<p>The problem is the maximization of profit of a firm that produces a differentiated good $i$ using labor $N_i$ as unique input, under the demand function of the consumers (or final goods producers) and the production function.</p>

<p>The first three steps are computed by substituting the two constraints and the market clearing condition for goods market $Y_t=C_t$ into the objective function in order to have a free maximization problem in one unknown, $P_{it}$.</p>

<p>The FOC for $P_{it}$ is right and the same is true for the next step, where the left hand side is positive because he has divided for $-(1-\epsilon)$, i.e. $(\epsilon-1)$ that you see at the denominator of $\epsilon$. The highlighted part is correct too: this is the derivative of the function nested in the square brackets; he added $\frac{1}{P_t}$ because he kept $\frac{P_{it}}{P_t}$ all elevated at $-\epsilon-1$, this means that $P_t^{-1}$ and $P_t$ goes away. You could also have written
\begin{gather}
\epsilon\frac{\frac{P_{it}^{-\epsilon-1}}{P_t^{-\epsilon}}C_t}{A_t}=\epsilon\frac{\big(\frac{P_{it}}{P_t}\big)^{-\epsilon-1}C_t}{A_t}\frac{1}{P_t}
\end{gather}</p>

<p>After the "Continue...", he just sum the two expressions in square brackets because they are exponentials with the same base and in the second step he notice that 
\begin{gather}
\bigg[\frac{\frac{P_{it}}{P_t}^{-\epsilon}C_t}{A_t}\bigg]^{\frac{1}{1-\alpha}}=N_t
\end{gather}
using again the demand for $Y_{it}$, $Y_t=C_t$ and the production function. </p>

<p>The last step is a rearrangement of the previous one (simple algebra) and it is still the FOC for $P_{it}$. If you rearrange it as in Gali, you obtain the optimal pricing condition for the monopolistic firm in flexible price setting.</p>

CC BY-SA 3.0
2017-01-24T11:28:50.893Z
I think there is problem in first derivation in last bracket, you can see more detailed in new representation (bold). Second one is "continue.." section; left side Nt *(1-α ) and right side Nit..and final these are converted to Nt *(-α )..How?
Engin YILMAZ
CC BY-SA 3.0
2017-01-24T14:39:26.930Z
@EnginYILMAZ I added to my answer the part concerning the FOC. Then, for what concern $N_t$ I would say there shouldn't be the subscript $i$ because the production function is $Y_{it}=A_tN_t^{1-\alpha}$. It is a typo
PostDocing
CC BY-SA 3.0
2017-01-25T07:40:30.063Z
Grazie mil*mille !
Engin YILMAZ
CC BY-SA 3.0
2017-01-24T08:49:59.770Z
2017-01-24T14:35:30.837Z
2017-01-24T14:35:30.837Z
PostDocing
<p>Jordi Gali defined price, marginal cost and mark up relationship as </p>

<p><a href="https://i.sstatic.net/nu2FH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/nu2FH.png" alt="enter image description here"></a></p>

<p>Drago Bergholt tried to reach gali's representation from firm maximization problem.</p>

<p>I think that there is error in derivation.</p>

<p>Can you help me correcting derivation process and obtaining gali's representation ?</p>

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

<p>Now, is there any derivation error below (last part) ?</p>

<p><a href="https://i.sstatic.net/4UGVp.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4UGVp.png" alt="enter image description here"></a></p>

<p><strong>is this part ?</strong></p>

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

<p>Here, in the left side need to be "-" negative  ?</p>

<p><a href="https://i.sstatic.net/vI9jH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/vI9jH.png" alt="enter image description here"></a></p>

<p>Continue....</p>

<p><a href="https://i.sstatic.net/O7nlN.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/O7nlN.png" alt="enter image description here"></a></p>

<p><strong>Nt and Nit are different?</strong></p>

<p><a href="https://i.sstatic.net/9wOjm.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9wOjm.png" alt="enter image description here"></a></p>

CC BY-SA 3.0
2017-01-23T14:24:57.440Z
The derivation is correct. You have to pay attention to the fact that Gali is computing the economy's average mark-up $\mu_t$, whereas the notes are deriving $\mu$, the constant mark-up a monopolistic producer puts on his cost in order to rise its price
PostDocing
CC BY-SA 3.0
2017-01-23T18:25:07.610Z
@Alessandro I have edited my question, but in derivation there are problem. Can you control this derivation ?
Engin YILMAZ
CC BY-SA 3.0
2017-01-24T08:37:16.440Z
Sure, I'll write you an answer
PostDocing
<macroeconomics><new-keynesian-economics>
Jordi Gali Markup Marginal Cost Relation
Engin YILMAZ
# Jordi Gali Markup Marginal Cost Relation

(Asked by: username_0 on 2017-01-23)

Jordi Gali defined price, marginal cost and mark up relationship as

[![enter image description here](https://i.sstatic.net/nu2FH.png)](https://i.sstatic.net/nu2FH.png)

Drago Bergholt tried to reach gali's representation from firm maximization problem.

