Friday, May 29, 2015

The political method

Let's say someone thinks the Fed should raise rates sooner rather than later. Let's say that person is known as a hawk on monetary policy. Now let's say that person put together a model [pdf] that says this:

We conclude that in economies where the key friction is NSCNC and the net nominal interest rate threatens to encounter the zero lower bound, monetary policymakers may wish to respond with a price level increase. A chief rival to this response observed in actual economies—forward guidance on the length of time the economy will remain at the zero lower bound beyond the time when that bound is actually binding—would be inappropriate in the theory presented here.

Lo and behold the model comes out with not only that person's preferred policy but also says the policy that person opposes is bad. What a coincidence!

And now that person sends a copy of that study to the most prominent advocate of the the specific model.

Lo and behold, that advocate loves it!

Isn't politics the scientific method grand?

I bet you didn't think you'd be laughing when you read an economics paper ... laughing so hard you couldn't breathe ...

 The private credit market completely solves the cross-sectional income inequality problem. It's quite awesome. I swear. (And can someone maybe label an axis, preferably two ... and what's with the Mathmatica 8 default formatting? Good enough for government work, I guess. What exactly does "non-stochastic" mean anyway? I used to teach a lab class and if one of the students presented this graph in their lab report it would have been all marked up in red. [Taken from the linked pdf, above]

2015 Q1 NGDP revised downward ...

... and I guess people (including myself) should take back all the praise heaped on the Atlanta Fed forecast. Hypermind gets more wrong and their implicit forecast for Q3 and Q4 (shown as a dotted line segment below) appears to be rather high 4.4% (that's what's needed if the Q2 forecast is correct in order to achieve their annual number of 3.4%).

The arrow and the dark blue dot indicate the new downward revision from the BEA. I'll just quote from what I said when the first estimate came out a month ago:
Overall: a continuing a trend of lukewarm economic performance, largely still in line with just about any model of the economy.

Thursday, May 28, 2015

Resolving the Cambridge capital controvery with abstract algebra

The title is a bit of a joke, and for the controversy see here. Looking into the Solow model bumps you into the question of what "capital" (K) is, and that met with the titular controversy awhile back where Cambridge, MA said you could add up different stuff in a sensible way while Cambridge, UK said you couldn't.

The information equilibrium (IE) model calls the argument for the UK (and Joan Robinson), but allows (at least) two possibilities for definitions of capital that are sensible. These sensible definitions weren't advocated by Solow/Samuelson at MIT, hence why I say that the UK won the debate: you can't just add stuff up and get a sensible answer.

First, two quick "proofs". I already showed IE is an equivalence relation (you can use it to define a set of things in IE with some economic aggregate), but I need a bit more: IE is a group under multiplication.

If $A \rightarrow K$ (with IT index $a$), then $A^{x} \rightarrow K$ (with IT index $a x$) because:

$$\frac{d}{dK} A^{x} = x A^{x - 1} \frac{dA}{dK} = a x \frac{A^{x}}{K}$$

(I show this because it applies for real exponents rather than just natural numbers and that might be important for some reason in the future; for natural numbers $x$ the following result would suffice.)

If $A \rightarrow K$ and $B \rightarrow K$ (with IT indices $a$ and $b$), then $A B \rightarrow K$ (with IT index $a + b$) because:

$$\frac{d}{dK} AB = \frac{dA}{dK} B + A \frac{dB}{dK} = (a + b) \frac{AB}{K}$$

So we have the set of all things that are in IE with $K$, and the product of any two of those things is another thing in the set -- therefore she's a witch it's a group. It is not, however, a ring -- the set isn't closed under addition:

$$\frac{d}{dK} (A + B) = \frac{dA}{dK} + \frac{dB}{dK} = \frac{a A + b B}{K}$$

so $A + B$ is not in IE with $K$ unless $a = b$.

This basically was Joan Robinson's point -- unless $A$ and $B$ are the same thing, you're comparing apples and oranges. Money doesn't help us either and if you introduce it, the relative prices of the capital goods become important **.

Sensible definitions of capital

Of course, this points to the first sensible solution to the capital controversy: instead of adding up capital items, use the geometric mean. Using the two results above, you can show that if $A \rightarrow K$, $B \rightarrow K$, $C \rightarrow K$, etc, then

$$(A B C \; ... \;)^{1/n} \rightarrow K$$

The geometric mean is also the only sensible mean for capital goods measured either as indices or in terms of money.

The second sensible solution to the controversy is a partition function approach (as I've done here) where we simply define capital to be the expected value of the capital operator, which is just the sum of the individual capital goods operators:

$$\langle K \rangle \equiv \langle A + B + C + \; ... \; \rangle$$

In that sense, "capital" would be more like NGDP than, say, a stock index.

