SFC / DSGE Hybrid

I did a little exercise recently trying to blend DSGE models with stock-flow consistent (SFC) models.  The basic idea was to take the accounting framework and policy set-up of an SFC model but use the behavioural assumptions of a DSGE model.  This is not because I think the behavioural assumptions of DSGE are better (I certainly don't); I'm just interested in seeing how it affects the results.

This post describes a simple hybrid model based on the SFC model called Model PC in Chapter 4 of Godley & Lavoie.  This consists of three sectors: households, the government and the central bank, and two assets: interest-bearing bonds and non-interest bearing money.  The flow of funds matrix for this model is shown below (for nominal amounts, with production shown as a separate column).  All rows and columns sum to zero.

	Households	Government	Central Bank	Production
Consumption	-C			C
Government spending		-G		G
Income	Y			-Y
Taxes	-T	T
Bond interest	R.Bh	-R.B	R.Bc
CB profit		CP	-CP
Change in bonds	-ΔBh	ΔB	-ΔBc
Change in money	-ΔH		ΔH

As with G&L, fiscal policy consists of setting the level of real government expenditure and the tax rate applied to GDP.  Monetary policy consists of setting the interest rate which applies to bonds.  The change in household wealth depends on household saving (which must be equal to the government deficit).

Household behavioural assumptions are needed to determine two things: a) how much households save; and b) how much of their savings they want to hold as money rather than bonds.  All of the other flows then follow, given the policy assumptions and the accounting constraints.

This is where I depart from G&L.  Rather than using the usual SFC behavioural assumptions, what we want to do here is use functions  that are consistent with what appears in a typical New Keynesian DSGE model.  So the starting point will be to assume that households maximise expected utility over time where utility for period t is given by:

                β^t  [ ln (c_t) + ψ ln ( H_t / p_t ) ]

That is utility depends on real consumption in the period and money holdings in the period.  from this we can derive functions for household expenditure and portfolio allocation.

The DSGE expenditure function depends critically on the expected real interest rate, which is derived from the nominal interest rate and expected inflation.  This is not a feature of Model PC in G&L, but if this exercise is to be at all meaningful, we need to have an inflation mechanism here.  So I've used the standard New Keynesian Phillips curve, which sets current inflation as a function of expected future inflation and current output.

A typical DSGE model will also include a central bank reaction function to set the nominal interest rate.  I can do this in the model (and might look at this a subsequent post) but here I want to take the nominal interest rate as fixed. 

With the interest rate fixed, I then wanted to look at the impact of a 5% permanent real increase in government spending.  The results are shown below (showing deviations from baseline values):

The first chart shows the increase in nominal GDP (relative to baseline).  This actually looks pretty similar to what you might expect from a regular SFC model, with a slow progression to a higher steady state.

Looking at what is going on behind this, however, we can see that the increase in real GDP is temporary, just as there is a temporary rise in inflation.  The fact that NGDP continues to rise after the first period therefore purely reflects the fact that the price level is rising.  As with a typical SFC model, the initial impact on real consumption is that it increases.  However, unlike in a typical SFC model, rather than then continuing to rise, real consumption here falls back to lower than its original level.

This is entirely a result of the assumption about consumer spending.  In particular, the DSGE assumption about infinite horizons means that the changing balance of household assets has no feedback effect on spending, something that is key to typical stock-flow dynamics.

Interestingly, the balance of household assets is doing the opposite here to what it would do in a SFC model.  Rather than a slow accumulation of assets, the real level of bonds falls to a new lower level.  This decline is mainly due to higher inflation eroding the real value of bonds (although the nominal level of bonds also falls slightly)

It is important to note that this fall in the real value of bonds must happen if the system is to arrive at a new steady state.  In steady state, the government budget must be balanced.  With a permanent increase in spending and taxes pegged by the natural level of output, this requires that the real interest service cost must fall.  As the real interest rate is dictated here by the natural rate, the thing that has to give is the real level of bonds.

Model Specification

I have consolidated the government and central bank here for simplicity.  It makes no difference to the results.

Output is the sum of consumption and government spending.

(1)          y_t = c_t + g_t

The behavioural equations for consumption expenditure and money holdings are derived from the household utility function and budget constraint.

(2)          c_t = E[c_t+1] / ( β ( 1 + E[r_t] ) )

(3)          H_t = ψ [ 1 + R_t ( 1 - τ_t ) ] / [R_t ( 1 - τ_t ) ] . c_t . p_t

The level of bonds held by households is given by the consolidated government budget constraint.

(4)          Bh_t = Bh_t-1 [ 1+ R_t ( 1 - τ_t ) ] + ( g_t - τ_t . y_t ) . p_t - ( H_t - H_t-1 )

Inflation is based on expected future inflation and a measure of the output gap.  The price level is derived from this.

(5)          π_t = ( E[π_t+1] )^β . ψ y_t^ε

(6)         p_t = p_t-1 . π_t

The real interest rate is based on the nominal interest rate and inflation.

(7)          r_t = [ 1 + R_t ( 1 - τ_t ) ] / π_t - 1

Expected values are set to be equal to actual outcomes, with the exception of the period when government spending is first changed.  The equations allow solutions where the real value of household assets tends to infinity (either positive or negative).   The solution with infinite negative household assets is excluded on a no Ponzi condition.  The solution within infinite household assets is excluded as it not consistent with the utility maximisation assumption.

