Consumption Smoothing - Model

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]

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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