[Paper Review] Constructive Identification of Heterogeneous Elasticities in the Cobb-Douglas Production Function

Tong Li, Yuya Sasaki|arXiv (Cornell University)|Nov 27, 2017

Economic Theory and Policy15 references3 citations

TL;DR

This paper provides a constructive identification method for heterogeneous elasticities in the Cobb-Douglas production function using a flexible input cost ratio as a control function under non-collinear heterogeneity. It derives closed-form formulas for firm-specific elasticities of labor, capital, and flexible inputs, and identifies additive productivity, enabling direct estimation without panel data or strong invertibility assumptions.

ABSTRACT

This paper presents the identification of heterogeneous elasticities in the Cobb-Douglas production function. The identification is constructive with closed-form formulas for the elasticity with respect to each input for each firm. We propose that the flexible input cost ratio plays the role of a control function under "non-collinear heterogeneity" between elasticities with respect to two flexible inputs. The ex ante flexible input cost share can be used to identify the elasticities with respect to flexible inputs for each firm. The elasticities with respect to labor and capital can be subsequently identified for each firm under the timing assumption admitting the functional independence.

Motivation & Objective

• To address the challenge of identifying firm-specific elasticities in the Cobb-Douglas production function when elasticities are heterogeneous and not functionally independent.
• To overcome identification failures due to functional dependence and instrument irrelevance, as highlighted by Ackerberg et al. (2015) and Gandhi et al. (2017).
• To develop a method that does not rely on panel data or invertible mappings between observed choices and unobserved technologies.
• To enable identification of both non-additive (elasticity) and additive (neutral productivity) components of firm-level productivity.
• To provide transparent, closed-form formulas for elasticities and productivity that are robust to subtle structural issues in identification.

Proposed method

• Introduces the flexible input cost ratio $ r^{l,m}_t = \frac{p^l_t \exp(l_t)}{p^m_t \exp(m_t)} $ as a control function for unobserved latent technology.
• Employs a non-collinear heterogeneity assumption between elasticities of two flexible inputs, replacing the need for invertible mappings.
• Uses ex ante flexible input cost shares $ s^l_t $ and $ s^m_t $ to identify elasticities of flexible inputs in closed form: $ \beta_l(\omega_t) = s^l_t $, $ \beta_m(\omega_t) = s^m_t $.
• Identifies the capital elasticity $ \beta_k(\omega_t) $ via the partial derivative of the conditional expectation of output net of flexible input effects.
• Derives additive productivity $ \beta_0(\omega_t) $ as the conditional expectation of the residual output after accounting for all input elasticities.
• Relies on a timing assumption that ensures functional independence, consistent with Ackerberg et al. (2015), to avoid rank deficiency in identification.

Experimental results

Research questions

• RQ1 Can firm-specific elasticities in the Cobb-Douglas production function be identified when they are heterogeneous and not functionally independent?
• RQ2 How can identification be achieved without relying on panel data or invertible mappings between observed choices and unobserved technologies?
• RQ3 What role does the flexible input cost ratio play in enabling constructive identification under non-collinear heterogeneity?
• RQ4 How can the functional dependence problem and instrument irrelevance issue be circumvented in elasticity identification?
• RQ5 Under what conditions can closed-form formulas be derived for all elasticity and productivity parameters in a heterogeneous production function model?

Key findings

• The flexible input cost ratio $ r^{l,m}_t $ serves as a valid control function under the non-collinear heterogeneity assumption, enabling identification without requiring invertibility of choice functions.
• The elasticities with respect to flexible inputs $ l $ and $ m $ are identified in closed form as the ex ante flexible input cost shares: $ \beta_l(\omega_t) = s^l_t $, $ \beta_m(\omega_t) = s^m_t $.
• The capital elasticity $ \beta_k(\omega_t) $ is identified via the derivative of the conditional expectation of the residual output with respect to capital input.
• Additive productivity $ \beta_0(\omega_t) $ is identified as the conditional expectation of the output residual after accounting for all input elasticities.
• The identification strategy does not require panel data, relying instead on first-order conditions and ex ante cost shares, which is a key distinction from traditional panel-based approaches.
• The method is robust to functional dependence and instrument irrelevance problems, provided the timing assumption for functional independence is satisfied.