spread of contagious diseases. In the simulation presented in this paper, the exponential/Poisson assumptions made about the nature of child behavior and child caregiver resolutions of off-task activities, such as aimless wandering, were all biased toward the need for lower child–caregiver ratios. In addition, the sample used is also likely to bias the results in favor of lower child–caregiver ratios. Relaxing these assumptions and basing our results on a more representative sample would strengthen the conclusion that child–caregiver ratios in some states appear to be lower than justifiable on a cost–benefit basis. The main policy implication of this research, however, is that regulation of child–staff ratios, group

11

sizes and caregiver qualifications is too blunt an instrument for improving the overall quality level of child care. This finding is not unique in the literature. Our contribution to this area of research has been the clarification of an aspect of the debate on the cost and benefits of regulation that has been to date at best ad hoc. We have shown that the focus on inputs and not outputs may limit child care centers’ options in achieving lower cost solutions. By focusing on outcomes rather than inputs, child care centers would be free to choose among different combinations of these inputs. At present child care centers are not free to take advantage of higher ability caregivers capable of supervising larger groups of children engaged in a richer array of activities at lower overall cost. Acknowledgements—The authors gratefully acknowledge the many helpful suggestions of Stephen Hoenack, the associate editor.

NOTES 1. More than five million preschool children are currently receiving child care outside their own home. The child care industry has grown into a large and diverse industry with annual revenues approaching $50 billion. 2. The model can also be applied to other learning environments. Mulligan (1984b) provides an application to computer-assisted instruction. In this application the teacher assists students engaged in selfpaced learning with the assistance of feedback from the computer software. Even with sophisticated software programs, there is still a role for the teacher in helping students. 3. According to Gormley (1990), state regulatory handbooks average 16.6 pages for family child care homes and 30.2 pages for group child care centers. 4. It also proposed maximum group sizes of six to eight for infants, six to 12 for 1- and 2-year-olds, 14 to 20 for 3-year-olds, and up to 20 for 4- or 5-year-olds. 5. Lowenberg and Tinnin (1992) apply Stigler’s Capture Theory of Regulation to the child care market. They suggest that child care providers are likely to be better organized than parents and may use their influence to have regulations passed that will impede entry into the market for child care. Their principle finding is that regulations raise costs and lower the quantity supplied of child care. 6. The effect on input demands and costs depends on how R is chosen. Clearly, the lower is R, then the greater is the effect of the regulatory constraint; in Figure 1, this would correspond to moving the constraint out in a parallel fashion to the right. In contrast, where R is high enough, it may have no effect. In the figure, this would be the case when the constraint intersected the isoquant to the left of point A. 7. Vandel et al. (1988) and Abt Associates (1980) also use aimless wandering as a negative measure of quality in their empirical analyses of center-based programs. 8. For a general introduction to queuing theory see Gross and Harris (1985). Carmichael (1987) provides detailed numerical examples of the queuing models with engineering applications in construction and mining. 9. Another aspect of the queuing process concerns the nature of the interaction between a caregiver and the children. When students are engaged in individually paced activities, free play or other non-group activities, the caregiver is most likely to interact on an individual basis. However, queuing theory can be applied to cases when a caregiver chooses to intervene with more than one child at a time. Even if the activities are initially individually paced, a caregiver may be forced to change the activity to a group activity if the number of children in need of individual attention exceeds an acceptable level. Queuing models also require a queue discipline that dictates the order the provider follows when attending to arrivals. Among the possibilities are first-come, first-served, last-come, last-served, and random service. In our model, which focuses on the expected length of time that children wait in a queue, the queue discipline is unimportant. 10. The coefficient of variation, V, equals (Var[T])1/2/E[T] = (r␭2) ⫺ 1/2/(1/␭) = (r)1/2. As a result, r = 1/V2 = E2[T]/Var[T]. 11. For details on the study design and findings, see Abt Associates (1980). The NDCS data are not publicly available, so we have relied on published tabulations. 12. Activities were coded using the Child Focus Instrument which was based on the Prescott Child Observation System. See Abt Associates (1980 p. 124) for further information. 13. The model used here is an M/M/c/K queuing process, where M/M indicates that the breakdown (or arrival) rate and the service rate follow Poisson distributions, c is the number of servers (caregivers) and K is the size of the finite population (number of children). See Gross and Harris (1985) or Mulligan (1984a) for a more detailed discussion of the properties of this model. 14. To illustrate, for ␭ = 5, ␮ = 20, and C = 3, (i.e. a group of three children and one caregiver where

