here are my solutions for problem set 3.
Class A: Please note the difference in deriving the Euler Equation in exercise 2(a). I was a bit too sloppy when diving both sides by epsilon (y_t+1 + p_t+1)/ p_t as Sara rightly pointed out during the class.
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4 comments:
Dear Andreas,
Why do you actually need the Envelope theorem to state that
V'(a)=(1+r)u'(c) on page 1?
Be careful, you can't simply differentiate through the max operator in the Bellman Equation.
Look at the Bellman Equation. a_t+1 is actually a function of the state a_t. Let's call the policy function a_t+1 = g(a_t). Then
V(a_t) = u[(1+r)a_t + y_t - g(a_t)] + beta E_t V(g(a_t))
Now if I calculate V'(a_t), not only do I have the direct effect of a_t on the value function
(1+r)u'(c_t)
but also the indirect effect through the adjustment of the optimal choice of a_t+1 in response to a change in the state variable ( that's where the max operator comes into play )
[u'(c_t) - beta E_t V'(a_t+1) ] g'(a_t)
But by our first order condition for a_t+1, we know that this indirect effect is zero ( the term in the square brackets is zero for optimal choices of a_t+1). This is known as an Envelope Condition.
Hi Andreas,
could you give us your gmail email address?
Thks
Andreas,
In your answer to question 3 on this set how do you get the differences in expectations for future income levels as E(t)y(t+j) - E(t-1)y(t+j) = Rho^j v(t)?
Is it derivable from the maths or does it just come from the structure of the model?
Thanks,
James
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