Question

For this exercise, use the data in AIRFARE, but only for the year \(1997 .\) (i) A simple demand function for airline seats on routes in the United States is \(\log (\)passen\()=\beta_{10}+\alpha_{1} \log (\) fare \()+\beta_{11} \log (d i s t)+\beta_{12}[\log (d i s t)]^{2}+u_{1}\) where $$\begin{aligned} \text {passen} &=\text { average passengers per day, } \\\ \text {fare} &=\text { average airfare, and } \\ d i s t &=\text { the route distance (in miles) } \end{aligned}$$ If this is truly a demand function, what should be the sign of \(\alpha_{1} ?\) (ii) Estimate the equation from part (i) by OLS. What is the estimated price elasticity? (iii) Consider the variable concen, which is a measure of market concentration. (Specifically, it is the share of business accounted for by the largest carrier.) Explain in words what we must assume to treat concen as exogenous in the demand equation. (iv) Now assume concen is exogenous to the demand equation. Estimate the reduced form for log(fare) and confirm that concen has a positive (partial) effect on log(fare). (v) Estimate the demand function using IV. Now whate? How does it compare with the OLS estimate? (vi) Using the IV estimates, describe how demand for seats depends on route distance.

Short answer

(i) \(\alpha_{1}\) should be negative. (ii) Estimate the equation to find price elasticity. (iii) Assume \(concen\) is uncorrelated with error term. (iv) Reduced form shows positive effect of \(concen\) on \(\log(fare)\). (v) Use IV to compare with OLS. (vi) IV shows distance's effect on demand.

Step-by-step solution

Understanding the Expected Sign of \(\alpha_{1}\)

For a demand function where the dependent variable is the logarithm of quantity demanded and the independent variable is the logarithm of price, a basic economic principle suggests that the coefficient of the price variable should be negative. This is because as the price increases, the quantity demanded typically decreases, indicating a negative relationship.

Estimating Using OLS

To estimate the demand equation given in the question using Ordinary Least Squares (OLS), first collect data on average passengers per day, average airfare, and route distance for the year 1997. Use these variables to fit the log-linear regression model provided in the question. The estimated coefficient of \(\log(fare)\) will give us the price elasticity of demand.

Analyzing Exogeneity of \(concen\)

To treat \(concen\) (market concentration measure) as exogenous in the demand equation, we must assume it is not correlated with the error term \(u_{1}\). This implies that any variations in \(concen\) are not due to changes in unobserved factors that affect demand for airline seats.

Estimating Reduced Form for \(\log(fare)\)

Assume \(concen\) is exogenous and estimate the reduced form equation for \(\log(fare)\). This involves regressing \(\log(fare)\) on \(concen\) and other exogenous variables, if any. The goal is to verify that \(concen\) has a statistically significant positive impact on \(\log(fare)\) when controlling for other factors.

Estimating Using IV (Instrumental Variables)

To estimate the demand function using IV, use \(concen\) as an instrument for \(\log(fare)\). This involves two-stage least squares (2SLS): First, regress \(\log(fare)\) on \(concen\) to get the predicted values, then regress \(\log(passen)\) on these predicted values. Compare the coefficient of \(\log(fare)\) from the IV estimation to the OLS estimate to assess any differences.

Analyzing Route Distance Effect with IV Estimates

Using the IV estimates, look at the coefficients of \(\log(dist)\) and \([\log(dist)]^2\) to determine the relationship between route distance and airline seat demand. Check if the signs and magnitudes of these coefficients change as compared to OLS estimates, indicating how demand varies with changes in route distance.

Key concepts

Demand Function

A demand function is a mathematical equation that describes how the quantity demanded of a good or service relates to various factors, like its price, income levels, or other influential variables. In this exercise, the demand function is specified in a log-linear form, which means that we express the relationship using logarithms of the variables. This approach is particularly helpful as it allows us to interpret the coefficients as elasticities, simplifying the understanding of percentage changes.
The demand function being analyzed involves variables such as average passengers per day, average airfare, and route distance. These are common variables considered for airline seat demand, capturing key demand drivers. Importantly, the coefficient of the log of airfare, \(\alpha_1\), represents the price elasticity of demand. In principle, for a typical demand function, this coefficient should be negative, indicating that an increase in price leads to a decrease in quantity demanded, all else being equal.

Ordinary Least Squares (OLS)

Ordinary Least Squares (OLS) is a method used widely in econometrics to estimate the parameters of a linear regression model. The fundamental goal of OLS is to minimize the sum of the squared differences between the observed and predicted values. This results in a line of best fit through the data points.
In this exercise, OLS is used to estimate the demand equation provided. With available data on passengers, fare, and distance, OLS allows for estimation of how changes in fare influence airline seat demand. A key output from the OLS estimation in our context is the coefficient on \(\log(\text{fare})\), which directly gives us the price elasticity of demand.

• OLS assumes that the relationship between dependent and independent variables is linear in the parameters.
• It also assumes that the error term, \(u_1\), is normally distributed and uncorrelated with the independent variables.

Instrumental Variables (IV)

Instrumental Variables (IV) estimation is an econometric technique used when an explanatory variable is correlated with the error term, leading to potential bias in an OLS estimate. The IV approach helps address this issue by using instruments - variables that are correlated with the endogenous explanatory variables but uncorrelated with the error term.
In the given problem, the variable \(\text{concen}\) - representing market concentration - is used as an instrument for \(\log(\text{fare})\). Concerns about endogeneity arise if airfare and unobservable factors affecting demand are related. By employing IV, we can obtain unbiased and consistent estimates of the demand function.

• The IV process involves first estimating a reduced form equation to predict the problematic variable using the instrument.
• Next, this predicted value replaces the endogenous variable in the primary equation, allowing for accurate estimation of the coefficients.

Price Elasticity of Demand

Price elasticity of demand measures how the quantity demanded of a good responds to changes in its price. Specifically, it is the percentage change in quantity demanded resulting from a one percent change in price. This concept is crucial in understanding consumer behavior and how sensitive demand is to price variations.
In the context of airline seats, the price elasticity enables us to quantify how sensitive passengers are to changes in airfare. The coefficient \(\alpha_1\) in the demand function directly provides this elasticity when using a log-linear specification. Typically, we expect this elasticity to be negative, signifying that higher prices lead to lower demand for airline seats.

• An elasticity greater than 1 in absolute value signifies elastic demand.
• If it is less than 1 in absolute value, demand is considered inelastic.

Understanding whether demand is elastic or inelastic helps airlines in setting prices to optimize revenue.