Why production functions?

• Production functions are essential to understand how firms behave
  • Profit maximization, \(\pi = PQ - wL - rK\) where \(Q = f(K, L)\)
  • \(w/P = f_L\) and \(r/P = f_K\)
• Production functions are main building blocks in many economic models
• Analyzing growth dynamics at the country, industry or firm level
  • Level and growth rate of (total factor) productivity
  • Increasing returns, economies of scale
  • Complementarity / substitutability between inputs
  • Economics of scope
  • R&D and innovation
  • Resource allocation

Production functions

• \(Q* = f(K, L), Q ≤ Q*\)
  
• \(Q = f(K, L, E, M, S)\)
  
• \(Q = f(K, L, T)\)

Production functions - one input

Type of production functions

• Production functions
• Production frontiers

Some concepts

• Average product / partial productivity, \(AP = Q/L\)
• Marginal product, \(MP = ∂Q/∂L\)
• Output elasticity, \(ε_L = (∂Q/Q)/(∂L/L) = (∂lnQ/∂lnL) = ∂q/∂l = MP/AP\)
• Scale elasticity, returns to scale, \(ε = \Sigma ε_i\)
• Economies of scale
• Marginal rate of technical substitution (slope of the isoquant) \(MRTS_{ij} = -MP_j/MP_i\)
  • For profit maximization, \(w/r = MRTS = -dK/dL|_{Q=Q*}\)
• Elasticity of substitution \(σ_{KL} = (dKL/dMRTS)/(KL/MRTS)\)

Some concepts - CD case

Production function \(Q = e^{beta_0}K^{\beta_K}L^{\beta_L}\)
or \(q = \beta_0 + \beta_Kk + \beta_Ll\)

• Labor elasticity of output, \(ε_L = ∂q/∂l = \beta_L\)
  • Perfectly competitive markets and constant returns to scale, \(ε_L = wL/pQ\)
  • What if \(ε_L > wL/pQ\)
• Capital elasticity of output, \(ε_K = ∂q/∂k = \beta_K\)
• Constant returns to scale, \(ε = ε_K + ε_L = 1\)
  • \(q = \beta_0 + \beta_1k + (1 - \beta_1)l\)
  • \(q-l = \beta_0 + \beta_1(k-l)\)
• Marginal rate of technical substitution \(MRTS = -dK/dL = -f_L/f_K = -MPL/MPK = - (\beta_L/\beta_K)(K/L)\)
• Elasticity of substitution, \(σ_{KL} = 1\)
  • Profit maximization, \(w/r = dK/dL\)
  • If the wage rate increases by 1% (w/r increases by 1%), K/L increases by 1% and \(wL/rK\) remains constant

Estimation of production functions

• Parametric methods
  • Econometric methods
• Non-parametric methods
  • Data envelopment analysis

Estimation methods

• OLS (Ordinary least squares)
  • \(q_{it} = X_{it}\beta + ε_{it}\)
  • \(E[ε | X] = 0\)
  • \(ε \sim N(0, \sigma^2)\)
  • MLE - minimize the sum of squares of error terms
    • OLS is the BLUE
• GMM (Generalized method of moments)
  • \(q_{it} = X_{it}\beta + ε_{it}\)
  • Moment conditions, \(E[X ' ε] = 0\)
  • \(E[X ' (q - X \beta)] = 0\)

Estimation issues

• Data
  • Measurement errors
    • Firm level prices
  • Unobserved firm-specific effects
• Functional form
• Technological change
  • Putty-clay technology
• Endogeneity/omitted variable bias
• Selection
• Heterogeneity

Functional forms

Cobb-Douglas function

\(Q = AK^{\beta_K}L^{\beta_L}\)
\(q = a + \beta k + \alpha l\)

Leontieff function

\(Q = min(aK, bL)\)

