# Find Lerner Index from Cobb Douglas regression

I'm currently following a Master's in Finance, and we have to calculate the Lerner Index.

To do that, we are given the following Cobb- Douglas regression, where the alphas have to sum up to 1:

$$\ln(TC) = c_{0} + \alpha \ln(P_{L}) + (1-\alpha )\ln(P_{k})+\beta \ln(TA) + \varepsilon$$

For the Lerner Index, we need the marginal costs. According to our professor, we can calculate these costs with the $$\beta$$ of $$\ln(TA)$$. However, he said we have to convert this beta to marginal costs because if we used this beta, all companies would have the exact marginal costs. How can this be done?

• By alphas, do you mean $\alpha$ and $1-\alpha$, or are you also considering $\beta$? Also, can you provide more information on your model? What does each variable mean? Commented Feb 16, 2023 at 16:47
• TC are total costs, PL and PK are input prices for labour and capital and TA are total assets. The coefficients for both PL and PK have to sum up to 1 due to linear homogeneity in prices. That is why it says alpha and (1 - alpha).
– Lars
Commented Feb 18, 2023 at 9:40
• To estimate the Lerner Index, we need to first estimate the marginal cost of production. However, the β coefficient in the Cobb-Douglas regression does not directly provide us with the marginal cost. We need to make additional assumptions about the production function and cost structure to calculate the marginal cost. One common assumption is that the production function is a constant elasticity of substitution (CES) function, which is a generalization of the Cobb-Douglas function. In a CES production function, the elasticity of substitution between labor and capital can be different from one.
– 0x90
Commented Feb 18, 2023 at 12:09

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