# RBC model with monopolistic competition

Hello everyone,

I have a question on RBC model with monopolistic competition. All the textbook material about RBC model I have seen assumes a perfect competition market where the general price level is normalized to 1.

While with monopolistic competition, you have one more variable: marginal cost (MC), so what is the additional first-order condition should I add to the model?

I guessed that the condition should be p = markup* MC, but it seems not correct. If the price p is normalized to 1, then MC would be constant over time. But this is weird since a positive TFP shock should drive down the MC.

Thanks!

Hi ttc,

The condition you probably have in mind is actually the pricing rule of the form
P_{it}=\frac{\epsilon}{\epsilon-1}MC_{t}P_{t}
Dividing through by P_{t} yields
\frac{P_{it}}{P_t}=\frac{\epsilon}{\epsilon-1}MC_{t}
In the absence of nominal rigidities all firms adjust their price to the same optimal price, so price dispersion is always zero, i.e. \frac{P_{it}}{P_t}=1 in all periods. It does not prevent the marginal cost from falling in response to a positive TFP shock .

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Good afternoon @cmarch ,

Iâ€™m jumping on this subject as I have a closely related question. If I have different firms that supply competitive goods, but I want the price of these goods to be different and reflect the MC of each of these firms (different in reality), should I integrate monopolistic competition in the model?

If I follow your argument, perfect competition among goods would translate into same prices for the goods of my competitive firms.

Thanks !

Hi hipa,

If all identical firms can adjust their price to changing conditions in factor markets and demand, independently of whether we are in a competitive or imperfectly competitive setting, they are all gonna charge the same price, which is the point I made in my previous post, that is that monopolistic competition and fully flexible prices can co-exist.
If your model allows you to have different marginal costs for different firms, then firms will not be identical and they can charge different prices, but if the goods they sell are identical, all firms that cannot produce at the lowest price (i.e. the price that firms with the lowest marginal costs can charge) will be driven out of the market in a competitive setting.

Thank you @cmarch

Let me explain very quickly the research question. I have three types of durable goods : a newly produced one, a repaired one and a shared one. They all give the same final service to the households, which is mobility service. Households will make decision on which durable goods they prefer given preferences and the price. The firms producing these three durable goods have different MC.

My question now is that can I have different prices for these different goods witch having a mark-up? Because, at the end in your two cases mentionned, if I have full flexibility, no rigidities, prices are gonna be the same for the three durable goods anyways â€¦ whether I have competition (monopolistically or perfectly) and fully flexible, or if I have different firms with different MC and identical goods that give the same service.

Maybe I should go with three monopolistic firms. At the end, my household can have a preference for a particular car, and hence a firm so that the good is not at the end 100% homogenous.

Thatâ€™s trickyâ€¦

You will not need monopolistic competition if, for example, you choose to have the household value consumption of all three goods in the utility function. If the household values each good, each good has some degree of in-substitutability.
This will allow you to have the household pay a different price for each good.

@cmarch great. I indeed have imperfect complementarity between the three durable goods in the aggregate consumption function. Is it ok if I bundle them in an aggregated CES consumption function such that the utility depends on this aggregate function:

u(aggregate consumption) = CRRA (aggregate consumption) with aggregate consumption being a CES bundle of the three different cars

OR should the utility value differently these goods such that:

u(carrepaired, carshared, newcar)= param1 * CRRA(carrepaired) + param2 * CRRA (carshared) + param3 * CRRA(newcar)

Thank you for the clarification. It helps a lot

If the goods deliver the same service, which you can call â€śconsumptionâ€ť, then it makes sense to aggregate them. If one of the goods served as, say, collateral for loans, then it would be probably better to keep it separate.

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thanks. Really appreciate it @cmarch

Good evening @cmarch ,

Regarding our discussion, can I assume that the price charged by the three firms is the same as they are in perfect competition, but the return on the three goods is different for the household because of in-substituability between the three goods, hence different valuation from households?

Even if my household values each good, I do not see how I can impose that each firms charges different price for their goods in perfect competition. For me, the taste of households for each good, and willingness to pay a different price, will come from my substitution elasticities rather than the formulation of different prices for the three different goods. I donâ€™t know if Iâ€™m clear enough. Trying to resolve the steady state of my model with three different prices for each goods but I hit the wall. Thought of supposing the same price for each good prices by the firm, but different valuation from households.

Thanks !

