1 Multinomial Logit and Demand Estimation
1.1 Theoretical foundation
- Let \(C=\left\{0,1,2,...,J \right\}\) be a consumers choice set of goods.
- All goods are mutually exlusive, that is, consumers only choose one good \(j.\)
- The goods are exhaustive, that is, all goods are known to the consumer.
- The 0 option is referred to as the “outside option”, and is the option of not buying anything in the choice set.
- The choice set \(C<\infty\), this is it is finite.
- Individuals attribute a utility to each element in \(C\):
\[u_{ij}=v_{ij}+\epsilon_{ij}\]
where \(u_{ij}\) is the utility for individual \(i\) from good \(j.\) Utility is decomposed into two parts:
- \(v_{ij}\): Is observed value of characteristics given by good \(j.\)
- \(\epsilon_{ij}\): Is an, to the econometrician, unobserved (but observed by the consumer) taste shock.
- Assume: \(\boldsymbol{\epsilon}_i=(\epsilon_0,\epsilon_1,...,\epsilon_J)\) has known joint density \(g(\boldsymbol{\epsilon}_i).\)
- The consumer problem:
- The consumer maximize his utility for a sequence of goods: \[\max_{j\in C}\left\{u_{ij}\right\}_{j=0}^J.\]
- Solution: By revealed preferences, if a consumer choose good \(j,\) it has to be such that \(u_{ij}\geq u_{ik},\) for all other goods \(k.\) Formally, \(j\) is chosen if \(v_{ij}+\epsilon_{ij}\geq v_{ik}+\epsilon_{ik}\) for all \(k\neq j, k\in C.\) Equivalently, the consumer has the following solution choice set: \[\left\{\epsilon_{ik}-\epsilon_{ij}\leq v_{ij}-v_{ik}, \forall k\neq j, k\in C \right\}.\]
- The probability of this happening is simply the probability of chosing \(j\) given \(v_i\): \[\mathbb{P}(a_i=j\mid v_i)=\mathbb{P}(\epsilon_{ik}-\epsilon_{ij}\leq v_{ij}-v_{ik}, \forall k\neq j, k\in C).\] Given \(g(\boldsymbol{\epsilon}_i)\) we can compute this probability for all \(j\in C.\)
Example 1.1 Binary choice. Consider the case where the choice set includes only two options, the outside option and good \(1\). That is, let $C={0,1}, and let the utility for both goods be linear combination of characteristics, hence, \[\begin{align} u_{i0}&=v_{i0}+\epsilon_{i0}=x_{i0}\beta+\epsilon_{i0}\\ u_{i1}&=v_{i1}+\epsilon_{i1}=x_{i1}\beta+\epsilon_{i1} \end{align}\] The consumer will chose the option \(j=1\) as long as \(x_{i1}\beta +\epsilon_{i1}\geq x_{i0}\beta +\epsilon_{i0}\), which in probability terms becomes \[\mathbb{P}(a_i=1\mid \mathbf{x})=\mathbb{P}(\epsilon_{i0}-\epsilon_{i1}\leq (x_{i1}-x_{i0})\beta) \tag{1.1}.\] This can typically be estimated using a logit model given that we have a distribution of the tast shock. Suppse that \(\epsilon_{ij}\) is iid distributed with a extreme value type I distribution. That is, the CDF is given by \(G(\epsilon_{ij})=\exp\left\{-\exp[-\mu(\epsilon_{ij}-\eta)]\right\}.\) Here \(\mu>0\) is the scale paramter, and \(\eta\) is the location parameter. We can write \(\epsilon_{ij}\sim EV1(\mu,\eta).\) If \(\mu=1,\eta=0,\) the distribution will be a Gumble distribution. Given the structure on the taste shock, we can compute the probabilit accordingly: \[\begin{align} \mathbb{P}(a_i=1\mid x_i)&=\mathbb{P}(\epsilon_0\leq x_i^*\beta +\epsilon_{ij}\mid x_i)\\ &=\int_{-\infty}^{+\infty}\mathbb{P}(\epsilon_0\leq x_i^*\beta +\epsilon_{ij}\mid x_i)g(\epsilon_{ij})\\ &=\int_{-\infty}^{+\infty}\exp\left(-\exp\left\{-x^*_{ij}\beta-\epsilon_{i1}\right\}\right)\exp(-\epsilon_{i1})\exp\left\{-\exp(\epsilon_{ij}) \right\}d\epsilon_{ij}. \end{align}\] which after some simplifications become \[ \boxed{\mathbb{P}(a_i=j\mid x_i)=\frac{\exp(x_{i1}\beta)}{\exp(x_{i1}\beta)+\exp(x_{i0}\beta)}.} \tag{1.2}\]
1.2 Multinomial Logit
- Now we just extend Example 1.1 with having more choices than two. The derivation remains the same.
