2. ( 40 points) External consumption habit (one that is not internalized). Which of the habit formation is the best in explaining the data? Take the model as follows max At In(c - yet-1) (1) (2) s.t.:kt = (1 - 6)kt-1+ hit (3) Notice that now the consumption of a household i is denoted as of, with the aggregate consumption begin ct = fo cidi. So when you derive the equations you have to treat c as exogenous and treat of as your choice variable! y is the habit parameter, @ is the productivity parameter with and autoregressive process Of = (1-pe)0 + poet-1 +et and the investment specific technology shock is yt = (1-py)7+ py 7-1+ 5t. Notice there are 2 shocks now, it is not exactly the model of the lecture. A. (10 points) Set up a Lagrangian or a Bellman equation. Take the first order conditions and set equal to zero. Derive Euler equation. Do the count of equations and variables, does it fit? B. (10 points) Calculate steady states by dropping time indexes and expressing variables as of parameter values only. C. (10 points) What do you think, how does the habit that households do not take into account differ from the one where they do take it into account? (5 points) How does the habit where the effect is not internalize differ from the case without habit at all? (5 points) D. (10 points) How would you set up a test between three models of habit: a) without a habit, b) with external habit, c) with internal habit in real world data? (5 points) What do you think is the likely result, why? (5 points)