Why is capital often not included in New Keynesian models? Is there a reason other than modeling difficulty?

In New Keynesian models, like the ones in Gali's simple New Keynesian model or even Mankiw-Reis NK model on sticky information, capital is often not included.

Now people do say that there are modeling difficulties and that's why capital ($K$) is not included, but is there another justifiable reason other than modeling difficulties?

Capital is included in all the big estimated New Keynesian models (Smets-Wouters, Christiano-Eichenbaum-Evans), etc. But you're absolutely right that the stylized core NK model does not have capital - which is hard to defend on empirical grounds, since capital investment is a very important part of business cycle fluctuations and the response to monetary policy. Ultimately, the reason does basically boil down to the "modeling difficulties" that you mention.

First, there is an obvious way in which capital makes the NK model more complicated: at an absolute minimum, it introduces at least one additional backward-looking state variable $K$. In contrast, the two core equations (the intertemporal Euler equation and New Keynesian Phillips curve) of the ordinary log-linearized NK model are completely forward-looking. Adding $K$ to the mix eliminates this nice analytical feature. Still, on its own, this is not such a compelling reason to leave $K$ out of the standard presentation of the model, since the increase in complexity would still be tolerable and possibly justified by the added realism.

The additional complications that make capital much more difficult to include are the following.

Capital adjustment costs are needed to avoid absurd results. Suppose that there are no capital adjustment costs, and that firms rent capital each period on competitive markets. Suppose also that there are no shocks today. The real rental cost of capital today will be approximately $r+\delta$, where $r$ is the real interest rate that was expected yesterday and $\delta$ is depreciation. For each firm $i$, we have $MC(i)=(r+\delta)/MPK(i)$: real marginal cost equals the real cost of producing another unit of output by increasing capital.

Now, suppose for simplicity that firms are identical and there is no price dispersion. (Moving away from this, everything I say will still be approximately true.) Then we can eliminate the $i$ and just write $MC=(r+\delta)/MPK$. Furthermore, real marginal cost is just the inverse markup, so we can rewrite this in terms of markup $\mathcal{M}$ as $MPK=\mathcal{M}(r+\delta)$. Finally, if we assume that production is Cobb-Douglas with capital share $\alpha$ this becomes $$\frac{K}{Y}=\frac{\alpha}{\mathcal{M}}\frac{1}{r+\delta}$$ Pick some reasonable parameters: say, $\alpha=0.3$, $\mathcal{M}=1.33$, $r=0.04$, and $\delta=0.06$. Given these we have $K/Y\approx 2.26$. Suppose that we exist in this steady-state world for a while, and then the Fed pushes down the projected path of interest rates such that the expected real rate declines to $r=0.02$. Then leaving $\mathcal{M}$ constant, the value of $K/Y$ given by the expression above increases to $K/Y\approx 2.82$. The NK model implies that this value should hold in the period after the shock.

This is an immense increase in $K/Y$, and the model tells us that it should happen in one period. If our period is a quarter, and $Y$ was not expected to take a sudden dive, then we'd have to invest well above the entire usual level of GDP to accomplish this. (In the continuous time limit, it becomes simply infeasible.) And although the assumption than $\mathcal{M}$ is constant is not right ($\mathcal{M}$ is determined endogenously in the NK model from the relationship of sticky prices to costs), relaxing this could easily make the puzzle more extreme: if $r=0.02$ is expected to prevail for a while, then it will imply higher-than-usual output $Y$ and lower-than-usual markups $\mathcal{M}$, which raise the implied $K$ even further.

The model simply doesn't work in this form: you need some form of capital adjustment costs. And these, of course, make the model still more complicated. (By the way, the problem here isn't so much the NK model as the fact that an assumption of no adjustment costs is generally absurd: seemingly small changes in the real interest rate must be accompanied by massive swings in the capital-output ratio, which we never see in practice. The NK model simply brings this absurdity, which is found in the basic RBC model as well, into sharper relief because exogenous interest rate shocks are such an important feature of the NK environment.)

Firm-specific capital is needed for strategic complementarity. Even if we fix the problem above by including capital adjustment costs of some form, we run into another awkward feature of NK models: taken alone, the Calvo price rigidity is not plausibly large enough to make the NKPC as flat as we think it is.

The most popular fix is some form of strategic complementarity, where firms try not to set prices too far from the aggregate price level. And the most popular way to get strategic complementarity is to assume that firms face both a high elasticity of demand and a steeply upward-sloping marginal cost curve. That way, for instance, any firm that sets its price too far below the average price will receive a flood of demand that causes its marginal cost to spike - and this discourages the firm from setting such a low price in the first place. (Yes, this sounds a little ridiculous, but it's how the models work.)

When the model excludes capital altogether, it's easy to just write a declining-returns-to-scale production function for each firm with labor as the only input. This makes each firm's marginal cost curve slope upward. But when we include both capital and labor in the model, the firm's production function probably should be much closer to constant-returns-to-scale. And this means, if the firm can rent any amount of capital from a competitive market on demand, that the firm's marginal cost curve is much closer to flat. This limits strategic complementarity.

To get around this, you need to dispense with the assumption of a common rental market for capital, and start talking about firm-specific capital accumulation. But then the model becomes much more complicated, and you're reduced to opaque quantitative exercises like ACEL.

Given all this, you can imagine how economists in insight-building mode often just dispense with capital altogether.

• One defensive argument I heard was that the model mostly aims to help with short-run policy analysis, in which capital and its adjustments wouldn't really have a big role. – FooBar Dec 18 '14 at 11:00
• It makes somewhat sense. Due to linearization and Non-Lucas-Critique-resistent price adjustments, we can't take it seriously too far off the steady state anyways. – FooBar Dec 18 '14 at 12:33
• @FooBar, yes, this is a pretty good argument for us to be able to abstract away from the effects of changes in $K$ on, say, the production function. The question is whether we can also set aside changes in $I$ as a component of production, and that's a lot harder, since $I$ often fluctuates much more than $C$. The usual approach here is to think about $C$ in the basic model as "really" including changes in $I$ as well -- assuming a higher EIS than would be justified by consumer behavior alone to make this realistic. Woodford has a passage about this. – nominally rigid Dec 18 '14 at 17:08
• (+1). This is a great answer. – Alecos Papadopoulos Dec 18 '14 at 17:32