Rotemberg term in the market clearing condition

Hello everyone,

I have a question regarding Rotemberg adjustment term in the market clearing condition. Let’s set our baseline model be Gali and Monacelli (2005) or the relevant chapters in Gali (2015).

According to their formulation, we can write the market clearing condition as follows:
\displaystyle Y_t(i) = C_{H,t}(i)+X_{t}(i)
\displaystyle Y_t(i)=(1-\nu)\left(\frac{P_{H,t}(i)}{P_{H,t}}\right)^{-\epsilon}\left(\frac{P_{H,t}}{P_t}\right)^{-\eta}C_t+\nu\left(\frac{P_{H,t}(i)}{P_{H,t}}\right)^{-\epsilon}\left(\frac{P_{H,t}}{P_{F,t}}\right)^{-\eta}C^*_t
(assuming PCP holds, complete exchange rate pass-through)
The above two equations valid in “goods piecewise”, and aggregately, we have
Y_t=C_{H,t}+X_t=\left[(1-\nu)p_{H,t}^{-\eta}C_t+\nu p_{H,t}^{*-\eta}C_t^*\right]\Delta_t
by integrating across i on both sides, if I remember the procedure correctly. \Delta_t is a price dispersion term under the Calvo setting.

Now my question is, in many Rotemberg related papers, I often found that authors directly write out market clearing condition below:
Y_t=C_{H,t}+X_t + Rotemberg.
So in “goods piecewise” level, how to write the clearing condition?
Y_t(i) = C_{H,t}(i)+X_{t}(i)+Rotemberg(i)?
Because if this is the case, I wonder that
\displaystyle C_{H,t}(i)=(1-\nu)\left(\frac{P_{H,t}(i)}{P_{H,t}}\right)^{-\epsilon}\left(\frac{P_{H,t}}{P_t}\right)^{-\eta}C_t
\displaystyle X_t(i)=\nu\left(\frac{P_{H,t}(i)}{P_{H,t}}\right)^{-\epsilon}\left(\frac{P_{H,t}}{P_{F,t}}\right)^{-\eta}C^*_t
may no longer hold.

Thank you in advance!

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I think people usually do the following. In a Rotemberg framework, there are monopolistically-competitive intermediate firms that produce differentiated goods. These differentiated goods are not purchased by domestic and foreign households. These goods are purchased by final-good firms that operate in perfectly competitive markets. Final good firms transform the differentiated good in a final good, which is then sold to domestic and foreign households. This implies that households consume a unique final good, and not a continuum of differentiated goods. So the last two equations that you mention do not hold, because households do not consume intermediate goods: the objects C_H(i) and X_t(i) do not exist in this framework. But maybe I am missing something, so if you find a counterexample, please point it out.

Valerio

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Also note that people consider symmetric equilibria, so the i will drop out.

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This is a good question. May I ask a further one?

When we say “symmetric equilibria”, we equate P_{H,t}(i)=P_{H,t}. And I foundd in some papers, authors said the price dispersion (in Calvo settings) performs like a negative technology shock, so in a symmetric equilibrium, dispersion is removed, agent’s welfare can be ameliorated. This thought can be seen in Ippei Fujiwara and Jiao Wang (2017), a very short remark.

Then my question is: in Calvo setting, can we really achieve a symmetric equilibrium?
Since only a small proportion of firms can vary their prices in each time period, while I believe in Rotemberg, this symmetry can be achieved.

Sorry, I’m a completely new beginner in macro and dynare.

Regards!

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In the Calvo framework, the firms that reset the price in the same period will charge the same price. But firms that reset prices in different periods they have different prices: hence, the equilibrium is not symmetric across firms and price dispersion may arise in equilibrium.

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Thank you all of you!

I’ve seen this arrangement in many papers as well. Yet I guess there are some correspondences between what you mentioned and what appears in Gali’s paper or book, such that the CES expressions of Y_t, Y_{Ht} mirror CES of C_t and C_{Ht}, equivalently, CES from the output side or from the consumption side is indifference.

I’ll check them. But in
C_{H,t}(i)=\displaystyle(1-\nu)\left(\frac{P_{H,t}(i)}{P_{H,t}}\right)^{-\epsilon}...
The label i already shows the uniqueness?


My point is: usually, authors first list the piecewise (i) clearing condition, then integrating across i. In Calvo settings, we can clearly figure out where the price dispersion term \Delta_t comes from during the integration process, approximately \displaystyle\int\left(\frac{P_{Ht}(i)}{P_{Ht}}\right)^{-\epsilon} di.

But when it comes to the Rotemberg case, they directly write out the aggregate clearing condition. So how can we obtain Rotemberg through a micro-level condition by integration \int across i? Or where the Rotemberg term comes from is vague!

Thank you for the supplementary question @HelloDynare !

@valerio88 You reconfirm what I thought.

Yes, with Rotemberg, you first compute the FOCs and then impose a symmetric equilibrium. As all firms can adjust, the i-index will drop out. That contrast with Calvo where you get price dispersion as only some firms can adjust.

