Demand Estimation

Introduction

Setting

Oligopoly Supply

• firms produce differentiated goods/products

• selling to consumers with heterogeneous preferences

• static model, complete information

  • products are given

  • equilibrium: NE for each product/market

Cost Function

Variable cost of product $j$: $C_j (Q_j , w_{jt} , \mathbb \omega_{jt}, \gamma)$

• $Q_j$: total quantity of good $j$ sold

• $w_{jt}$ observable cost shifters; may include product characteristics $x_{jt}$ that will affect demand (later)

• $\omega_{jt}$ unobserved cost shifters (“cost shocks”); may be correlated with latent demand shocks (later)

• $\gamma$: parameters

Notes

• for multi-product firms, we’ll assume variable cost additive across products for simplicity
• we ignore fixed costs: these affect entry/exit/innovation but not pricing, conditional on these things

Notation

Some other variables

• $J_t$: products/goods/choices in market $t$ (for now $J_t = J$)
• $P_t = (p_{1t},…,p_{Jt})$: prices of all goods
• $\boldsymbol X_t = ( \boldsymbol x_{1t} , … , \boldsymbol x_{Jt})$ : other characteristics of goods affecting demand (observed and unobserved to us)

In general

• I use bold for arrays in dimensions that are not $i$ (consumers), $j$ (firms) or $t$ (markets)
  • For example product characteristics $\boldsymbol x_{jt} = \lbrace x_{jt}^1,, …, x_{jt}^K \rbrace$
• I use CAPS for variables aggregated over $j$ (firms)
  • For example vector of prices in market $t$: $P_t = (p_{1t},…,p_{Jt})$

Equilibrium Pricing

• Demand system:

  $$ q_{jt} = Q_j ( P_t, \boldsymbol X_t) \quad \text{for} \quad j = 1,…,J. $$

• Profit function

  $$ \pi_{jt} = Q_j (P_t, \boldsymbol X_t) \Big[p_{jt} − mc_j (w_{jt}, \omega_{jt}, \gamma) \Big] $$

• FOC wrt to $p_{jt}$:

  $$ p_{jt} = mc_{jt} - Q_j (P_t, \boldsymbol X_t) \left(\frac{\partial Q_j}{\partial p_{jt}}\right)^{-1} $$

• Inverse elasticity pricing (i.e., monopoly pricing) against the “residual demand curve” $Q_j (P_t, \boldsymbol X_t)$:

  $$ \frac{p_{jt} - mc_{jt}}{p_{jt}} = - \frac{Q_j (P_t, \boldsymbol X_t)}{p_{jt}} \left(\frac{\partial Q_j}{\partial p_{jt}}\right)^{-1} $$

What do we get?

1. Holding all else fixed, markups/prices depend on the own-price elasticities of residual demand. Equilibrium depends, further, on how a change in price of one good affects the quantities sold of others, i.e., on cross-price demand elasticities

2. If we known demand, we can also perform a small miracle:

   • Re-arrange FOC

     $$ mc_{jt} = p_{jt} + Q_j (P_t, \boldsymbol X_t)\left(\frac{\partial Q_j}{\partial p_{jt}}\right)^{-1} $$

   • Supply model + estimated demand $\to$ estimates of marginal costs!

3. If we know demand and marginal costs, we can”predict” a lot of stuff - i.e., give the quantitative implications of the model for counterfactual worlds

Issues

• Typically we need to know levels/elasticities of demand at particular points; i.e., effects of one price change holding all else fixed

• The main challenge: unobserved demand shifters (“demand shocks”) at the level of the good×market (e.g., unobserved product char or market-specific variation in mean tastes for products)

• demand shocks are among the things that must be held fixed to measure the relevant demand elasticities etc.

• explicit modeling of these demand shocks central in the applied IO literature following S. Berry, Levinsohn, and Pakes (1995) (often ignored outside this literature).

