The Lerner Index is a measure of market power that can be calculated as the difference between price and marginal cost, divided by price:
$$L = (P - MC) / P$$
To calculate the Lerner Index, we need to estimate the marginal cost of production, which is the cost of producing an additional unit of output.
To compute the marginal cost using the Cobb-Douglas equation in the form you provided, we first need to obtain the expression for the total cost function ($TC$) from the equation. Taking the exponential of both sides of the equation, we have:
$$TC = e^{c_0} P_L^\alpha P_k^{1-\alpha} \text{TA}^\beta e^\epsilon$$
where $e$ is the mathematical constant $e \approx 2.718$.
To obtain the marginal cost, we need to take the partial derivative of the total cost function with respect to the quantity of output ($Q$), assuming that the firm is a price-taker in the market, and thus faces a constant market price ($P$).
The expression for marginal cost given by:
$$MC = \frac{\partial TC}{\partial Q}$$
is a general formula that applies to any production function. It tells us that the marginal cost is the derivative of total cost with respect to the quantity of output.
However, in the specific case of the Cobb-Douglas production function that we are considering, we can substitute the expression for total cost in terms of $Q$, $P_L$, $P_k$, and $\text{TA}$, which is:
$$TC = e^{c_0} P_L^\alpha P_k^{1-\alpha} \text{TA}^\beta e^\epsilon$$
and then use the chain rule of differentiation to find the partial derivatives of total cost with respect to the inputs $P_L$, $P_k$, and $\text{TA}$:
$$\frac{\partial TC}{\partial P_L} = \alpha e^{c_0} P_L^{\alpha - 1} P_k^{1 - \alpha} \text{TA}^\beta e^\epsilon$$
$$\frac{\partial TC}{\partial P_k} = (1-\alpha) e^{c_0} P_L^{\alpha} P_k^{-\alpha} \text{TA}^\beta e^\epsilon$$
$$\frac{\partial TC}{\partial \text{TA}} = \beta e^{c_0} P_L^{\alpha} P_k^{1-\alpha} \text{TA}^{\beta - 1} e^\epsilon$$
Substituting these partial derivatives into the expression for marginal cost, we obtain:
$$MC = \frac{\partial TC}{\partial Q} = \frac{\partial TC}{\partial P_L} \frac{\partial P_L}{\partial Q} + \frac{\partial TC}{\partial P_k} \frac{\partial P_k}{\partial Q} + \frac{\partial TC}{\partial \text{TA}} \frac{\partial \text{TA}}{\partial Q}$$
Since we are assuming that the firm is a price-taker and faces a constant market price $P$, we can use the production function to express $Q$ in terms of the input prices and technology as:
$$Q = \left(\frac{P}{e^{c_0}}\right)^{1/(1-\alpha-\beta)} \frac{P_L^\alpha}{\text{TA}^\beta} \frac{P_k^{1-\alpha}}{\text{TA}}$$
We can then use this expression to obtain the partial derivatives with respect to $Q$:
$$\frac{\partial P_L}{\partial Q} = \frac{\alpha}{Q P_L}$$
$$\frac{\partial P_k}{\partial Q} = \frac{1-\alpha}{Q P_k}$$
$$\frac{\partial \text{TA}}{\partial Q} = -\frac{\beta}{Q \text{TA}}$$
Substituting the expressions we obtained earlier for $\frac{\partial P_L}{\partial Q}$, $\frac{\partial P_k}{\partial Q}$, and $\frac{\partial \text{TA}}{\partial Q}$ into the expression for marginal cost, we get:
$$MC = \frac{\partial TC}{\partial Q} = \frac{\alpha}{P_L} TC + \frac{1-\alpha}{P_k} TC + \frac{\beta}{\text{TA}} TC$$
where we have replaced the partial derivatives with the expressions we derived earlier.
Simplifying, we get:
$$MC = TC \left[\frac{\alpha}{P_L}\left(\frac{P_k}{P}\right)^{1-\alpha} + \frac{1-\alpha}{P_k}\left(\frac{P_L}{P}\right)^{\alpha} + \beta \frac{1}{\text{TA}}\right]$$
where we have used the fact that the market price is equal to the firm's average revenue ($P = TR/Q = TC/Q$), and we have multiplied and divided the first two terms by $P$ to obtain expressions that involve only the relative prices of labor and capital.
This expression shows that the firm's marginal cost depends on the relative prices of labor and capital, the share of each input in the production function, and the level of technology. The term in brackets represents the inverse elasticity of the input mix with respect to the relative price of labor, which measures the sensitivity of the firm's input mix to changes in the relative prices of labor and capital.
some code to simulate a luxury winery industry (we get about 0.6 lerner index)
import numpy as np
import pandas as pd
import statsmodels.api as sm
# Define the number of observations
n = 1000
# Define the prices and total assets
price_labor = np.random.uniform(low=20, high=30, size=n)
price_capital = np.random.uniform(low=10, high=20, size=n)
price_product = np.random.uniform(low=30, high=50, size=n)
total_assets = np.random.uniform(low=1000, high=5000, size=n)
# Define the coefficients for the Cobb-Douglas production function
alpha = 0.4
beta = 0.3
gamma = 0.3
# Calculate the total cost using the Cobb-Douglas production function
total_cost = (
gamma
* (price_labor**alpha)
* (price_capital ** (1 - alpha))
* (total_assets**beta)
)
# Calculate the marginal cost using the expression you provided
marginal_cost = total_cost * (
(alpha / price_labor) * ((price_capital / price_product) ** (1 - alpha))
+ ((1 - alpha) / price_capital) * ((price_labor / price_product) ** alpha)
+ beta * (1 / total_assets)
)
# Calculate the Lerner index using the Cobb-Douglas regression model
X = sm.add_constant(
np.column_stack((np.log(price_labor), np.log(price_capital), np.log(total_assets)))
)
y = np.log(total_cost)
model = sm.OLS(y, X)
result = model.fit()
lerner_index = np.mean((price_product - marginal_cost) / price_product)
print(f"The Lerner index is {lerner_index:.2f}")