** Update 5/29/2015: I thought I'd add in the details of the sentence I marked with ** above. We assume two goods markets (information equilibrium conditions) $p_{a} : N \rightarrow A$ and $p_{b} : N \rightarrow B$ where $N$ is aggregate demand/nominal output and the $p_{i}$ are prices. That gives us:

$$k_{a} p_{a} A = N \; \text{and} \; k_{b} p_{b} B = N$$

Substituting into the formula above

$$\frac{d}{dK} (A + B) = \frac{a A + b B}{K} = \left( \frac{a}{k_{a} p_{a}} + \frac{b}{k_{b} p_{b}} \right) \; \frac{N}{K}$$

which basically shows that $K$ is in information equilibrium with aggregate demand. Note the appearance of the prices in the information transfer index.

Wednesday, May 27, 2015

The basic asset pricing equation as a maximum entropy condition

Commenter LAL has brought up the basic asset pricing equation a couple of times, and so I had a go at looking at it as a maximum entropy/information equilibrium model. Turns out it works out. In Cochrane's book (updated with link) the equation appears as:

$$\text{(1) }\; p_{t} = E \left[ \beta \frac{u'(c_{t+1})}{u'(c_{t})} x_{t+1} \right]$$

Where $p_{t}$ is the price at time $t$, $c_{t}$ is consumption at time $t$, $u$ is a utility function, and $\beta$ is a future discount factor. Now $x_{t}$ is also the price at time $t$ (although it's called the payoff) and of course there is the funny business of the $E$ that essentially says all the terms at a time $t+1$ exist only in the minds of humans (and turns an $x$ into a $p$). Rational expectations is the assumption that the $E$ is largely meaningless on average (i.e. approximately equal to the identity function).

As a physicist, I'm not particularly squeamish about the future appearing in an equation (or time dropping out of the model altogether), so I will rewrite equation (0) as:

$$\text{(1) }\; p_{i} = \beta \frac{u'(c_{j})}{u'(c_{i})} p_{j}$$

It turns out much of the machinery is the same as the Diamond-Dybvig model, so I'll just adapt the beginning of that post for this one.

The asset pricing equation is originally a model of consumption in two time periods, but we will take that to be a large number of time periods (for reasons that will be clear later). Time $t$ will be between 0 and 1.

Let's define a utility function $U(c_{1}, c_{2}, ...)$ to be the information source in the markets

$$MU_{c_{i}} : U \rightarrow c_{i}$$

for $i = 1 ... n$ where $MU_{c_{i}}$ is the marginal utility (a detector) for the consumption $c_{i}$ in the $i^{th}$ period (information destination). We can immediately write down the main information transfer model equation:

$$MU_{c_{i}} = \frac{\partial U}{\partial c_{i}} = \alpha_{i} \; \frac{U}{c_{i}}$$

Solving the differential equations, our utility function $U(c_{1}, c_{2}, ...)$ is

$$U(c_{1}, c_{2}, ...) = a \prod_{i} \left( \frac{c_{i}}{C_{i}} \right)^{\alpha_{i}}$$

Where the $C_{i}$ and $a$ are constants. The basic timeline we will consider is here:

Period $i$ is some "early" time period near $t = 0$ with consumption $c_{i}$ while period $j$ is some "late" time period near $t = 1$ with consumption $c_{j}$. We'll only be making changes in these two time periods. The "relevant" (i.e. changing) piece of the utility function is (taking a logarithm):

$$\text{(2) }\; \log U \sim \;\; ... + \alpha_{i} \log c_{i} + ... + \alpha_{j} \log c_{j} + ... + \log U_{0}$$

where all the various $C_{i}$'s, $\alpha_{i}$'s and $a$ ended up in $\log U_{0}$.

Now the derivation of the asset pricing equation sets up a utility maximization problem where normal consumption in period $i$ (called $e_{i}$) is reduced to purchase $\xi$ of some asset at price $p_{i}$, and added back to consumption in period $j$ at some new expected price $p_{j}$. So we have:

$$\text{(3a) }\; c_{i} = e_{i} - p_{i} \xi$$

$$\text{(3b) }\; c_{j} = e_{j} + p_{j} \xi$$

Normally, you'd plug these into the utility equation (2), and maximize (i.e. take a derivative with respect to $\xi$ and set equal to zero). The picture appears in this diagram (utility level curves are in gray):

The change in the amount $\xi$ of the asset held represents wiggling around the point $(e_{i}, e_{j})$ along a line with slope defined by the relative size of the prices $p_{i}$ and $p_{j}$ to reach the point labeled with an 'x': the utility maximum constrained to the light blue line.