Key Challenge

The demand of product $j$

$$ q_{jt} (\boldsymbol X_{t}, P_t, \Xi_t) $$

depends on:

• $P_t$: $J$-vector of all goods’ prices in market $t$

• $\boldsymbol X_t$: $J \times k$ matrix of all non-price observables in market $t$

• $\Xi_t$: J-vector of demand shocks for all goods in market $t$

Key insight: we have an endogeneity problem even if prices were exogenous!

Price Endogeneity Adds to the Challenge

• all $J$ endogenous prices are on RHS of demand for each good

• equilibrium pricing implies that each price depends on all demand shocks and all cost shocks

  • prices endogenous

  • control function generally is not a valid solution

• clear that we need sources of exogenous price variation, but

  • what exactly is required?

  • how do we proceed?

BLP: Model

Goals of BLP

Model of S. Berry, Levinsohn, and Pakes (1995)

1. parsimonious specification to generate the distribution $F_U (\cdot| P, \Xi)$ of random utilities
2. sufficiently rich heterogeneity in preferences to permit reasonable/flexible substitution patterns
3. be explicit about unobservables, including the nature of endogeneity “problem(s)”
4. use the model to reveal solutions to the identification problem, including appropriate instruments
5. computationally feasible (in early 1990s!) algorithm for consistent estimation of the model and standard errors.

Utility Specification

Utility of consumer $i$ for product $j$

$$ u_{ijt} = \boldsymbol x_{jt} \boldsymbol \beta_{it} - \alpha p_{jt} + \xi_{jt} + \epsilon_{ijt} $$

Where

• $\boldsymbol x_{jt}$: $K$-vector of characteristics of product $j$ in market $t$

• $\boldsymbol \beta_{it} = (\beta_{it}^{1}, …, \beta_{it}^K)$: vector of tastes for characteristics $1,…,K$ in market $t$

  • $\beta_{it}^k = \beta_0^k + \sigma_k \zeta_{it}^k$

    • $\beta_0^k$: fixed taste for characteristic $k$ (the usual $\beta$)

    • $\zeta_{it}^k$: random taste, i.i.d. across consumers and markets $t$

• $\alpha$: price elasticity

• $p_{jt}$ price of product $j$ in market $t$

• $\xi_{jt}$: unobservable product shock at the level of products $j$ $\times$ market $t$

• $\epsilon_{ijt}$: idiosyncratic (and latent) taste

Exogenous and Endogenous Product Characteristics

Utility of consumer $i$ for product $j$

$$ u_{ijt} = \boldsymbol x_{jt} \beta_{it} - \alpha p_{jt} + \xi_{jt} + \epsilon_{ijt} $$

• exogenous characteristics: $\boldsymbol x_{jt} \perp \xi_{jt}$

• endogenous characteristics: $p_{jt}$ (usually a scalar, price)

  • typically each $p_{jt}$ will depend on whole vector $\Xi_t = (\xi_{1t} , . . . , \xi_{Jt} )$
    • and on own costs $mc_{jt}$ and others’ costs $mc_{-jt}$
  • we need to distinguish true effects of prices on demand from the
  • effects of $\Xi_t$ ; this will require instruments
  • of course the equation above is not an estimating equation ($u_{ijt}$ not observed)
  • because prices and quantities are all endogenous - indeed determined - simultaneously, you may suspect (correctly) that instruments for prices alone may not suffice.

Utility Specification, Rewritten

Rewrite

$$ \begin{align} u_{ijt} &= \boldsymbol x_{jt} \boldsymbol \beta_{it} - \alpha p_{jt} + \xi_{jt} + \epsilon_{ijt} = \newline &= \delta_{jt} + \nu_{ijt} \end{align} $$

where

• $\delta_{jt} = \boldsymbol x_{jt} \boldsymbol \beta_0 - \alpha p_{jt} + \xi_{jt}$
  • mean utility of good $j$ in market $t$
• $\nu_{ijt} = \sum_{k} x_{jt}^{k} \sigma^{k} \zeta_{i t}^{k} + \epsilon_{ijt} \equiv \boldsymbol x_{jt} \tilde{\boldsymbol \beta}{it} + \epsilon{ijt}$
  • We split $\beta_{it}$ into its random ($\tilde{\beta}_{it}$) and non-random ($\beta_0$) part