Translog function

\(Q = AK^{\beta_K + beta_{KK}k + beta_{KL}l}L^{\beta_L + beta_{LL}l + beta_{KL}k}\)
\(q = \beta_0 + \beta_Kk + \beta_Ll + \beta_{KK}k^2 + \beta_{LL}l^2 + 2\beta_{KL}kl\)

Constant elasticity of substitution (CES) function

\(Q = A[\theta (a_KK)^\gamma + (1 - \theta) (a_LL)^\gamma]^{\rho / \gamma}\)
\(\rho\) degree of homogeneity
\(\gamma\) degree of substitutability of inputs
Elasticity of substitution \(\sigma = 1/(1 - \gamma)\)

Taylor series expansion

Production function in log form, \(q = f(k, l)\)

1st order Taylor series expansion of \(f(k, l)\) at point \((a,b)\)

\(q ≈ f(a, b) + f_k(a, b)(k - a) + f_l(a, b)(l - b)\)

\(q ≈ \beta_0 + \beta_Kk + \beta_Ll\)

2nd order Taylor series expansion of \(f(k, l)\) at point \((a,b)\)

\(q ≈ f(a, b) + f_k(a, b)(k - a) + f_l(a, b)(l - b) +\)
\(½[f_{kk}(a, b)(k -a)^2 + 2f_{kl}(a, b)(k -a)(l - b) + f_{ll}(a, b)(l - b)^2]\)

\(q ≈ \beta_0 + \beta_Kk + \beta_Ll + \beta_{KK}k^2 + \beta_{LL}l^2 + \beta_{KL}kl\)

CD vs translog

Taylor series expansion, 1st, 2nd and 3rd order

CD functions

• Extensively used in theoretical models (with constant returns to scale)
• Emprical support for CD but
• There is as a “publication bias”, the mean elasticity is 0.3.
  • Gechert, Sebastian, Tomas Havranek, Zuzana Irsova and Dominika Kolcunova (2022), “Measuring capital-labor substitution: The importance of method choices and publication bias”, Review of Economic Dynamics, 45: 55-82.
• CD is a special case of the translog function
  • Test \(H_0\): \(\beta_{KK} = \beta_{LL} = \beta_{KL} = 0\)
  • It is rejected in most of the cases

Technological change

CD function and technological change

Deterministic technological change

• Total factor productivity growth, \(Q = AK^{\beta_K}L^{\beta_L}, A = A_0e^{\lambda t}\)
  • \(q = a_0 + \lambda t + \beta_Kk + \beta_Ll\), \(\lambda\) = TFP growth rate
  • Hick neutral technological change
• Increase in capital efficiency, \(K = Ke^{\gamma t}\)
  • \(q = a_0 + \gamma t + \beta_Kk + \beta_Ll\), \(\lambda\) = TFP growth rate
• Increase in labor efficiency, \(L = Le^{\omega t}\)
  • \(q = a_0 + \omega t + \beta_Kk + \beta_Ll\), \(\lambda\) = TFP growth rate
• All of them together
  • \(q = a_0 + (\lambda + \gamma + \omega) t + \beta_Kk + \beta_Ll\), \(\lambda\) = TFP growth rate

We cannot identify TFP, capital efficiency and labor efficiency effects in a CD function. They are all observationally equivalent.

CD functions and technological change

Stochastic technological change

• Production function, \(q_{it} = \beta_Kk_{it} + \beta_Ll_{it} + \omega_{it}\)
• Technological change, \(\omega_{it} = \rho \omega_{it-1} + ε_{it}\)

Translog function and technological change

Deterministic technological change

\(q = \beta_0 + \beta_Kk + \beta_Ll + \beta_{KK}k^2 + \beta_{LL}l^2 + \beta_{KL}kl\)
\(+ \beta_Tt + \beta_{TT}t^2 + \beta_{TK}tk + \beta_{TL}tl\)

Stochastic technological change

\(q = \beta_0 + \beta_Kk + \beta_Ll + \beta_{KK}k^2 + \beta_{LL}l^2 + \beta_{KL}kl + \omega_{it}\)
\(\omega_{it} = \rho \omega_{it-1} + ε_{it}\)

