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:

                C^m = α₁ . ( W  -   L_t-1 . r_t-1 )  -  α₂ . L_t-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 . M_t-1 . r_t-1   +  α₄ . M_t-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.