Variables

Name	Description
c	Consumption expenditure
g	Government expenditure
p	Price level
r	Real interest rate
y	Real GDP
Bh	Household bond holdings
H	Household money holdings
R	Nominal interest rate on bonds
π	Inflation
τ	Tax rate

 

Solution Technique

The first stage of solution is finding the steady state real values.  Each period is then solved sequentially using the steady state values as expected values for everything except inflation.  Each period is then solved again using the previous results for expected values (apart from inflation).  In each case expected inflation is estimated using a version of the fiscal theory of the price level, where expected future surpluses are discounted at the expected future effective rate on government debt and compared with the nominal value outstanding.  The whole process is repeated until the expectation errors on all variables is sufficiently small.

Jakab and Kumhof on Banks and Loanable Funds

Zoltan Jakab and Michael Kumhof  (JK) have produced a working paper for the Bank of England on whether banks should be viewed as money creators or simple intermediaries.  One of their main claims is that the mainstream modelling of banks is flawed because it views banks as the latter, when in reality they are the former.

I don't find their reasoning very convincing.  They produce some detailed models in which they compare results where banks are intermediaries of loanable funds (ILF) with those where there is finance through money creation (FMC).  By introducing various frictions into the ILF versions, they dampen the impact of shocks so that the FMC versions display greater volatility.

JK's argument is that most mainstream models with banks implicit assume that they those banks operate under the ILF model.  Presumably this means that those models are somehow reflecting an unrealistic dampening of shocks due to implicit frictions.  I find this line of argument odd.  On the whole, mainstream models are constructed to exclude all frictions other than those under consideration.  It is not clear to me what frictions JK would remove in those models to make them more realistic.

Part of the problem is that when JK compare the results of their different models, they are not comparing like with like.  There are some important structural differences between their ILF models and their FMC models to do with which agents do what, and it is this that is giving their results rather than anything to do with bank operation.

Take for example their ILF Model 1 and FMC Model 1.  In ILF Model 1, they have two types of household - borrowers and lenders.  Borrowers have real capital assets which they use as collateral to get loans from banks.  Lenders just hold bank deposits.  Lenders are then assumed to face a transaction cost friction which depends on their holding of deposits.  The greater the level of deposits the less this friction.

In FMC Model 1, these two types of household are folded into one.  The representative household has loans, deposits and real capital.  It still faces the same transaction costs, but is now in a position to mitigate this.  It has no need to borrow to fund a holding of capital assets, but it can use the collateral to borrow to raise its holding of deposits and reduce the transaction cost friction.  In the ILF Model 1, this cannot happen because lenders need the deposits but do not hold the collateral.

Now this may reflect a genuine friction that arises in the real world, but it seems to me to be all about heterogeneity and distribution and nothing to do with the operation of banks.  Their entire result here depends on there being two classes of agent in one model, but only one in the other.  JK seemed to have compared two different structural set-ups and concluded rather arbitrarily that the different results are all to do with loanable funds.

Even from the start, when they give a simple overview of what they see as the different models, they make this mistake of not comparing like with like (pp 11, 12 and figures 2 and 3).  To make a proper comparison, we need to use the same assumptions about who is involved and what they are trying to achieve and then examine how the results might differ if we restrict what steps can be involved in getting to the result.  

It's helpful to illustrate this by rejigging their examples slightly.  So, we start with three non-bank entities (which I'll call A, B and C) and the Bank.

- A holds gravel (as per their example), which it does not wish to consume currently - it wishes to hold a bank deposit instead.

- B wants to invest in machinery, but has no current resources so needs to borrow.

- C has machinery to sell and would like to acquire gravel

- Each of A, B and C will take credit risk on the Bank, but only the Bank is willing to take credit risk on any of A,B or C.

We then have two different ways in which these objectives can be reconciled, the ILF route or the FMC route.  These are illustrated in the diagrams below.

In the ILF model deposits and loans are contracted and settled in real goods rather than monetary payments (or ledger entries).  (Odd as this sounds, it is a common assumption in mainstream models of banking.  However, it is usually possible to reconstruct these into monetary models that are structurally equivalent - see here for example.)  So, here, the steps are:

ILF1  -  A deposits gravel with Bank

ILF2  -  Bank lends gravel to B

ILF3  -  B trades gravel with C in exchange for machinery

This scenario is the same as the ILF one that JK describe in the paper.

In the FMC scenario, we have monetary deposits so the steps are:

FMC1  - Bank lends to B (creating a deposit for B in doing so).

FMC2  - B buys machinery from C (by "transferring" the deposit).

FMC3  - C buys gravel from A (by "transferring" the deposit)

This is not equivalent to the FMC scenario in JK's paper, which lacks the final step here.

As I have set them out, we can see that in both scenarios we get the same end result.  The issue then becomes whether the process allows these steps to take place as set out or not.  In JK's alternative scenarios, they end up with completely different outcomes, so they don't even get near this issue.

I think there are some legitimate points to be made about how the operation of banks might impact on transaction frictions and what this means for volatility and response to shocks.  Unfortunately, I do not think this paper has very much of interest to say on the topic.