<p>It is difficult to provide definitive answer, if there is possibility that something is missing.</p>
<ul>
<li><p>it could be a trick question and <span class="math-container">$w$</span> could be zero given the assumptions, indeed if that is the production function and we assume perfect competition <span class="math-container">$w_t=0$</span>.</p>
<p>Note that with <span class="math-container">$w_t=0$</span> the problem still solves and it is even simplified, you will have <span class="math-container">$r=A$</span> and end up with:</p>
</li>
</ul>
<p><span class="math-container">$$\sum \beta^t \ln(c_t), \quad s.t. (1+A)s_t =c_t+s_{t+1}$$</span></p>
<p>it is not realistic but as a practice problem you can solve this.</p>
<ul>
<li><p>if the 'hint' was correct you could just substitute the expression <span class="math-container">$w=(1-\alpha) A$</span> and <span class="math-container">$r=(1-\alpha)A$</span> into the budget constraint, and then solve the problem, it does not seem to be consistent with the previous assumptions but taking the teaching assistant at his word maybe its just poorly worded problem (I saw some professors making homework asking students to derive supply curve for monopoly which is actually not possible but what the problem meant was just to derive expression for optimum quantity).</p>
</li>
<li><p>does the problem perchance says that production and capital are measured in per capita terms? In such case the output relation would actually be <span class="math-container">$Y/L=AK$</span> and you can get to 'regular' production function.</p>
</li>
</ul>
<p>I think one of the options above is probably correct, but I would recommend contacting the professor.</p>

CC BY-SA 4.0
2023-07-20T01:22:42.883Z
Thank you for your answer! I got advised by a classmate to use the fact that “all income in the economy is used to pay production factors” for the non-perfect competition case, which happened to be the official solution, which I just posted as an answer.
Nicolas Torres
CC BY-SA 4.0
2023-07-18T00:37:13.430Z
2023-07-18T00:37:13.430Z
1muflon1
<p>For the perfect competition case, it was my trivial solution: set price equal to marginal productivity.</p>
<p><span class="math-container">$Y_t = A_t K_t$</span></p>
<p><span class="math-container">$r_t = \frac{\partial Y_t}{\partial K_t} = A_t$</span></p>
<p><span class="math-container">$w_t = \frac{\partial Y_t}{\partial L_t} = 0$</span></p>
<p>In the next exercise, suppose perfect competition no longer holds, such that capital is paid a fraction <span class="math-container">$\alpha \in (0,1)$</span> of its <span class="math-container">$MP$</span>.</p>
<p><span class="math-container">$\implies r_t = \alpha \frac{\partial Y_t}{\partial K_t} = \alpha A_t$</span></p>
<p>Now here is the key part: We use the equation that states that all income in the economy is used to pay production factors, to solve for <span class="math-container">$w_t$</span>.</p>
<p><span class="math-container">$Y_t = w_t L_t + r_t K_t$</span></p>
<p><span class="math-container">$\implies A_t K_t = w_t L_t + \alpha A_t K_t$</span></p>
<p><span class="math-container">$\implies w_t L_t = (1-\alpha) A_t K_t$</span></p>
<p><span class="math-container">$\therefore w_t = (1-\alpha) A_t \kappa_t$</span></p>
<p>Here <span class="math-container">$\kappa_t$</span> is capital per capita.</p>
<p>I know a rational firm wouldn’t hire workers at that salary when their <span class="math-container">$MP$</span> is always <span class="math-container">$0$</span>, but that was the official solution for my particular problem set.</p>