I am not sure if I have understood the first question. What I mean is that in a Rotemberg settings, I’ve never seen the households consuming different C(i). Usually, in a Rotemberg setting there is a final-good firm that puts together all the differentiated inputs and sell the final good to households. So households have to choose only the amount of final good consumption: the differentiated goods are inputs for the final good firms, not consumption goods for households.

Now you ask where the adjustment costs come out of the blue in the market clearing condition. Take a closed economy with no capital and public spending, where a final good firms produces the final consumption goods using differentiated inputs. Differentiated inputs are produced by monopolistically competitive firms that pay price adjustment costs. Logic says that the final output is either consumed or used to pay adjustment costs. If you want to see the math, you can do the following. Consider the budget constraint in nominal terms, after imposing the bond clearing condition:

P_tC_t = W_t H_t +\int_0^1\Pi_t\left(i\right)di

where profits are

\Pi_t\left(i\right) = P_t\left(i\right)Y_t\left(i\right)-W_tH_t\left(i\right)-Adj_t\left(i\right)

Integrate over i:

\int_0^1\Pi_t\left(i\right)di=P_tY_t-W_tH_t-Adj_t

and use the latter condition in the budget constraint. In this way you can see where Adjustment costs come from. But I think there is no micro-level condition. Which condition would you expect?

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Wow. This is a real eye-opener to me. I have always wondered why adjustment costs show up in the aggregate resource constraint, but no other costs, e.g. labor costs. The three equations above show it clearly. Labor costs cancel out in the resource constraint because, essentially, they represent expenditures by one sector (firms) that another sector (private households) receives as income. In the aggregate, they do not matter.

This confirms to me that Rotemberg adjustement costs are a really strange thing from an economic point of view because these costs are paid by firms, but nobody receives them as income. This stands in contrast to any other costs, like labor costs, capital costs, etc. In the real world, any expenditure by one agent represents income for another agent. Rotemberg adjustment costs, by contrast, are lost.

Or am I missing something?

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The Rotemberg costs are usually treated as a kind of intermediate input. Note also that capital or investment adjustment costs usually are subtracted from the capital stock and are also not income. Also, at first order these costs are 0.

What do you mean by “treated as a kind of intermediate input”? In the case of intermediate inputs, the expenditure by the final good firm to purchase intermediates represents the income for the intermediate firms. Adjustment costs aren’t anybody’s income, right?

They are intermediate inputs that nobody produces. So they reduce value added, i.e cause a deadweight loss.

Thank you for all of you @HelloDynare, @valerio88, @jpfeifer, @dgroll

My further elaborated question would be as follows:
I plan to build an open-econ model involving some LCP features.

For the Home country, its INTERMEDIATE producers sell idiosyncratic products both domestically (via PCP) and internationally (via LCP, say, the USD).

The FINAL producers assemble domestically produced intermediate goods with imported foreign produced intermediate goods, and then sell them exclusively in Home.

The Rotemberg term appears during the intermediate producers selling to final producers.


Let’s speak through math.
For each of intermediate goods Y_t(\omega),
Y_t(\omega)=A_tK^{\alpha}_{t-1}(\omega)N^{1-\alpha}_t(\omega),
Y_{t}(\omega) \equiv Y_{Ht}(\omega)+ Y^*_{Ht}(\omega),
in which Y_{Ht}(\omega) is specialized for Home final producers,
while Y^*_{Ht}(\omega) specialized for Foreign final goods firms.

Then assume still in the intermediate firms,

\displaystyle Y_{Ht}=\left[\int Y_{Ht}(\omega)^{\frac{\varepsilon-1}{\varepsilon}}d\omega\right]^{\frac{\varepsilon}{\varepsilon-1}}, \quad \displaystyle Y^*_{Ht}=\left[\int Y^*_{Ht}(\omega)^{\frac{\varepsilon-1}{\varepsilon}}d\omega\right]^{\frac{\varepsilon}{\varepsilon-1}}

\displaystyle P_{Ht}=\left[\int P_{Ht}(\omega)^{1-\varepsilon}d\omega\right]^{\frac{1}{1-\varepsilon}},\quad P^*_{Ht}={\left[\int P^*_{Ht}(\omega)^{1-\varepsilon}d\omega\right]}^{\frac{1}{1-\varepsilon}}

We have the demands of Y_{Ht}(\omega) and Y^*_{Ht}(\omega) respectively,

\displaystyle Y_{Ht}(\omega)= \left(\frac{P_{Ht}(\omega)}{P_{Ht}}\right)^{-\epsilon}Y_{Ht}, \quad Y^*_{Ht}(\omega)=\left(\frac{P^*_{Ht}(\omega )}{P^*_{Ht}}\right)^{-\varepsilon}Y^*_{Ht},

or the demand of Y_{t}(\omega) in total.

