<p>Dividing equation 25 by <span class="math-container">$\gamma$</span> gives:
<span class="math-container">$$
Y_t = C_h + \frac{\gamma}{1-\gamma} C_h^\ast.
$$</span>
Then using 7 and 7a to substitute for <span class="math-container">$C_h$</span> and <span class="math-container">$C_h^\ast$</span>, we get:
<span class="math-container">$$
Y_t = (1-\gamma) \frac{P_t}{P_h} C_t + \frac{\gamma}{1-\gamma}(1-\gamma) \frac{P_t^\ast}{P_h^\ast} C_t^\ast.
$$</span>
Then multiply both sides by <span class="math-container">$P_h$</span> and use <span class="math-container">$C_t^\ast = C_t$</span> to get:
<span class="math-container">$$
\begin{align*}
P_h Y_t &amp;= (1-\gamma)P_t C_t + \gamma \left(\frac{P_t^\ast}{P_t}\frac{P_h}{P_h^\ast}\right) P_t C_t,\\
&amp;= P_t C_t \left(1- \gamma + \gamma \left(\frac{P_t^\ast}{P_t}\frac{P_h}{P_h^\ast}\right)\right)
\end{align*}
$$</span>
This is equal to equation 28 if:
<span class="math-container">$$
\frac{P_t^\ast}{P_t} \frac{P_h}{P_h^\ast} = 1.
$$</span>
Now <span class="math-container">$\dfrac{P_t^\ast}{P_t} = \dfrac{1}{\varepsilon}$</span>, which would require that:
<span class="math-container">$$
\frac{P_h}{P_h^\ast} = \varepsilon.
$$</span></p>
<p>Dividing 7 by 7a gives:
<span class="math-container">$$
\frac{P_h}{P_h^\ast} \frac{C_h}{C_h^\ast} = \frac{P_t}{P_t^\ast} = \varepsilon.
$$</span>
So this means that <span class="math-container">$C_h$</span> has to equal <span class="math-container">$C_h^\ast$</span>. I don't know if this makes sense (as I don't know the meaning of all these variables).</p>
<p>Similarly, you can show that 29 is satisfied if <span class="math-container">$C_f = C_f^\ast$</span>.</p>

