<p>Your calculation has two typos a) you type two times the minus sign related to $f'$ and b) you write $(1+\delta)$ instead of $(1-\delta)$. If we correct for these we have </p>

<p>$$u_{c}(c_{t},m_{t}) + \beta V_{\omega}(\omega_{t+1})\left[\frac{f'(\omega_{t}-c_{t}-m_{t}-b_{t})}{1+n}(-1)+\frac{1-\delta}{1+n}(-1)\right]$$</p>

<p>Taking out the minus sign and $1/(1+n)$, and compacting $f'(\omega_{t}-c_{t}-m_{t}-b_{t})=f_k(k_t)$  we have </p>

<p>$$u_{c}(c_{t},m_{t}) - \frac {\beta}{1+n}V_{\omega}(\omega_{t+1})\left[f_k(k_t)+1-\delta\right]$$</p>

<p>which is the expression in the book. </p>

<p><strong>As regards the definition of output per worker:</strong><br>
Given the clarification in the comments, aggregate output during period $t$ is</p>

<p>$$Y_t = F(K_{t-1}, N_{t})$$</p>

<p>while $N_t = N_{t-1}(1+n)$.</p>

<p>I guess constant returns to scale are assumed so per capita magnitudes are</p>

<p>$$y_t = \frac{Y_t}{N_t}  = F\left (\frac{K_{t-1}}{N_t}, \frac{N_{t}}{N_t}\right ) = F\left (\frac{K_{t-1}}{N_{t-1}(1+n)}, 1\right )$$</p>

<p>Now, consider the notational/conceptual problem here: we will tend "automatically" to write $K_{t-1}/N_{t-1} \equiv k_{t-1}$ due of the same index, <em>but $K_{t-1}/N_{t-1}$ is economically meaningless because $K_{t-1}$ is not combined in production with $N_{t-1}$</em>. This ratio describes "capital used in period $t$ per worker in period $t-1$".  </p>

<p>Anyway, <em>if</em> we clearly declare in building the model that we define $k_{t-1}$ in this way, then we end up with </p>

<p>$$y_t = f\left(\frac {k_{t-1}}{1+n}\right)$$</p>

<p>and the budget constraint for period $t$ is correct while the one for period $t+1$ is not. How this affects the first-order condition with respect to consumption?</p>

<p>(Notice that with these notational coventions, "capital per worker during period $t$" is $k_{t-1}/(1+n)$ -<a href="https://economics.stackexchange.com/a/6003/61">I have argued elsewhere</a> why it is preferable for modellers to adopt the meaning that $k_{t-1}$ represents the value <em>at the beginning</em> of period $t-1$ and so is used in the production of period $t-1$).</p>

CC BY-SA 3.0
2017-07-08T11:46:05.357Z
Thank you so much for the help, much appreciated. Do you have any thoughts about the output per worker notation? Walsh seems to be switching between two different expressions and I don't know if I am just misunderstanding him or if it's a typo.
user11767
CC BY-SA 3.0
2017-07-08T12:01:35.330Z
@BenBernke What does $k_{t-1}$ stands for in this book? Capital _at the end_ or _at the beginning_ of period $t-1$?
Alecos Papadopoulos
CC BY-SA 3.0
2017-07-08T13:09:54.487Z
$K_{t-1}$ is the aggregate stock of capital in the beginning of period $t$ and $k_{t-1}$ is the per capita capital stock in the beginning of period $t$.
user11767
CC BY-SA 3.0
2017-07-08T11:32:55.333Z
2017-07-08T13:42:06.797Z
2017-07-08T13:42:06.797Z
Alecos Papadopoulos
<p>I am reading Walsh's (2003) book on monetary economics. Specifically the chapter on money in utility function. I understand the basics of a value functions but I can't seem to get the same results as the author. </p>

<p><a href="https://i.sstatic.net/aqRTb.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/aqRTb.png" alt="enter image description here"></a> </p>

<p>Where <a href="https://i.sstatic.net/7c16e.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/7c16e.png" alt="enter image description here"></a> </p>

<p>I.e the per capita budget constraint. He then finds an expression for $w_{t+1}$:<a href="https://i.sstatic.net/IOz1n.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/IOz1n.png" alt="enter image description here"></a></p>

<p>This is the first source of my confusion. Previously he defines output per worker as a function of capital per worker, i.e $y_{t}=f(\frac{k_{t-1}}{1+n})$ where $n$ is the population growth rate. But all of a sudden he changes it to $\frac{f(k_{t-1})}{1+n}$. I am aware that there are many typos in this book, is this just one of them or am I missing something trivial?</p>

<p>Either way, he uses the budget constraint to express $k_{t}$ as $w_{t}-c_{t}-m_{t}-b_{t}$ and the definition of $w_{t+1}$ to express the value function as:</p>

<p><a href="https://i.sstatic.net/16Zcu.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/16Zcu.png" alt="enter image description here"></a></p>

<p>Now, I don't know if it is because I am sleep deprived or because there is a typo, but I just can't seem to get the same results as Walsh. E.g, differentiating w.r.t $c_{t}$ I get:</p>

<p>$u_{c}(c_{t},m_{t}) + \beta*V_{w}(w_{t+1})[\frac{-f'(w_{t}-c_{t}-m_{t}-b_{t})}{1+n}(-1)+\frac{1+\delta}{1+n}(-1)]$</p>

<p>While Walsh gets </p>

<p><a href="https://i.sstatic.net/PvRdW.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/PvRdW.png" alt="enter image description here"></a></p>

<p>Am I missing something obvious or is there a typo?</p>

<p>Thanks in advance!</p>

<dynamic-optimization>
Money in utility function - Value function
user11767
# Money in utility function - Value function

(Asked by: username_0 on 2017-07-07)

I am reading Walsh's (2003) book on monetary economics. Specifically the chapter on money in utility function. I understand the basics of a value functions but I can't seem to get the same results as the author.

