# Discrete choice

## Random utility

### Continuous utility functions

With non-discrete choice a consumer chooses how much of product $$x$$ to consume.

The utility function is of the form:

$$U_i(x_1,...,x_m;d)$$

And the consumer chooses how much of $$x_i$$ to consume to maximise this, subject to the budget constraint.

We include features relating to the individual, $$d$$.

### Discrete choice

The utility customer $$i$$ gets from product $$j$$ is:

$$U_{ij}=f_i(p,d)+\epsilon_{ij}$$

The customer chooses the product with the highest utility.

### The outside option

We need to know the market share of the outside option. Do this theoretically. Eg number of customers in area, and 1 per day.

## Linear random utility functions

### Price preferences

We start with a simple model, where the customer has price preference.

$$U_{ij}=-\beta_i p_{ij} +\epsilon_{ij}$$

### Product characteristics

$$U_{ij}=\alpha_i x_j -\beta_i p_{ij} +\epsilon_{ij}$$

### Individual characteristics

$$U_{ij}=\alpha_i x_j -\beta_i p_{ij} + \theta_j d_i +\epsilon_{ij}$$

### The general form

We can convert this to the form:

$$U_{ij}=\Theta z_{ij} + \epsilon_{ij}$$

## Modelling homogeneous preferences with multinomial logit

### Recap

Our model is:

$$U_{ij}=\Theta z_{ij} + \epsilon_{ij}$$

$$U_{ij}=\alpha_i x_j -\beta_i p_{ij} + \theta_j d_i +\epsilon_{ij}$$

### Homogeneous model

We model all customers as having the same preferences.

$$U_{ij}=\alpha x_j -\beta p_{ij} + \theta_j d_i +\epsilon_{ij}$$

### The multinomial logit assumption

If errors are IID and extreme we get:

$$P_{ij}=\dfrac{e^{\Theta z_j}}{\sum_k e^{\Theta z_k }}$$

### The outside option

A user has the option of not buying anything.

$$U_0=0$$

This gives us the following shares:

$$P_{ij}=\dfrac{e^{\Theta z_j}}{e^0+\sum_{k=1} e^{\Theta z_k }}$$

$$P_{ij}=\dfrac{e^{\Theta z_j}}{1+\sum_{k=1} e^{\Theta z_k }}$$

### Own-price elasticity of demand

$$P_{ij}=\dfrac{e^{\Theta z_j}}{1+\sum_{k=1}} e^{\Theta z_k }$$

$$P_{ij}=\dfrac{e^{\alpha x_j -\beta p_j + \theta_j d_i}}{1+\sum_{k=1} e^{ \alpha x_k -\beta p_k +\theta_k d_i}}$$

$$\dfrac{\delta P_{ij} }{\delta p_j}\dfrac{p_j}{P_{ij}}=-\beta p_j(1-P_{ij})$$

$$\dfrac{\delta P_{ij} }{\delta p_k}\dfrac{p_j}{P_{ij}}=\beta p_kP_{ij}$$

This means that the lower the price, the lower the own price elasticity of demand.

This means that mark ups are higher for cheaper goods, which doesnâ€™t always match reality.

This can be adjusted by changing the form. For example we could use $$\ln p$$ or $$p^2$$.

However, we are still getting the shape by assumption.

### Getting aggregate market shares

$$s_j = \dfrac{1}{n}\sum_i P_{ij}$$

## Modelling homogeneous preferences with nested logit

### Nested logit

With IID all goods are equal substitutes. If prices rise customers will switch to others in proportion to market size.

In practice if the price of a cheap car rises, there will be more substitution to other cheap cars than expensive cars.

### Estimating the nested logit model

As with the multinomial logit model, we have prices, market shares and product characteristics.

## Modelling heterogeneous preferences with individual characteristics

### Recap

In the multinomial logit model we had:

$$P_{ij}=\dfrac{e^{\Theta z_j}}{1+\sum_{k=1} e^{\Theta z_k }}$$

If $$z_j$$ just includes product characteristics then we have homogeneous preferences.

We can include customer level data in $$z_j$$, for example individual income, location, age etc.

### Estimation

As before, we want product characteristics and prices.

However rather than market share we instead use customer level information.

## Modelling heterogeneous preferences with mixed logit

### Modelling heterogeneous preferences with mixed logit

Using demographic data