Stochastic production frontiers

\(q = \beta_0 + \beta_Kk + \beta_Ll + ε - u\)

\(u\) is non-negative technical efficiency term, \(TE = exp^{-u}\), therefore \(0 ≤ TE ≤ 1\)

\(u \sim N^+(\mu, \sigma^2)\)

• \(\mu = 0\)
• \(\mu = Z\delta\)

Endogeneity/omitted variable bias

• Cobb-Douglas production function
  • \(Q = e^{\beta_0}K^{\beta_K}L^{\beta_L}e^{\omega + ε}\)
  • \(q = \beta_0 + \beta_Kk + \beta_Ll + \omega + ε\)
• Profit maximization
  • \(\pi = PQ - wL - rK\)
  • Decision variables?
  • Does the firm observe \(\omega\) before its decision?

Endogeneity/omitted variable bias

• The firm does not observe \(ε\) before K and L decisions
• The firm observes \(\omega\) before L and after K decisions
  • Assume competitive markets
• \(\pi = Pe^{\beta_0}\bar K^{\beta_K}L^{\beta_L}e^{\omega} - wL - rK\)
  • \(L = [(p/w) \beta_Le^{\beta_0 + \omega}\bar K^{\beta_K}]^{1/(1-\beta_L)}\)
  • \(L\) and \(\omega\) are positively correlated
    • Omitted variable bias: \(\beta_L\) will be over estimated

Endogeneity/omitted variable bias

How to solve the endogeneity bias?

• Fixed effects, if assume \(\omega_{it} = \omega_i\)
  • If there are measurement errors, fixed-effect estimator can generate higher biases than OLS
  • Fixed effects may explain most the of the variation in output, especially if the panel is short
    • Returns to scale parameter tend to be low
• First-order conditions for inputs
  • Needs additional assumptions on the structure of factor markets and firm behavior
• Instrumental variables
  • Find instruments that are correlated with \(L\) but not \(\omega\)
    • Input prices? Usually low or no cross-sectional variation
    • Lagged inputs? If \(\omega\) is auto-correlated…

Endogeneity/omitted variable bias

• Proxy variables to control for unobservables (control function approach)
  • Olley & Pakes - use investment, define \(i = f(k, \omega)\), then \(\omega = f^{-1}(k, i)\)
    • What about L and prices?
  • Levinsohn & Petrin - use intermediate inputs as proxy
  • Ackerberg, Caves & Frazer - addresses the identification issues in the first stage
  • Wooldridge - one step GMM estimation
• Dynamic panel approach
  • Blundell & Bond
• GMM and measurement errors (revenue based markup estimation)
  • De Ridder, Grassi and Morzenti
    • If there is a measurement error, variance is high, productivity if measured as \(\omega + \eta\), and estimates are not consistent if \(\omega\) follows a non-linear AR process.
    • First purge the measurement error (purging equation includes fixed effects, price and controls for the mark up like the firm’s market share)

Blundell & Bond (2000)

\(q_{it} = \beta_Ll_{it} + \beta_Kk_{it} + \gamma_t + \eta_i + v_{it} + m_{it}\)

\(v_{it} = \rho v_{it-1} + e_{it}\)

\(e_{it} \ sim N(0, \sigma_e), m_{it} \sim N(0, \sigma_m)\)

Both employment (l)and capital (k) are potentially correlated with the firm-specific effects (\(\eta\)), and with both productivity shocks (e)and measurement errors (m).

