Recent studies of delayed reproduction, including papers in this issue, often focus on causes and mechanisms, on individual choices, and on technological developments (e.g., advances in medically assisted reproduction and egg freezing) that are providing new choices that were not previously available. However, delayed fertility also has effects at the population level – on growth rates, age distributions, and population size – that deserve consideration.

A majority of the world’s population now lives in countries with below-replacement fertility levels. ² The resulting concerns about population decline and ageing (surveyed by Skirbekk, 2022) have spilled from the pages of academic demography journals to become a major cultural and political issue and sources of inspiration for pro-natalist movements. This paper aims to present some possibly surprising results about the effects of delayed reproduction on population growth and structure in the current demographic situation of declining populations. We will see that delayed reproduction, all else being equal, may increase, rather than decrease population growth.

The effects of delayed reproduction have long been of interest to researchers. The earliest study on this topic seems to be that of Dublin and Lotka (1925). Analysing U.S. life tables from 1920, they found that a fixed postponement of reproduction by five years reduced the intrinsic rate of increase from r = 0.0055 to r = 0.0036 . Coale (1956) and Coale and Tye (1961) also concluded that delayed fertility, modelled as a shift in the fertility function along the age axis, would reduce the population growth rate and lead to older populations. Coale and Tye (1961) did note in passing that if r were “sufficiently negative” the effect of delayed reproduction might be positive, but they did not explore this possibility further.

Population biologists and evolutionary demographers also addressed the question of the timing of reproduction, perhaps because of the much greater range of life histories, and thus of development patterns, found across the plant and animal kingdoms. ³ Early contributions include those of Cole (1954, 1965), Smith (1954), and Slobodkin (1961). The focus of this research was on the ways in which earlier reproduction could increase the population growth rate, and thus increase fitness. One of the most influential studies was that of Lewontin (1965), which was presented in a symposium on the genetics of colonising species. A colonising species, in this usage, might be a new arrival on a newly created island (MacArthur and Wilson, 1967) or a weed taking root in a newly ploughed field. In such cases, natural selection favours rapid population growth to take advantage of the situation.

Lewontin approached the problem by creating an idealised triangular net maternity function and shifting that function along the age axis in various ways. This allowed him to calculate the response of the intrinsic rate of increase r to changes in the timing of reproduction. His conclusion, supported by empirical examples of insect populations was that “the general point holds true that small absolute changes in developmental rates of the order of 10% are roughly equivalent to large increases in fertility of the order of 100%” (Lewontin, 1965, p. 85).

Caswell and Hastings (1980) were the first to use perturbation (i.e., sensitivity) analysis to explore the effects of reproductive timing on the population growth rate. They focused on advanced reproduction by shifting fertility one or more years earlier. In that case, if f x is fertility at age x , then Δ f x = f x + 1 − f x . (1) They calculated the marginal effect on the population growth rate λ as Δ λ = ∑ x ∂ λ ∂ f x Δ f x , (2) where λ is related to the continuous-time intrinsic rate of increase r by r = log λ . For the most part their results echoed Lewontin’s, showing that earlier reproduction had a strongly positive effect on population growth, and that delayed reproduction would reduce population growth.

The conclusion that delayed reproduction would reduce the population growth rate became such a widely held belief that it became a basis for the “later, longer, fewer” birth planning campaign in China. Introduced in the early 1970s, this policy aimed to slow population growth by mandating later marriage (age 25 for women, age 27 or 28 for men) and longer birth intervals of at least four years (Whyte et al., 2015; Zhang, 2017). It was replaced by the one-child policy in 1979. The “later, longer, fewer” program was an explicit attempt to enforce delayed reproduction precisely because of its negative effects on population growth.

However, Caswell and Hastings (1980) had noted that in equilibrium or declining populations, delayed reproduction could increase the population growth rate. They speculated about the evolutionary implications of this observation, but did not pursue the issue further.

The current trend of declining fertility, the prevalence of below-replacement demographic rates, and the resulting concern about depopulation and population ageing suggest that it is time to re-examine the effects of delayed reproduction on population growth and structure. This paper brings to bear a much more powerful set of perturbation analyses to examine the effects of delayed reproduction.

