Equations and Variable Definitions from Coal Mining Study
Main Equations
Equation (1) - Cobb-Douglas Production Function
\[ q_{ft} = \beta_{l} l_{ft} + \beta_{m} m_{ft} + \beta_{k} k_{ft} + \omega_{ft} \tag{1} \]
Equation (2) - TFP Transition Process
\[ \omega_{ft} = \rho \omega_{ft-1} + u_{ft} \tag{2} \]
Labor Supply Function - Equation (3)
\[ W^{l}_{it} = L_{it}^{\Psi^{l}} \eta_{it} \tag{3} \]
Cost Minimization Objective - Equation (4)
\[ \min_{L_{ft}, M_{ft}} \left\{ \sum_{g \in F_{i(f)t}} \left( \lambda_{fgt} (L_{gt} W^{l}_{gt} + M_{gt} W^{m}_{gt}) \right) - MC_{ft} \left( Q(L_{ft}, M_{ft}, K_{ft}, \Omega_{ft}; \beta) - Q_{ft} \right) \right\} \tag{4} \]
No Collusion Cost Minimization - Equation (5)
\[ \min_{L_{ft}, M_{ft}} \left( L_{ft} W^{l}_{ft} + M_{ft} W^{m}_{ft} \right) - MC_{ft} \left( Q_{ft} - Q(L_{ft}, M_{ft}, K_{ft}, \Omega_{ft}; \beta) \right) \tag{5} \]
First-Order Condition for Labor (No Collusion) - Equation (6)
\[ L_{ft} \frac{\partial W^{l}_{it}}{\partial L_{it}} + W^{l}_{it} = \frac{\partial Q_{ft}}{\partial L_{ft}} \frac{P_{ft}}{\mu_{ft}} \tag{6} \]
Markdown Under No Collusion - Equation (7)
\[ \widetilde{m}^{l}_{ft} = 1 + s^{l}_{ft} \Psi^{l} \tag{7} \]
Perfect Collusion Cost Minimization - Equation (8)
\[ \min_{L_{ft}, M_{ft}} \left\{ \sum_{g \in F_{i(f)t}} \left( L_{gt} W^{l}_{gt} + M_{gt} W^{m}_{gt} \right) - MC_{ft} \left( Q_{ft} - Q(L_{ft}, M_{ft}, K_{ft}, \Omega_{ft}; \beta) \right) \right\} \tag{8} \]
First-Order Condition for Labor (Perfect Collusion) - Equation (9)
\[ L_{it} \frac{\partial W^{l}_{it}}{\partial L_{it}} + W^{l}_{it} = \frac{\partial Q_{ft}}{\partial L_{ft}} \frac{P_{ft}}{\mu_{ft}} \tag{9} \]
Markdown Under Perfect Collusion - Equation (10)
\[ \overline{m}^{l}_{ft} = 1 + \Psi^{l} \tag{10} \]
General First-Order Condition - Equation (11)
\[ W^{l}_{it} + \widetilde{\lambda}_{ft} \frac{\partial W^{l}_{it}}{\partial L_{it}} L_{it} = \frac{\partial Q_{ft}}{\partial L_{ft}} \frac{P_{ft}}{\mu_{ft}} \tag{11} \]
General Markdown Expression - Equation (12)
\[ m^{l}_{ft} = 1 + \widetilde{\lambda}_{ft} \Psi^{l} \tag{12} \]
Markup from Materials - Equation (13)
\[ \mu_{ft} = \frac{\beta_{m}}{\alpha^{m}_{ft}} \tag{13} \]
Core Identification Equation - Equation (14)
\[ m^{l}_{ft} = 1 + \widetilde{\lambda}_{ft} \Psi^{l} = \frac{\beta_{l} \alpha^{m}_{ft}}{\beta_{m} \alpha^{l}_{ft}} \tag{14} \]
Conduct Parameter Definition - Equation (15)
\[ \widehat{\lambda}_{ft} = \frac{m^{l}_{ft} - \widetilde{m}^{l}_{ft}}{\overline{m}^{l}_{ft} - \widetilde{m}^{l}_{ft}} \tag{15} \]
Moment Conditions for Production Function - Equation (16)
\[ E[u_{ft} | (l_{fr-1}, m_{fr-1}, k_{fr}, w^{agri}_{r-1})_{r \in \{2,...,t\}}] = 0 \tag{16} \]
GMM Moment Conditions - Equation (17)
\[ E[q_{ft} - \rho q_{ft-1} - \beta_{0}(1-\rho) - \beta_{l}(l_{ft} - \rho l_{ft-1}) - \beta_{m}(m_{ft} - \rho m_{ft-1}) - \beta_{k}(k_{ft} - \rho k_{ft-1}) | (l_{ft-1}, m_{ft-1}, k_{ft}, k_{ft-1}, w^{agri}_{t-1})] = 0 \tag{17} \]
Additional Key Definitions
Markdown Definition
\[ m^{l}_{ft} \equiv \frac{MRPL_{ft}}{W^{l}_{ft}} \]
Percentage Markdown Definition
\[ d^{l}_{ft} \equiv \frac{MRPL_{ft} - W^{l}_{ft}}{MRPL_{ft}} = \frac{m^{l}_{ft} - 1}{m^{l}_{ft}} \]
Markup Definition
\[ \mu_{ft} \equiv \frac{P_{ft}}{MC_{ft}} \]
Inverse Labor Supply Elasticity (Firm-level)
