### Production Functions and Household Choices

#### ECON 7366 Health Economics (Elaine Liu)

Class Presentation

Tuesday, November 12th, 2019

#### 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)