[![enter image description here](https://i.sstatic.net/aqRTb.png)](https://i.sstatic.net/aqRTb.png)

Where [![enter image description here](https://i.sstatic.net/7c16e.png)](https://i.sstatic.net/7c16e.png)

I.e the per capita budget constraint. He then finds an expression for $w\_{t+1}$:[![enter image description here](https://i.sstatic.net/IOz1n.png)](https://i.sstatic.net/IOz1n.png)

This is the first source of my confusion. Previously he defines output per worker as a function of capital per worker, i.e $y\_{t}=f(\frac{k\_{t-1}}{1+n})$ where $n$ is the population growth rate. But all of a sudden he changes it to $\frac{f(k\_{t-1})}{1+n}$. I am aware that there are many typos in this book, is this just one of them or am I missing something trivial?

Either way, he uses the budget constraint to express $k\_{t}$ as $w\_{t}-c\_{t}-m\_{t}-b\_{t}$ and the definition of $w\_{t+1}$ to express the value function as:

[![enter image description here](https://i.sstatic.net/16Zcu.png)](https://i.sstatic.net/16Zcu.png)

Now, I don't know if it is because I am sleep deprived or because there is a typo, but I just can't seem to get the same results as Walsh. E.g, differentiating w.r.t $c\_{t}$ I get:

$u\_{c}(c\_{t},m\_{t}) + \beta\*V\_{w}(w\_{t+1})[\frac{-f'(w\_{t}-c\_{t}-m\_{t}-b\_{t})}{1+n}(-1)+\frac{1+\delta}{1+n}(-1)]$

While Walsh gets

[![enter image description here](https://i.sstatic.net/PvRdW.png)](https://i.sstatic.net/PvRdW.png)

Am I missing something obvious or is there a typo?

Thanks in advance!

**Tags:** dynamic-optimization


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## 1 answers

### Answer by username_1 on 2017-07-08. Score: 1 | Accepted answer

Your calculation has two typos a) you type two times the minus sign related to $f'$ and b) you write $(1+\delta)$ instead of $(1-\delta)$. If we correct for these we have

$$u\_{c}(c\_{t},m\_{t}) + \beta V\_{\omega}(\omega\_{t+1})\left[\frac{f'(\omega\_{t}-c\_{t}-m\_{t}-b\_{t})}{1+n}(-1)+\frac{1-\delta}{1+n}(-1)\right]$$

Taking out the minus sign and $1/(1+n)$, and compacting $f'(\omega\_{t}-c\_{t}-m\_{t}-b\_{t})=f\_k(k\_t)$ we have

$$u\_{c}(c\_{t},m\_{t}) - \frac {\beta}{1+n}V\_{\omega}(\omega\_{t+1})\left[f\_k(k\_t)+1-\delta\right]$$

which is the expression in the book.

**As regards the definition of output per worker:**  
Given the clarification in the comments, aggregate output during period $t$ is

$$Y\_t = F(K\_{t-1}, N\_{t})$$

while $N\_t = N\_{t-1}(1+n)$.

I guess constant returns to scale are assumed so per capita magnitudes are

$$y\_t = \frac{Y\_t}{N\_t} = F\left (\frac{K\_{t-1}}{N\_t}, \frac{N\_{t}}{N\_t}\right ) = F\left (\frac{K\_{t-1}}{N\_{t-1}(1+n)}, 1\right )$$

Now, consider the notational/conceptual problem here: we will tend "automatically" to write $K\_{t-1}/N\_{t-1} \equiv k\_{t-1}$ due of the same index, *but $K\_{t-1}/N\_{t-1}$ is economically meaningless because $K\_{t-1}$ is not combined in production with $N\_{t-1}$*. This ratio describes "capital used in period $t$ per worker in period $t-1$".

Anyway, *if* we clearly declare in building the model that we define $k\_{t-1}$ in this way, then we end up with

$$y\_t = f\left(\frac {k\_{t-1}}{1+n}\right)$$

and the budget constraint for period $t$ is correct while the one for period $t+1$ is not. How this affects the first-order condition with respect to consumption?

(Notice that with these notational coventions, "capital per worker during period $t$" is $k\_{t-1}/(1+n)$ -[I have argued elsewhere](https://economics.stackexchange.com/a/6003/61) why it is preferable for modellers to adopt the meaning that $k\_{t-1}$ represents the value *at the beginning* of period $t-1$ and so is used in the production of period $t-1$).

> **Comments:**
> **username_0**: Thank you so much for the help, much appreciated. Do you have any thoughts about the output per worker notation? Walsh seems to be switching between two different expressions and I don't know if I am just misunderstanding him or if it's a typo.  
> **username_1**: @BenBernke What does $k_{t-1}$ stands for in this book? Capital _at the end_ or _at the beginning_ of period $t-1$?  
> **username_0**: $K_{t-1}$ is the aggregate stock of capital in the beginning of period $t$ and $k_{t-1}$ is the per capita capital stock in the beginning of period $t$.  

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