- Let \(C=\left\{0,1,2,...,J \right\}\) be the choice set, and let \(u_{ij}=v_{ij}+\epsilon_{ij}\) be the utility for individual \(i\) and good \(j,\) with \(\epsilon_{ij}\) being extreme value type I distributed. Then the corresponding probability computation becomes: \[\begin{align} \mathbb{P}(a_i=j\mid \mathbf{x})&=\mathbb{P}(u_{ij}\geq u_{ik}, \forall k\neq j, k\in C)\\ &= \int_{-\infty}^{+\infty}\mathbb{P}(\epsilon_{ik}\leq v_{ij}-v_{ik}+\epsilon_{ij}, \forall k\in C, k\neq j\mid v_i\epsilon_i)g(\epsilon_{ij})d\epsilon_{ij}\\ &=\int_{-\infty}^{+\infty} \prod_{k\neq j}\mathbb{P}(\epsilon_{ik}\leq v_{ij}-v_{ik}+\epsilon_{ij}\mid v_i\epsilon_i)g(\epsilon_{ij})d\epsilon_{ij}, \end{align}\] which after some simplification turns into the multinomial logit demand for good \(j, j\in C\): \[\boxed{ \mathbb{P}(a_i=j\mid \mathbf{x})=\frac{\exp(v_{ij})}{\sum_{k=0}^J\exp(v_{ik})}} \tag{1.3}\]
1.3 Estimation
- Assume: \(v_{ij}=\sum_{\ell =1}^L\beta_{\ell}x_{\ell i},\) with \(\left\{x_{\ell j} \right\}_{\ell=1}^L\) is the value of characteristics for option \(j.\) This assumption just means that we impose a linear structure on the observed part of the utility function.
- The solution follows directly from equation \((1.3)\) then:
\[\boxed{ \mathbb{P}(a_i=j\mid \mathbf{x})=\frac{\exp\left(\sum_{\ell=1}^L\beta_{\ell}x_{\ell j} \right)}{\sum_{k=0}^J\exp\left(\sum_{\ell=1}^L\beta_{\ell}x_{\ell j}\right)}=S_j(\mathbf{x}).} \tag{1.4}\]
- Here equation \((1.4)\) is the market share of good \(j\).
- We estimate \((1.4)\) using data and product characteristics. Note that the value of the estimates \(\widehat{\beta}_{\ell}\) are the same for all individuals \(i.\)
- In practice, we differentiate between two situations when we have:
- aggregate data,
- individual level data.
1.3.1 Aggregate data
- To extend model a bit, include a choice (or year) fixed effect in the specification. So the market share just becomes \[S_{jt}(\mathbf{x}_t)=\frac{\exp\left(\sum_{\ell=1}^L\beta_{\ell}x_{\ell j}+\xi_{jt} \right)}{\sum_{k=0}^J\exp\left(\sum_{\ell=1}^L\beta_{\ell}x_{\ell j}+\xi_{kt}\right)}.\]
- The goal here is to derive a useful expression that we can estimate using data. To do this, we utilize a trick where we divide \(S_{jt}(\mathbf{x}_t)\) by \(S_{j't}(\mathbf{x}_t)\) for any \(j'\neq j\) and take the logs on both sides: \[\ln S_{jt}(\mathbf{x}_t)-\ln S_{j't}(\mathbf{x}_t)=\sum_{\ell=1}^L\beta_{\ell}(x_{\ell jt}-x_{\ell j't})+\underbrace{(\xi_{jt}-\xi_{j't})}_{\text{structural error}}.\]
- If we observe (in the data) \(\left\{q_{jt} \right\},\) number of times \(j\) was chosen at time \(t\) (or location etc), it is true that \[S_{jt}(\mathbf{x}_t)=\frac{q_{jt}}{\sum_{j'=0}^J q_{j't}}.\]
- In practice we normalize \(v_{0,t}\equiv 0\), so we get the very convenient expression
\[\boxed{\ln S_{jt}(\mathbf{x}_t)-\ln S_{0t}(\mathbf{x}_t)=\sum_{\ell=1}^L\beta_{\ell}x_{\ell jt}+\xi_{jt}} \tag{1.5},\]
which can be estimated using OLS.