CC BY-SA 4.0
2023-07-20T01:20:07.157Z
2023-07-20T01:52:23.440Z
2023-07-20T01:52:23.440Z
Nicolas Torres
<p>I'm trying to solve this exercise for Macroeconomics III (the course covers long-run growth models).</p>
<p>Find the wage <span class="math-container">$w_t$</span> and the interest rate <span class="math-container">$r_t$</span> in the <span class="math-container">$AK$</span> model, i.e., the production function is</p>
<p><span class="math-container">$Y_t = A_t K_t$</span></p>
<p>Assume the factor markets are in perfect competition.</p>
<p>Since the production function doesn't depend on labor, shouldn't the salary <span class="math-container">$w$</span> just be <span class="math-container">$0$</span>, since the <span class="math-container">$MP$</span> of labor is <span class="math-container">$0$</span>?</p>
<p>The question can't be that trivial so there must be something I'm missing.</p>
<p>By <span class="math-container">$MP$</span> logic, the interest rate <span class="math-container">$r_t$</span> should just be <span class="math-container">$r_t = A_t$</span>, but again, this seems too trivial for my class.</p>
<p>I just asked the course's teacing assistant, and got told that the way they'd do it is to take the interest rate as a fraction of the capital's <span class="math-container">$MP$</span>, i.e. <span class="math-container">$r_t = \alpha A_t$</span>, where <span class="math-container">$\alpha \in (0,1)$</span>, and the wage as its complement, i.e. <span class="math-container">$w_t = (1-\alpha) A_t$</span>, but they're not actually sure.</p>
<p>However, I can't make sense of this.</p>
<p>Why would a firm hire workers at a (positive) salary, when they make zero contribution to production?</p>
<p>I'd appreciate any insight on how to find the factor prices for this case. (Every production function in the course depended on both factors, and was concave in both of them, so standard <span class="math-container">$\text{f.o.c.}$</span> logic used to work).</p>
<p>In a next exercise I'm asked to assume perfect competition no longer holds, so factors are only paid a fraction <span class="math-container">$\alpha \in (0,1)$</span> of their respective <span class="math-container">$MP$</span>'s, so wouldn't the fraction <span class="math-container">$\alpha$</span> thing apply for this next exercise and not the one I asked above?</p>
<p>In case it helps, consumers want to maximize their lifetime utility given by</p>
<p><span class="math-container">$W = \sum_{t=0}^{\infty} \beta^t \ln(c_t)$</span></p>
<p>subject to</p>
<p><span class="math-container">$w_t + (1+r_{t+1}) s_t = c_t + s_{t+1}$</span></p>
<p>where <span class="math-container">$c_t$</span> and <span class="math-container">$s_t$</span> are consumption and savings at time <span class="math-container">$t$</span>, respectively.</p>

CC BY-SA 4.0
2023-07-18T08:35:43.303Z
Do we need a new, separate tag for the AK model? If so, would you (as the tag's creator) care to write a tag Wiki and its excerpt?
Richard Hardy
CC BY-SA 4.0
2023-07-20T13:30:18.933Z
@RichardHardy Just added some info, waiting for it to be reviewed.
Nicolas Torres
<macroeconomics><economic-growth><endogenous-growth><long-run-macroeconomics><ak-model>
Find the salary and interest rate in the $AK$ model
Nicolas Torres
# Find the salary and interest rate in the $AK$ model

(Asked by: username_0 on 2023-07-17)

I'm trying to solve this exercise for Macroeconomics III (the course covers long-run growth models).

Find the wage $w\_t$ and the interest rate $r\_t$ in the $AK$ model, i.e., the production function is

$Y\_t = A\_t K\_t$

Assume the factor markets are in perfect competition.

Since the production function doesn't depend on labor, shouldn't the salary $w$ just be $0$, since the $MP$ of labor is $0$?

The question can't be that trivial so there must be something I'm missing.

By $MP$ logic, the interest rate $r\_t$ should just be $r\_t = A\_t$, but again, this seems too trivial for my class.

I just asked the course's teacing assistant, and got told that the way they'd do it is to take the interest rate as a fraction of the capital's $MP$, i.e. $r\_t = \alpha A\_t$, where $\alpha \in (0,1)$, and the wage as its complement, i.e. $w\_t = (1-\alpha) A\_t$, but they're not actually sure.

