<p>Two points:</p>
<ol>
<li><p>As Richard Hardy points out <a href="https://economics.stackexchange.com/questions/48627/wealth-in-the-utility-function#comment78534_48627">in the comment</a>, there is not really any depiction of wealth in the question.</p>
</li>
<li><p>I am very curious which model this is. I can not think of any models that include lagged capital in the utility function. The form of the Euler is however very similar to a Cash-in-Advance model (which is interesting).</p>
</li>
</ol>
<p>Following the way you have written the Bellman, but getting rid of the multiplier, since according to the problem, the constraint binds.</p>
<p><span class="math-container">$V[k_t, k_{t-1}] = U[AF[k_t] +(1-\delta)k_t, k_{t-1}]+\beta V[k_{t+1}, k_t]$</span></p>
<p>FOC(with respect to <span class="math-container">$(k_{t+1})$</span>):</p>
<p><span class="math-container">$U_1[c_t, k_{t-1}]=\beta V_1[k_{t+1}, k_t]$</span></p>
<p>Envelope Condition 1 (with respect to <span class="math-container">$k_t$</span>):</p>
<p><span class="math-container">$V_1[k_t, k_{t-1}] = U_1[c_t, k_{t-1}](AF_1[k_t] + 1-\delta) + \beta V_2[k_{t+1}, k_t]$</span></p>
<p>Envelope Condition 2 (with respect to <span class="math-container">$k_{t-1}$</span>):</p>
<p><span class="math-container">$V_2[k_t, k_{t-1}] = U_2[c_t, k_{t-1}]$</span></p>
<p>Iterate Envelope conditions forward by one period and use the FOC to get (assuming finite consumption):</p>
<p><span class="math-container">$U_1[c_t, k_{t-1}]/U_1[c_{t+1}, k_{t}] = \beta  (AF_1[k_{t+1}] + 1-\delta) + \beta^2 U_2[c_{t+2}, k_{t+1}]/U_1[c_{t+1}, k_{t}]$</span></p>
<p>Now impose steady state (denoting at steady state <span class="math-container">$k_{t+1}=k_t=k$</span>) and you have:</p>
<p><span class="math-container">$1 = \beta  (AF_1[k] + 1-\delta) + \beta^2 \frac{U_2[AF[k]-\delta k, k]}{U_1[AF[k]-\delta k, k]} $</span></p>

CC BY-SA 4.0
2021-12-07T10:35:06.350Z
Thank you for your input. This question is taken from https://lhendricks.org/econ720/ih1/IH1_PS-answers.pdf, maybe it will be clearer now
Maybeline Lee
CC BY-SA 4.0
2021-12-06T17:53:04.103Z
2021-12-06T17:53:04.103Z
erik
<p>Suppose there is a representative household of unit mass who lives forever. Preferences are given as: <span class="math-container">$$\sum\beta^tu(c_t,k_{t-1})$$</span>
Technology is given as: <span class="math-container">$$k_{t+1}=AF(K_t,L_t)+(1-\delta)K_t-c_t$$</span> The production function has constant returns to scale.</p>
<p>The household’s dynamic program is: <span class="math-container">$$V (k, z) = \max u(c, z) + \beta V (k', k) + \lambda [R k + w − c − k']$$</span> or alternatively <span class="math-container">$L=\beta^tU(c_t,k_{t-1})+\lambda_t[R_t k_t + w_t − c_t − k_{t+1}]$</span></p>
<p>FOC: <span class="math-container">$U_c(c,z)=\beta R'U_c(c',z')+\beta^2U_z(c'',z'')$</span>, where <span class="math-container">$z'=k$</span>.</p>
<p>My question is why is steady-state capital stock written like this?
<span class="math-container">$$1=\beta[f'(k)+1-\delta]+\beta^2\frac{u_z(f(k)-\delta k,k)}{u_c(f(k)-\delta k,k)}$$</span></p>