- Just calculate the market share for each good \(j,\) subtract the market share for the outside option, \(1-S_{jt}\), and then run the regression as usual.
- Note: It is likely that estimating equation \((1.5)\) using OLS will lead to biased estimates, since the equation likely suffers from endogeneity problems. One solution to this is to use IV, where the instruments are “cost shifters”, that is, factors that affects firms pricing but that does not affect demand directly (such as production costs).
1.3.2 Individual data
- With individual level data, the estimation process is different.
- Let \(D=\left\{a_i,\mathbf{x}_i\right\}_{i=1}^N\) be the sequence of individuals that makes a choice \(a_i\) with characteristics \(\mathbf{x}_i.\)
- To estimate the demand, we use the maximum likelihood estimator (MLE).
- The likelihood function is just
\[L(\boldsymbol{\beta})=\prod_{i=1}^N\prod_{j=1}^J\mathbb{P}(a_i=j\mid \mathbf{x}_j\boldsymbol{\beta})^{y_{ij}},\]
where \(y_{ij}=\begin{cases}1, \quad \text{if $a_i=j$}\\ 0, \quad \text{otherwise} \end{cases}\), and \(\mathbb{P}(a_i=j\mid \mathbf{x}_j\boldsymbol{\beta})\) is as before.
- Taking logs on both sides gives \[l(\boldsymbol{\beta})=\sum_i \sum_j y_{ij}\ln\left\{\mathbf{x}_j\boldsymbol{\beta}\right\}.\]
- The usual MLE results hold, thus, \(\beta^{ML}\) orders \(\left\{\frac{\partial l(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}_{\ell}} \right\}_{\ell=1}^L=0,\) which is just the score equal to zero condition.
- Endogeneity with individual data:
- Here we do not use IV (2SLS) to solve the endogeneity problem, instead we use a control function approach, which requires us to make some explicit assumptions about the structure in which the endogeneity problem arise.
- We have the same as before, but we explicitly write out the price since we assume that it is the only endogenous regressor:
\[u_{ij}=\sum_{\ell=1}^L\beta_{\ell}x_{\ell j}+\alpha p_j+\xi_j+\epsilon_{ij},\]
and the corresponding choice probability is
\[\mathbb{P}(a_i=j\mid \mathbf{x})=\frac{\exp\left(\sum_{\ell=1}^L \beta_{\ell}x_{\ell j}+\alpha p_j+\xi_j \right)}{\sum_k \exp\left( \sum_{\ell=1}^L \beta_{\ell}x_{\ell k}+\alpha p_k+\xi_k \right)},\]
moreover, we assume that:
- \(p_j=\gamma z_j+\eta_j,\) where \(\xi_j\) and \(\eta_j\) are unobserved and correlated (\(eta_j\) is the source of endogeniety) but independent of \((\mathbf{x},\mathbf{z}).\)
- Furthermore, assume that the control function can be written as \[\xi_j=\psi_1\eta_j+\psi_2\tilde{\eta}_j,\] in which the choice probability becomes \[\mathbb{P}(a_i=j\mid \mathbf{x},\boldsymbol{\eta},\tilde{\boldsymbol{\eta}})=\frac{\exp\left(\sum_{\ell=1}^Lx_{\ell j}\beta_\ell +\alpha p_j+\psi_1\eta_j+\psi_2\tilde{\eta}_k \right)}{\sum_k\exp\left(\sum_{\ell=1}^Lx_{\ell k}\beta_\ell +\alpha p_k+\psi_1\eta_k+\psi_2\tilde{\eta}_k \right)}.\]
- Assume that \(\tilde{\eta}_j\) is iid standard normal, then \[ \mathbb{P}(a_i=j\mid \mathbf{x}\boldsymbol{\eta})=\int \mathbb{P}(a_i=j\mid \mathbf{x}, \boldsymbol{\eta},\tilde{\boldsymbol{\eta}})\phi(\boldsymbol{\eta})d\boldsymbol{\eta}. \tag{1.6} \]
- Equation \((1.6)\) is estimated a two-step procedure (usual control function approach):
- Use \[p_j=\gamma z_j+\eta_j,\] and get \(\eta_j\) by computing the residual.
