Continuum of firms and Calvo pricing

[glspisso]

Hi, first post here.

I am trying to build a model wich has a continuum of monopolistically competitive firms and Calvo pricing for said firms.

I can work out all math conditions, but as far as I have seen model with monopolistic competition are solved in Dynare making use of the trick that the aggregate production function will be the same of the generic (and symmetric firm).

If I had Calvo pricing to the recipe this means that prices for some firm might be dispersed at any given time, hence I am not sure if I can aggregate the production function. Anybody can help me out/has experience with such a model or can point me to some similar looking paper (the issue is in writing down the simulation).

Thanks

[jpfeifer]

Seems you are talking about the standard Calvo approach. Please have a look at the NK_baseline.mod in the Dynare examples folder and the associated references. They contain all the math. You can also find it here https://github.com/DynareTeam/dynare/blob/master/examples/NK_baseline.mod
The trick is that due to constant returns to scale all firms will use the same capital to labor ratio and thus aggregation is possible. There will be a price dispersion term showing up in the aggregate production function, but because this term has a recursive representation, it can be easily entered to Dynare.

[glspisso]

Thank you, this has been very helpful.

If you don't mind, I'd ask you something which is not strictly related to dynare, as I am facing problems with computation of marginal costs. I fear this might be due to having posed a wrong production/profit function.

I have the following profit function


AK^(a)(LT)^(1-a)+Z(1-T)*L-RK-WN[c*T+p(1-T)]

K is capital, L is labor, T is a share between zero and one allocating labor between a market (T) and underground sector (1-T). R is rental rate, W is wage and c,p are parameters related to taxation.

I am not sure if given this function I am able to compute marginal cost following the procedure illustrated in the paper.

[jpfeifer]

In this case, derive marginal costs as the Lagrange multiplier in the firm's cost minimization problem. See e.g. Walsh (2003) Monetary Theory and Policy, Second Edition, p. 235

[glspisso]

Ok, I've considered this before and I have been able to obtain an expression for MC. I am only left with one problem: in my model I have two sector in the economy a domestic and a foreign one. This implies that prices aggregator P_t=int p_it/p_t appear quite often in the model by themselves and play a relevant role.

As far as I have seen in the baseline model the trick consists in obtaining inflation expression and price dispersions terms which are then made operative by the Calvo inflation dynamics .

I am wondering if it is instead possible to have the price aggregator P_t appear in the model, if this will cause any issues or there is any trick to apply to obtain this.

Thank you for the help so far =)

[jpfeifer]

It depends. The reason we express everything in terms of inflation is that the aggregate price level is indeterminate and we need a relation defining aggregate inflation. If your model uniquely determines the price level, you should be fine with defining the whole integral as the price level variabe and keep it in the equations.

[glspisso]

Hi, I have more or less completed the model and wrote code for Dynare.

I have computed SS by hand, but when I ran the model using solve_algo=4 it is unable to find the steady state, naming infinite terms in some equations, but I am unable to identify the issue.

I attach the code, the model is quite huge, so I understand you might not have the time to look through it, but maybe you can give me a hint on where to look for to get it to work.

As usual, thanks!

[jpfeifer]

The most important problem is

MODEL_DIAGNOSTICS: The following endogenous variables aren't present at the current period in the model:
q
d

The steady state found was

STEADY-STATE RESULTS:

c 0.776943
cd 21.9563
cf 0.00957176
n 0.0243576
thet 0.735
M 0.235854
P 0.7742
PD 0.0312371
PF 14.0735
I 0.0248386
ID 0.701935
IF 0.000306006
Q 49
B -87
PI 223.575
varpi 0.041918
w 6.879
k 0.993545
r 0.0540573
lt 5.56402
S 0.98
i 0.0204082
TR 0
Delta 543.184
g1 2.31519
g2 8.23519
T 0
q 63.2912
d 1
Y 0.370471
vp 1
eta 0.333
PP 0.0312371
PPbar 0.0312371
E 1
mc 0.236153
A 1
Z 1
Bshock -87
Astar 1
varsigmac 0.18
varsigmai 0.18
varsigmap 0.18
varsigmaex 0.18
tauv 0.16
taud 0.16
taun 0.16
tauk 0.16
deval 1
cstar 99.1914
cdstar 8691.31
cfstar 6.81146
nstar 0.107545
Mstar 14.6442
Pstar 1.94175
PDstar 0.0217637
PFstar 11.8217
Istar 99.6227
IDstar 8729.1
IFstar 6.84107
Qstar 25.235
qstar 25.235
Bstar 87
PIstar 0
varpistar 34.4662
wstar 667.667
kstar 3984.91
rstar 0.0454082
mu 5.27699e-07
Sstar 0.98
istar 0.0204082
x1 0.175459
x2 0.00164913
Deltastar 378.45
Ystar 287.218
vpstar 1
PPstar 11.8217
PPbarstar 11.8217
Tstar 0
mcstar 106.395