Demand curve from logit model

Is it possible to derive a demand curve from a discrete choice multinomial model. For instance, a logit model would give the probability of purchasing each good. But can I translate that into a demand curve.

We cannot obtain "demand" in the usual sense, because demand is a random variable. The "best" we can do is first, to obtain the Conditional Expectation of individual demand (conditional on the variables that determine the probability of whether the consumer will demand/buy).
Let a situation where consumers decide to buy or not to buy a single (for simplicity) item of a good $A$. We can model this underlying utility framework by

$$u_i(A) = \alpha p_i+\mathbf x_i'\beta +e_i$$

with $i$ denoting the consumer, $p_i$ is the good's price, faced by consumer $i$, $\mathbf x_i$ containing various other variables pertinent to the case, and $e_i$ representing a "random" preference shock, assumed, say, to follow the standard logistic distribution, conditional on the $x$'s. The consumer will demand/buy, $q^d_i=1$, if (by convention) $u_i(A)>0$, so we can model the conditional probability of buying by

$$E(q_i^d =1\mid p, \mathbf x_i) = \Lambda(\alpha p_i+\mathbf x_i'\beta)$$

Then the conditional expectation of Market demand is

$$Q_d = \sum_{i=1}^nq^d_i = \sum_{i=1}^n\Lambda(\alpha p_i+\mathbf x_i'\beta)$$

This is still a random variable, and with unknown parameters. To obtain something one dimensional, given a sample of size $n$ on $\mathbf x_i$ and on relevant transactions (and so on $p_i$), one can estimate $\hat \alpha, \; \hat \beta$, say by maximum likelihood. Then we can obtain some concept of "average market demand curve" by using the sample means of the $x$'s as

$$\hat Q_d(p\,; \mathbf {\bar x}) = \sum_{i=1}^n\Lambda(\hat \alpha p+ \mathbf {\bar x'}\hat \beta) = n\cdot \Lambda(\hat \alpha p+ \mathbf {\bar x'}\hat \beta)$$

where here the price $p$ is not indexed anymore, and we vary it to obtain a curve.

As a toy numerical example, assume there are $100$ consumers, that ${\bar x'}\hat \beta =1$, and $\hat \alpha = -2$. Then $\hat Q_d(p\,; \mathbf {\bar x_i})$ will look like MULTINOMIAL LOGIT

In a multinomial logit where we examine the choice to buy "this or this or this" (mutually exclusive choices $k=1,..,K$), we end up with (after the usual normalization)

$$\hat E(q^d_{ik} \mid p_{ik}, \mathbf x_{ik}) = \frac {\exp\{\hat \alpha p_{ik} + \mathbf x'_{ik}\hat \beta\}}{1+\sum_{k=2}^K\exp\{\hat \alpha p_{ik} + \mathbf x'_{ik}\hat \beta\}}$$

Then we can obtain $K$ demands (each conditional on the rest) using (say, for good $1$)

$$\hat Q^{(1)}_d(p_1) = \frac {n\cdot \exp\{\hat \alpha p_1 + \mathbf {\bar x}_{1}'\hat \beta\}}{1 + \sum_{k=2}^{K}\exp\{\hat \alpha \bar p_{k} + \mathbf {\bar x}_k'\hat \beta\}}$$

Note that here we have to average the prices of the other goods also.

• It's a good answer. If I extend the binary logit to a multinomial logit, I would just need to aggregate the probability of purchasing each good over the entire sample. Am I undestanding it correctly. Sep 10 '15 at 19:07
• @YanSong Are you using multinomial logit to model alternative, mutually exclusive choices? Sep 10 '15 at 19:21
• Yes. I just want to relate it to the standard demand curve to explain how to calculate the consumer surplus. Sep 10 '15 at 19:30
• @YanSong I extended the answer to cover multinomial logit. Sep 10 '15 at 20:06
• I hope this answer gets the upvotes it deserves. Sep 10 '15 at 20:48