Instead of doing that, we will use entropy maximization to find the 'equilibrium'. In that case, we can actually be more general, allowing for the case that e.g. you don't (in period $j$) sell all of the asset you acquired in period $i$ -- i.e. any combination below the blue line is allowed. However, if there are a large number of time periods (a high dimensional consumption space), the most probable values of consumption are still near the blue line (more on that here, here). Yes, that was a bit of a detour to get back to the same place, but I think it is important to emphasize the generality here.

If the states along the blue line are all equally probable (maximum entropy assumption), then the average state will appear at the midpoint of the blue line. I won't bore you with the algebra, but that gives us the maximum entropy equilibrium:

$$\xi = \frac{e_{i} p_{j} - e_{j} p_{i}}{2 p_{i} p_{j}}$$

If we assume we have an "optimal portfolio", i.e we are already holding as much of the asset as we'd like, we can take $\xi = 0$, which tells us $e_{k} = c_{k}$ via the equations (3) above, and we obtain the condition:

$$\text{(4) }\; p_{i} = \frac{c_{i}}{c_{j}} p_{j}$$

Not quite equation (1), yet. However, note that

$$\frac{1}{U} \frac{\partial U}{\partial c_{i}} = \frac{\partial \log U}{\partial c_{i}} = \frac{\alpha_{i}}{c_{i}}$$

So we can re-write (4) as (note that the $j$, i.e. the future, and $i$, i.e. the present, flip from numerator and denominator):

$$\text{(5) }\; p_{i} = \frac{\alpha_{i}}{\alpha_{j}} \frac{\partial U/\partial c_{j}}{\partial U/\partial c_{i}} p_{j}$$

Which is formally similar to equation (1) if we identify $\beta \equiv \alpha_{i}/\alpha_{j}$. You can stick the $E$ and brackets around it if you'd like.

I thought this was pretty cool.

Now just because you can use the information equilibrium model and some maximum entropy arguments to arrive at equation (5) doesn't mean equation (1) is a correct model of asset prices -- much like how you can build the IS-LM model and the quantity theory of money in the information equilibrium framework, this is just another model with a information equilibrium description. Actually equation (4) is more fundamental in the information equilibrium view and basically says that the condition you'd meet for the optimal portfolio is simply that the ratio of the current to expected future consumption is equal to the ratio of the current to the expected price of that asset. Essentially if you think the price of some asset is going to go up 10%, you will adjust your portfolio so your expected future consumption goes up by 10%.

Paul Romer feels misunderstood

After an initial misunderstanding of his definition of mathiness, I think passed Romer's reading comprehension quiz. Romer answers "False" to each of these questions ...
1. T/F: Romer thinks that economists should not try to use the mathematics of Debreu/Bourbaki and should instead use math in the less formal way that physicists and engineers use it.
I think this (and Mark Buchanan approving thinks Romer would answer true), but Romer answers false.
2. T/F: Romer thinks that abstract mathematical models that could turn out to be of no use in understanding data and evidence are examples of mathiness.

This captures my initial misunderstanding (I linked the original, and here is the corrective from the next day). Overall, Romer should have left off the word empirical when he said: "Like mathematical theory, mathiness uses a mixture of words and symbols, but instead of making tight links, it leaves ample room for slippage between statements in natural versus formal language and between statements with theoretical as opposed to empirical content." (I crossed out the offending clause -- Romer's idea of mathiness is completely independent of data, so I'm not sure why he mentioned it.)

3. T/F: Romer thinks that errors in mathematical arguments are examples of mathiness.

As I said, it's lack of rigor (or "tight links" as Romer phrases it). A lack of rigor can be associated with errors, but are not identical to them.
4. T/F: Romer says that the economists he has accused of mathiness are using it to promote a right-wing political agenda designed to influence national politics.

The academic politics seems to line up with national politics, but as I mention here (in the PS) it's mostly about tribes of graduate students revolving around big names.
5. T/F: Romer thinks that economists should use less math.
I personally think economics should use less formal math, but I never attributed this to Romer.
6. T/F: Romer is angry.

I think the emotional states I attributed to Romer were being "upset", "zeal" and being "weird". I insinuated Lucas and Moll might be a bit annoyed with Romer.

Tuesday, May 26, 2015

Dynamics of the savings rate and Solow + IS-LM

Hello! I'm back from a short vacation and slowly getting to the comments.

As I mentioned here, there might be a bit more to the information equilibrium picture of the Solow model than just the basic mechanics -- in particular I pointed out we might be able to figure out some dynamics of the savings rate relative to demand shocks.

In the previous post, we built the model:

$$Y \rightarrow K \rightarrow I$$

Where $Y$ is output, $K$ is capital and $I$ is investment. Since information equilibrium (IE) is an equivalence relation, we have the model:

$$p: Y \rightarrow I$$

with abstract price $p$ which was described here (except using the symbol $N$ instead of $Y$) in the context of the IS-LM model. If we write down the differential equation resulting from that IE model

$$\text{(1) }\;\; p = \frac{dY}{dI} = \frac{1}{\eta} \; \frac{Y}{I}$$

There are a few of things we can glean from this ...