CC BY-SA 4.0
2021-12-05T06:25:22.910Z
Have you looked at the Envelope condition?
erik
CC BY-SA 4.0
2021-12-05T12:04:48.487Z
@erik yes, the EC are: $V_k(k,z)=\beta V_z(k',z')+\lambda R$ and $V_z(k,z)=U_z(c,z)$
Maybeline Lee
CC BY-SA 4.0
2021-12-06T10:34:00.627Z
The title concerns wealth, but does that reflect the body of your question?
Richard Hardy
CC BY-SA 4.0
2021-12-07T10:34:28.680Z
I am attaching the reference for this exercise: https://lhendricks.org/econ720/ih1/IH1_PS-answers.pdf you can read the details here
Maybeline Lee
<macroeconomics><wealth><bellman-equations>
Wealth in the utility function
Maybeline Lee
# Wealth in the utility function

(Asked by: username_0 on 2021-12-04)

Suppose there is a representative household of unit mass who lives forever. Preferences are given as: $$\sum\beta^tu(c\_t,k\_{t-1})$$
Technology is given as: $$k\_{t+1}=AF(K\_t,L\_t)+(1-\delta)K\_t-c\_t$$ The production function has constant returns to scale.

The household’s dynamic program is: $$V (k, z) = \max u(c, z) + \beta V (k', k) + \lambda [R k + w − c − k']$$ or alternatively $L=\beta^tU(c\_t,k\_{t-1})+\lambda\_t[R\_t k\_t + w\_t − c\_t − k\_{t+1}]$

FOC: $U\_c(c,z)=\beta R'U\_c(c',z')+\beta^2U\_z(c'',z'')$, where $z'=k$.

My question is why is steady-state capital stock written like this?
$$1=\beta[f'(k)+1-\delta]+\beta^2\frac{u\_z(f(k)-\delta k,k)}{u\_c(f(k)-\delta k,k)}$$

**Tags:** macroeconomics, wealth, bellman-equations
> **Comments:**
> **username_1**: Have you looked at the Envelope condition?  
> **username_0**: @erik yes, the EC are: $V_k(k,z)=\beta V_z(k',z')+\lambda R$ and $V_z(k,z)=U_z(c,z)$  
> **username_2**: The title concerns wealth, but does that reflect the body of your question?  
> **username_0**: I am attaching the reference for this exercise: https://lhendricks.org/econ720/ih1/IH1_PS-answers.pdf you can read the details here  

---

## 1 answers

### Answer by username_1 on 2021-12-06. Score: 2 | Accepted answer

Two points:

1. As Richard Hardy points out [in the comment](https://economics.stackexchange.com/questions/48627/wealth-in-the-utility-function#comment78534_48627), there is not really any depiction of wealth in the question.
2. I am very curious which model this is. I can not think of any models that include lagged capital in the utility function. The form of the Euler is however very similar to a Cash-in-Advance model (which is interesting).

Following the way you have written the Bellman, but getting rid of the multiplier, since according to the problem, the constraint binds.

$V[k\_t, k\_{t-1}] = U[AF[k\_t] +(1-\delta)k\_t, k\_{t-1}]+\beta V[k\_{t+1}, k\_t]$

FOC(with respect to $(k\_{t+1})$):

$U\_1[c\_t, k\_{t-1}]=\beta V\_1[k\_{t+1}, k\_t]$

Envelope Condition 1 (with respect to $k\_t$):

$V\_1[k\_t, k\_{t-1}] = U\_1[c\_t, k\_{t-1}](AF\_1[k\_t] + 1-\delta) + \beta V\_2[k\_{t+1}, k\_t]$

Envelope Condition 2 (with respect to $k\_{t-1}$):

$V\_2[k\_t, k\_{t-1}] = U\_2[c\_t, k\_{t-1}]$

Iterate Envelope conditions forward by one period and use the FOC to get (assuming finite consumption):

$U\_1[c\_t, k\_{t-1}]/U\_1[c\_{t+1}, k\_{t}] = \beta (AF\_1[k\_{t+1}] + 1-\delta) + \beta^2 U\_2[c\_{t+2}, k\_{t+1}]/U\_1[c\_{t+1}, k\_{t}]$

Now impose steady state (denoting at steady state $k\_{t+1}=k\_t=k$) and you have:

$1 = \beta (AF\_1[k] + 1-\delta) + \beta^2 \frac{U\_2[AF[k]-\delta k, k]}{U\_1[AF[k]-\delta k, k]} $

> **Comments:**
> **username_0**: Thank you for your input. This question is taken from https://lhendricks.org/econ720/ih1/IH1_PS-answers.pdf, maybe it will be clearer now  

---

