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

Resources:
http://www.econ2.jhu.edu/people/ccarroll/HabitsEconLett.pdf
http://www.isaacbaley.com/uploads/6/7/3/5/6735245/lecture_7_baley.pdf