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In economics, dynamic inconsistency, or time inconsistency, describes a situation where a decision-maker's preferences change over time, such that what is preferred at one point in time is inconsistent with what is preferred at another point in time. It is often easiest to think about preferences over time in this context by thinking of decision-makers as being made up of many different "selves", with each self representing the decision-maker at a different point in time. So, for example, there is my today self, my tomorrow self, my next Tuesday self, my year from now self, etc. The inconsistency will occur when somehow the preferences of some of the selves are not aligned with each other.
One type of inconsistency is more closely affiliated with game theory, and "dynamic inconsistency" is the more commonly used terminology in this case. Another type of inconsistency is more closely affiliated with behavioral economics, and "time inconsistency" is the more commonly used terminology there.
In game theory
In the context of game theory, dynamic inconsistency is a situation in a dynamic game where a player's best plan for some future period will not be optimal when that future period arrives. A dynamically inconsistent game is subgame imperfect. In this context, the inconsistency is primarily about commitment and credible threats.
For example, a firm might want to commit itself to dramatically dropping the price of a product it sells if a rival firm enters its market. If this threat were credible, it would discourage the rival from entering. However, the firm might not be able to commit its future self to taking such an action because if the rival does in fact end up entering, the firm's future self might determine that, given the fact that the rival is now actually in the market and there is no point in trying to discourage entry, it is now not in its interest to dramaticatically drop the price. As such, the threat would not be credible. The present self of the firm has preferences that would have the future self be committed to the threat, but the future self has preferences that have it not carry out the threat. Hence, the dynamic inconsistency.
In behavioral economics
In the context of behavioral economics, time inconsistency is related to how much each different self of a decision-maker cares about herself and all of the selves that will then follow her, relative to each other. Generally, the view is that now has especially high value for any decision-maker at any point in time, so that then any given present self will care too much about herself and not enough about her future selves. In this context, the inconsistency is primarily about self control, and it relates to a variety of topics including procrastination, addiction, efforts at weight loss, and saving for retirement.
Time inconsistency basically means that there is disagreement between a decision-maker's different selves about what actions should be taken. Formally, consider an economic model with different mathematical weightings placed on the utilities of each self. Consider the possibility that for any given self, the weightings that that self places on all the utilities could be different than the weightings that another given self places on all the utilities. The important consideration now is the relative weighting between two particular utilities. Will this relative weighting be the same for one given self as it is for a different given self? If it is, then we have a case of time consistency. If the relative weightings of all pairs of utilities are all the same for all given selves, then the decision-maker has time-consistent preferences. If there exists a case of one relative weighting of utilities where one self has a different relative weighting of those utilities than another self has, then we have a case of time inconsistency and the decision-maker will be said to have time-inconsistent preferences.
For example, consider having the choice between getting the day off of work tomorrow or getting a day and a half off of work one month from now. Suppose you would choose one day off tomorrow. Now suppose that you were asked to make that same choice ten years ago. That is, you were asked then whether you would prefer getting one day off in ten years or getting one and a half days off in ten years and one month. Suppose that then you would have taken the day and a half off. This would be a case of time inconsistency because your relative preferences for tomorrow versus one month from now would be different at two different points in time -- namely now versus ten years ago.
It is common in economic models that involve decision-making over time to assume that decision-makers are exponential discounters, and, in fact, this treatment of decision-making over time is generally what students are taught. Exponential discounting happens to have the characteristic of yielding time-consistent preferences. Often, exponential discounting and, more generally, time-consistent preferences are considered to be a part of rational choice theory because of the fact that these imply that all of a decision-makers' selves will agree with the choices made by each self. However, empirical research on hyperbolic discounting makes a strong case that time inconsistency is, in fact, standard in human preferences. This would imply disagreement by people's different selves on decisions made and a rejection of the time consistency aspect of rational choice theory.
One way that time-inconsistent preferences have been formally introduced into economic models is by first giving the decision-maker standard exponentially discounted preferences, and then adding an additional term that heavily discounts any time that is not now. Preferences of this sort have been called "present-biased preferences".
Stylized examples
Students, the night before an exam, often wish that the exam could be put off for one more day. If asked on that night, such students might agree to commit to paying, say, $10 on the day of the exam for it to be held the next day. Months before the exam is held, however, students generally do not care much about having the exam put off for one day. And, in fact, if the students were made the same offer at the beginning of the term, that is, they could have the exam put off for one day by committing during registration to pay $10 on the day of the exam, they probably would reject that offer. It is the exact same choice, just made a different points in time, and because the outcome would change depending on the point in time, the students would exhibit time inconsistency.
Each day smokers face a dynamic inconsistency: their best plan is to enjoy smoking today, but to quit tomorrow in order to get health benefits. However, the next day, the plan is the same; enjoy smoking today and quit tomorrow. This goes on, and they never give up, even though they plan to, hence the inconsistency.
Government policy makers also suffer from dynamic inconsistency, as they are best off promising that there will be lower inflation tomorrow. But once tomorrow comes lowering inflation may have negative effects, such as increasing unemployment, so they do not make much effort to lower it. This is why independent central banks can be advantageous for a country: they worry about making decisions for the greater good, not to keep government policy makers popular.
"Dynamic inconsistency"