In my last post, I explained how I like to think about the
relationship between exchange rates and international balances. The key point was that current flows do not
matter much in themselves. What matters
is the build up of balances, in particular where those involve entities having
to take positions in their nonfunctional currency.
I like models, so I'm using this post to set out a little
model of this process. This is based on
the models of Godley and Lavoie. The
most notable departure is that I am setting the expected exchange rate equal to
the actual outcome. I'm doing this
because the results of this type of model can depend heavily on how exchange
rate expectations are formed. To fully
understand this, we need to see how the models behave when we eliminate any
systematic expectation error.
Unfortunately, open economy models with floating exchange
rates generally require many more equations than closed economy models. To keep this manageable enough to contain in
a blog post, I have made the model as simple as I can whilst retaining enough
to show the key dynamics. The main point
here is that I have assumed a "small" economy, so that I can take
what goes on in the rest of the world as exogenous. As usual, I have relegated the equation
listing to the end.
There are three
sectors: a public sector, a domestic private sector and the rest of the
world. There are two financial assets  domestic
government bonds and foreign government bonds.
Both are held by both domestic and foreign investors. This is shown in the balance sheet matrix below. As I have consolidated foreign investors and
foreign issuers into one, foreign holdings of foreign bonds do not appear. Foreign bonds are recorded at their foreign currency
value and divided by the exchange rate so the matrix is all in the domestic
currency.
Private
Sector

Government

Rest
of World


Domestic bonds

Bd

B

Bw

Foreign bonds

Fd / e

 Fd / e


Total

V

B

NFI

The net wealth of each sector is determined by historical flows
to date (subject to valutaion at the prevailing exchange rate). The exchange rate must then adjust to ensure
that investors wish to hold domestic and foreign bonds in the proportions in
which they are in issue.
There are two portfolio decisions to be made here. First, the domestic private sector has to
decide how to allocate its financial wealth between domestic and foreign
bonds. This is assumed to be a function
of the domestic interest rate (r) and the expected return on foreign bonds (rrf). This latter return needs to take into account
expected exchange rate movements.
Fd / e = f_{1} ( V, r, rrf )
The amount of domestic bonds that foreign investors wish to
hold is also assumed to depend on rates.
Here the rates are the domestic rate adjusted for expected exchange rate
movements and the foreign rate. The
function also depends on total overseas financial wealth (Vf), converted to its
domestic equivalent value. As we are
assuming a small economy, Vf is treated as exogenous.
Bf = f_{2} ( Vf / e, rrd , rf )
The other behavioural equations required in the model are
those describing expenditure  total private expenditure, exports and imports 
and those describing how domestic prices adjust. I have used similar expenditure functions to
those that appear in G&L with prices determined by a Phillips curve type
relationship based on adaptive expectations.
(It would not be too difficult to adapt this to include some more
microfounded behavioural assumptions and a forward looking Phillips curve.) I have also assumed there is no intermediate
production (imports are not used in production of exports).
The accounting structure of the model is best captured by
the flow of funds matrix. (This includes
a production account so that all rows sum to zero, as well as all columns.)
Private
Sector