CC BY-SA 4.0
2021-05-07T18:43:25.413Z
2021-05-07T18:43:25.413Z
tdm
<p>I am facing some problems in deriving the two equations for trade balance (from which we derive that the trade balance is zero within each country) in the article &quot;A SIMPLE FRAMEWORK FOR INTERNATIONAL MONETARY POLICY ANALYSIS&quot; (2002) that you can find <a href="https://www.nber.org/system/files/working_papers/w8870/w8870.pdf" rel="nofollow noreferrer">here</a>.
My objective is to derive equations 28 and 29 of that paper. In order to do that, I used the following equations:</p>
<p><img src="https://latex.codecogs.com/gif.latex?equation%5C%3B%207%3A%20P_%7Bh%7DC_%7Bh%7D%3D%281-%5Cgamma%29P_%7Bt%7DC_%7Bt%7D" alt="formula" /></p>
<p><img src="https://latex.codecogs.com/gif.latex?equation%5C%3B%207a%3A%20P_%7Bh%7D%5E%7B*%7DC_%7Bh%7D%5E%7B*%7D%3D%281-%5Cgamma%29P_%7Bt%7D%5E%7B*%7DC_%7Bt%7D%5E%7B*%7D" alt="formula" /></p>
<p><img src="https://latex.codecogs.com/gif.latex?equation%5C%3B%208%3A%20P_%7Bf%7DC_%7Bf%7D%3D%28%5Cgamma%29P_%7Bt%7DC_%7Bt%7D" alt="formula" /></p>
<p><img src="https://latex.codecogs.com/gif.latex?equation%5C%3B%208a%3A%20P_%7Bf%7D%5E%7B*%7DC_%7Bf%7D%5E%7B*%7D%3D%28%5Cgamma%29P_%7Bt%7D%5E%7B*%7DC_%7Bt%7D%5E%7B*%7D" alt="formula" /></p>
<p><img src="https://latex.codecogs.com/gif.latex?LOOP%3A%20P_%7Bt%7D%3D%5Cepsilon%20P_%7Bt%7D%5E%7B*%7D" alt="formula" /></p>
<p><img src="https://latex.codecogs.com/gif.latex?C_%7Bt%7D%3DC_%7Bt%7D%5E%7B*%7D" alt="formula" /></p>
<p><img src="https://latex.codecogs.com/gif.latex?equation%20%5C%3B%2025%3A%20%281-%5Cgamma%29Y_%7Bt%7D%3D%281-%5Cgamma%29C_%7Bh%7D+%5Cgamma%20C_%7Bh%7D%5E%7B*%7D" alt="formula" /></p>
<p><img src="https://latex.codecogs.com/gif.latex?equation%20%5C%3B%2026%3A%20%5Cgamma%20Y_%7Bt%7D%5E%7B*%7D%3D%281-%5Cgamma%29C_%7Bf%7D+%5Cgamma%20C_%7Bf%7D%5E%7B*%7D" alt="formula" /></p>
<p>Firstly, I tried to substitute the value of <img src="https://latex.codecogs.com/gif.latex?C_%7Bh%7D%5E%7B*%7D" alt="formula" /> given by equation 7a in equation 25, but what I got was not right:</p>
<p><img src="https://latex.codecogs.com/gif.latex?P_%7Bh%7D%5E%7B*%7DY_%7Bt%7D%3DP_%7Bh%7D%5E%7B*%7DC_%7Bh%7D+%20%5Cgamma%20P_%7Bt%7D%5E%7B*%7DC_%7Bt%7D%5E%7B*%7D" alt="formula" /></p>
<p>Then I tried to get the value of <img src="https://latex.codecogs.com/gif.latex?C_%7Bh%7D" alt="formula" /> from equation 25 and substitute in equation 7, but this was not right either:</p>
<p><img src="https://latex.codecogs.com/gif.latex?P_%7Bt%7DC_%7Bt%7D%3D%281-%20%5Cgamma%29P_%7Bh%7DY_%7Bt%7D-%20%5Cgamma%20P_%7Bh%7DC_%7Bh%7D%5E%7B*%7D" alt="formula" /></p>
<p>I also tried to substitute for gamma, but the results were not correct. I cannot think of anything else. How would you do that?</p>
<p>The equations that I am looking for:</p>
<p>Equation 28: <img src="https://latex.codecogs.com/gif.latex?P_%7Bh%7DY_%7Bt%7D%3DP_%7Bt%7DC_%7Bt%7D" alt="formula" /></p>
<p>Equation 29: <img src="https://latex.codecogs.com/gif.latex?P_%7Bf%7D%5E%7B*%7DY_%7Bt%7D%5E%7B*%7D%3DP_%7Bt%7D%5E%7B*%7DC_%7Bt%7D%5E%7B*%7D" alt="formula" /></p>

CC BY-SA 4.0
2021-05-07T12:02:51.837Z
To make the question more self contained, perhaps also include equations 28 and 29?
Giskard
CC BY-SA 4.0
2021-05-07T12:03:55.173Z
@Giskard You are right, I will write them immediately.
Vallo
<macroeconomics>
Derivations of trade balance
Vallo
# Derivations of trade balance

(Asked by: username_0 on 2021-05-07)

I am facing some problems in deriving the two equations for trade balance (from which we derive that the trade balance is zero within each country) in the article "A SIMPLE FRAMEWORK FOR INTERNATIONAL MONETARY POLICY ANALYSIS" (2002) that you can find [here](https://www.nber.org/system/files/working_papers/w8870/w8870.pdf).
My objective is to derive equations 28 and 29 of that paper. In order to do that, I used the following equations:

![formula](https://latex.codecogs.com/gif.latex?equation%5C%3B%207%3A%20P_%7Bh%7DC_%7Bh%7D%3D%281-%5Cgamma%29P_%7Bt%7DC_%7Bt%7D)

![formula](https://latex.codecogs.com/gif.latex?equation%5C%3B%207a%3A%20P_%7Bh%7D%5E%7B*%7DC_%7Bh%7D%5E%7B*%7D%3D%281-%5Cgamma%29P_%7Bt%7D%5E%7B*%7DC_%7Bt%7D%5E%7B*%7D)

![formula](https://latex.codecogs.com/gif.latex?equation%5C%3B%208%3A%20P_%7Bf%7DC_%7Bf%7D%3D%28%5Cgamma%29P_%7Bt%7DC_%7Bt%7D)

![formula](https://latex.codecogs.com/gif.latex?equation%5C%3B%208a%3A%20P_%7Bf%7D%5E%7B*%7DC_%7Bf%7D%5E%7B*%7D%3D%28%5Cgamma%29P_%7Bt%7D%5E%7B*%7DC_%7Bt%7D%5E%7B*%7D)

![formula](https://latex.codecogs.com/gif.latex?LOOP%3A%20P_%7Bt%7D%3D%5Cepsilon%20P_%7Bt%7D%5E%7B*%7D)