I. General equilibrium

We can solve equation (1) under general equilibrium giving us $Y \sim I^{1/\eta}$. Empirically, we have $\eta \simeq 1$:

Combining that with the results from the Solow model, we have

$$Y \sim K^{\alpha} \; \text{,} \; K \sim I^{\sigma} \; \text{and} \; Y \sim I$$

which tells us that $\alpha \simeq 1/\sigma$ -- one of the conditions that gave us the original Solow model.

II. Partial equilibrium

Since $Y \rightarrow I$ we have a supply and demand relationship between output and investment in partial equilibrium. We can use equation (1) and $\eta = 1$ to write

$$I = (1/p) Y \equiv s Y$$

Where we have defined the saving rate as $s \equiv 1/(p \eta)$ to be (the inverse of) the abstract price $p$ in the investment market. The supply and demand diagram (including an aggregate demand shock) looks like this:

A shock to aggregate demand would be associated in a fall in the abstract price and thus a rise in the savings rate. There is some evidence of this in the empirical data:

Overall, you don't always have pure supply or demand shocks, so there might be some deviations from a pure demand shock view. In particular, a "supply shock" (investment shock) should lead to a fall in the savings rate.

III. Interest rates

If we update the model here (i.e. the IS-LM model mentioned above) to include the more recent interest rate ($r$) model written in terms of investment and the money supply/base money:

$$(r \rightarrow p_{m}) : I \rightarrow M$$

where $p_{m}$ is the abstract price of money (which is in IE with the interest rate), we have a pretty complete model of economic growth that combines the Solow model with the IS-LM model. The interest rate joins the already empirically accurate production function:

Since I inevitably get questions about causality, it is important to note that these are all IE relationships therefore all relationships are effectively causal in either direction. However it is also important to note that the direct impact of $M$ on $Y$ is neglected in the above treatment (including the interest rates) -- and the direct impact changes depending on the information transfer index in the price level model.

Summary

A full summary of the Solow + IS-LM model in terms of IE relationships is:

$$Y \rightarrow K \rightarrow I \; \text{,} \; K \rightarrow D$$

$$Y \rightarrow L$$

$$1/s : Y \rightarrow I$$

$$(r \rightarrow p_{m}) : I \rightarrow M$$

Update 5/27/2015: Forgot first graph; corrected.

Friday, May 22, 2015

The rest of the Solow model

Here, I mostly referred to the Cobb-Douglas production function piece, not the piece of the Solow model responsible for creating the equilibrium level of capital. That part is relatively straight-forward. Here we go ...

Let's assume two additional information equilibrium relationships with capital $K$ being the information source and investment $I$ and depreciation $D$ (include population growth in here if you'd like) being information destinations. In the notation I've been using: $K \rightarrow I$ and $K \rightarrow D$.

This immediately leads to the solutions of the differential equations:

$$\frac{K}{K_{0}} = \left( \frac{D}{D_{0}}\right)^{\delta}$$

$$\frac{K}{K_{0}} = \left( \frac{I}{I_{0}}\right)^{\sigma}$$

Therefore we have (the first relationship coming from the Cobb-Douglas production function)

$$Y \sim K^{\alpha} \text{ , }\;\;\;\; I \sim K^{1/\sigma} \text{ and }\;\;\;\; D \sim K^{1/\delta}$$

If $\sigma = 1/\alpha$ and $\delta = 1$ we recover the original Solow model, but in general $\sigma > \delta$ allows there to be an equilibrium. Here is a generic plot:

Assuming the relationships $K \rightarrow I$ and $K \rightarrow D$ hold simultaneously gives us the equilibrium value of $K = K^{*}$:

$$K^{*} = K_{0} \exp \left( \frac{\sigma \delta \log I_{0}/D_{0}}{\sigma - \delta} \right)$$

As a side note, I left the small $K$ region off on purpose. The information equilibrium model is not valid for small values of $K$ (or any variable). That allows one to choose parameters for investment and depreciation that could be e.g. greater than output for small $K$ -- a nonsense result in the Solow model, but just an invalid region of the model in the information equilibrium framework.

An interesting add-on is that $Y$ and $I$ have a supply and demand relationship in partial equilibrium with capital being demand and investment being supply (since $Y \rightarrow K$, by transitivity they are in information equilibrium). If $s$ is the savings rate (the price in the market $Y \rightarrow I = Y \rightarrow K \rightarrow I$), we should be able to work out how it changes depending on shocks to demand. There should be a direct connection to the IS-LM model as well.