## Wednesday, December 3, 2014

### An information transfer DSGE model

For fun (and bowing to the inherent superiority of economics over other fields), here's the information transfer (macro) model as a fancy log-linearized DSGE/RBC model with lots of neat $\LaTeX$ symbols:

$$\text{(1) } n_{t} = \sigma_{t} + \left( \frac{1}{\kappa} - 1 \right) (m_{t} - m_{t-1}) + n_{t-1}$$
$$\text{(2) } \pi_{t} = \left( \frac{1}{\kappa} - 1 \right) (m_{t} + m^{*}) + c_{m}$$
$$\text{(3a) } r^{l}_{t} = c_{1} (n_{t} - m_{t} - m^{*}) + c_{2}$$
$$\text{(3b) } r^{s}_{t} = c_{1} (n_{t} - b_{t} - b^{*}) + c_{2}$$
$$\text{(4) } \ell_{t} = n_{t} - \pi_{t} + c_{\ell}$$
$$\text{(5) } w_{t} = n_{t} + c_{w}$$

Here the log-linearized variables are $n$ (nominal output), $m$ (the currency base, M0), $b$ (monetary base, MB), $\pi$ (the price level), $\ell$ (labor supply), $w$ (nominal wages) and $r^{x}$ (interest rates with x = l, s for long- and short-term). The term $\sigma$ represents the 'nominal shocks' (see also here, mathematically no different from the RBC TFP shocks, but could include e.g. changes in government spending). The parameters are the $c_{i}$, $m^{*}$ and $b^{*}$ (the log of the equilibrium/starting values of the monetary base components) along with our old friend $\kappa$, the information transfer index. In a log-linearized model, $\kappa$ is a constant since we're considering small deviations from the variables in the log-linearization.

This representation explicitly shows that there aren't any $E_{t}$ terms (expectations) unlike e.g the RBC model [pdf]. A couple of additional interesting facts pop out if we set $\kappa$ to the 'IS-LM' regime (1.0) or quantity theory of money regime (0.5) (see here for more about the meaning of this). If $\kappa = 1.0$ we have:

$$\text{(1a) } n_{t} = \sigma_{t} + n_{t-1}$$
$$\text{(2a) } \pi_{t} = c_{m}$$

where nominal output is unaffected by monetary policy and inflation is constant. If we plug (1a) back into the interest rate formula, we see that monetary expansion will cause interest rates to fall (nominal output is unchanged by monetary expansion). If $\kappa = 0.5$ we have:

$$\text{(1b) } n_{t} = \sigma_{t} + m_{t} - m_{t-1} + n_{t-1}$$
$$\text{(2b) } \pi_{t} = m_{t} - m^{*} + c_{m}$$

which says the price level grows with the monetary base (the quantity theory of money). If we plug (1b) back into the interest rate formula, we see that monetary expansion will generally cause interest rates to rise (nominal output will increase because nominal growth will exceed monetary expansion). Explicitly:

$$r_{t} = c_{1} (\sigma_{t} + n_{t-1} - m_{t-1} - m^{*}) + c_{2} = c_{1} \sigma_{t} + r_{t-1} + c_{2}$$

where the nominal shocks $\sigma$ are observed to be generally positive (think growth in TFP or population) along with $c_{2}$ (which is $\sim \log NGDP/M0$).

Anyway, this is just the baseline model -- I'll put in some equilibrium conditions and policy targets and see if economics really is superior to other fields of inquiry. I mean this kind of thing was Nobel prize level work in the 1980s.

PS Where's the information theory in this, you ask? It's in the parameter $\kappa$ for starters which defines the relative information content of nominal output relative to the supply of money. Equations (1), (4) and (5) represent information equilibrium conditions while equations (2) and (3) represent measures of the rates of change of information transfer.

1. Side note: equation (4) is just a form of Okun's law.

2. I should probably say liquidity trap regime rather than IS-LM since we're looking at $\kappa = 1$ rather than close to 1.

