<p>To get back to the original production function just multiply both sides with capital. Here:</p>
<p><span class="math-container">$$Y/K = A (N/K)^{1.1} = F(N/K,1) \implies Y = AN^{1.1}K^{-0.1}  =F(N,K)$$</span></p>
<p>Also, the marginal product of labor here will be:</p>
<p><span class="math-container">$$F_N'(N,K) =  1.1 A N^{0.1}K^{-0.1} = 1.1 A (N/K)^{0.1}$$</span></p>

CC BY-SA 4.0
2020-11-17T12:32:38.393Z
Okay, I see right now. Thanks a lot. Now, I don’t want to have a direct derivative of the function $F$. As you know, MPL(marginal product of labor) is equal to the real wage (w/p). That’s, $F_N(N,K)=w/p$. So, in this case, $F_N(N/K,1)= b_0(w/p)^{b_1}$ then, the original one is $F_N(N,K)= b_0(w/p)^{b_1}K$. Am I right?
studentp
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2020-11-17T12:39:53.297Z
@B11b It is true that marginal product of labor, in competitive equilibrium, is equal to real wage but also marginal product of labor is by definition $\partial F/\partial N$. So in this case it would be appropriate to say that MPL = $1.1A(N/K)^{0.1} = w/p$ and this implies that $N/K = (\frac{1}{1.1A})^{1/0.1}(w/p)^{1/0.1}$ but you can simply state that $\frac{1}{1.1A})^{1/0.1} = \beta_0$ and that $1/0.1 = \beta_1$ and get $N/K = \beta_0 (w/p)^{\beta_1}$
1muflon1
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2020-11-17T13:58:58.417Z
But, in your case, $\beta_1$ is not negative. Yet, question says it is negative.
studentp
CC BY-SA 4.0
2020-11-17T14:17:38.103Z
@B11b that is actually good point, can you please give me reference for source where you got the model above? I think that there must be some explanation for this otherwise math checks out nicely. I cant see any obvious mistake in the marginal product derivation.
1muflon1
CC BY-SA 4.0
2020-11-17T15:13:36.927Z
I posted the full question. It is from Keynesian model problem set. I agree with you. Your math side is correct I think so.
studentp
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2020-11-17T15:16:05.140Z
But, $\beta_1$ must be negative such that the system can be stable.
studentp
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2020-11-17T15:16:33.517Z
@B11b but I really wanted to see the source because it might be there is some typo inconsistency somewhere. I mean the math in my answer is pretty straightforward so there must be something else at play - I wanted to check the book that should have more examples of this - already the example you show uses pretty non-standard production function so there could be some additional background assumption mentioned somewhere in the source.
1muflon1
CC BY-SA 4.0
2020-11-17T15:23:44.437Z
No, this is problem set provided by my instructor. There is no resource. That is, I don’t know it’s resource.
studentp
CC BY-SA 4.0
2020-11-17T15:27:14.297Z
@B11b ah ok, I will try to expand my answer above using Romer Advanced Macroeconomics as a reference perhaps that will help
1muflon1
CC BY-SA 4.0
2020-11-17T15:36:43.420Z
@B11b ok I checked the Romer advanced macroeconomics, and the marginal product of labor is $dF/dL$ (see pp 245). However, book also says that in some cases the demand for labor does not need to be equal to the marginal product of labor - if there is some markup or market power etc. However, this being said reading the full Q above is it possible this is just a trick question since it asks you to check for system stability so there might be inconsistency on purpose which would lead to no solution?
1muflon1
CC BY-SA 4.0
2020-11-17T15:52:04.247Z
@B11b I also investigated this further and I think something must be wrong here. Normally cobb-Douglas production function would be given by $AL^{\alpha}K^{\beta}$ where not just $\alpha + \beta = 1$ but also $0<\alpha, \beta <1$. Here you have some weird case when $\alpha>1$ even though $\alpha+\beta=1$ this is what creates the problems - note if $\alpha<1$ in your case also the $\beta_1$ would be negative... so in this case I think there is something not fully kosher with this model and the assumptions of the model might be inconsistent- this requires investigation beyond the original Q
1muflon1
CC BY-SA 4.0
2020-11-17T11:57:32.373Z
2020-11-17T11:57:32.373Z
1muflon1
<p>In The Keynesian economy,</p>
<p>I have the following model</p>
<p><a href="https://i.sstatic.net/ODsz4.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ODsz4.jpg" alt="enter image description here" /></a></p>
<p>Here, the production function <span class="math-container">$F(N,K) = Y/K = A(N/K)^{1.10}$</span> and by the labor demand function <span class="math-container">$F_N(K,N)= N/K= b_0(w/p)^{b_1}$</span>.</p>
<p>right?</p>
<p>That is, what is the F(K,N)=? and <span class="math-container">$F_N(N,K)=?$</span></p>
<p>Here , I mean that <span class="math-container">$F_N(N,K)$</span> is marginal product of labor.</p>
<p><a href="https://i.sstatic.net/5EGP7.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/5EGP7.jpg" alt="enter image description here" /></a></p>

