I did a little exercise recently trying to blend DSGE models
with stockflow consistent (SFC) models.
The basic idea was to take the accounting framework and policy setup 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: interestbearing bonds and
noninterest 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
stockflow 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_{t1 }[ 1+_{ }R_{t }( 1  τ_{t} ) ] + ( g_{t}
 τ_{t} . y_{t} ) . p_{t}  ( H_{t}  H_{t1}
)
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_{t1 }. 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.
Nick, two questions:
ReplyDelete is the tax rate constant?
 shouldn't eq. 7 be: rt = [ 1 + Rt ( 1  τt ) ] / ( 1 + πt)  1 ?
Anton
And a remark: it seems to me that the household budget constraint is given by the consolidated government budget constraint, as they add up to zero (and eq. 3 is a portfolio function)
DeleteAnton
And shouldn't it be: pt = pt1 . ( 1 + πt)
DeleteHi Anton,
DeleteThe tax rate is an exogenous variable here. I set it up to allow for this to be varied, but it is fixed in the simulation shown.
I have used inflation to mean p(t) / p(t1), rather than p(t) / p(t1) 1 here, because that's how it's usually expressed in the NK Phillips curve. I should have mentioned that in the post, though. Sorry.
Yes  the household budget constraint is exactly the same as the public sector budget constraint. They are really the same thing. Eq 3 is the behavioural equation for money holdings.