\displaystyle Y_{t}(\omega)= \left(\frac{P_{Ht}(\omega)}{P_{Ht}}\right)^{-\epsilon}Y_{Ht} + \left(\frac{P^*_{Ht}(\omega )}{P^*_{Ht}}\right)^{-\varepsilon}Y^*_{Ht},

Aggregately, either integrating across [0,1],
or due to symmetric equilibrium under the Rotemberg,

Y_{t}= Y_{Ht} + Y^*_{Ht}


For every final goods \mathcal{F}_t producer,

\displaystyle \mathcal{F}_{t} = \left[(1-\gamma)^{\frac{1}{\theta}}{Y_{Ht}}^{\frac{\theta-1}{\theta}}+\gamma^{\frac{1}{\theta}}{Y_{Ft}}^{\frac{\theta-1}{\theta}}\right]^{\frac{\theta}{\theta-1}}, \quad \mathcal{F}^*_{t} = %\displaystyle \left[\gamma^{\frac{1}{\theta}}{Y^{*}_{Ht}}^{\frac{\theta-1}{\theta}}+(1-\gamma)^{\frac{1}{\theta}}{Y^*_{Ft}}^{\frac{\theta-1}{\theta}}\right]^{\frac{\theta}{\theta-1}},

\displaystyle P_{t}=\left[(1-\gamma) {P_{Ht}}^{1-\theta}+ \gamma {P_{Ft}}^ {1-\theta}\right]^{\frac{1}{1-\theta}},\quad P^*_{t}=\left[\gamma {P_{Ht}^*}^{1-\theta}+ (1-\gamma) {P_{Ft}^*}^{1-\theta}\right]^{\frac{1}{1-\theta}}.

Similarly, we have

\displaystyle Y_{Ht}=(1-\gamma)\left(\frac{P_{Ht}}{P_t}\right)^{-\theta}\mathcal{F}_{t}, \quad Y^*_{Ht}=\gamma\left(\frac{P^*_{Ht}}{P^*_t}\right)^{-\theta}\mathcal{F}^*_{t}.

or in together, the expression below relates intermediate goods Y_t with final goods \mathcal{F}_t

\displaystyle Y_{t}(\omega)=(1-\gamma)\left(\frac{P_{Ht}(\omega)}{P_{Ht}}\right)^{-\varepsilon}\left( \frac{P_{Ht}}{P_t}\right)^{-\theta}\mathcal{F}_{t} + \gamma\left(\frac{P^*_{Ht}(\omega )}{P^*_{Ht}}\right)^{-\varepsilon}\left(\frac{P^*_{Ht}}{P^*_t}\right)^{-\theta}\mathcal{F}^*_{t}.

\displaystyle Y_t = (1-\gamma)\left(\frac{P_{Ht}}{P_t}\right)^{-\theta}\mathcal{F}_{t}+ \gamma\left(\frac{P^*_{Ht}}{P^*_t}\right)^{-\theta}\mathcal{F}^*_{t}

The above are what I said “micro-level conditions”.


I partially agree what @valerio88 said:

So for example, we add one equation to illustrate that:
\mathcal{F}_t = C_t

But when considering the firms’ pricing behaviors, I’m confused.

Home intermediate goods firms who sell Y_{Ht} to Home final goods firms are subject to, e.g.

\displaystyle \left(P_{Ht}-\mathcal{MC}_t \right)\left(\frac{P_{Ht}(i)}{P_{Ht}}\right)^{-\epsilon}Y_{Ht}- (1-\gamma)\frac{\delta}{2}\left( \pi_{Ht}-\pi_H\right)^2P_{Ht}Y_{Ht}

Analogously, Home intermediate firmes selling Y_{Ht}^* to Foreign final firms have to pay

\displaystyle \gamma\frac{\delta}{2}\left( \pi^*_{Ht}-\pi^*_H\right)^2\mathcal{E}_tP^*_{Ht}Y^*_{Ht}


QUESTION: does the following equation hold again?

\displaystyle \mathcal{F}_{t} = \left[(1-\gamma)^{\frac{1}{\theta}}{Y_{Ht}}^{\frac{\theta-1}{\theta}}+\gamma^{\frac{1}{\theta}}{Y_{Ft}}^{\frac{\theta-1}{\theta}}\right]^{\frac{\theta}{\theta-1}}

I guess at least for this equation Y_{Ht} should be smaller due to the Rotemberg friction.

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Mm…you may want to ask among all “micro-level conditions” that you listed, which will no longer hold, if taking the Rotemberg frictions into account? :face_with_monocle:

(1-\gamma)\frac{\delta}{2}\left( \pi_{Ht}-\pi_H\right)^2Y_{Ht}

and

\gamma\frac{\delta}{2}\left( \pi^*_{Ht}-\pi^*_H\right)^2Y^*_{Ht}

that have Y_{Ht} and Y_{Ht}^* terms.

My thought: Nothing wrong, except:
\displaystyle \mathcal{F}_t = C_t + (1-\gamma)\frac{\delta}{2}\left( \pi_{Ht}-\pi_H\right)^2Y_{Ht}+\gamma\frac{\delta}{2}\left( \pi_{Ft}-\pi_F\right)^2Y_{Ft}

So I regard Rotembergs as “garbage” terms, and do not enter into any previous piecewise conditions.