Friday, 11 April 2014

Eggertsson & Mehrotra and SFC Models



I'm very interested in the new paper from Gaudi Eggertsson and Neil Mehrotra (E&M) on secular stagnation.  What they have done is to adapt a simple overlapping generations model like Samuelson's consumption-loan model so that the financial asset is represented by a corresponding liability of other agents in the model.  This is very similar to something I myself posted on recently, but whereas I used land purchase as the motivation for private borrowing, they have households borrowing to finance consumption ahead of earning income.

I have reservations about some of the things they have done in this paper but, on the whole, I like it.  I think this sort of approach with differentiated borrowers and lenders is key to understanding the dynamic role of debt.

Other are less impressed.  Amongst various criticisms (some of with which I'd agree), Unlearning Economics dislikes the apparent omission of banks and believes that a stock-flow consistent (SFC) model would be preferable.  I find this quite interesting, because as far as I can see there are no stock-flow inconsistencies in their model and I can imagine a simple SFC model with banks that would be structurally identical to what E&M have produced.  So, I thought it would be useful to recast E&M's basic model in some more heterodox language. 

E&M don't talk about banks.  However, there is an important structural element in there where I think that banks fit the story quite well.  They are vague on the story, which is OK because it doesn't have to involve banks - it's a more general point.  But I'm going to use banks in my description.  (Note that, as a result, some of my variable names do not correspond to theirs)

So, in this model households live for three periods, then they die and are replaced by new households.  They consume in each period, but only work and earn in the middle period.  They therefore need to borrow to fund consumption in the first period and to save to pay for consumption in the second period.  Borrowing takes the form of loans from banks and savings take the form of bank deposits. 

The balance sheet at the end of each period, after the old have spent their savings, is as follows:


Young
Middle Aged
Old
Banks
Loans
- L


L
Deposits

M

- M


In each period a number of things are going on.  The young are borrowing from banks to fund consumption spending.  The old are funding their spending by drawing down their deposits (together with interest).  The middle aged are earning wages*, spending on consumption and repaying with interest the loans they incurred when young.  As they are spending less than they earn, they are also accruing deposit balances.  Banks are assumed to pay the same rate of interest on deposits as they charge on loans.  All this is shown below:



And more formally, in a flow of funds matrix.


Young
Middle Aged
Old
Banks
Wages

W


Loan interest

- Lt-1 . rt-1

Lt-1 . rt-1
Deposit interest


Mt-1 . rt-1
- Mt-1 . rt-1
Consumption
-Cy
-Cm
-Co

Loans
L
- Lt-1

- ( L - Lt-1 )
Deposits

- M
Mt-1
M - Mt-1

Every row and column here sums to zero, providing a budget constraint for each cohort.  These correspond to equations (1), (2) and (3) of E&M.

The spending behaviour of the young and old is dictated by their budget constraints.  The middle-aged have to choose between how much to spend and how much to save.  E&M derive this decision from some assumptions about household utility functions, but it turns out that middle-aged consumption can be given by:

                Cm = α1 . ( W  -   Lt-1 . rt-1 )  -  α2 . Lt-1              with α1 = α2

Interestingly, the current interest rate doesn't factor into this, because with the utility function E&M use, the income effect and the substitution effect cancel out.  The equation for aggregate consumption (taking into account that L = M) is then of the form:

                C = α1 . W  +  α3 . Mt-1 . rt-1   +  α4 . Mt-1  +  L        

This looks very similar to the sort of function you might see in a standard SFC model, and indeed it does generate steady state stock-flow ratios. 

It is a critical feature of this model that banks make independent lending decisions.  In other words, the decision by middle-aged about how much to save is not the same as the decision about how much is loaned to the young.  So we need a separate way of determining how much banks want to lend.

We could do this by modelling profit-maximising banks and things like asymmetric information, but is not really warranted by the model and is probably unrealistic anyway.  The best approach I think, which is the one taken by E&M, is to put it down to what can best be described as animal spirits.  The level of bank lending is therefore treated as exogenous.

Although the lending decision and the saving decision are independent, the accounting dictates that loans and deposits will be equal.  The actual amount of deposits that the middle-aged accumulate is determined by the extent of lending.  For a given interest rate, it is the level of consumption spending that has to change to ensure that they are saving as they intend.

E&M want to use the model to explain secular stagnation.  I would agree with Unlearning Economics that there it doesn't have enough going on to do that.  I also think that the way the interest rate is working within this model is unusual and they should say more about that.  However, I believe that the basic model they have constructed here involves very much the sort of approach that I think is needed to address this issue.

Even though they do not use the same language, their model actually incorporates features, like stock-flow consistency and the direction of causation between lending and saving, that are key elements of a post-Keynesian approach.  There is more than enough disagreement between economists on methodology, so it's nice when seemingly different approaches are in fact quite similar.

[edited for a typo in the FoF matrix spotted by dsquared]

* In E&M's basic model, the middle-aged just get an endowment, but since this must equal total consumption, I've just tied it in to income from production.

