Notes
These are my personal notes on my understanding of demand system and its estimation (especially focusing on logit-like model) in IO. Note that this is a living document and will be updated as I learn more things.
What is a structural model?
This section is not for rigorously defining the meaning of structural model. It is more about listing factors that I think are useful to think about in the back of your head when dealing with structural model. This is important because lot of what we do in modeling demand system in IO is about specifying a structural model.
- At the end of the day, the goal of the structural model is to specify a system of economic environment with features that the researcher believe is important for understanding certain human behavior or outcome.
- For example, why does some people choose insurance A while others choose insurance B (which is an outcome observed in the data). Depending on the empirical context, it could be about unobserved product quality or it could be about some search friction.
- Data (observed variation) is what we use to pin down our features (e.g. “deep” parameters) of our interest.
- Having a probablistic model (and thinking about unobservable) is important for two reasons:
- It is realistic. Econometrician cannot observe everything.
- We cannot fit the data if we do not have them. Data has weird variation that we cannot explain and we need to assume unobservable shock to justify it.
- In the end, it’s all about finding the identifying variation for your parameters of interest (e.g. If you have a utility function with price variable and unobserved product quality, you cannot identify the coefficient on the price variable unless you have some temporal variation in price).
What is IO?
This section is not for formally defining the field of IO. I am just writng down some things IO research seem to focus on.
IO is about understanding market structure and how it affects the equilibrium outcomes such as price, quantity, etc. By market structure, we mean mainly the various features of the supply-side of the market (the firm). IO tries to understand the characteristics of the market structure and how it affects welfare (consumer surplus, profit) in the equilibrium.
Old IO used to analyze empirical association of market structures across industries. But this was problematic because market structure is endogenous. This lead to the period of game theory. Game theory was useful because it provided useful tools to characterize the strategic interactions of individuals that make up the market.
New empirical IO seems to focus their research within some specific industry and also employ economic theory and econometric methods to fully characterize the market of their interests. During this process, they developed many useful tools (demand estimation, production function estimation, dynamic model, etc) that are also helpful to non-IO researchers.
Why demand system?
So why care about demand system? Understanding demand system itself is also crucial and helpful. But demand can tell us more than just demand per se.
In IO, people are interested about market structure and competition. One quantitative measure that is useful in this respect is markups. This is because markups tell us about the market power the firm has in the market. In the simple perfect competition, firms will set price equal to marginal cost. But many markets are not perfectly competitive. In this case, measuring the markups will give us some sense of the competitive nature of the market.
The problem is we cannot directly observe markup. Markup is basically \(\frac{p - mc}{mc}\). From this expression you can clearly see the issue: we don’t know the marginal cost. We usually do not have data on cost for producing a marginal product (This is different from accounting cost. Also, accounting cost usually is not sufficient to tell us about the marginal cost).
So where does estimating demand fit in this context? Surprisingly, estimating demand can help us recover marginal cost and thus markup.
Differentiated product Bertrand
Statistical model of product differentiation: discrete choice model
btw, Why do we start with indirect utility in Discrete choice model?
Actually, this can be derived from the usual utility function. Suppose consumer maximizes:
\[ \max_{q_1, q_2, c} U(q_1, q_2, c) \]
subject to
\[\begin{align} p_1 q_1 + p_2 q_2 + c = m \\ q_1 q_2 = 0 \end{align}\]
\(c\) is the numeraire good with its price normalized to 1. To make this more simple, we will assume that quantity has to be either 1 or 0.
Then we can get conditional indirect utility function which is (we condition on \(q_1 = 0\))
\[ \max_{q_2, c} U(0, q_2, c) \]
subject to
\[ p_2 q_2 + c = m \]
There is a standard solution for this demand functions \(q_2(p_2, m)\) and \(c(p_2, m)\). So plug this into the demand function and we get the indirect utility function:
\[ V_2(p_2, m) = U(0, q_2(p_2, m), c(p_2, m)) = U(0, 1, m-p_2) \]
where the last equation holds because we are only considering that consumer can buy at most one good for \(q_1, q_2\).
Doing the same conditioning on \(q_2 = 0\), we can see that the solution to the discrete choice part of the decision problem is then given by a choice between the wo conditional utility functions:
\[ \max_{j=\{1,2 \}} V_j (p_j, m) \]
This is the discrete choice model we usually see. We then usually add some random shocks to accomodate the changes in the people’s choice in the real-world data.