3
$\begingroup$

I am solving practice problems in Walsh. This is from 3rd edition page 87. Question 1.

I have max: $$U=\sum\beta^iU(c_{t+i},\frac{M_{t+i}}{P_{t+i}})$$

s.t. $$\omega_t=f(k_{t-1})+(1-\delta)k_{t-1}+(1+r_{t-1})b_{t-1}+m_t$$

I use a value function:

$$V(\omega_t,m_t)= max [U(c_{t},m_t)+\beta V(\omega_{t+1},m_{t+1})]$$

I then take F.O.C. w.r.t:$\space\space\space c_t,k_t,b_t,m_{t+1}$

My issue is when taking F.O.C. wrt $k_t$ I am getting $V_{\omega}(V_{\omega_{t+1}},m_{t+1})=\lambda_t$

but I should be getting my time subscript as t here instead of t+1. This is throwing off the next part of the question:

Show that $$\ \frac{U_m(c_{t+1},m_{t+1})}{U_c(c_{t+1},m_{t+1})}=i_t$$

Proving this ^^^ is easily done by rearranging and substituting my FOC. However, I need to correct my FOC for $k_t$ before the solution actually works since I am equating: $$U_c(c_t,m_t)=V_{\omega}(\omega_t,m_t)$$ and $$U_m(c_t,m_t)=V_{m}(\omega_t,m_t)$$

to achieve my solution. For now, my time subscript issue with k is creating a time subscript issue with my solution.

Can anyone help me with the time subscript issue?

*****I can add more of my solution if necessary but it is just FOC. Also - I have solved the problem and checked against a solutions manual. I have done every other part correctly - just need to fix the time subscripts for k*****

$\endgroup$
2
  • $\begingroup$ I don't remember writing a book... $\endgroup$ Mar 4, 2016 at 22:54
  • $\begingroup$ What is w here? Also, do you have an expression for f(k_t-1) in the constraint? Do you know why consumption doesn't feauture in the constraint? $\endgroup$
    – BB King
    Mar 21, 2016 at 0:13

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.