# Money in utility function - Value function

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.

Where

I.e the per capita budget constraint. He then finds an expression for $w_{t+1}$:

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:

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

Am I missing something obvious or is there a typo?

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 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 $t-1$).

• 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. Jul 8, 2017 at 11:46
• @BenBernke What does $k_{t-1}$ stands for in this book? Capital at the end or at the beginning of period $t-1$? Jul 8, 2017 at 12:01
• $K_{t-1}$ 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 $t$. Jul 8, 2017 at 13:09