However, I can't make sense of this.

Why would a firm hire workers at a (positive) salary, when they make zero contribution to production?

I'd appreciate any insight on how to find the factor prices for this case. (Every production function in the course depended on both factors, and was concave in both of them, so standard $\text{f.o.c.}$ logic used to work).

In a next exercise I'm asked to assume perfect competition no longer holds, so factors are only paid a fraction $\alpha \in (0,1)$ of their respective $MP$'s, so wouldn't the fraction $\alpha$ thing apply for this next exercise and not the one I asked above?

In case it helps, consumers want to maximize their lifetime utility given by

$W = \sum\_{t=0}^{\infty} \beta^t \ln(c\_t)$

subject to

$w\_t + (1+r\_{t+1}) s\_t = c\_t + s\_{t+1}$

where $c\_t$ and $s\_t$ are consumption and savings at time $t$, respectively.

**Tags:** macroeconomics, economic-growth, endogenous-growth, long-run-macroeconomics, ak-model
> **Comments:**
> **username_1**: Do we need a new, separate tag for the AK model? If so, would you (as the tag's creator) care to write a tag Wiki and its excerpt?  
> **username_0**: @RichardHardy Just added some info, waiting for it to be reviewed.  

---

## 2 answers

### Answer by username_2 on 2023-07-18. Score: 2 | Accepted answer

It is difficult to provide definitive answer, if there is possibility that something is missing.

* it could be a trick question and $w$ could be zero given the assumptions, indeed if that is the production function and we assume perfect competition $w\_t=0$.

  Note that with $w\_t=0$ the problem still solves and it is even simplified, you will have $r=A$ and end up with:

$$\sum \beta^t \ln(c\_t), \quad s.t. (1+A)s\_t =c\_t+s\_{t+1}$$

it is not realistic but as a practice problem you can solve this.

* if the 'hint' was correct you could just substitute the expression $w=(1-\alpha) A$ and $r=(1-\alpha)A$ into the budget constraint, and then solve the problem, it does not seem to be consistent with the previous assumptions but taking the teaching assistant at his word maybe its just poorly worded problem (I saw some professors making homework asking students to derive supply curve for monopoly which is actually not possible but what the problem meant was just to derive expression for optimum quantity).
* does the problem perchance says that production and capital are measured in per capita terms? In such case the output relation would actually be $Y/L=AK$ and you can get to 'regular' production function.

I think one of the options above is probably correct, but I would recommend contacting the professor.

> **Comments:**
> **username_0**: Thank you for your answer! I got advised by a classmate to use the fact that “all income in the economy is used to pay production factors” for the non-perfect competition case, which happened to be the official solution, which I just posted as an answer.  

---

### Answer by username_0 on 2023-07-20. Score: 1

For the perfect competition case, it was my trivial solution: set price equal to marginal productivity.

$Y\_t = A\_t K\_t$

$r\_t = \frac{\partial Y\_t}{\partial K\_t} = A\_t$

$w\_t = \frac{\partial Y\_t}{\partial L\_t} = 0$

In the next exercise, suppose perfect competition no longer holds, such that capital is paid a fraction $\alpha \in (0,1)$ of its $MP$.

$\implies r\_t = \alpha \frac{\partial Y\_t}{\partial K\_t} = \alpha A\_t$

Now here is the key part: We use the equation that states that all income in the economy is used to pay production factors, to solve for $w\_t$.

$Y\_t = w\_t L\_t + r\_t K\_t$

$\implies A\_t K\_t = w\_t L\_t + \alpha A\_t K\_t$

$\implies w\_t L\_t = (1-\alpha) A\_t K\_t$

$\therefore w\_t = (1-\alpha) A\_t \kappa\_t$

Here $\kappa\_t$ is capital per capita.

I know a rational firm wouldn’t hire workers at that salary when their $MP$ is always $0$, but that was the official solution for my particular problem set.


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