\[ \theta^{l}_{ft} \equiv \frac{\partial W^{l}_{ft}}{\partial L_{ft}} \frac{L_{ft}}{W^{l}_{ft}} \]
Inverse Materials Supply Elasticity (Firm-level)
\[ \theta^{m}_{ft} \equiv \frac{\partial W^{m}_{ft}}{\partial M_{ft}} \frac{M_{ft}}{W^{m}_{ft}} \]
Variable Definitions
| Variable | Definition |
|---|---|
| \(Q_{ft}\) | Output (tonnage of coal) extracted by firm \(f\) in year \(t\) |
| \(L_{ft}\) | Amount of effective labor throughout the year for firm \(f\) in year \(t\) |
| \(M_{ft}\) | Amount of intermediate inputs purchased by firm \(f\) in year \(t\) |
| \(K_{ft}\) | Capital stock (steam engines) used by firm \(f\) in year \(t\) |
| \(q_{ft}\) | Logarithm of output \(Q_{ft}\) |
| \(l_{ft}\) | Logarithm of labor \(L_{ft}\) |
| \(m_{ft}\) | Logarithm of materials \(M_{ft}\) |
| \(k_{ft}\) | Logarithm of capital \(K_{ft}\) |
| \(\beta_{l}\) | Output elasticity of labor |
| \(\beta_{m}\) | Output elasticity of materials |
| \(\beta_{k}\) | Output elasticity of capital |
| \(\omega_{ft}\) | Log total factor productivity |
| \(u_{ft}\) | Unexpected productivity shock |
| \(\rho\) | Serial correlation parameter in productivity process |
| \(W^{l}_{it}\) | Wage in labor market \(i\) in year \(t\) |
| \(W^{l}_{ft}\) | Wage for firm \(f\) in year \(t\) |
| \(W^{m}_{ft}\) | Price of materials for firm \(f\) in year \(t\) |
| \(L_{it}\) | Market-level employment in market \(i\) in year \(t\) |
| \(\Psi^{l}\) | Inverse market-level labor supply elasticity |
| \(\eta_{it}\) | Market-specific residual in labor supply |
| \(\lambda_{fgt}\) | Collusion weight that firm \(f\) puts on firm \(g\)’s costs |
| \(\widetilde{\lambda}_{ft}\) | Conduct parameter (firm-level aggregate of bilateral conduct parameters) |
| \(\widehat{\lambda}_{ft}\) | Normalized conduct parameter ranging from 0 to 1 |
| \(MC_{ft}\) | Marginal cost for firm \(f\) in year \(t\) |
| \(P_{ft}\) | Coal price for firm \(f\) in year \(t\) |
| \(MRPL_{ft}\) | Marginal revenue product of labor for firm \(f\) in year \(t\) |
| \(m^{l}_{ft}\) | Wage markdown (ratio of MRPL to wage) |
| \(\widetilde{m}^{l}_{ft}\) | Wage markdown under no collusion |
| \(\overline{m}^{l}_{ft}\) | Wage markdown under perfect collusion |
| \(d^{l}_{ft}\) | Percentage wage markdown |
| \(\mu_{ft}\) | Price markup (ratio of price to marginal cost) |
| \(s^{l}_{ft}\) | Labor market share of firm \(f\) |
| \(\alpha^{l}_{ft}\) | Revenue share of labor for firm \(f\) in year \(t\) |
| \(\alpha^{m}_{ft}\) | Revenue share of materials for firm \(f\) in year \(t\) |
| \(\theta^{l}_{ft}\) | Inverse firm-level labor supply elasticity |
| \(\theta^{m}_{ft}\) | Inverse firm-level materials supply elasticity |
| \(F_{i(f)t}\) | Set of firms in market \(i\) (where firm \(f\) operates) in year \(t\) |
| \(w^{agri}_{t-1}\) | Agricultural wages in Belgium in year \(t-1\) (instrument) |
Tab-Separated Variable Definitions
Variable Definition
Q_ft Output (tonnage of coal) extracted by firm f in year t
L_ft Amount of effective labor throughout the year for firm f in year t
M_ft Amount of intermediate inputs purchased by firm f in year t
K_ft Capital stock (steam engines) used by firm f in year t
q_ft Logarithm of output Q_ft
l_ft Logarithm of labor L_ft
m_ft Logarithm of materials M_ft
k_ft Logarithm of capital K_ft
beta_l Output elasticity of labor
beta_m Output elasticity of materials