From Consumer Utility to Demand

With a continuum of consumers in each market: market shares = choice probabilities

• P.S. continuum not needed, enough that sampling error on choice probs negligible compared to that of moments based on variation across products/markets

$$ s_{jt} (\Delta_t, \boldsymbol X_t, \boldsymbol \sigma) = \Pr (y_{it} = j) = \int_{\mathcal A_j (\Delta_t)} \text d F_{\nu} \Big(\nu_{i0t}, \nu_{i1t}, … , \nu_{iJt} \ \Big| \ \boldsymbol X_t, \boldsymbol \sigma \Big) $$

• where $$ \mathcal A_j(\Delta_t) = \Big\lbrace (\nu_{i0t}, \nu_{i1t}, … , \nu_{iJt} ) \in \mathbb{R}^{J+1}: \delta_{jt} + \nu_{ijt} \geq \delta_{kt} + \nu_{ikt} \ , \ \forall k \Big\rbrace $$ In words: market share of firm $j$ is the frequency of consumers buying good $j$

Demand is just shares $s_{jt}$ per market size $M_t$ $$ q_{jt} = M_t \times s_j (\Delta_t, \boldsymbol X_t, \boldsymbol \sigma) $$

Why Random Coefficients?

Without random coefficients $$ \begin{aligned} u_{ijt} &= \underbrace{\boldsymbol x_{jt} \boldsymbol \beta_0 - \alpha p_{jt} + \xi_{jt}} + \epsilon_{ijt} \newline &= \hspace{3.4em} \delta_{jt} \hspace{3.4em} + \epsilon_{ijt} \end{aligned} $$ If $\epsilon_{ijt}$ are iid and independent of $(\boldsymbol X_t, P_t)$, e.g. as in the multinomial logit or probit models,

• products differ only in mean utilities $\delta_{jt}$
• $\to$ market shares depend only on the mean utilities
• $\to$ price elasticities (own and cross) depend only on mean utilities too

Implication: two products with the same market shares have the same cross elasticities w.r.t. all other products

Does this matter?

Yes!

• Mercedes class-A and Fiat Panda might both have low market shares
• But realistically should have very different cross-price elasticities w.r.t. BMW series-2

What is the issue?

• Models (like MNL) that have only iid additive taste shocks impose very restrictive relationships between the levels of market shares and the matrix of own and cross-price derivatives

  • Impact on counterfactuals!

• Restrictions only coming from model assumptions (analytical convenience)

• Models always imporse restrictions

  • necessary for estimation
  • but must allow flexibility in the relevant dimensions

How do random coefficients help?

In reality:

• goods differ in multiple dimensions
• consumers have (heterogeneous) preferences over these differences

How do random coefficients capture it?

• large $\beta_i^k$ $\leftrightarrow$ strong taste for characteristic $k$
  • e.g., maximum speed for sport car
• Consumer $i$’s first choice likely to have high value of $x^k$
• $i$’s second choice too!
  • Mark: cross elasticities are always about 1st vs. 2nd choices

Incorporating this allows more sensible substitution patterns

• competition is mostly “local”
• i.e., between firms offering products appealing to the same consumers.

Which random coefficients?

Which characteristics have random coefficients?

• dummies for subsets of products?
  • S. T. Berry (1994): covers the nested logit as a special case
• certain horizontal or vertical characteristics?
  • parts of $(\boldsymbol X_t, P_t)$?