<macroeconomics><self-study><new-keynesian-economics>
Keynesian model functions
studentp
# Keynesian model functions

(Asked by: username_0 on 2020-11-17)

In The Keynesian economy,

I have the following model

[![enter image description here](https://i.sstatic.net/ODsz4.jpg)](https://i.sstatic.net/ODsz4.jpg)

Here, the production function $F(N,K) = Y/K = A(N/K)^{1.10}$ and by the labor demand function $F\_N(K,N)= N/K= b\_0(w/p)^{b\_1}$.

right?

That is, what is the F(K,N)=? and $F\_N(N,K)=?$

Here , I mean that $F\_N(N,K)$ is marginal product of labor.

[![enter image description here](https://i.sstatic.net/5EGP7.jpg)](https://i.sstatic.net/5EGP7.jpg)

**Tags:** macroeconomics, self-study, new-keynesian-economics

---

## 1 answers

### Answer by username_1 on 2020-11-17. Score: 1 | Accepted answer

To get back to the original production function just multiply both sides with capital. Here:

$$Y/K = A (N/K)^{1.1} = F(N/K,1) \implies Y = AN^{1.1}K^{-0.1} =F(N,K)$$

Also, the marginal product of labor here will be:

$$F\_N'(N,K) = 1.1 A N^{0.1}K^{-0.1} = 1.1 A (N/K)^{0.1}$$

> **Comments:**
> **username_0**: Okay, I see right now. Thanks a lot. Now, I don’t want to have a direct derivative of the function $F$. As you know, MPL(marginal product of labor) is equal to the real wage (w/p). That’s, $F_N(N,K)=w/p$. So, in this case, $F_N(N/K,1)= b_0(w/p)^{b_1}$ then, the original one is $F_N(N,K)= b_0(w/p)^{b_1}K$. Am I right?  
> **username_1**: @B11b It is true that marginal product of labor, in competitive equilibrium, is equal to real wage but also marginal product of labor is by definition $\partial F/\partial N$. So in this case it would be appropriate to say that MPL = $1.1A(N/K)^{0.1} = w/p$ and this implies that $N/K = (\frac{1}{1.1A})^{1/0.1}(w/p)^{1/0.1}$ but you can simply state that $\frac{1}{1.1A})^{1/0.1} = \beta_0$ and that $1/0.1 = \beta_1$ and get $N/K = \beta_0 (w/p)^{\beta_1}$  
> **username_0**: But, in your case, $\beta_1$ is not negative. Yet, question says it is negative.  
> **username_1**: @B11b that is actually good point, can you please give me reference for source where you got the model above? I think that there must be some explanation for this otherwise math checks out nicely. I cant see any obvious mistake in the marginal product derivation.  
> **username_0**: I posted the full question. It is from Keynesian model problem set. I agree with you. Your math side is correct I think so.  
> **username_0**: But, $\beta_1$ must be negative such that the system can be stable.  
> **username_1**: @B11b but I really wanted to see the source because it might be there is some typo inconsistency somewhere. I mean the math in my answer is pretty straightforward so there must be something else at play - I wanted to check the book that should have more examples of this - already the example you show uses pretty non-standard production function so there could be some additional background assumption mentioned somewhere in the source.  
> **username_0**: No, this is problem set provided by my instructor. There is no resource. That is, I don’t know it’s resource.  
> **username_1**: @B11b ah ok, I will try to expand my answer above using Romer Advanced Macroeconomics as a reference perhaps that will help  
> **username_1**: @B11b ok I checked the Romer advanced macroeconomics, and the marginal product of labor is $dF/dL$ (see pp 245). However, book also says that in some cases the demand for labor does not need to be equal to the marginal product of labor - if there is some markup or market power etc. However, this being said reading the full Q above is it possible this is just a trick question since it asks you to check for system stability so there might be inconsistency on purpose which would lead to no solution?  
> **username_1**: @B11b I also investigated this further and I think something must be wrong here. Normally cobb-Douglas production function would be given by $AL^{\alpha}K^{\beta}$ where not just $\alpha + \beta = 1$ but also $0<\alpha, \beta <1$. Here you have some weird case when $\alpha>1$ even though $\alpha+\beta=1$ this is what creates the problems - note if $\alpha<1$ in your case also the $\beta_1$ would be negative... so in this case I think there is something not fully kosher with this model and the assumptions of the model might be inconsistent- this requires investigation beyond the original Q  

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