Government

Rest
of World

Production


Factor income

Y

 Y


Taxes

 T

T


Domestic interest

r . Bd

 r . B

r . Bw


Foreign interest

rf. Fd / e

 rf . Fd / e


Domestic consumption

 d . p

d . p


Government spending

 g . p

g . p


Exports

 x . p

x . p


Imports

 m . pf / e

m .
pf / e


Change in domestic bonds

 Δ Bd

Δ B

 Δ Bw


Change in foreign bonds

 Δ Fd / e

Δ Fd / e


Total

0

0

0

0

There are lots of things we can explore with this model,
including many of those typically explored within MundellFleming style models. As an example, I have looked at what happens
when there is a sudden unexpected change in the preferences of domestic
investors towards foreign assets. The
charts below show some of the results.
The immediate impact is a sharp drop in the exchange
rate. Domestic investors purchase more
foreign bonds (reflected in the jump in gross overseas investment), but in the
short run the current account flows are insufficient to finance this. The counterpart must therefore be purchase by
foreign investors of the domestic bonds that domestic investors are selling. The exchange rate falls until foreign
investors expect sufficient future currency gains to make them want the
additional domestic bonds. The initial
impact on net foreign investment is principally due to the upwards revaluation
of overseas investment due to the exchange rate movement.
The drop in the exchange rate gives a boost to net exports, which
together with rising import prices gives a brief increase in domestic inflation. However, having initially fallen, the
exchange rate now rises (which delivers the higher returns expected by foreign
investors). The combined effect erodes
the price advantage in foreign trade and the initial increase in the trade
balance is soon reversed. However, by
then the net foreign investment position has improved enough that net interest
income from abroad outweighs the negative trade balance.
In the longterm steadystate position, lower domestic
inflation is matched by currency appreciation, so the real exchange rate is
constant. The higher current account
balance is offset by the impact of currency appreciation on the capital losses
to domestic investors on foreign bondholdings.
There are many interesting dynamics that are brought out by this model. I might look at some more in subsequent posts.
Equation Listing
Real output is the sum of domestic private expenditure,
government expenditure and exports.
(1) y = d + g + x
The real value of domestic private expenditure is total
nominal private expenditure less nominal imports, divided by the price of
domestic goods.
(2) d = ( C 
m . p_{f} / e ) / p
Total private expenditure is based on disposable income and
the stock of bonds held:
(3) C = α_{1}
. YD + α_{2} . ( Bd_{1}
+ Fd_{1} / e )
where disposable income is based on current flows, but
excludes capital gains and losses (part of the reason for doing this is to
avoid blips in disposable income due to revaluations in foreign bonds when the
exchange rate jumps.)
(4) YD = y . p
 T + r_{1} . BD_{1} + rf . Fd_{1} / e
Taxes are levied as a proportion (τ) of production income
and nominal interest income.
(5) T = τ
. ( y . p + r_{1} . Bd_{1}
+ rf . Fd_{1} / e )
The ratio of real imports to real domestic private expenditure
is based on the terms of trade.
(6) m = d . µ .
( e . p / pf )^{σ1}
Real exports are based on the terms of trade.
(7) x = x_{0}
. ( e . p / pf )^{σ2}
The definition of disposable income implies that private
wealth evolves according to the following accounting identity:
(8) V = YD  C
+ Bd_{1} + Fd_{1} / e
The domestic private sector allocates a portion of its
wealth to foreign bonds based on relative expected returns.
(9) Fd = e . λ_{1}
. V. ( rrf / r )^{κ1}
Foreign holdings of domestic bonds is based on the relative
rates of return and the stock of the rest of the world's savings (converted
into the domestic currency equivalent).
(10) Bw = λ_{2}
. Vf / e . ( rrd / rf )^{κ2}
The stock of domestic bonds evolves in line with the
government budget constraint.
(11) B = B_{1}
. ( 1 +r_{1} ) + g . p  T
The balance of wealth is made up of domestic bonds
(12) Bd = V  Fd
/ e
The balance of domestic bonds are held by overseas
investors.
(13) Bw = B  Bd
(We already have an equation for Bw. For the purposes of solving the model, this equation (10) is rearranged to define the exchange rate)
The expected rate of return (expressed in foreign currency
terms) on domestic bonds depends on the nominal domestic interest rate and
expected exchange rate movements.
(14) rrd = ( 1 +
rd ) . E[ e_{+1} ] / e  1
(15) rrf = ( 1 +
rf ) . e / E[ e_{+1} ]  1
Price setting here is based on inflation expectations and a
measure of the output gap.
(16) p = p_{1}
. E[π] . ( y / y* )^{ψ}
Inflation expectations are adaptive.
(17) E[π] = E[π]_{1}
+ ε . ( π_{1}  E[π]_{1} )
Where inflation is taken as the change in the price of
domestic output (expressed as 1 plus the change).
(18) π = p / p_{1}
Variables
The variables are listed below. Uppercase variables denote nominal values.
B

Domestic bonds

Bd

Domestic bonds held by domestic investors

Bw

Domestic bonds held by overseas investors

D

Domestic expenditure by the domestic private sector

e

Exchange rate (units of foreign currency per unit of domestic
currency)

Fd

Overseas bonds held by domestic investors (in foreign currency terms)

g

Real government expenditure

m

Real imports

p

Price of domestic output

pf

Price of foreign output

r

Interest rate on domestic bonds

rf

Interest rate on foreign bonds

rrd

Expected return to foreign investors on domestic bonds

rrf

Expected return to domestic investors on foreign bonds

T

Taxes

V

Net financial wealth of domestic private sector

Vf

Financial wealth of rest of world (in foreign currency terms)

X

Real exports

x0

Base level of real exports

y

Real output

YD

Disposable income of domestic private sector

π

Inflation of domestic output prices

Policy variables are the level of government expenditure,
the tax rate (τ) and the domestic interest rate. Foreign variables (Vf, x0 and rf) are taken
as exogenous in accordance with the "small" economy assumption.
"I might look at some more in subsequent posts."
ReplyDeletePlease do! I find these models a great tool for thinking about these complex dynamics.
Thank you. I think these models are invaluable for thinking about the dynamics. With many SFC models, I have a good sense of what they will do before I run the simulation, but ones like this are rich enough that they often surprise me and I learn something new by working through why the results are what they are.
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