15 comments:

  1. Isn't the problem not so much about stock-flow consistency in equilibrium, but that the model cannot adjust from one equilibrium to a post-shock equilibrium in a stock-flow consistent way. This is the fundamental problem of all comparative static equilibrium models.

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    1. I think that can be a problem for comparative static models, but it doesn't seem to be a problem here. We have a comprehensive accounting structure, together with the assumptions we need to determine behaviour period to period. I could run a simulation here to show the adjustment to shocks (although I don't think it would be a very interesting one).

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  2. Hi Nick,

    Nice post. However, my charge is not that the EM model is *not* stock-flow consistent - I think that all DSGE models are. My point is quite simply that the 'microfoundations' of the model, such as utility maximising agents, force EM to omit important variables or make them exogenous in order to prevent the model from being too complicated. I do not see the advantages of having different generations optimising intertemporally over having simple 'hydraulic' consumption functions etc, which could allow us to focus on the more important mechanics at play here. I also disagree that we can just make the deleveraging shock exogenous and attribute it to 'animal spirits', because this implies there is no logic to when a bust begins. It would be less arbitrary to have it happen when eg debt levels exceed income, but EM don't seem to want to go down this route and I'm not sure how their framework would let them.

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    1. Thank you.

      I'd planned to write something on this paper anyway, because it tied in to stuff I'd been looking at recently. Your own post just happened to raise one or two of the issues I was interested in, so I used it as a prompt. Sorry.

      I'm happy to use "hydraulic" consumption functions, but I'm also interested in what the underlying behaviour is. I want to know what shapes the parameters and I've been particularly interested in how things like access to credit and changes in asset prices might affect these. I think that understanding life cycle saving patterns play quite an important role in this because pension schemes and mortgage paydown are such an important component of household savings behaviour.

      OLG models provide quite a good way of exploring these issues, but you do need to make some assumptions about how people plan their lifetime spending. Usually, I just assume something simple, but I have no problem with using some kind of conventional inter-temporal optimisation element if that works. It actually makes more sense, yet makes less difference to the results, in these models than in the infinite horizon versions. E&M would have got the same results if they simply assumed that households aimed to spend the same amount in each period of their life.

      That said, I absolutely agree with you that the mainstream's obsessive commitment to inter-temporal optimisation is wrong. If it fits fine; if it means you to have to compromise other elements, ending up with a less realistic picture, or if you have to choose assumptions on the basis of tractability rather than plausibility, then what's the point? Too often the latter seems to be the case.

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  3. I think that it is clear that this and the E&M model is for one household. One household for three periods of life. Your model includes a bank as holder of money and perhaps decision maker of loan quantity.

    You also make it clear that there is an end period with Old running out of funds. I presume there is also a start period where Young have no loans. I must also assume that there is an interim period where Middle have both loans and money; the alternative would be that there is some sort of transition period where Middle begins with all the loans and ends with all the money.

    Several relationships come to mind here:

    1. The bank, as decision maker, would likely expect that all loans were paid off while wages were still being earned. This would be a severe constraint on loans It would also preclude Old from having any loans remaining because they would be borrowing from themselves (the single household restraint). My conclusion here is that we should expect this model to result in stagnation.

    2. If we allow Middle to have both loans and money, then Middle owes Middle which would allow for loan carryover into Old. I think this is what you are depicting in the flow of funds matrix, but the problem here is that all players can not become Old at the same time. i.e., the flow of funds matrix is reflecting more than a simple one household situation.

    3. The aggregate consumption formula clearly indicates that consumption increases as loans increase (and money increases). To me, this correctly indicates that money is a key to improved consumption even if we are borrowing from ourselves to be repaid in the future. The borrowing becomes a promise to work harder and the byproduct will be an improved lifestyle--even if we ARE simply borrowing from ourselves.

    4. While with this model we might not be able to move loans to Old (due the the banks wage limitation), we could move paid assets into Old. This could be the land tie you previously used (and urged by Nick Rowe), or could be a debt-free residence. The use of other non-monetary assets easily makes the life period transition from Middle to Old.

    5. To me, in this model, money can not be a store of value between age periods because of the single household and bank induced wage limitation. Money here functions as a key to make things happen. We promise to repay the money, even if it is to ourselves, but hopefully we will create some long lived and useful assets.

    I think both your post and the E&M post give valuable economic insight. Thanks to all of you for sharing this thinking.



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    1. Roger,

      The model involves several households. Each has three stages of life and they are overlapping. So in each period there are young, middle-aged and old households. Then they all move on to the next stage for the next period. E&M (and Samuelson) have a number of households in each period with the number in each stage growing over time. I've been vague on this.

      The timing works as follows (note this is different to what I did in my version with the land). The young start their first period with no loans. They then borrow during the period (and spend), so end the period with loans outstanding. During the next period, they repay these and build up deposits, to end their second period with no loans but with money balances. Dyuring their last period they run these down to end with nothing.

      1. Loans must be paid off here during the middle stage of life when income is being earned. The old have no income to be able to repay the loans.

      2. Middle could be said to have both loans and money, but not at the same time. They start with loans, pay these down, then accumulate money.

