Model
Robert Hall (1978) formalized Friedman's idea. By taking into account the diminishing returns to consumption, and therefore, assuming a concave utility function, he showed that agents optimally would choose to keep a stable path of consumption.
With (cf. Hall's paper)
- being the mathematical expectation conditional on all information available in
- being the agent's rate of time preference
- being the real rate of interest in
- being the strictly concave one-period utility function
- being the consumption in
- being the earnings in
- being the assets, apart from human capital, in .
agents choose the consumption path that maximizes:
Subject to a sequence of budget constraints:
The first order necessary condition in this case will be:
By assuming that we obtain, for the previous equation:
Which, due to the concavity of the utility function, implies:
Thus, rational agents would expect to achieve the same consumption in every period.
Hall also showed that for a quadratic utility function, the optimal consumption is equal to:
![c_{t}=\left \left[ E_{t}\sum_{i=0}^{\infty
}\left( \frac{1}{1+r}\right) ^{i}y_{t+i}+A_{t}\right]](http://upload.wikimedia.org/math/4/a/7/4a792c13f7c40464834ef87d825d1101.png)
This expression shows that agents choose to consume a fraction of their present discounted value of their human and financial wealth.
Read more about this topic: Consumption Smoothing
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