In this paper, matrices are written as upper case bold characters (e.g.,  U ) and vectors as lower case bold characters (e.g.,  a ). Vectors are column vectors by default; x ⊤ is the transpose of x . The i th unit vector (a vector with a one in the i th location and zeros elsewhere) is e i . The vector 1 is a vector of ones, and the matrix I is the identity matrix. When necessary, subscripts are used to denote the size of a vector or matrix; e.g.,  I ω is an identity matrix of size ω × ω .

The symbol ⊗ denotes the Kronecker product. The vec operator stacks the columns of a m × n matrix into a m n × 1 column vector. The notation ∥ x ∥ denotes the 1-norm of x , i.e., the sum of the absolute values of the entries. Matlab notation will be used to refer to rows and columns of a matrix, e.g.,  F ( i , ) and F ( : , j ) refer to the i th row and j th column of the matrix F , respectively.

The effects of changes in the reproductive timing are obtained by calculating the derivative of some demographic outcome with respect to changes in some aspect of the fertility schedule. These derivatives are calculated using matrix calculus. This technique has been used in a long series of demographic papers published over the last 15 years (e.g., Caswell, 2008, 2010; Engelman et al., 2014; Van Raalte and Caswell, 2013). The reader is assumed to have some familiarity with the approach. A complete mathematical presentation can be found in Magnus and Neudecker (1988), and a more accessible treatment can be found in Abadir and Magnus (2005). There is also a book (Caswell, 2019) containing an elementary introduction to matrix calculus and presenting many demographic applications that is available for free download. The appendix of this paper gives some basic facts.

Briefly, if y and x are vectors, the derivative of y with respect to x must include the derivatives of all entries of y with respect to each of the entries of x . If y is of length n and x is of length m , the derivative is the n × m matrix d y d x T = ( d y i d x j ) . (3) The derivative of a matrix Y with respect to a matrix X has even more entries. It is obtained by first applying the vec operator to transform the matrices to vectors, leading to d vec Y d vec T X . (4) If Y is m × n and X is p × q , then this matrix is m n × p q . These derivatives satisfy the chain rule, which we will use frequently, so that d y d z T = d y d x T d x d z T . (5)

The goal of this paper is to explore the population-level effects of delayed reproduction. We start with the population projection matrix A = U + F , (6) where U contains survival probabilities on the subdiagonal and zeros elsewhere, and F contains age-specific fertilities on the first row and zeros elsewhere. Let f be the vector of age-specific fertilities, in units of female children per female. Delayed reproduction will be modelled by a change in f .

Let ξ denote some demographic outcome of interest (e.g., the population growth rate) that can be calculated from A and that depends on the fertility schedule. Using the chain rule of equation (A.5), the sensitivity of ξ to changes in the fertility schedule is d ξ d f T = ( d ξ d vec T A ) ( d vec A d vec T F ) ( d vec F d f T ) . (7) The marginal effect of a change Δ f in reproduction on ξ is, to first order, Δ ξ = d ξ d f T Δ f . (8)

We need to create a change Δ f in fertility that represents a specified degree of delay. Because we want the effect of delay alone, all else being equal, we need a formulation that changes only the timing of reproduction. Simultaneously delaying and reducing fertility would reveal nothing about the effects of delay per se. Hence, the manipulations here hold TFR fixed.

One way to delay reproduction is to apply a fixed delay to the fertility schedule, so that all fertility at age x is shifted to a specified later age (e.g., age x + 1 ). This approach was used by Dublin and Lotka (1925) and Caswell and Hastings (1980). Here we consider a more flexible description of delayed reproduction by creating a distributed delay. In this calculation, some of the fertility at age x remains at age x , while some is delayed to x + 1 , some is delayed to x + 2 , and so on. We do this using a diffusion process. We define a continuous-time biased random walk with transition rate matrix Q given by (supposing that ω = 4 ) Q = ( − 1 0 0 0 1 − 1 0 0 0 1 − 1 0 0 0 1 0 ) . (9) The component of fertility at age x is delayed to age x + 1 at the rate 1. The random walk is absorbing at the upper end, since fertility cannot be delayed beyond age ω .