In practice

• Choice depends on the application and data set, including instruments
• Too many RC’s (w.r.t quantity of data available) $\to$ imprecise estimates of $\boldsymbol \sigma$

BLP: Estimation

Setting

Observables

• $\boldsymbol X_t$: product characteristics
• $P_t$: prices
• $S_t$: observed market shares
• $\boldsymbol W_t$: observable cost shifters
• $\boldsymbol Z_t$: excluded instruments

Sketch of procedure

1. start with demand model alone
2. suppose $ F_{\nu{=tex}} (\cdot {=tex} |  \boldsymbol {=tex}X, \boldsymbol {=tex}\sigma {=tex})$ is known (i.e., $\sigma$ known)
3. for each market $t$, find mean utilities $\Delta_t \in \mathbb R$ such that $s_{jt} (\Delta_t, \boldsymbol X_t, \boldsymbol \sigma) = s^{obs}_{jt} \ \forall j$
   • i.e.,“invert” model at observed market shares to find mean utilities $\boldsymbol \delta$
   • where $s^{obs}_{jt}$ are the observed market shares
4. using IV ,e.g. $\mathbb E [\boldsymbol z_{jt} | \xi_{jt} ] = 0$, estimate the equation

Issues

1. What instruments?
2. Will the “inversion” step actually work?
3. What about $\boldsymbol \sigma$??
4. Formal estimator?
5. Computational algorithm(s)?
6. Supply side
   • additional restrictions (moment conditions)
     • help estimation of demand
   • additional parameters: marginal cost function
     • why? may care directly
     • and needed for counterfactuals that change equilibrium quantities unless $mc$ is constant

Instruments

We need intruments for all endogenous variables—prices and quantities—independently.

1. Excluded cost shifters $\boldsymbol W_t$ (classic)

   • Usually: wages, material costs, shipping cost to market $t$, taxes/tariffs, demand shifters from other markets

2. Or proxies for them

   • Usually: price of same good in another mkt (“Hausman instruments”)

3. Markup shifters:

   • Usually: characteristics of “nearby” markets (“Waldfogel instruments”)

   • Logic: income/age/education in San Francisco might affect prices in Oakland but might be independent fo Oakland preferences

4. Product characteristics of other firms in the same market $\boldsymbol X_{-jt}$

   • “BLP instruments”
   • affect quantities directly; affect prices (markups) via equilibrium only

Inversion

How do we get from market shares to prices??

Given x,σ and any positive shares sh, define the following mapping $\Phi : \mathbb R^j \to \mathbb R^j$ $$ \Phi (\Delta_t) = \Delta_t + \log\Big( \hat S^{obs}_t \Big) - \log \Big( S_t (\Delta_t, \boldsymbol X_t, \boldsymbol \sigma) \Big) $$ S. T. Berry (1994): for any nonzero shares sh, Φ is a contraction

• under mild conditions on the linear random coefficients random utility model
• extreme value and normal random coeff not necessary

What does it imply?

• It has a unique fixed point: we can compute $\delta_{jt} = \delta (S_t, \boldsymbol X_t, \boldsymbol \sigma)$
• We can compute the fixed point iterating the contraction from any initial guess $\Delta_{0t}$

What about $\sigma$?

What we we got?

• inversion result: for any market shares and any $\boldsymbol \sigma$, we can find a vector of mean utilities $\Delta_t$ that rationalizes the data with the BLP model
• a non-identification result? there is no information about $\boldsymbol \sigma$ from market shares?

What are we forgetting?

• Cross-market variation!
• We can get the mean utilities $\delta_{jt} = \boldsymbol x_{jt} \boldsymbol \beta_0 - \alpha p_{jt} + \xi_{jt}$
• As in OLS, use $\boldsymbol z_{jt} \perp \xi_{jt}$ to get identification of $(\alpha, \boldsymbol \beta_0, \boldsymbol \sigma)$

Identification of $\sigma$

We are trying to estimate $(\alpha, \boldsymbol \beta_0, \boldsymbol \sigma)$ from $$ \mathbb E \Big[ \xi_{jt} (\alpha, \boldsymbol \beta_0, \boldsymbol \sigma) \cdot \boldsymbol z_{jt} \Big] = \mathbb E \Big[ \big( \delta_{jt}(\boldsymbol \sigma) - \boldsymbol x_{jt} \boldsymbol \beta_0 + \alpha p_{jt} \big) \cdot \boldsymbol z_{jt} \Big] $$ What kind of intruments $\boldsymbol z_{jt}$ do we need?