beta_k Output elasticity of capital
omega_ft Log total factor productivity
u_ft Unexpected productivity shock
rho Serial correlation parameter in productivity process
W_l_it Wage in labor market i in year t
W_l_ft Wage for firm f in year t
W_m_ft Price of materials for firm f in year t
L_it Market-level employment in market i in year t
Psi_l Inverse market-level labor supply elasticity
eta_it Market-specific residual in labor supply
lambda_fgt Collusion weight that firm f puts on firm g's costs
lambda_tilde_ft Conduct parameter (firm-level aggregate of bilateral conduct parameters)
lambda_hat_ft Normalized conduct parameter ranging from 0 to 1
MC_ft Marginal cost for firm f in year t
P_ft Coal price for firm f in year t
MRPL_ft Marginal revenue product of labor for firm f in year t
m_l_ft Wage markdown (ratio of MRPL to wage)
m_l_tilde_ft Wage markdown under no collusion
m_l_bar_ft Wage markdown under perfect collusion
d_l_ft Percentage wage markdown
mu_ft Price markup (ratio of price to marginal cost)
s_l_ft Labor market share of firm f
alpha_l_ft Revenue share of labor for firm f in year t
alpha_m_ft Revenue share of materials for firm f in year t
theta_l_ft Inverse firm-level labor supply elasticity
theta_m_ft Inverse firm-level materials supply elasticity
F_i_f_t Set of firms in market i (where firm f operates) in year t
w_agri_t_minus_1 Agricultural wages in Belgium in year t-1 (instrument)Table 1
| log(Output) | log(Output) | log(Output) | ||||
|---|---|---|---|---|---|---|
| A. Production Function, log(Output) | ||||||
| log(Labor) bl | .794 | .699 | .661 | |||
| (.034) | (.327) | (.041) | ||||
| log(Materials) bm | .275 | .222 | .237 | |||
| (.028) | (.138) | (.080) | ||||
| log(Capital) bk | 2.008 | .153 | .102 | |||
| (.140) | (.075) | (.088) | ||||
| Serial correlation TFP r | .866 | .853 | ||||
| (.198) | (.157) | |||||
| Method | OLS | GMM | GMM | |||
| RTS | Free | Free | Fixed at 1.05 | |||
| R2 | .941 | .938 | .826 | |||
| Hansen J-test | 2.34 | 2.72 | ||||
| Hansen J-test p-value | .126 | .255 | ||||
| Number of firms | 166 | 159 | 159 | |||
| Observations | 4,480 | 4,005 | 4,005 | |||
| B. Markdowns and Markups | ||||||
| Median markdown | 1.541 | 1.680 | 1.486 | |||
| (.193) | (.450) | (.330) | ||||
| Average markdown | 1.676 | 1.828 | 1.616 | |||
| (.224) | (.491) | (.361) | ||||
| Median markup | .884 | .714 | .763 | |||
| (.112) | (.494) | (.287) | ||||
| Average markup | .946 | .764 | .816 | |||
| (.120) | (.535) | (.315) | ||||
| Method | OLS | GMM | GMM | |||
| RTS | Free | Free | Fixed at 1.05 | |||
| C. Labor Supply | ||||||
| log(Wage) | log(Wage) | |||||
| Est. | SE | Est. | SE | |||
| log(Employment) | .066 | .006 | 1.009 | .265 | ||
| Method | OLS | IV | ||||
| First-stage F-statistic | 462 | |||||
| Hansen J-test | 5.92 | |||||
| Hansen J-test p-value | .014 | |||||
| Observations | 1,990 | 1,990 | ||||
| Firm-level elasticity | 155.56 | 10.172 | ||||
Note.—Panels A and B are estimated at the firm-year level, and panel C is estimated at the market-year level. Standard errors (SEs) in panels A and B are block-bootstrapped with 200 iterations. Standard errors in panel C are estimated using the Driscoll and Kraay (1998) correction to allow for both cross-sectional (i.e., intratemporal) and intertemporal dependence, using the STATA command ivreg2,draay(2).