The discrete delay matrix is the matrix exponential ⁴ of c Q P = e c Q . (10) The value of c ≥ 0 determines the mean reproductive delay. The distributions of reproductive delay for c = 1 and c = 3 are shown in Figure 1. Delayed reproduction is defined by the transformation f ↦ P f (11) and then Δ f = P f − f (12) which maintains the TFR unchanged, as desired.

As an example of the trend towards increased survival and declining fertility in high income countries, I will analyse the case of Japan from 1947 to 2019 (rates from HMD 2022 and HFD 2022). Over this time span, period life expectancy increased from 54 to 87 years, the TFR declined from 4.6 to 1.3 children per woman, and the population growth rate declined from λ = 1.020 to 0.987. Both indices declined to below the replacement level, as shown in Figure 2. This sequence provides a convenient example, but similar results would follow from any similar time series. Comparative studies will be interesting.

Delayed reproduction may, of course, be more complicated than just a shift of age-specific fertility. Fertility is also affected by parity, and age-specific parity transition matrices capture those effects. Caswell (2020) developed a multistate age × parity model as part of a matrix kinship model. The details of the multistate models are laid out in Caswell et al. (2018). Very briefly, individuals are classified into ω age classes and s parity stages (in the HFD, s = 6 ). The state of the population is given by the block-structured vector n ˜ = ( n 11 ⋮ n s 1 ⋮ n 1 ω ⋮ n s ω ) . (30) The vector is projected by a projection matrix constructed from four sets of matrices, which specify transitions among parity classes, survival from one age class to the next, fertility as the result of a transition from one parity class to the next and the assignment of new births to age class 1 and parity 0.

Using the protocol given by Caswell (2012) and Caswell et al. (2018) yields a block-structured population projection matrix A ˜ = U ˜ + F ˜ (31) with blocks arranged in the same form as the familiar age-classified Leslie matrix, which is a familiar structure in multiregional demography (e.g., Feeney, 1970; Rogers, 1995). The population growth rate λ and stable structure w ˜ ( age × parity ) are calculated from A ˜ in the usual way.

Caswell (2020) developed an age × parity model using demographic data from the HMD and HFD for Slovakia from 1950 to 2014. The case of Slovakia was chosen because it had one of the longest sequence of parity information and steepest declines in fertility in the database. As shown in Figure 8, the population growth rate in Slovakia dropped below the replacement level in 1989 and remained below the replacement level thereafter.

Of the several ways that delayed reproduction could be implemented in this model, we consider here a distributed delay, with a specified value of c , applied to the entire age-specific parity transition structure. The following figures, which are more suggestive than definitive, show some results.

Figure 8 shows the impact of a delay in reproduction on the population growth rate. Because the effects for Slovakia are smaller in magnitude, for clarity the results are shown for c = 5 . When population growth rate dropped below the replacement level, the effect of delayed reproduction changed from negative to positive, just as it did in the analyses based on age-specific, as opposed to age × parity-specific, fertility.

Figure 9 shows the effect of delayed reproduction on the stable age distribution (the marginal age distribution from the stable age × parity distribution), expressed as the proportion of the population over age 65. When population growth was above the replacement level, delayed reproduction led to an older population. Conversely, when population growth was below the replacement level, delayed reproduction had the opposite effect.

Finally, Figure 10 shows the components of the marginal parity distribution for parity 0 and parity 5 (the highest parity included in the HFD data). Delayed reproduction increased the proportion of parity 0, and had little effect on the proportion of parity 5.

“Everyone knows” that delayed reproduction can be expected to reduce population growth and population size, and make age distributions older. But this is not always the case. The effects of delayed reproduction differ qualitatively depending on whether the population is increasing or decreasing. Early authors sometimes recognised this, but their attention was focused on situations of positive, and even rapid, growth. Today, when much of the world has demographic rates below the replacement level, this received wisdom must be challenged. The methods presented here have potential applications to other populations, other models, and other response variables.