I. Module 3 Homework 1 [practice]: Habits

In this assignment you'll explore a simple adjustment to the standard power-utility model. The power utility model needed variation in expected consumption growth or the variance of consumption growth to generate variation in the risk free rate or the risk premium. The utility function in this assignment generates variation in interest rates and risk premiums internally -- with no variation at all in the expected value or variance of consumption growth! Moreover, it captures the basic facts of business cycles and panics. 

Suppose the utility function is

 may represent an amount that the investor has borrowed, and there are huge costs of bankruptcy if he does not repay it. Or the investor could be a university endowment that has to pay  to its tenured professors, or a defined-benefit pension fund that must pay to retirees. could also represent a backstop level of consumption that the investor is simply not willing to risk no matter what ("I'd rather die than fly commercial, honey" -- overheard at hedge fund cocktail party.)

This small variation gives you practice with the transformations we did in lecture. It is also very interesting. The power utility model needed variation in expected consumption growth or the variance of consumption growth to generate variation in the risk free rate or the risk premium. This utility function generates variation in interest rates and risk premiums internally -- with no variation at all in the expected value or variance of consumption growth! Moreover, it captures the basic facts of business cycles and panics.

(This is a special case of " habit persistence" utility. In general, we also let  adapt slowly to past consumption, yielding a bit more realism in the dynamics.)