\(q_{it} = \rho q_{it-1} + \beta_Ll_{it} - \rho \beta_Ll_{it-1} + \beta_Kk_{it} - \rho \beta_Kk_{it-1} + \gamma_t - \rho \gamma_{t-1} + \eta_i (1 - \rho) + e_{it} + m_{it} - \rho m_{it-1}\)

\(q_{it} = \pi_1 q_{it-1} + \pi_2l_{it} - \pi_3l_{it-1} + \pi_4k_{it} - \pi_5k_{it-1} + \gamma_t^* + \eta_i^* + \omega_{it}\)

Blundell & Bond (2000)

Take the first difference to eliminate firm specific effects

\(\Delta q_{it} = \pi_1 \Delta q_{it-1} + \pi_2 \Delta l_{it} - \pi_3 \Delta l_{it-1} + \pi_4 \Delta k_{it} - \pi_5 \Delta k_{it-1} + \Delta \gamma_t^* + \Delta \omega_{it}\)

Moment conditions

\(E[x_{it-s} \Delta \omega_{it}] = 0\)

where \(x_{it} = (l_{it}, k_{it}, q_{it})\) for \(s \ge 3\)

• GMM estimation: Use suitably lagged levels as instruments
  • First-differenced GMM estimator have poor finite sample properties (bias and imprecision)
• System-GMM estimation: Add level equation
  • \(E[\Delta_{i,t-s}(\eta_i^* + \omega_{it}] = 0\) for \(s = 2\)

De Ridder, Grassi and Morzenti (2025)

Case 1. Quantity data, no persistence in productivity

\(q_{it} = \alpha v_{it} + \omega_{it}\)

The firms sets v after observing \(\omega\)

GMM estimator: \(E[\omega_{it}v_{it-1}] = 0\)
\(v_{it-1}\) is a valid instrument if \(E[v_{it} v_{it-1}] ≠ 0\)

Case 2. Measurement error

\(\tilde q_{it} = \alpha v_{it} + \omega_{it} + \eta_{it}\)

GMM is still consistent but has a higher variance if \(E[\eta_{it}v_{it-1}] = 0\)

De Ridder, Grassi and Morzenti (2025)

Case 3. Revenue as a proxy for output output

\(r_{it} = q_{it} + p_{it}\)

GMM is not consistent if \(E[p_{it}v_{it-1}] ≠ 0\)

Under imperfect competition, they are likely to be correlated

Case 4. Persistent productivity

\(q_{it} = \alpha v_{it} + \omega_{it}\)
\(\omega_{it} % |rho \omega_{it-1} + ε_{it}\)

GMM estimator: \(E[e_{it}v_{it-1}] = 0\) and \(E[e_{it}\omega_{it-1}] = 0]\)
\(\omega_{it} = q_{it} - \alpha v_{it}\) and \(e_{it} = q_{it} - \alpha c_{it} - \rho \omega_{it-1}\)

If there is a persistence in productivity, there may be two solutions, one is unbiased, the other one is biased!

Selection

• If less efficient firms exit (low \(\omega\) value), and if capital intensive firms stay in the market even for low \(\omega\), then, conditional on being in the market, there is a (negative) correlation between K and \(\omega\)
• Solution
  • Use balanced panel data
  • Two stage estimation
  • Use a method that takes into account the selection process (like OP)
  • If selection is based on fixed effects (\(\eta_i\)), panel GMM is fine

Heterogeneity

Do all firms use the same technology?

• Differences in products, processes, patented technology, etc.
  • \(q_{it} = \beta_{0,it} + \beta_{K,it}k_{it} + \beta_{L,it}l_{it} + \omega_{it}\)
• Putty-clay technology
  • \(q_{it} = \beta_{0,i} + \beta_{K,i}k_{it} + \beta_{L,i}l_{it} + \omega_{i}\)

Good practice

• Check returns to scale parameter, is it around 0.9-1.1?
• Check output elasticities
• Check the sensitivity of your assumptions
  • Make simulations
• Check different models

Good practice

Food and Beverages Industry in Belgium

Source: Ilke Van Beveren (2010), “Total Factor Productivity Estimation: A Practical Review”, Journal of Economic Surveys, 1-38.

Good practice

Food and Beverages Industry in Belgium

Source: Ilke Van Beveren (2010), “Total Factor Productivity Estimation: A Practical Review”, Journal of Economic Surveys, 1-38.