12

Economics of Education Review the average child needs attention five times an hour and the average breakdown requires 3 min of attention), P0 is 1/[1 + 0.75 + 0.375 + 0.09375] = 0.451 and Lq = 3 ⫺ (0.549·5) = 0.255. Thus, 45% of the time no child will need attention and the average length of the queue is about one-fourth of a child (that is, on average, one child will be in the queue every four time periods). 15. Using the figures above for C = 3, we find that Q = 0.915. Thus, on average, children would spend 91.5% of the time either not needing assistance or receiving it when they do need it. Q is thus the percent of time that the child spends on task. 16. “Aimless wandering” is defined as “child wanders around center with no apparent purpose to his movement. He may be sitting or standing doing nothing, looking around the area with no apparent focus” (Abt Associates, 1980, p. 129). Adding the proportion of time spent asking for attention, information, assistance or help, or comfort plus time spent crying to aimless wandering would reduce ontask time to about 0.930. It is not clear from the data, however, whether these behaviors represent time spent in a queue, so we have elected to use both values for OTT. 17. These are the values of ␭/␮ for the case of 14 children and two caregivers, and the two on-task rates. 18. For these simulations we are modeling the child care environment using the steadystate properties of the queuing model. We believe that this is a reasonable assumption, since children are likely to arrive at the center and immediately require attention. On the other hand, relaxing this assumption to allow periods of time where there are no queues (such as at the beginning of the day) suggests that observations of child care environments later in the day overstate the degree of breakdowns in the system. To the extent that this is true our simulations understate the actual amount of on-task time that children experience during a given day. Relaxing the steadystate assumption, therefore, strengthens the argument that increasing the child–staff ratio does not significantly harm quality. The model shown here assumes that child care workers attend to each child individually. As indicated in the previous section, one might argue that there were situations where two or more children were handled at the same time. For example, a worker breaking up a fight between two children will deal with both children at the same time. While the total elapsed time spent with both children may be the same as it would if they were attended to individually, the time spent in non-productive activities would be less. If child care workers can handle more than one child at a time, there would be higher average on-task rates that are higher than our empirical results suggest. As a result, child–staff ratios could be even higher than suggested by our simulations. 19. A child carefully supervised at home in a 1:1 child–staff ratio might have an on-task time close to 1. A child who was largely unsupervised would have a low on-task time. 20. Below we discuss the case of more skilled caregivers, in which ␭/␮ is much lower. 21. See Mulligan (1983) for a more detailed description of this property of queuing models. 22. The results are very similar for ␭/␮ = 0.132. 23. By “type” of child we mean children distinguished by their average value of ␭. For example, “types” could be children of different ages and/or backgrounds, as well as children with different emotional or learning needs. 24. The curve shown in Figure 5 looks like an isoquant. Note, though, that on-task time increases by movement closer to the origin (that is, by decreases in the child–staff or decreases in ␮/␭). By inverting both ratios one can create a map of “isoquants” showing increases in on-task time as a result of increases in the number of staff (holding the number of children constant) and/or increasing the average ability level of daycare workers. 25. This property of queuing models is developed in detail in Mulligan (1986). 26. For an example of the use of criterion-referenced testing in a computerized learning environment see Project SYNERGY (1995).

REFERENCES Abt Associates (1980) Children at the Center. Abt Associates, Cambridge, MA. Carmichael, D. (1987) Engineering Queues in Construction and Mining. Wiley, New York, NY. Fuchs, V. (1990) Economics applies to child care too. Wall Street Journal, April 2. Fuchs, V. and Coleman, M. (1991) Small children, small pay: why child care pays so little. American Prospect Winter, 74–79. Gormley, W. (1990) Regulating Mister Rogers’ neighborhood: the dilemmas of day care regulation. Brookings Review 8, 21–29. Gormley, W. (1991) State regulations and the availability of child-care services. Journal of Policy Analysis and Management 10, 78–96. Gross, D. and Harris, C. (1985) Fundamentals of Queuing Theory, 2nd edn. Wiley, New York, NY. Hayes, C., Palmer, J. and Zaslow, M., eds. (1990) Who Cares for America’s Children? Child Care for the 1990s. National Academy Press, Washington, DC. Howes, C. (1990) Can the age of entry into child care and the quality of child care predict adjustment in kindergarten? Developmental Psychology 26, 292–303. Lowenberg, A. and Tinnin, T. (1992) Professional versus consumer interests in regulation: the case of the U.S. child care industry. Applied Economics 24, 571–580. Mulligan, J. (1983) Economies of massed reserves. American Economic Review 73, 725–734. Mulligan, J. (1984a) A classroom production function. Economic Inquiry 22, 218–226. Mulligan, J. (1984b) A cost function for computer-assisted programmed instruction. Journal of Economics and Education 15, 275–281. Mulligan, J. (1986) Technical change and scale economies given stochastic demand and production. International Journal of Independent Organizations 4, 189–201. National Research Council (1991) Who Cares for American’s Children? National Reseach Council, Washington, DC.

13

Daycare Quality and Regulation Project SYNERGY: software support for underprepared students. Year four report, May, 1995. MiamiDade Community College, Division of Educational Technologies, Miami, FL. Richardson, C. and Marx, E. (1990) A Welcome for Every Child: How France Achieves Quality in Child Care. Report of the Child Care Study Panel of the French–American Foundation, New York. Vandel, D., Henderson, V. K. and Wilson, K. (1988) A longitudinal study of children with day-care experiences of varying quality. Child Development 59, 1286–1292.

APPENDIX With more than one caregiver (i.e S⬎1) and C children, the length of the queue is:

冘 (c ⫺ S)P C

Lq(S) =

c

c=S

where Pc, the probability that exactly c children are either in the queue or receiving attention is: Pc =

P0·(␭/u)c·C!/[(C ⫺ c)!c!], P0·(␭/u) ·C!/[(C ⫺ c)!S!S c

c⫺S

for c = 1...S ], for S ⱕ c ⱕ C

and

冘 C!·(␭/u) /((C ⫺ c)!c!) + 冘 C!·(␭/u) /((C ⫺ c)!S!S

S⫺1

P0 = 1/[

C

c

c=0

c

c=S

c⫺S

)].