## Introduction#

This note goes over the Random Coefficient Logit model, often referred to as the “BLP” model, proposed by Berry et al. (1995). This model offers a method for estimating demand that considers the varying demand responses to changes in price while addressing issues related to price endogeneity.

## Outline#

1. Begin by establishing a indirect utility function for consumer i consuming product j in market m.
2. Transform the indirect utiltiy function into a composite of mean utility, which applies uniformly to all consumers using product j in market m, and individual-specific random components that capture consumer tastes (preferences).
3. Assuming that consumers are utility maximizers and that errors entering the indirect utility function follows a Type I Extreme Value distribution, the market share of product j in market m is simply the individualistic probability that product j yields higher utility than any other products in market m, integrated across consumer types weighted by their probabilities in the population.
4. However, solving the model analytically becomes challenging due to the integration over nonlinear parameters. BLP addresses this issue by a contraction mapping.
5. In the contraction mapping, you start with a guesses for taste parameters and mean utility. From there, you compute the predicted market share by using (3).
6. Update teh mean utility by adding the differences between the observed market share and the predicted market share to its previous value.
7. Iterate this process until the observed and predicted market shares closely match, indicating convergence to the estimated mean utility.
8. Note that mean utility estimate is composed of average preferences and unobserved product shocks. Use 2SLS to estimate the average preferences.
9. Subtract the average preferences from the mean utility to isolate the unobserved product shock component. This component is used to construct sample moments.
10. Use the sample moments and the GMM estimator (or more specifically, its weighing matrix, in which it assigns lower weights to moments with high variance) to search over random taste parameters that most closely satisfy moment conditions.

### Indirect Utility function#

Note that $$i, j, m$$ represent the consumer, product, market dimensions respectively.

Let individual taste parameters consist of average preferences and individual taste observed by consumer specific characteristics:
$$\beta_i = \bar{\beta} + \eta_i,$$ where $$\bar{\beta}$$ is `average preferences in the population’, $$\eta_i$$ is individual taste parameters drawn from some known distribution with variance $$\Sigma$$ multiplied by observable consumer characteristics.

Let indirect utility function defined by \begin{aligned} u_{ijm} &= x_{jm}^\prime(\bar{\beta} + \eta_i) + \xi_{jm} + \varepsilon_{ijm} \newline &= x_{jm}^\prime\bar{\beta} +x_{jm}^\prime \eta_i+ \xi_{jm} + \varepsilon_{ijm} \newline &= x_{jm}^\prime\bar{\beta} + \xi_{jm} +x_{jm}^\prime \eta_i+ \varepsilon_{ijm} \newline &= \delta_{jt}(\bar{\beta}, \xi_{jm}) + x_{jm}^\prime \eta_i+ \varepsilon_{ijm}, \end{aligned}

where $$\delta_{jm}(\bar{\beta}, \xi_{jm})$$ represents mean-utiltiy for product j in market m, $$x_{jm}$$ is observable product characteristics of product j in market m, $$\xi_{jm}$$ is unobservable product characteristics, $$\eta_i$$ is individual taste parameters drawn from some known distribution with variance $$\Sigma$$ multiplied by observable consumer characteristics, and lastly $$\varepsilon_{ijm}$$ is error following Type-1 Extreme Value distribution.

### Unobservables#

• $$\xi_{jm}$$: unobservable product characteristics of product j in market m
• $$\varepsilon_{ijt}$$: all other unobserables affecting consumer i’s utility

### Observables#

• $$x_{jt}$$: observable product characteristics of product j in market m (including price information)
• $$\eta_i$$: observable consumer characteristics (from guess)
• market shares (from data)

### Guess#

• $$\Sigma$$: variances of the distribution of consumer tastes
• $$\delta$$: mean-utility

### Parameters to estimate#

• $$\delta$$: mean-utilty
• $$\Sigma$$: consumer taste parameters
• $$\beta$$: consumer’s preferece for product characteristics

• $$\delta$$: mean-utilty
• $$\Sigma$$: consumer taste parameters

** These parameters are estimated by a contraction mapping and GMM, respectively.

### Subsequently computed#

• $$\bar{\beta}$$: average preferences, estimated by regressing estimated $$\delta$$ on product observable characteristics using 2SLS – (a)
• $$\xi_{jm}$$: unobservable product characteristics of product j in market m,obtained by calculating residuals from (a)
• $$\beta$$: consumer’s preferece for product characteristics

## Instruments#

1. Cost shifters: transportation cost, manufacturing costs, taxes/tariffs, labor costs, material costs
2. Hausman instruments: price of same good in other markets
3. BLP instruments: For product j that are made by firm k, the BLP instruments include
• product j’s characteristics made by firm k (its own product characteristics),
• the sum of all other products’ characteristics that are NOT product j made by firm k (characteristics of its other products),
• the sum of all other products’ characteristics that are NOT product j and are NOT made by firm k. (characteristics of its rivals’ all other products)