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

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

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

<p>I use a value function: </p>

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

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

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

<p>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: </p>

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

<p>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 $$U_m(c_t,m_t)=V_{m}(\omega_t,m_t)$$ </p>

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

<p>Can anyone help me with the time subscript issue?  </p>

<p>*****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*****</p>

CC BY-SA 3.0
2016-03-04T22:54:20.457Z
I don't remember writing a book...
Graeme Walsh
CC BY-SA 3.0
2016-03-21T00:13:46.057Z
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?
BB King
<macroeconomics>
Question about M.I.U. model - Walsh
123
# Question about M.I.U. model - Walsh

(Asked by: username_0 on 2016-03-04)

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\*\*\*\*\*

**Tags:** macroeconomics
> **Comments:**
> **username_1**: I don't remember writing a book...  
> **username_2**: 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?  

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## 0 answers

