# Can intermediate good producer's profit ever be zero in new keynesian model?

In monopolistic competition, firms are said to have zero long-run economic profit.

This makes me wonder whether profits of intermediate goods producers in New Keynesian model ever be zero.

For reference we can assume Gali's simple New Keynesian model.

Yes. In the core New Keynesian model in Galí, it is possible for the aggregate profits of intermediate good producers to be zero or negative; and it is even more likely that the profits of individual intermediate good producers, who may be stuck at prices far from the optimum, will be negative.

Since it's easier to characterize the aggregate, I'll discuss the conditions there. Note that the aggregate profits of intermediate good producers are just aggregate revenues minus the aggregate wage bill, or $$\Pi =PY-WN$$ where $P$ is the aggregate price index, $Y$ is aggregate output, $W$ is the nominal wage (common to all producers thanks to the competitive labor market) and $N$ is aggregate labor input.

Now, consider the special case where intermediate producers have constant returns to scale and productivity is normalized to 1. There we have, to first order, $Y\approx N$ (there is a second order difference due to price dispersion). Hence, again to first order, we have $$\Pi \leq 0 \Longleftrightarrow P\leq W$$ When is $P\leq W$? Well, using lowercase variables to denote logs, we know from household optimization that $w - p = \sigma c + \varphi n$ (see Gali p.43 equation (2), for instance), where $\sigma$ is the inverse EIS and $\varphi$ is the inverse Frisch elasticity. And since market clearing and our simple production function imply that $y=c=n$ to first order, this becomes $$w-p = (\sigma + \varphi)y$$ Finally, we observe that at the "natural level" of output $y^n$, markups should equal their desired level, which is $\mathcal{M}=\varepsilon/(\varepsilon-1)$, where $\varepsilon$ is the elasticity of firm demand. Writing $\mu =\log \mathcal{M}$ and substituting into the equation for $w-p$, this implies $-\mu=(\sigma+\varphi)y^n$. Finally, subtracting from the equation for $w-p$, we have $$w-p = (\sigma+\varphi)(y-y^n) - \mu$$ We'll have negative aggregate profits $P\leq W \Longleftrightarrow w-p \geq 0$ when $(\sigma+\varphi)(y-y^n) \geq \mu$. Note that this happens when output is high, since profits in the basic NK model are countercyclical.

Interestingly, under some not-totally-crazy calibrations this can happen. For instance, if we assume that both the EIS and Frisch are 1/2, so that $\sigma=\varphi=2$, and also assume $\varepsilon = 10$, so that $\mu = 0.105$, then exceeding the natural level of output by even 3% is enough to deliver negative aggregate profits, since $4\times 0.03 > 0.105$. And again, note that it is much more likely that we'll have individual firms with negative profits; in the case where aggregate profits are even close to negative, any firm unlucky enough to be stuck at a much lower-than-average price will have negative profits.

Finally, I should clarify that all of this is unrelated to the concept of zero long-term economic profit that you mention. This concept is often embedded in models of monopolistic competition in the form of a free-entry condition (where firms pay some fixed cost ex ante to exist), but this is very rare in New Keynesian models - presumably because NK models focus on business-cycle frequencies where the entry margin is limited.

We like to think that profits for monopolistic competition have to decrease because of some usually not-modelled interactions, most importantly that households respond to the pricing markup in a way that makes the setup more costly for these firms.

In NK models, such a behavior is not modeled. Moreover, we assume there to be no entry/exit, and no other sectors. Hence, there is no way for households to respond to this in the long run, as the intuition dictates.

The existence of the Calvo fairy will induce heterogeneity within the firms. Some firms might - sometimes - make zero profits, because they cannot adjust prices. However, the firms that can adjust will take the non-responsiveness of the others into account. All the firms, on aggregate, will always make positive profits.

• Agreed with almost everything here, but I don't think that aggregate profits are always positive; if there is enough of a boom we'll have aggregate negative profits. I do a calculation in my answer. – nominally rigid Dec 27 '14 at 7:43