Production Functions and Household Choices

ECON 7366 Health Economics (Elaine Liu)

Class Presentation

Tuesday, November 12th, 2019

Fan Wang

Production Function Questions

  • Which inputs matter?
    • Protein vs. no-protein
    • Time vs. in-kind
  • When do they matter? (Todd and Wolpin 2003)
    • $\beta_1^1$: Effects of period one input on period one
    • $\beta_1^2$: Does period 1 input have effects in period two?
    • $\beta_2^2$: Is the impact of period 2 input on period two the same as $\beta_1^1$
    • $y_{t}$ and $y_{t-1}$: can lagged output $y_{t-1}$ summarize the effects of past inputs in a value-added specification?

Functional Form Assumptions

  • Linear
    • Inputs are substitutes
    • In an optimal choice problem, would only choose one input, even in a dynamic context
    • But easier to estimate (Puentes et al. 2016)
  • Cobb-Douglas
    • Linear in log
    • Decreasing returns in single input (Wang et al. 2019a)
    • Inputs are complementary if have multiple inputs
    • Complementarity of inputs within and across period the same (Wang et al. 2019b)
  • Constant Elasticity of Substitution

Production Function Estimations: OLS

  • Are parents aware of the production function?
  • Can parents choose inputs?
  • Are there shocks observable to parents but not to econometrician?
  • If no to any of the three questions above: OLS might be able to identify input effects
  • OLS allows for the estimation of many inputs concurrently (Wang et al. 2019b)
  • In Wang et al. (2019b), we use profiles of height to absorb error term

Production Function Estimations: Instruments

  • With multiple inputs, potential problem with weak instruments
  • IVs from prices (Puentes et al. 2016)
  • IVs based on differenced moments (Mani et al. 2018)
  • LATE, can you extrapolate to production function?
  • Difficult to estimate parameters for non-linear and non-separable production functions with multiple inputs

Production Function Estimations: Experiments

  • Ideally, there are different levels of input variations
  • Difference in difference to get the treatment effect
  • Impose functional form assumptions, extrapolate from differential effects between treatment quantities to production function parameters (Wang et al. 2019a)
  • Even single experimental input level might be OK if individuals have existing variations in inputs and the treatment is a subsidy

Production Function Estimations: Model Choices

  • Model household choices directly (Wang et al. 2019a)
  • There is no endogeneity issue, households in model choose given shocks
  • What is the model assumption about the relationship between the unobservable (to econometrician) shock term across periods?
  • What is the model assumption about the relationship between the unobservable (to econometrician) shock term with initial conditions?
  • Model provides "causal" results, identification in this case is largely a statistical concept. In OLS, statistical identical is trivial given invertibility.

Given Estimation Results, What to do With them?

  • Know which inputs matter, how much it matters, and how inputs matter jointly
  • As a part of a structural model
  • How to optimally allocate resources, across children, across periods, and across inputs (Wang 2019)