Kimball aggregator derivation

Hi all,

I know this is not directly related to Dynare, but I have a question related to some derivation. I am pretty sure that it must be relatively straightforward but I am stuck for too long and maybe somebody can help me out.
It is about deriving the Kimball demand. I know that @wmutschl already asked a similar question here, which was very helpful, but my problem is a bit different. As there it was about a final good producer and in my case the consumer.
I am trying to replicate Casas et al. (2016),(Dominant Currency Paradigm). My question is how to derive equation (5) from the paper linked. What we normally do is to calculate the household expenditure minimization problem in order to derive the demand function. But when I do that I am just not able to get to that solution.
A very similar form, but with less notation, can be found in Klenow and Willis (2016). There, my problem ist basically identical.

I am happy to provide my derivations, as far as I got, in case anyone is interested.

Would be amazing if somebody could help me solve this mystery :slight_smile:

Cheers

PS: In case this is not the place to ask, just let me know.

Hello DoubleBass,

I’m happy someone asks exactly the same question that I asked my supervisor a few months ago :joy: Here is his brief answer:

Denote H = Home, U = USA, R = RoW (Rest of the World). Home is an SOE.

Reformulate the question:
Cost minimization:
PC=\displaystyle\min_{\{C_{iH}(\omega)\}} \sum_i \int_{\omega\in\Omega_i} P_{iH}(\omega)C_{iH}(\omega)d \omega, i\in\{H,U,R\}
s.t. Kimball aggregator:
\displaystyle\sum_i\frac{1}{|\Omega_i|}\int_{\omega\in\Omega_i}\gamma_i\Upsilon\left(\frac{|\Omega_i|C_{iH}(\omega)}{\gamma_iC}\right) d \omega = 1

Solution:
FOC wrt C_{iH}(\omega):
\displaystyle P_{iH}(\omega)=\lambda\Upsilon'(\cdot)\frac{1}{C}
where \lambda is the Lagrandgian multiplier ahead of the Kimball aggregator.
Envelope theorem wrt C:
\displaystyle P=\lambda\sum_i\int_{\Omega_i} \Upsilon'(\cdot)\frac{C_{iH}(\omega)}{C^2}d\omega
Then we combine the two equations, eliminating \lambda.
By the first one,
\displaystyle\frac{|\Omega_i|C_{iH}(\omega)}{\gamma_iC}=\psi\left(\frac{P_{iH}(\omega)C}{\lambda}\right)
where \psi = \Upsilon'^{-1}(\cdot). Call this equation (AA).
The second implies:
\lambda =\displaystyle \frac{PC}{D}, D = \sum_i\int_{\Omega_i}\Upsilon'\left(\frac{|\Omega_i|C_{iH}(\omega)}{\gamma_iC}\right)\frac{C_{iH}(\omega)}{C}d\omega
Finally, plug this \lambda into (AA). We arrive at the (5) in the paper.

BTW, Gopinath has written many papers using Kimball, e.g. the paper that titled “The Macroeconomics of Border Taxes”. You will be comfortable with the notations.

Hope it helps.

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Hi @HelloDynare,

thank you so much it helps tremendously :slight_smile: This was exactly what I was looking for. Just totally missed the envelope theorem step. Now that I see it, it makes so much sense.
I have seen her other papers, but never managed to get around the derivation.

Thank you again and have a wonderful day.

Cheers

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I am able to follow the derivations of the two first order conditions. But I am curious how you get rid of the |\Omega_i| term on the left hand side of the equation \frac{|\Omega_i| C_{iH}(\omega) }{\gamma_i C} = \psi \Big( \frac{P_{iH}(\omega) C}{\lambda} \Big) after substituting in \lambda. When I attempt the steps you suggest, I end up with (5) from the paper, but with an extra |\Omega_i| term multiplying C_{iH}(\omega).

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Thank you for pointing out this :+1: You are right!

In my professor’s reply, he indeed added this term :grinning_face_with_smiling_eyes:

That is,

\displaystyle C_{iH}(\omega) = \frac{\gamma_i}{|\Omega_i|}\psi \left(\frac{P_{iH}(\omega)D}{P}\right)C_t\equiv \frac{\gamma_i}{|\Omega_i|}\psi (Z_{iH,t}(\omega))C_t.

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Thanks for following up, I really appreciate it!