![formula](https://latex.codecogs.com/gif.latex?C_%7Bt%7D%3DC_%7Bt%7D%5E%7B*%7D)

![formula](https://latex.codecogs.com/gif.latex?equation%20%5C%3B%2025%3A%20%281-%5Cgamma%29Y_%7Bt%7D%3D%281-%5Cgamma%29C_%7Bh%7D+%5Cgamma%20C_%7Bh%7D%5E%7B*%7D)

![formula](https://latex.codecogs.com/gif.latex?equation%20%5C%3B%2026%3A%20%5Cgamma%20Y_%7Bt%7D%5E%7B*%7D%3D%281-%5Cgamma%29C_%7Bf%7D+%5Cgamma%20C_%7Bf%7D%5E%7B*%7D)

Firstly, I tried to substitute the value of ![formula](https://latex.codecogs.com/gif.latex?C_%7Bh%7D%5E%7B*%7D) given by equation 7a in equation 25, but what I got was not right:

![formula](https://latex.codecogs.com/gif.latex?P_%7Bh%7D%5E%7B*%7DY_%7Bt%7D%3DP_%7Bh%7D%5E%7B*%7DC_%7Bh%7D+%20%5Cgamma%20P_%7Bt%7D%5E%7B*%7DC_%7Bt%7D%5E%7B*%7D)

Then I tried to get the value of ![formula](https://latex.codecogs.com/gif.latex?C_%7Bh%7D) from equation 25 and substitute in equation 7, but this was not right either:

![formula](https://latex.codecogs.com/gif.latex?P_%7Bt%7DC_%7Bt%7D%3D%281-%20%5Cgamma%29P_%7Bh%7DY_%7Bt%7D-%20%5Cgamma%20P_%7Bh%7DC_%7Bh%7D%5E%7B*%7D)

I also tried to substitute for gamma, but the results were not correct. I cannot think of anything else. How would you do that?

The equations that I am looking for:

Equation 28: ![formula](https://latex.codecogs.com/gif.latex?P_%7Bh%7DY_%7Bt%7D%3DP_%7Bt%7DC_%7Bt%7D)

Equation 29: ![formula](https://latex.codecogs.com/gif.latex?P_%7Bf%7D%5E%7B*%7DY_%7Bt%7D%5E%7B*%7D%3DP_%7Bt%7D%5E%7B*%7DC_%7Bt%7D%5E%7B*%7D)

**Tags:** macroeconomics

> **Comments:**
> **username_1**: To make the question more self contained, perhaps also include equations 28 and 29?  
> **username_0**: @username_1 You are right, I will write them immediately.  

---

## 1 answers

### Answer by username_2 on 2021-05-07. Score: 2 | Accepted answer

Dividing equation 25 by $\gamma$ gives:
$$
Y\_t = C\_h + \frac{\gamma}{1-\gamma} C\_h^\ast.
$$
Then using 7 and 7a to substitute for $C\_h$ and $C\_h^\ast$, we get:
$$
Y\_t = (1-\gamma) \frac{P\_t}{P\_h} C\_t + \frac{\gamma}{1-\gamma}(1-\gamma) \frac{P\_t^\ast}{P\_h^\ast} C\_t^\ast.
$$
Then multiply both sides by $P\_h$ and use $C\_t^\ast = C\_t$ to get:
$$
\begin{align\*}
P\_h Y\_t &= (1-\gamma)P\_t C\_t + \gamma \left(\frac{P\_t^\ast}{P\_t}\frac{P\_h}{P\_h^\ast}\right) P\_t C\_t,\\
&= P\_t C\_t \left(1- \gamma + \gamma \left(\frac{P\_t^\ast}{P\_t}\frac{P\_h}{P\_h^\ast}\right)\right)
\end{align\*}
$$
This is equal to equation 28 if:
$$
\frac{P\_t^\ast}{P\_t} \frac{P\_h}{P\_h^\ast} = 1.
$$
Now $\dfrac{P\_t^\ast}{P\_t} = \dfrac{1}{\varepsilon}$, which would require that:
$$
\frac{P\_h}{P\_h^\ast} = \varepsilon.
$$

Dividing 7 by 7a gives:
$$
\frac{P\_h}{P\_h^\ast} \frac{C\_h}{C\_h^\ast} = \frac{P\_t}{P\_t^\ast} = \varepsilon.
$$
So this means that $C\_h$ has to equal $C\_h^\ast$. I don't know if this makes sense (as I don't know the meaning of all these variables).

Similarly, you can show that 29 is satisfied if $C\_f = C\_f^\ast$.


---