Derivation of (14)
Looking at equation (14) and the context provided, I can explain how the wage markdown is derived by combining the two markup estimates from the production approach.
Cost minimization
From the cost minimization problem, the first-order condition for labor (equation 11) is: \[W^l_{it} + \tilde{\lambda}_{ft} \cdot \frac{\partial W^l_{it}}{\partial L_{it}} \cdot L_{it} = \frac{\partial Q_{ft}}{\partial L_{ft}} \cdot \frac{P_{ft}}{\mu_{ft}}\]
Where the right side is the marginal revenue product of labor (MRPL).
Rearranging and using: - \(\beta_l\) = output elasticity of labor - \(\alpha^l_{ft} = \frac{W^l_{ft} \cdot L_{ft}}{P_{ft} \cdot Q_{ft}}\) = revenue share of labor
The markup derived from the labor first-order condition becomes: \[\mu_{ft} = \frac{\beta_l}{\alpha^l_{ft} \cdot (1 + \tilde{\lambda}_{ft} \cdot \Psi^l)}\]
From (11)
\[W^l_{it} + \tilde{\lambda}_{ft} \frac{\partial W^l_{it}}{\partial L_{it}} L_{it} = \frac{\partial Q_{ft}}{\partial L_{ft}} \frac{P_{ft}}{\mu_{ft}}\]
From the Cobb-Douglas production function: \(Q_{ft} = e^{\beta_l l_{ft} + \beta_m m_{ft} + \beta_k k_{ft} + \omega_{ft}}\)
Taking the derivative with respect to labor: \[\frac{\partial Q_{ft}}{\partial L_{ft}} = \frac{\partial Q_{ft}}{\partial l_{ft}} \cdot \frac{\partial l_{ft}}{\partial L_{ft}} = \beta_l \cdot Q_{ft} \cdot \frac{1}{L_{ft}} = \beta_l \frac{Q_{ft}}{L_{ft}}\]
Substitute this into the RHS of equation (11) \[\frac{\partial Q_{ft}}{\partial L_{ft}} \frac{P_{ft}}{\mu_{ft}} = \beta_l \frac{Q_{ft}}{L_{ft}} \frac{P_{ft}}{\mu_{ft}} = \beta_l \frac{P_{ft} Q_{ft}}{\mu_{ft} L_{ft}}\]
Multiply both sides by \(L_{ft}\) \[L_{ft} W^l_{it} + \tilde{\lambda}_{ft} \frac{\partial W^l_{it}}{\partial L_{it}} L_{it}^2 = \beta_l \frac{P_{ft} Q_{ft}}{\mu_{ft}}\]
From equation (3): \(W^l_{it} = L_{it}^{\Psi^l} \eta_{it}\)
Taking the derivative: \[\frac{\partial W^l_{it}}{\partial L_{it}} = \Psi^l L_{it}^{\Psi^l - 1} \eta_{it} = \Psi^l \frac{W^l_{it}}{L_{it}}\]
Substitute this back to the above \[L_{ft} W^l_{it} + \tilde{\lambda}_{ft} \Psi^l \frac{W^l_{it}}{L_{it}} L_{it}^2 = \beta_l \frac{P_{ft} Q_{ft}}{\mu_{ft}}\]
\[L_{ft} W^l_{it} + \tilde{\lambda}_{ft} \Psi^l W^l_{it} L_{it} = \beta_l \frac{P_{ft} Q_{ft}}{\mu_{ft}}\]
\[W^l_{it} L_{ft} (1 + \tilde{\lambda}_{ft} \Psi^l) = \beta_l \frac{P_{ft} Q_{ft}}{\mu_{ft}}\]
Markup \[\mu_{ft} = \frac{\beta_l P_{ft} Q_{ft}}{W^l_{it} L_{ft} (1 + \tilde{\lambda}_{ft} \Psi^l)}\]
Define the revenue share of labor: \(\alpha^l_{ft} = \frac{W^l_{ft} L_{ft}}{P_{ft} Q_{ft}}\), therefore: \(W^l_{ft} L_{ft} = \alpha^l_{ft} P_{ft} Q_{ft}\)
\[\mu_{ft} = \frac{\beta_l P_{ft} Q_{ft}}{\alpha^l_{ft} P_{ft} Q_{ft} (1 + \tilde{\lambda}_{ft} \Psi^l)} = \frac{\beta_l}{\alpha^l_{ft} (1 + \tilde{\lambda}_{ft} \Psi^l)}\]
This gives us the markup expression derived from the labor first-order condition, which incorporates the wage markdown term \((1 + \tilde{\lambda}_{ft} \Psi^l)\) in the denominator.
