# Loglinearization of balance of payment

**URL:** <https://forum.dynare.org/t/loglinearization-of-balance-of-payment/20738>\
**Category:** General DSGE Modeling\
**Created:** [15 July 2022 16:53 UTC](https://forum.dynare.org/t/loglinearization-of-balance-of-payment/20738 "2022-07-15T16:53:29Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![YoungY](https://forum.dynare.org/letter_avatar_proxy/v4/letter/y/87869e/32.png) [@YoungY](https://forum.dynare.org/u/YoungY)\
**Post date:** [15 July 2022 16:53 UTC](https://forum.dynare.org/t/loglinearization-of-balance-of-payment/20738/1 "2022-07-15T16:53:29Z")

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Hi everyone,

I’m trying to log-linearize the balance of payment equation in my model. I have read some papers which linearize the BoP equation, but the results of them are different in the coefficient of interest rate. I attach the file that shows my procedure and result, but it is different from others. I don’t know whether it is wrong or not. The confusing point is the deviation form of interest rate. Could you please give me some hints?

Thanks.  
[Log-linearization .pdf](https://forum.dynare.org/uploads/short-url/yDGvHGruYfLRKpc5diPTQBb8Owc.pdf) (99.5 KB)

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**Author:** ![ManuelTorres](https://forum.dynare.org/user_avatar/forum.dynare.org/manueltorres/32/11115_2.png) [@ManuelTorres](https://forum.dynare.org/u/ManuelTorres)\
**Post date:** [16 July 2022 20:01 UTC](https://forum.dynare.org/t/loglinearization-of-balance-of-payment/20738/2 "2022-07-16T20:01:54Z")

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In my case I use a Taylor approximation of order 1 around the steady state and the coefficient of \widehat{r}\_{t} is r\_{ss}, the steps are as follows:

E\_{t}B\_{t+1}=\left(1+r\_{t}\right)B\_{t}+\frac{X\_{t}}{Q\_{t}}-M\_{t},

\left(B\_{t+1}-B\_{ss}\right)=B\_{ss}\left(r\_{t}-r\_{ss}\right)+\left(1+r\_{ss}\right)\left(B\_{t}-B\_{ss}\right)+\frac{1}{Q\_{ss}}\left(X\_{t}-X\_{ss}\right)-\frac{X\_{ss}}{Q\_{ss}^{2}}\left(Q\_{t}-Q\_{ss}\right)-\left(M\_{t}-M\_{ss}\right),

B\_{ss}\widehat{B}\_{t+1}=B\_{ss}r\_{ss}\widehat{r}\_{t}+\left(1+r\_{ss}\right)B\_{ss}\widehat{B}\_{t}+\frac{X\_{ss}}{Q\_{ss}}\widehat{X}\_{t}-\frac{X\_{ss}}{Q\_{ss}}\widehat{Q}\_{t}-M\_{ss}\widehat{M}\_{t},

\widehat{B}\_{t+1}=r\_{ss}\widehat{r}\_{t}+\left(1+r\_{ss}\right)\widehat{B}\_{t}+\frac{1}{B\_{ss}}\left[\frac{X\_{ss}}{Q\_{ss}}\left(\widehat{X}\_{t}-Q\_{t}\right)-M\_{ss}\widehat{M}\_{t}\right]

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**Author:** ![YoungY](https://forum.dynare.org/letter_avatar_proxy/v4/letter/y/87869e/32.png) [@YoungY](https://forum.dynare.org/u/YoungY)\
**Post date:** [18 July 2022 10:10 UTC](https://forum.dynare.org/t/loglinearization-of-balance-of-payment/20738/3 "2022-07-18T10:10:34Z")

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Hi Manuel,

Thanks for your reply. The deviation form of net interest rate should be r\_hat= r\_t-rss instead of r\_hat= (r\_t-r\_ss)/r\_ss because net interest rate is a very small value. So the coefficient of r\_hat should be 1 in your case. Is that right?

Btw, how do you put maths equations in the text?

Thanks,  
Young

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**Author:** ![ManuelTorres](https://forum.dynare.org/user_avatar/forum.dynare.org/manueltorres/32/11115_2.png) [@ManuelTorres](https://forum.dynare.org/u/ManuelTorres)\
**Post date:** [18 July 2022 22:09 UTC](https://forum.dynare.org/t/loglinearization-of-balance-of-payment/20738/4 "2022-07-18T22:09:55Z")

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Yes, you are right, I forget that detail; this will make the coefficient of the net interest rate equal to 1. However, on the other hand, operating in the same way but using the gross interest rate, I get the following:

\widehat{B}\_{t+1}=R\_{ss}\left(\widehat{R}\_{t}+\widehat{B}\_{t}\right)+\frac{X\_{ss}}{B\_{ss}Q\_{ss}}\left(\widehat{X}\_{t}-\widehat{Q}\_{t}\right)-\frac{M\_{ss}}{B\_{ss}}\widehat{M}\_{t}

which is similar to the result you got. In fact,

R\_{ss}\widehat{R}\_{t} = (1 + r\_{ss})(\frac{1 + r\_{t} - 1 - r\_{ss}}{1 + r\_{ss}}) = \widehat{r}\_{t}.

Note: To use mathematical expressions, just include the $ sign before and at the end of each expression and use the LaTeX language.

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**Author:** ![YoungY](https://forum.dynare.org/letter_avatar_proxy/v4/letter/y/87869e/32.png) [@YoungY](https://forum.dynare.org/u/YoungY)\
**Post date:** [19 July 2022 11:22 UTC](https://forum.dynare.org/t/loglinearization-of-balance-of-payment/20738/5 "2022-07-19T11:22:47Z")

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Thanks, Manuel. It seems that most of papers I’ve read ignore the detail.