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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.

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    $\begingroup$ 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? $\endgroup$ Jul 18, 2023 at 8:35
  • $\begingroup$ @RichardHardy Just added some info, waiting for it to be reviewed. $\endgroup$ Jul 20, 2023 at 13:30

2 Answers 2

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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.

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  • $\begingroup$ 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. $\endgroup$ Jul 20, 2023 at 1:22
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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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