# How can I motivate a dynamic model using utility and consumer preference where only one good is affected by past consumption?

I've been looking how I can motivate the model I want to estimate using a utility optimization framework. Basically, I want to write a model using two goods: x and y, in two periods: 1 and 2, where the optimal amount of $x_2$ is negatively affected by the amount $x_1$ that was consumed in the previous period. For example, a person's preference between Italian (x) and Thai (y) restaurant at period t should depend on whether or not a person Italian food at period t-1, so consuming $Italian_{t-1}$ should negatively affect the optimal amount of $Italian_{t}$ by some proportional amount. The nature of the problem rules out the Discounted Utility Model because it assumes consumption independence. I read about alternative such as habit forming models,but these models describe the opposite of what I want and can lead to intractable. Any reference or help will be greatly appreciated.

I just wanted to clarify on Starfall's comment a bit. $\phi$ can be a bit confusing.

Let me appeal to Chris Carroll who uses the CRRA utility function $v(\frac{c}{x^{\gamma}})$ where $c$ is consumption, $x$ is the habit, and $\gamma$ is the "importance" of habits. In this framework, $\phi$ can be seen as the speed at which habits catch up to consumption. If $\phi = 0$, then the consumer is constantly maximizing their utility with a constant multiplicative factor in the denominator of the function's argument. If $\phi=1$, then habits collapse to last period's consumption.

There are a number of interesting features from this model. For instance, if persistence is low enough then the intertemporal elasticity of substitution tends to zero. It is used famously in the asset pricing literature to explain the excess return premium puzzle. The development of habit formation theory in microeconomics (both theory and econometrics) is slow but finally arriving.

Habit forming models should work fine for this if you just switch the sign with which the habit enters the utility function of the consumer. The usual habit model looks something like

$$\Delta X_t = \phi (C_t - X_t)$$

with $\phi > 0$ where $X_t$ is the consumer's habit at time $t$ and $C_t$ is consumption. Habit forming models usually take a standard utility function $u$ and say that consumer's time $t$ utility from consumption is given by $u(C_t - X_t)$. For your purposes, assuming the utility from consumption at time $t$ is given by $u(C_t + X_t)$ will do. With multiple goods instead of one, you may specify a moving habit for each good and put each habit in the same way into the multivariate utility function. Habit models may also assume that habit formation is "external", i.e the consumer makes decisions without taking into account that consumption decisions affect the habit process. For your purposes, you want the exact opposite: the consumer should notice the effect consumption decisions have on habit formation and act accordingly.

The point of this modification is that for large values of $X_t$ it reduces the marginal utility of consumption, so (for instance) if the consumer has to work and suffer some marginal disutility to acquire income and there are no savings, they will work less in times of high $X_t$ compared to times of low $X_t$. If there are savings, then the consumer will still consume less in times of high $X_t$, because the marginal utility gain from doing so is small.