3. deeply confused by your problem with taking expectations...so it is impossible for output or the price level to change in response to anticipated changes in the monetary base? that is a consequence of your equations and i think flies in the face of reality.

...I dont object to not believing in taking the expectations in the way DSGEs usually do, but the S in the DSGE stands for stochastic....you have so far a D and a G...

1. Hi LAL,

The random shock $\sigma_{t}$ makes the system of equations stochastic. It is true there are no equilibrium conditions in the DSGE sense, so you could say the E is still missing. However the model above is based on information equilibrium, so in a sense equations (1), (4) and (5) are equilibrium conditions.

Regarding expectations, I have no problem with expectations moving market variables in the short term. Expectations of monetary expansion could cause markets to rise or inflation to rise in the short term. However, eventually the expected monetary expansion would actually have to take place (creation of base money), or else inflation and/or market would have to return to the previous equilibrium.

One way to see my issue with expectations is that expectations assumes it has the correct fundamental economic theory. Imagine a world where markets believed fiscal policy raised NGDP, but the unknown fundamental economic theory actually has monetary offset and fiscal policy has no effect. Markets would respond positively and NGDP would go up due to expectations, but how long would they stay up? In our thought experiment, the fundamental theory says that fiscal policy does nothing.

There are two choices here:

1) The market doesn't return to the previous NGDP level. That implies that the fundamental theory is incorrect, a logical contradiction.

2) The market returns to the previous NGDP level per the fundamental economic theory. In that case expectations had only a temporary effect on markets.

Markets respond with how people think economics works right now, but the economic "theory of everything" is not known, so those responses (expectations) may or may not be correct. Therefore a given theory that uses expectations, unless the underlying theory is known perfectly, could use the wrong expectations.

Another way to say this is that expectations let you get the answer you want regardless of how economics actually works.

But that is not saying expectations have no effect -- in fact, I think expectations have powerful effects:

http://informationtransfereconomics.blogspot.com/2014/05/the-effect-of-expectations-in-economics.html

2. Here are a couple more links on expectations:

Expectations tend to destroy information unless they are perfectly accurate:

http://informationtransfereconomics.blogspot.com/2014/05/expectations-destroy-information.html

However, there is another interpretation of the information transfer model where expectations are everything. This is still different from the way modern economics uses expectations, because it averages over all possible expectations weighted equally.

http://informationtransfereconomics.blogspot.com/2014/12/meeting-expectations-halfway.html

3. Maybe I'm being too rigid in my definitions of stochastic...I guess really though how does the timing work: when would agents be allowed to make a guess at sigma? your model looks like people just passively respond to shocks, they don't seem to make any plans for the future at all... combined with this route of leaving all these ci's undefined is going to lead to some classic lucas critique territory, in that only when people do not anticipate sigma_t and then respond passively to it, can the ci's remain constant. what would be the implications to information theory if the cis are changing? does it change the nature of information equilibrium.

In: "http://informationtransfereconomics.blogspot.com/2014/05/expectations-destroy-information.html"

You show that planning for the average value leads to inefficiencies since only one outcome is realized, but I am having trouble fitting it into the context of any model.

The inefficiency is not expectations but aggregate uncertainty. With aggregate certainty all this loss could be traded away by holding the right portfolio, thus everyone's expectations would be coordinated and there would be no information loss. Since in the real world we do face aggregate uncertainty, I can see how some expectations could be worse than others, but what is the rule that would fix it? and still why should people care at all about information inefficiency? I guess I need a proof that an economy in equilibrium is necessarily in information equilibrium and/or vice-versa.

4. Hi LAL,

You said:

"your model looks like people just passively respond to shocks, they don't seem to make any plans for the future at all"

On the contrary, I make no assumptions about and place no limits on the plans agents make. Another way to put it is that I, the theoretical model-maker, am making the assumption of maximum ignorance about agent behavior. The information transfer model should work for any set of agent behaviors that aren't highly correlated.