- Control for \(\widehat{\eta}_j\) in \(\mathbb{P}(a_i=j\mid \mathbf{x}\boldsymbol{\eta})\) and estimate \((\alpha. \beta, \psi_1, \psi_2)\) by simulated MLE. That is, approximate \(\mathbb{P}(a_i=j\mid \mathbf{x}\boldsymbol{\eta})\) using numerical integration and from this form the likelihood function; maximize and find the parameters.
1.4 Elasticities
Price elasticity: \(\epsilon^{\ell}_{ij}=\frac{\partial \mathbb{P}(a_i=j\mid \mathbf{x})}{\partial x_{\ell j}}\frac{x_{\ell j}}{\mathbb{P}(a_i=j\mid \mathbf{x}}=x_{j\ell}\cdot \beta_{\ell}[1-\mathbb{P}(a_i=j\mid \mathbf{x})].\)
Cross-price elasticity: \(\epsilon_{jk}=-x_{k\ell}\cdot \beta_{\ell}\mathbb{P}(a_i=j\mid \mathbf{x}).\)$
Note that the cross-price elasticity is the same across all goods. This can potentially be problematic if products are related to each other in different ways.
2 Nested Logit and Demand Estimation
2.1 Theoretical foundation
- Now we extend the model by including mutually exclusive partitions of the choice set \(C.\)
- These are referred to as nests.
- Mathematically we write the choice set as \[C\equiv \bigcup_{m=0}^{M}C_m,\] with \(C_m\cap C_{\ell}=\emptyset, \forall \ell\neq m.\)
- The utility function (exclude index \(i\) for simplicity) is given by:
\[u_{jm}=v_{jm}\xi_m+\xi_{jm}\],
where \(v_{jm}\) is value of characteristics for good \(j\) in nest \(m,\) \(\xi_m\) is nest fixed effects, and \(\xi_{jm}\) is good in nest fixed effect.
- The corresponding choice probability is
\[\underbrace{\mathbb{P}(a=j_m)}_{\text{Probability of certain good in nest}}=\underbrace{\mathbb{P}(a\in C_m)}_{\text{Nest choice}}\times \underbrace{\mathbb{P}(a=j_m\mid a\in C_m)}_{\text{Product choice given nest choice}}.\]
- The goal, again, is to derive an expression this choice probability. Lets derive both RHS expressions separately.
- The corresponding choice probability is
\[\underbrace{\mathbb{P}(a=j_m)}_{\text{Probability of certain good in nest}}=\underbrace{\mathbb{P}(a\in C_m)}_{\text{Nest choice}}\times \underbrace{\mathbb{P}(a=j_m\mid a\in C_m)}_{\text{Product choice given nest choice}}.\]
2.1.1 Deriving \(\mathbb{P}(a=j_m\mid a\in C_m)\)
- Assume that \(\xi_{jm}\) is iid for attributes in the same nest with distribution \(EV(\mu_m,0).\) Then
\[\begin{align}
\mathbb{P}(a=j_m\mid a\in C_m)&=\mathbb{P}(u_{jm}\geq u_{km}, \forall k_m\neq j_m\mid j_m,k_m\in C_m)\\
&=\mathbb{P}(v_{jm}+\xi_m+\xi_{jm}\geq v_{km}+xi_m+\xi_{km}\mid j_m,k_m\in C_m)\\
&\vdots \quad \quad \quad \quad (\text{same as in multinomial logit})\\
&\vdots\\
&=\frac{\exp(\mu_m,v_{jm})}{\sum_{k\in C_m}\exp(\mu_m,v_{km})}
\end{align}\]
- The derivation from the second to the last equality follows the same structure as the derivation in the multinomial logit.