Markup from materials (following De Loecker & Warzynski 2012)
For materials, since firms are price-takers (competitive input market), the standard markup formula applies: \[\mu_{ft} = \frac{\beta_m}{\alpha^m_{ft}}\]
Where \(\alpha^m_{ft}\) is the revenue share of materials.
From the cost minimization problem, the first-order condition for materials is: \[W^m_{ft} = \frac{\partial Q_{ft}}{\partial M_{ft}} \frac{P_{ft}}{\mu_{ft}}\]
From the Cobb-Douglas production function: \(Q_{ft} = e^{\beta_l l_{ft} + \beta_m m_{ft} + \beta_k k_{ft} + \omega_{ft}}\)
Taking the derivative with respect to materials: \[\frac{\partial Q_{ft}}{\partial M_{ft}} = \frac{\partial Q_{ft}}{\partial m_{ft}} \cdot \frac{\partial m_{ft}}{\partial M_{ft}} = \beta_m \cdot Q_{ft} \cdot \frac{1}{M_{ft}} = \beta_m \frac{Q_{ft}}{M_{ft}}\]
Substitute this into the RHS
\[W^m_{ft} = \beta_m \frac{Q_{ft}}{M_{ft}} \frac{P_{ft}}{\mu_{ft}} = \beta_m \frac{P_{ft} Q_{ft}}{\mu_{ft} M_{ft}}\]
\[W^m_{ft} M_{ft} = \beta_m \frac{P_{ft} Q_{ft}}{\mu_{ft}}\]
Solve for the markup \[\mu_{ft} = \frac{\beta_m P_{ft} Q_{ft}}{W^m_{ft} M_{ft}}\]
Define the revenue share of materials: \(\alpha^m_{ft} = \frac{W^m_{ft} M_{ft}}{P_{ft} Q_{ft}}\)
Therefore: \(W^m_{ft} M_{ft} = \alpha^m_{ft} P_{ft} Q_{ft}\)
\[\mu_{ft} = \frac{\beta_m P_{ft} Q_{ft}}{\alpha^m_{ft} P_{ft} Q_{ft}} = \frac{\beta_m}{\alpha^m_{ft}}\]
This gives us the standard De Loecker & Warzynski (2012) markup formula: \[\mu_{ft} = \frac{\beta_m}{\alpha^m_{ft}}\]
Equate the two markup expressions
Since both expressions equal \(\mu_{ft}\): \[\frac{\beta_l}{\alpha^l_{ft} \cdot (1 + \tilde{\lambda}_{ft} \cdot \Psi^l)} = \frac{\beta_m}{\alpha^m_{ft}}\]
Rearranging: \[1 + \tilde{\lambda}_{ft} \cdot \Psi^l = \frac{\beta_l \cdot \alpha^m_{ft}}{\beta_m \cdot \alpha^l_{ft}}\]
From equation (12), we know that: \[m^l_{ft} = 1 + \tilde{\lambda}_{ft} \cdot \Psi^l\]
Therefore: \[m^l_{ft} = \frac{\beta_l \cdot \alpha^m_{ft}}{\beta_m \cdot \alpha^l_{ft}}\]
This is equation (14) in the paper.
The wage markdown \(m^l_{ft}\) can be calculated in two equivalent ways:
- Labor supply approach: \(1 + \tilde{\lambda}_{ft} \cdot \Psi^l\) (depends on conduct parameter and labor supply elasticity)
- Production approach: \(\frac{\beta_l \cdot \alpha^m_{ft}}{\beta_m \cdot \alpha^l_{ft}}\) (depends only on production function parameters and cost shares)