      3. Yes. It is a key feature of this model that increased lending leads to increased spending.

      5. Money is definitely a store of value here. In this version (as opposed to mine), deposits (and the underlying loans) are the only store of value. It is what enables each household to consume in three different time periods even though they only produce in one.

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  4. Thanks for the additional explanation. Your comments serve to smooth the model for me, filling some logic gaps.

    I think the model could be improved by using not three time periods but four. This would allow for a loan peak AND a money peak, both of equal size so that the loan would always be the negative of the money quantity.

    I would suggest naming the four periods, Young, Young-Middle, Middle-Old, and Old. Loans would grow from zero at the beginning of Young, peak at the intersection of Young to Young-Middle, and taper to zero at the intersection of Young-Middle to Middle-Old. Money would begin at zero at the beginning of Middle-Old, peak at the intersection of Middle-Old to Old, and taper to zero at end of Old.

    With the four sector model, a constant supply of loans and money is easily seen to transfer between the overlapping households numbering in the millions and billions.

    With the four sector model, increased money supply from bank loans is easily accommodated (within the wage payoff restriction) with loans from Middle-Old to Young-Middle. The penalty to the Young-Middle borrowers would be an increased likelihood of entering Old with no monetary assets.

    Would the four period model be more complex? Yes, there would be more terms but each term would be balanced with a corresponding term in another column, making the model easier to understand.

    We might want to expand the model into even more time sectors but I would suggest that we always do the expansion in groups of two, in order to maintain balance. For example, we might want to add a Baby period on the Young side and a Decrepit period on the Old side. The Baby period would be an investment period and the Decrepit period a charity period, with both additional periods becoming consumption.

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    1. The simplest OLG models use just two period lives (as my own land based version of this). E&M need three period lives to have both borrowers and savers, given that borrowing is for consumption purposes.

      You can of course have more and to get a more realistic picture you need many more. I sometimes use a structure with 40 periods of work and 20 periods of retirement, which sort of reflects an average life cycle in years. You can do this in simulation models, but then it's a bit more complex to explain what's going on. The bare minimum is best for illustrating the dynamic relationships.

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  5. Nick E, off topic... WAY off actually, but I've been playing with something called "Scilab." I don't know if you're familiar with Matlab or not, but it's similar except that it's free. My educational background is in the field of feedback control systems, although the only part of that education I use on a regular basis is building/designing optimal estimators (normally both a controller and an estimator are part of a feedback control system). It's always seemed to me that the central bank problem of attempting to target some nominal variable (inflation, price levels, NGDP, etc) amounted to a kind of feedback control problem. Of course I've found some good literature on that recently... some of it quite old! I have yet to really dig in, but in the course of dusting off my unused control system knowledge from eons ago, I started playing a bit with Scilab: I don't think it helps with interactivity, but in some ways it's much easier to deal with than Excel. If you go down to the bottom of this page I put some plots up (having nothing really to do with economics) and an *ugly* listing of a Scilab script I used to generate them:

    http://banking-discussion.blogspot.com/p/partial-endogeneity-and-exogeneity.html

    My goal is to one day understand the kinds of systems that macro econ people model well enough to put a system into some sort of a feedback control formulation, just to get a better feel for some of the challenges that confront a CB. Perhaps that's a foolish goal, I don't know. :D

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    1. The reason I tell you all that is that I think doing the simple analysis I do there (which I hope to someday apply to economic models perhaps) would have been a nightmare in Excel. Maybe not using VisualBasic (I'm don't know much about VisualBasic), but I still suspect it would have been difficult.

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    2. Tom,

      It all looks very impressive but, to be honest, it would take me quite a long time to get to grips with all that. I might look at the Scilab thing, but I had the advantage of having learned a bit of VBA way back, so it was easier for me to sort of relearn it rather than try to get to grips with something new.

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    3. I understand completely. In fact Excel and Visual Basic may be best for the kinds of things you put together. If you're not already familiar with Matlab it may be a bigger headache than it's worth. What I should really do is translate one of your other models into Scilab and see if there's any advantage.

      Just thought I'd make you aware of it!

      BTW, I cleaned up the general purpose spreadsheet based Newton's method equation solver a bit and I now make use of the circular references iteration facility in Excel (which they preserved in their online version). Still it would need a fancy VBA script to really be useful for adapting it to general purpose problems (off-line) to preserve the interactive solving capability online.

      http://banking-discussion.blogspot.com/p/non-linear-equation-solver-with-up-to.html

      A few bugs in Excel drove me nuts though in getting it to work! In theory I would be able to eliminate almost all those purple intermediate result cells, but in practice the functions which would allow that don't always work correctly when circular references are enabled and used.

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    4. Nick, you ever hear Rowe draw an analogy between the CB's job and guiding a broom balanced upright in the middle of your hand? What about three brooms, balanced one atop the other?:
      http://pragcap.com/forums/topic/pendulums#post-64011

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  6. typo in the second line of the flow of funds matrix isn't there - the "Banks" shouldn't have a minus sign as they are receiving loan interest not paying it?

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    1. Yes - thanks for the spot. I'll edit that.

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