2.1.2 Deriving \(\mathbb{P}(a\in C_m)\)
- This derivation is a bit harder to do. First, note that \[\begin{align} \mathbb{P}(a\in C_M)=\mathbb{P}\left[ \max_{j_m\in C_m} \left\{ v_{jm}+\xi_m+\xi_{jm}\right\}\geq \max_{j_n\in k} \left\{ v_{jk}+\xi_k+\xi_{jk} \right\},\forall k\neq m \right]. \end{align}\]
- The next step is to derive the distribution of \(\max_{j_m\in C_m}\left\{v_{jm}+\xi_{jm} \right\}.\) Thus, \[\begin{align} \mathbb{P}\left( \max_{j_m\in C_m}\left\{ v_{jm}\xi_{jm} \right\}<\xi\right)&=\mathbb{P}(v_{jm}+\xi_{jm}<\xi, \forall j_m\in C_m)\\ &=\mathbb{P}(\xi_{jm}<\xi-v_{jm}, \forall j_m\in C_m)\\ &=\prod_{j_m\in C_m}\exp\left[ -\exp(-\mu(\xi-v_{jm})) \right] \quad \left(\text{from $\xi_{jm}\overset{iid}{\sim}EV$}\right)\\ &=\prod_{j_m\in C_m}\exp\left[ -\exp(-\mu -\xi)\cdot \exp(\mu v_{jm}) \right]\\ &=\exp\left[ -\sum_{j_m\in C_m}\exp(-\mu \xi)\cdot \exp(\mu v_{jm}) \right]\\ &=\exp\left[ -\exp(-\mu \xi)\cdot \sum_{j_m\in C_m}\exp(\mu_m v_{jm}) \right] \quad \left(\text{let $\phi_m=\sum_{j_m\in C_m}\exp(\mu v_{jm})$}\right)\\ &=\exp\left[ -\exp(-\mu \xi)\cdot \exp(\ln \phi_m) \right]\\ &=\underbrace{\exp\left[ -\exp\left( - \mu_m \left( \xi-\frac{1}{\mu_m}\ln \phi_m \right)\right) \right]}_{\text{Extreme value distribution with scale $\mu_m$ and location $\xi-\frac{1}{\mu_m}\ln \phi_m$}} \end{align}\]
- After some more manipulations (that I will not show here), we can arrive to the following expression for this part of the choice probability: \[\mathbb{P}(a\in C_m)=\frac{\exp\left(\tilde{v}_m \right)}{\sum_{k\neq m}\exp\left(\tilde{v}_k \right)},\] where \(\tilde{v}_{m}=1/\mu_m \ln \left[\sum_{j_m\in C_m}\exp(\mu_m v_{jm}) \right].\)
- So the choice probability becomes: \[\boxed{\mathbb{P}(a=j_m)= \frac{\exp\left(\tilde{v}_m \right)}{\sum_{k\neq m}\exp\left(\tilde{v}_k \right)}\cdot \frac{\exp(\mu_m,v_{jm})}{\sum_{k\in C_m}\exp(\mu_m,v_{km})}} \tag{2.1}\]
2.2 Estimation
- For individual data we use MLE based on the choice probability derived above (eq \((2.1)\)).
