Details

Autor(en) / Beteiligte

• Gordon, Grey

Titel

Computing Dynamic Heterogeneous-Agent Economies: Tracking the Distribution

Ist Teil von

• Economic quarterly - Federal Reserve Bank of Richmond, 2020-03, Vol.106 (2), p.61-95

Ort / Verlag

Richmond: Federal Reserve Bank of Richmond

Erscheinungsjahr

2020

Link zum Volltext

Quelle

PAIS Index

Beschreibungen/Notizen

• The evolution of prices in dynamic heterogeneous-agent economies typically depends on the state of every agent, thereby requiring that a distribution be a state variable. The contribution of this paper is to introduce a method for computing equilibrium in these models by including an entire distribution if finite-dimensional-or a fine approximation of it if infinite-dimensional-as a state variable. The insight of Krusell and Smith (1997, 1998) is that this approach is not necessary if a model features quasi-aggregation, the condition where prices can be accurately forecasted using just a few state variables. However, not all economies feature quasi-aggregation, and I show that the method presented in this paper is capable of accurately computing equilibrium in at least one of these: Huffmans (1987) overlappinggenerations (OLG) economy paired with an extreme calibration used in Krueger and Kubler (2004). Even when quasi-aggregation obtains, including a distribution as a state variable may be desirable from a conceptual or purely pragmatic perspective. I show that the method accurately computes equilibrium in an economy of this type also: a version of Krusell and Smiths (1998) (KS) economy where households occasionally binding constraints. The method is feasible for these two economies with equilibrium for both computed in just a few minutes in Matlab.1 As discussed momentarily, Smolyak's (1963) sparse-grid interpolation algorithm introduced to economics by Krueger and Kubler (2004) makes this possible.Smolyak's algorithm is a projection method that uses collocation on a very sparse grid.2 The algorithm approximates a function by interpolating its value at a set of predefined gridpoints (collocation points) using weighted sums of polynomials. The fineness of the approximation is controlled by using different "levels of approximation." For the lowest level of approximation, which is the only one used in this paper, the number of gridpoints grows only linearly in dimension. More specifically, given a function of dimension d, Smolyak's algorithm gives 2d + 1 points that the function must be evaluated at in order to approximate it. In contrast, linear interpolation or any tensor-product interpolation method would require at least 2d points. To see the difference this makes, consider that the distributions (and hence state spaces) used in this paper have up to 200 elements: to approximate a function of this dimension using linear interpolation would require more than 1048 trillion function evaluations compared with only 401 for Smolyak interpolation. Not only is the Smolyak algorithm computationally efficient, but Barthelmann, Novack, and Ritter (2000) prove the approximation has nearly optimal error bounds for smooth functions.3 The disadvantage of Smolyak's algorithm is that approximations to nonsmooth functions may be quite poor.The application of Smolyak's algorithm presented in this paper leverages the strengths of the Smolyak algorithm, computational efficiency and accuracy for smooth functions, while avoiding its main weakness, poor approximation of nonsmooth functions. While many heterogeneous-agent models feature policy functions that are kinked in individual wealth or income and hence are not smooth, as long as they are smooth in the aggregate state, they can be approximated well by the Smolyak algorithm. This is accomplished through indexing policy functions by individual states and constructing a Smolyak approxima tion to each indexed policy function. For example, given a capital policy function k'(k,^), where k is a household's current capital holdings and ^ is a distribution of holdings across households, the Smolyak approximation to k'(k,^) would likely be poor if k! were kinked in k. However, if k! is fairly smooth in ^ for fixed k, then the indexed policy function k0 (^) could be accurately approximated using Smolyak interpolation.4 By indexing policies in this way, the resulting Smolyak approximations may be accurate even if the policies are "not smooth." I refer to this approach as the Smolyak method.Gender ideology represents an "underlying concept of an individual's level of support for a division of paid work and family responsibilities that is based on the notion of separate spheres."1 A wide range of family-related outcomes, including relationship formation, quality, and dissolution, the number and timing of births, and divisions of labor, are influenced by gender ideology.2 Gender ideology also shapes labor market decisions and outcomes, such as labor force participation, entry into gender-atypical occupations, work hours, and earnings.3 Examining change in gender ideology therefore has far-reaching implications for future gender equality in society.4The current study investigates changes in gender attitudes about work and family across birth cohorts in China. In doing so, it contributes to the literature in two primary ways. First, this study highlights the multidimensionality of gender ideology by examining public beliefs about proper gender arrangements in the public and private spheres separately. Broadly speaking, the private sphere refers to the private family, whereas the public sphere involves political, economic, and educational domains.5 Following recent research that theorizes the understanding of new gender dynamics in post-reform China, in this study we focus on the labor market as the public sphere and the private family as the private sphere.6 In China and elsewhere, progress toward gender equality has been more substantial in the public sphere than in the private sphere.7 In light of the uneven gender revolution, scholars call attention to the importance of understanding gender and gender ideology as multidimensional constructs.8 Prior research in the Chinese context, however, either measured gender ideology as a one-dimensional concept,9 or did not distinguish between attitudes toward gender equality in the workforce and beliefs about gendered family roles.10 We take a first step forward toward systematically investigating trends and variation in "the public's multidimensional understanding of gender roles" in China.11Second, while recent research has suggested rising public support for traditional gender role attitudes in China between 1990 and 2010,12 we adopt a cohort perspective to examine changes in gender ideology across cohorts born between the 1950s and the 1990s. In addition, although prior studies show that women tend to hold more egalitarian gender ideology than men,13 it remains unclear whether this gender gap in ideology has widened or narrowed across cohorts. This study thus contributes to the literature by investigating how the differences between men and women in terms of gender attitudes about work and family have changed across Chinese cohorts. In fact, the concept of cohort has long been considered crucial to the study of social change.14 International empirical studies have shown that cohort replacement-the replacement of older cohorts by younger cohorts-plays a pivotal role in explaining aggregate change in gender role attitudes.15 Thus, examining cohort differences in gender ideology will advance our understanding of recent trends and future change in gender egalitarianism in China.