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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*****

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  • $\begingroup$ 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

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