Nonparametric estimation of a primary care production function in urban Brazil

Abstract

Background The Brazilian public health system is one of the largest health systems in the world, with a mandate to deliver medical care to more than 200 million Brazilians. The objective of this study is to estimate a production function for primary care in urban Brazil. Our goal is to use flexible estimates to identify heterogeneous returns and complementarities between medical capital and labor. Methods We use a large dataset from 2012 to 2016 (with more than 400 million consultations, 270 thousand physicians, and 11 thousand clinics) to nonparametrically estimate a primary care production function and calculate the elasticity of doctors’ visits (output) to two inputs: capital stock (number of clinics) and labor (number of physicians). We benchmark our nonparametric estimates against estimates of a Cobb-Douglas (CD) production function. The CD model was chosen as a baseline because it is arguably the most popular parametric production function model. By comparing our nonparametric results with those from the CD model, our paper shed some light on the limitations of the parametric approach, and on the novelty of nonparametric insights. Results The nonparametric results show significantly heterogeneity of returns to both capital and labor, depending on the scale of operation. We find that diseconomies of scale, diminishing returns to scale, and increasing returns to scale are possible, depending on the input range. Conclusions The nonparametric model identifies complementarities between capital and labor, which is essential in designing efficient policy interventions. For example, we find that the response of primary care consultations to labor is steeper when capital level is high. This means that, if the goal is to allocate labor to maximize increases in consultations, adding physicians in cities with a high number of clinics is preferred to allocating physicians to low medical infrastructure municipalities. The results highlight how the CD model hides useful policy information by not accounting for the heterogeneity in the data.

Optimal interventions in networks during a pandemic

“…Our choice of the Cobb-Douglas function as a parametric estimate of the production function is motivated by our empirical analyses in Section 5 . Our consideration is also more in line with several studies that argue that the Cobb-Douglas function is a standard parameterization of production functions in the literature (Douglas 1976 ), especially in the context of primary care (Wichmann and Wichmann 2020 ), and nursing homes (Reyes-Santías et al 2020 ). Using the recent data collected by Chen et al ( 2021 ) on US nursing homes, we approximate a typical nursing home’s production function as , where y i is the total number of residents (proxies the nursing home’s output) who receive care, k i is the total number of beds (proxies the capital), and h i is the number of occupied beds (proxies the labor supply).…”

Optimally Targeting Interventions in Networks during a Pandemic: Theory and Evidence from the Networks of Nursing Homes in the United States

“…Our choice of the Cobb-Douglas function as a parametric estimate of the production function is motivated by our empirical analyses in Section 5. Our consideration is also more in line with several studies that argue that the Cobb-Douglas function is a standard parameterization of production function in the literature (Douglas, 1976), and especially in primary care (Wichmann & Wichmann, 2020), and nursing homes (Reyes-Santías et al, 2020). Using the recent data collected by Chen et al ( 2021) on U.S. nursing homes, we approximate a typical nursing home's production function as…”