I think that there is error in derivation.

Can you help me correcting derivation process and obtaining gali's representation ?

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

Now, is there any derivation error below (last part) ?

[![enter image description here](https://i.sstatic.net/4UGVp.png)](https://i.sstatic.net/4UGVp.png)

**is this part ?**

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

Here, in the left side need to be "-" negative ?

[![enter image description here](https://i.sstatic.net/vI9jH.png)](https://i.sstatic.net/vI9jH.png)

Continue....

[![enter image description here](https://i.sstatic.net/O7nlN.png)](https://i.sstatic.net/O7nlN.png)

**Nt and Nit are different?**

[![enter image description here](https://i.sstatic.net/9wOjm.png)](https://i.sstatic.net/9wOjm.png)

**Tags:** macroeconomics, new-keynesian-economics
> **Comments:**
> **username_1**: The derivation is correct. You have to pay attention to the fact that Gali is computing the economy's average mark-up $\mu_t$, whereas the notes are deriving $\mu$, the constant mark-up a monopolistic producer puts on his cost in order to rise its price  
> **username_0**: @Alessandro I have edited my question, but in derivation there are problem. Can you control this derivation ?  
> **username_1**: Sure, I'll write you an answer  

---

## 1 answers

### Answer by username_1 on 2017-01-24. Score: 0 | Accepted answer

The problem is the maximization of profit of a firm that produces a differentiated good $i$ using labor $N\_i$ as unique input, under the demand function of the consumers (or final goods producers) and the production function.

The first three steps are computed by substituting the two constraints and the market clearing condition for goods market $Y\_t=C\_t$ into the objective function in order to have a free maximization problem in one unknown, $P\_{it}$.

The FOC for $P\_{it}$ is right and the same is true for the next step, where the left hand side is positive because he has divided for $-(1-\epsilon)$, i.e. $(\epsilon-1)$ that you see at the denominator of $\epsilon$. The highlighted part is correct too: this is the derivative of the function nested in the square brackets; he added $\frac{1}{P\_t}$ because he kept $\frac{P\_{it}}{P\_t}$ all elevated at $-\epsilon-1$, this means that $P\_t^{-1}$ and $P\_t$ goes away. You could also have written
\begin{gather}
\epsilon\frac{\frac{P\_{it}^{-\epsilon-1}}{P\_t^{-\epsilon}}C\_t}{A\_t}=\epsilon\frac{\big(\frac{P\_{it}}{P\_t}\big)^{-\epsilon-1}C\_t}{A\_t}\frac{1}{P\_t}
\end{gather}

After the "Continue...", he just sum the two expressions in square brackets because they are exponentials with the same base and in the second step he notice that
\begin{gather}
\bigg[\frac{\frac{P\_{it}}{P\_t}^{-\epsilon}C\_t}{A\_t}\bigg]^{\frac{1}{1-\alpha}}=N\_t
\end{gather}
using again the demand for $Y\_{it}$, $Y\_t=C\_t$ and the production function.

The last step is a rearrangement of the previous one (simple algebra) and it is still the FOC for $P\_{it}$. If you rearrange it as in Gali, you obtain the optimal pricing condition for the monopolistic firm in flexible price setting.

> **Comments:**
> **username_0**: I think there is problem in first derivation in last bracket, you can see more detailed in new representation (bold). Second one is "continue.." section; left side Nt *(1-α ) and right side Nit..and final these are converted to Nt *(-α )..How?  
> **username_1**: @EnginYILMAZ I added to my answer the part concerning the FOC. Then, for what concern $N_t$ I would say there shouldn't be the subscript $i$ because the production function is $Y_{it}=A_tN_t^{1-\alpha}$. It is a typo  
> **username_0**: Grazie mil*mille !  

---

