BLPDemand.jl
Estimate random coefficients demand models in the style of Berry, Levinsohn, and Pakes (1995).
- Function Reference
- Developer Notes
- BLPDemand.jl
- Model
- Introduction
- Getting started
- Problem 1: load and explore the data
- Problem 2: Logit Demand
- Problem 3: Demand Side Estimation
- Problem 4: Demand and Supply
- Problem 5: Elasticities
- Problem 6: Merger Simulation
- Simulations
Model
Demand
There are $J$ products available. We have data from $T$ markets on product shares, s[j,t]
, K
product characteristics, x[k,j,t]
, L
instruments, z[l,j,t]
, and C
cost shifters, w[c,j,t]
.
Market shares come from a random coefficients model.
\[s[j,t] = \int \frac{\exp(x[:,j,t]'(\beta + \nu.*\sigma) + \xi[j,t])}{1 + \sum_{\ell} \exp(x[:,\ell,t]'(\beta + \nu .* \sigma) + \xi[\ell,t])} dF_\nu\]
where $\beta$ and $\sigma$ are parameters to be estimates, $\xi$ are market level demand shocks, and $\nu$ represents heteregeneity in tastes for characteristics. Let
\[\delta[j,t] = x[:,j,t]'\beta + \xi[j,t]\]
Then we can write shares as
\[s[j,t] = \int \frac{\exp(\delta[j,t] + x[:,j,t]'(\nu .* \sigma)}{1 + \sum_{\ell} \exp(\delta[\ell,t] + x[:,\ell,t]'*\nu .* \sigma)} dF_\nu\]
the right hand side of this equation is computed by share(δ, σ, x,ν)
.
Conversely, given s[:,t]
we can solve for δ[:,t]
using delta(s, x, ν, σ)
.
To estimate $\theta=(\beta,\sigma)$, we assume that
\[E[\xi_{j,t} * z_{\cdot,j,t} ]= 0\]
and minimize a quadratic form in the corresponding empirical moments,
m = [mean(xi.*z[l,:,:]) for l in 1:L]
m'*W*m
See demandmoments
and estimateRCIVlogit
.
Supply
For the supply side of the model, we assume that x[1,:,:]
is price. Marginal costs are log linear and firms choose prices in Bertrand-Nash competition, so (for single product firms)
\[(p[j,t] - \exp(w[:,j,t]'\gamma + \omega[j,t])) \frac{\partial s}{\partial x[1,:,:]} + s[j,t] = 0\]
For estimation, as with demand, we assume that
\[E[\omega_{j,t} * z_{\cdot,j,t} ]= 0\]
and minimize a quadratic form in the corresponding empirical moments (along with the demand moments above).
See supplymoments
and estimateBLP