• $\boldsymbol x_{jt}$ (for $\boldsymbol \beta_0$)
• intruments for $p_{jt}$ (for $\alpha$)
• but also something for $\boldsymbol \sigma$!

BLP Estimation

Steps

1. Take guess of parameters $(\alpha, \boldsymbol \beta_0, \boldsymbol \sigma)$
2. From observed market shared $S^{obs}{t}$ and $\boldsymbol \sigma$ get mean utilities $\delta{jt} (\boldsymbol \sigma)$
3. Use also $(\alpha, \boldsymbol \beta_0)$ to get $\xi_{jt} (\alpha, \boldsymbol \beta_0, \boldsymbol \sigma)$
4. Compute empirical moments $\frac{1}{JT} \xi_{jt} (\alpha, \boldsymbol \beta_0, \boldsymbol \sigma) \cdot \boldsymbol z_{jt}$

The GMM estimator is $(\hat \alpha, \boldsymbol{\hat{\beta}_0}, \boldsymbol{\hat{\sigma}})$ that get the empirical moments as close to $0$ as possible.

Issues

1. Computing $S_t (\Delta_t, \boldsymbol X_t, \boldsymbol \sigma)$ involves a high dimensional integral
   • Use simulation to approximate distribution of random tastes $\zeta_{it}^k$
   • P.S. recall that we have decomposed random coefficients $\beta_{it}^k$ as $\beta_{it}^k = \beta_0^k + \sigma_k \zeta_{it}^k$
2. $\xi_{jt} (\alpha, \boldsymbol \beta_0, \boldsymbol \sigma)$ has no closed form solution
   • Compute it via contraction
   • MPEC?

Computation

Nested fixed point algorithm

Sketch of the algorithm

• Draw a vector of consumer tastes
• Until you have found a minimum for $\mathbb E \Big[ \xi_{jt} (\alpha, \boldsymbol \beta_0, \boldsymbol \sigma) \cdot \boldsymbol z_{jt} \Big]$ do
  • Pick a vector of parameter values $(\alpha, \boldsymbol \beta_0, \boldsymbol \sigma)$
  • Initialize mean utilities $\delta_{jt}^0$
  • Until $\big|\big| \Delta_{t}^{n+1} - \Delta_{t}^{n} \big|\big| < tolerance$ do
    • Compute implied shares: $s_{jt} (\Delta_{t}^{n}, \boldsymbol X_t, \boldsymbol \sigma) = \int \frac{\exp \left[ \boldsymbol x_{j t} \boldsymbol{\tilde{\beta}}{it}+\delta{j t}\right]}{1+\sum_{j^{\prime}} \exp \left[\boldsymbol x_{j^{\prime} t} \boldsymbol{\tilde{\beta}}{it}+\delta{j’ t}\right]} f\left( \boldsymbol{\tilde{\beta}}{it} \mid \theta\right) d \tilde{\beta}{i t}$
    • Update mean utilities: $\Delta_{t}^{n+1} = \Delta_{t}^{n} + \log\Big( \hat S^{obs}t \Big) - \log \Big( S_t (\Delta{t}^{n}, \boldsymbol X_t, \boldsymbol \sigma) \Big)$
  • Compute $\xi_{jt} = \delta_{jt} - \boldsymbol x_{jt} \boldsymbol \beta_0 + \alpha p_{jt}$
  • Compute $\mathbb E \Big[ \xi_{jt} (\alpha, \boldsymbol \beta_0, \boldsymbol \sigma) \cdot \boldsymbol z_{jt} \Big]$

Notes

• Important to draw shocks outside the optimization routine!

Appendix

References [references]