This actually seems to be more sensible that the way mainstream macro approaches this issue -- different agents are going to have different portfolios that respond differently to shocks. Some will have to change their behavior when inflation expectations shoot up because they are exposed to a lot of inflation risk, others will welcome those expectations. For any position relative to a shock (oil prices, financial risk), someone has to be taking the opposite position (the other side of that deal), so assuming a collection of economic agents anticipates a particular shock in a specific way

We don't know what those plans are -- they are fundamentally unobservable (many different sets of plans about the future lead to the same outcome) -- so making assumptions about them is not only ad hoc, but untestable empirically.

The kinds of shocks that don't have this binary quality (i.e. two sides of the deal) are essentially "NGDP shocks", i.e. recessions. You can make it so you are less exposed to NGDP shocks by buying e.g. government bonds that pay by law (hence your hedge takes the risk that the government won't collapse), but there is no real hedge against slower economic growth.

You said:

"combined with this route of leaving all these ci's undefined is going to lead to some classic lucas critique territory, in that only when people do not anticipate sigma_t and then respond passively to it, can the ci's remain constant. what would be the implications to information theory if the cis are changing? does it change the nature of information equilibrium."

The ci's are all defined, I just didn't write down the definitions in this post. And it is possible for them to change (one of the cases is a major monetary regime change) -- it appears that these parameters change suddenly in a process that is similar to a phase transition in thermodynamics.

http://informationtransfereconomics.blogspot.com/2013/08/the-liquidity-trap-and-information.html

This has been the only way I've seen parameters change, however. As an aside, I did write something specifically about the Lucas critique:

http://informationtransfereconomics.blogspot.com/2014/12/information-equilibrium-theories.html

5. [continued]

You said:

"You show that planning for the average value leads to inefficiencies since only one outcome is realized, but I am having trouble fitting it into the context of any model."

What I intended to show is that having a guess about the future (an informative prior) is worse (more inefficient) than assuming ignorance about the future (least informative prior) ... unless you have a stable economy (e.g. NGDP forecast errors are normally distributed).

A way to interpret this in terms of the "plans" at the top of this comment is that diverse plans lead to less inefficiency than coordinated plans. You can imagine that a set of coordinated plans are far more brittle than diverse (almost random) set of plans when subjected to shocks.

You continue:

"With aggregate certainty all this loss could be traded away by holding the right portfolio, thus everyone's expectations would be coordinated and there would be no information loss."

Actually, in the case of coordinated plans, there is maximal information loss. Coordination of economic agents results in recessions. You can imagine that the economy rolls a pair of dice (resuting in 2-12) every time period. Everyone who gets it right wins money; everyone who gets it wrong loses money. If everyone had a coordinated plan (3), then that would lose money much of the time. If everyone had a coordinated plan at the average (7), that would lose less money. However if everyone chose different plans, that loses the least amount of money over time ... it would correspond to the least average-case information loss.

There is more here:

http://informationtransfereconomics.blogspot.com/2014/10/coordination-costs-money-causes.html

David Glasner wrote a good piece about plans awhile ago, and I reference it at that link.

I'd like to thank you for your questions -- questions are one of the things that help me understand this theory better. It's still a work in progress and could well be wrong.

You said:

"I guess I need a proof that an economy in equilibrium is necessarily in information equilibrium and/or vice-versa."

That is a great idea -- I will try to work on that. One hypothesis that I have is that information equilibrium helps you select the right Arrow-Debreu equilibrium (of all of the equilibria, the maximum entropy equilibrium is the most likely).

6. thanks good stuff, i maintain though with aggregate certainty...a possibly irrelevant way of thinking (nevertheless they teach it in gradschool)... the idea is that some risks can be perfectly insured away...in such cases coordination of expectations is "good" i think as it allows the existence of markets for full insurance (ie leads to a pareto optimal outcome)...I think I need to dig into a model to see the relationship of information to choosing an asset portfolio

If I'm right there are definitely some benefits of coordinated expectations that should be traded off with as you suggest coordinated expectations about uncertain aggregates

4. Should Eq. (1) have (1/k) instead of (1/k - 1)? I think so ...