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

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