- Aggregate data:
- Define the outside option \(v_0=0\), so \[\mathbb{P}(a=0)=\frac{\exp(\tilde{v}_0)}{\sum_{k}\exp(\tilde{v}_k)}.\]
- Write \(\mathbb{P}(a=j)/\mathbb{P}(a=0)=\exp(\tilde{v}_m)\mathbb{P}(a=j_m\mid a\in C_m),\) and take logs \[\ln \mathbb{P}(a=j_m)-\ln \mathbb{P}(a=0)=\tilde{v}_m+\ln \mathbb{P}(a=j_m\mid a\in C_M) \tag{2.2}.\]
- Then from \[\mathbb{P}(a=j_m\mid a\in C_m)=\frac{\exp(\mu_mv_{jm})}{\sum_{k}\exp(\mu_mv_{km})},\] we have \[\ln \mathbb{P}(a=j_m\mid a\in C_m)=\mu_m v_{jm}-\ln\left(\sum_{k}\exp(\mu_m v_{km}) \right)\tag{2.3}.\]
- From \(\tilde{v}_{m}=1/\mu_m \ln \left[\sum_{j_m\in C_m}\exp(\mu_m v_{jm}) \right]\) we have \[\mu_m \tilde{v}_m=\ln\left(\sum_{m}\exp(\mu_m v_{jm}\right). \tag{2.4}\]
- From equation \((2.3)\) and \((2.4)\) we have that \[\tilde{v}_m=v_{jm}-\frac{1}{\mu_m}\ln \mathbb{P}(a=j_m\mid a\in C_m),\] and substitute this into \((2.2)\) \[\boxed{\begin{align} \ln \mathbb{P}(a=j_m)-\ln \mathbb{P}(a=0)&=\left( 1-\frac{1}{\mu_m} \right)\ln\mathbb{P}(a=j_m\mid C_m)+v_{jm}\quad \text{(assume linear $v_{jm}$)}\\ &=\underbrace{\left( 1-\frac{1}{\mu_m} \right)}_{\text{coefficient}}\underbrace{\ln\mathbb{P}(a=j_m\mid C_m)}_{\text{share in nest}}+\sum_{\ell}\beta_\ell x_{\ell j_m}+\underbrace{\eta_{jm}}_{\text{structural error}} \end{align}}\tag{2.5}\]
3 Random Coefficient Logit and Demand Estimation
3.1 Theoretical foundation
- The model is the same as before, \(u_{ij}=v_{ij}+\epsilon_{ij}\), but now we let the coefficients on regressors vary across individuals. In particular, we have \[v_{ij}=\sum_{\ell =1}^L \beta_{\ell i}x_{\ell i}.\]
- Assume that \(\boldsymbol{\beta}_i'\) is distributed in the population with density \(f(\boldsymbol{\beta}_i':\boldsymbol{\theta})\), where \(\boldsymbol{\theta}\) is a vector of parameters.
- For example, we can have \(\boldsymbol{\beta}_i=\boldsymbol{\theta}'D_i+\sigma\eta_i.\)
- Individuals know \((\boldsymbol{\beta}_i, \boldsymbol{\epsilon}_i)'\), but they are unobserved to the econometritian.
- Define the set of individuals with unobserved characteristics with \(a=j\): \[A_j=\left\{(\epsilon_i, \beta_i): \boldsymbol{\beta}_i'\mathbf{x}_j+\epsilon_{ij}\geq \boldsymbol{\beta}'_i\mathbf{x}_k+\epsilon_{ik}, \forall k\neq j \right\}.\] Therefore, arriving \(\boldsymbol{\epsilon}_i,\boldsymbol{\beta}_i\) independent: \[\begin{align} \mathbb{P}(a_i=j)&=\int \mathbb{I}\left[ (\boldsymbol{\epsilon}_i,\boldsymbol{\beta}_i)\in A_i \right]g(\boldsymbol{\epsilon}_i)f(\boldsymbol{\beta}_{ij};\boldsymbol{\theta})d(\boldsymbol{\epsilon}_i,\boldsymbol{\beta}_i)\\ &=\int \left\{ \left[ (\boldsymbol{\epsilon}_i,\boldsymbol{\beta}_i)\in A_i \mid \boldsymbol{\beta}_i \right]g(\boldsymbol{\epsilon}_i)d\boldsymbol{\epsilon}_i\right\}f(\boldsymbol{\beta}_i;\boldsymbol{\theta})d\boldsymbol{\beta}_i\\ &=\int \frac{\exp(\boldsymbol{\beta}'_i\mathbf{x}_j)}{\sum_{k}\exp(\boldsymbol{\beta}'_i\mathbf{x}_j)}f(\boldsymbol{\beta}_i;\boldsymbol{\theta})d\boldsymbol{\beta}_i. \end{align}\]
- As we can see, the choice probability is not easily estimated since it requires an integral over the unobserved values of the individual specific estimates.