How can there be perfect competition in the market for goods if the goods are not perfect substitutes?

To expand on @jpfeiferâ€™s point, you may want to consider the following option: there are three monopolistic intermediate good sectors, one for each good s.
For each of these intermediate goods sectors, there is a continuum of intermediated good producers, and each of them has monopoly power over their variety s_j because within each good there is a competitive final good producer who uses a CES aggregator to bundle these varieties, which yields a downward-sloping demand for each s_j within good s.
So now you have three goods, three monopolistically competitive intermediate goods sectors, and three final goods producers. This gives you three final price indices P_s and three final outputs Y_s.

At this point you can have the household consume a CES aggregate of the three final goods.

Thank you all.

@jpfeifer indeed, they are not perfectly substitutable because of different tastes for the three goods. Though their function/attribute is the same. The goods have different green gas houses emission content. Households can prefer one good (letâ€™s say a cleaner good) to another even though the three goods are perfect substitutes in reality.

@cmarch so how could you conceptualize this on Dynare or analytically if I do not have different varieties for each good but only one variety for my three goods ? If I have only one variety for each good then the intermediate good producer for each sector corresponds exactly to the CES aggregator of that sector, no ?

Thanks

The setup I have explained is a very standard formulation of production in New Keynesian, see for example, just to mention one, A Baseline DSGE model.
Assume the existence of a continuum of monopolists over their respective varieties within a certain good and assume the existence of perfectly competitive final good producer within each good who bundles the varieties into a final good, for each good, so you end up having three price indices P_{g1}, P_{g2}, P_{g3} and three final goods Y_{g1}, Y_{g2}, Y_{g3}.

As you have explained, your household consumes all three goods: therefore, you can have the household consume a a CES aggregate of the three goods, for example:
C = \left[\gamma_{1}^{1/\nu} C_{g1}^{\frac{\nu-1}{\nu}} + \gamma_{2}^{1/\nu} C_{g2}^{\frac{\nu-1}{\nu}} + \left( 1- \gamma_{1} -\gamma_{2}\right)^{1/\nu} C_{g3}^{\frac{\nu-1}{\nu}} \right]^{\frac{\nu}{\nu-1}}
and corresponding price index will be
P= \left[\gamma_{1} P_{g1}^{1-\nu} + \gamma_{2} P_{g2}^{^{1-\nu} } + \left( 1- \gamma_{1} -\gamma_{2}\right) P_{g3}^{1-\nu} \right]^{\frac{1}{1-\nu}}.

Then the household will have utility defined as U(C), the household has to purchase P_t C_t on the expenditure side of your budget constraint, and so on and so forth.

Thank you @cmarch

I see that the price should equal ((epsilon-1)/epsilon)*marginal cost. Most models assume that the price is equal to 1 for simplicity. However, they only have on producer aggregator and one sector.

In my case, what happens. Should I assume that the price in each sector is equal to 1 but the mark up, epsilon, is different for each sector ?

Thanks

So I guess the pricing rule should not be part of the model block right? Otherwise, we are going to have constant mc by construction?

Hi siriy ,

The pricing rule is embedded in the optimal reset price, so it does appear in the model. If the economy is subject to shocks, real marginal costs will not be constant, as they are affected by movements in wages and the rental rate of capital, which are determined in general equilibrium, and by exogenous movements in the level of stationary technology.
See pages 11 through 13 in Villaverdeâ€™s â€śA Baseline DSGE modelâ€ť (in particular the last equation of page 12).

I guess what is still confusing to me is that the RHS of the pricing rule equation under flexible price (monopolistic competition setup) always equal to 1 while the real mc varies, for example, w.r.t. technology shock (?) Which does not make much sense to me. Can you maybe help clarifying to me this pls?

Why not. It is well-known that with isoelastic demand, a monopolist will always charge a constant markup over marginal cost. The FOC tells you just that.

Definitely. I guess I got confused while implementing mc_t as one of the variables identifying the equilibrium of a perfectly flexible price model with monopolistic competition.

Unlike NK model, mc_t is an auxilary variable in an RBC setup. I wonder if this understanding is correct? If thatâ€™s the case, the real marginal cost then equal the marginal product. How can I retrieve the real marginal cost when the model has more than one factor of production (without having mc_t as one of the variable characterizing the equilibrium of the model) then?

Finally, I wonder if you could maybe recommend a replication file for a flex price economy with monopolistic setup. I cannot find one on your Github.