- BLP (1995) provides a way to estimate this.
3.2 BLP (1995)
- BLP (1995) provides an estimator of the random coefficient logit when we have aggregate data and endogenous variables.
3.2.1 Model
- Utility is gven by
\[u_{ij}=\alpha_i(y_i-p_j)+\boldsymbol{\beta}_i'\mathbf{x}_j+\xi_j+\epsilon_{ij} \tag{3.1}\]
where
- \(y_i\) is the income
- \(p_j\) is the price
- \(\xi_j\) is the unobserved choice
- \(\epsilon_{ij}\) is an \(EV1\) taste shock.
- \(\mathbf{x}_j\) is as before.
- Define \[\begin{bmatrix} \alpha_i \\ \beta_{i1} \\ \vdots\\ \beta_{iL} \end{bmatrix}=\begin{bmatrix} \alpha \\ \beta_{1} \\ \vdots\\ \beta_{L} \end{bmatrix}+\begin{bmatrix} \pi_{00} & \cdots & \pi_{0D} \\ \vdots & \ddots & \vdots \\ \pi_{L0} & \cdots & \pi_{LD} \end{bmatrix}\begin{bmatrix} D_{i1} \\ D_{i2} \\ \vdots\\ D_{iD} \end{bmatrix}+\begin{bmatrix} \sigma_0v_{i0} \\ \sigma_1v_{i1}\\ \vdots\\ \sigma_Lv_{iL} \end{bmatrix}\tag{3.2}\] where -\(D_{i}\)’s are observed individual characteristis, and \(v_i\)’s are unobserved individual characteristics.
- If we substitute \((3.2)\) into \((3.1)\) we have that \[u_{ij}=\alpha_iy_i-\left\{ \alpha+(\pi_{01}D_{i1}+\cdots +\pi_{0D}D_{iD})+\sigma_0v_{i0} \right\}p_j+\sum_{\ell=1}^L\left\{ \beta_\ell+(\pi_{\ell 1}D_{i1}+\cdots \pi_{\ell D}D_{iD})+\sigma_\ell v_{i\ell} \right\}x_{j\ell}+\xi_j+\epsilon_{ij}.\]
- Define \(\delta_j=\alpha p_j+\sum_{\ell=1}^L\beta_\ell x_{j\ell}+\xi_k\); and \[\mu_{ij}=(\pi_{01}D_{1i}+\cdots +\pi_{0D}D_{Di}+\sigma_0+v_{i0})+\sum_{\ell=1}^L(\pi_{\ell 1}D_{1i}+\cdots +\pi_{\ell D}D_{Di}+\sigma_0+v_{i0})x_{j\ell},\] so utility becomes \[u_{ij}=\alpha_i y_i+\delta_j+\mu_{ij}(D_i,v_i)+\epsilon_{ij}.\]
- Thus, in this model individuals will be characterized by \((D_i,v_i, \epsilon_i).\)
- That is, the set of individuals choosing good \(j\) becomes \[A_j=\left\{(D_i, v_i,\epsilon_i): u_{ij}\geq u_{ik}, \forall k\neq j\right\}.\]
- Assume that \((D_i,v_i, \epsilon_i)\) has CDF \(P(D_i,v_i,\epsiloon_i)\) in the population.
- Therefore, the share of product \(j\) in the population is \[s_j=\int \mathbb{I}[(D_i,v_i,\epsilon_i)\in A_j]dP(D_i,v_i,\epsilon_i).\]
- Assume that \(D_i,v_i,\epsilon\) are independent. Then \[\boxed{ \begin{align} s_j&=\int \int \left[ \int\mathbb{I}(D_i,v_i,\epsilon_i)\in A \right]d P_D(D_i)dP_v(v_i)\\ &=\int \int \frac{\exp(\delta_j+\mu_{ij}(D_i,v_i))}{\sum_k\exp(\delta_k+\mu_{ik}(D_k,v_k))}dP_D(D_i)dP_v(v_i) \end{align} } \tag{3.3}\]