Wealth Concentration and Loanable Funds

Jo Michell has an interesting post on loanable funds.  This was prompted by the question of whether wealth concentration has led to rising asset prices and falling yields.  Whilst sharing Jo's views in many respects, my own analysis here is slightly different, so I thought it worth setting out.

First of all, I would agree that increasing wealth concentration is likely to increase the propensity to save out of income.  However, this in itself is not enough to change interest rates.

The diagram below is the standard model of loanable funds (borrowed from Jo, and in turn from Nick Rowe).  The idea here is that if there is an increase in the desired quantity of saving at any given interest rate, this is represented by a rightward shift in the Sd curve.  This leads to a fall in the equilibrium interest rate (even if the Id curve is vertical).

 

However, an increased propensity to save is not the same thing as a desire to save an increased quantity.  In fact, an increased desire to save leads to lower income.  The actual quantity people end up wanting to save is unchanged; it simply represents a higher proportion of income.  This is the paradox of thrift.  So there is no movement in the Sd curve and no change in the (partial) equilibrium interest rate.

There may be temporary movements, if people are mistaken in what they expect will happen.  If one person tries to save more without realising others are also doing so, then they may overestimate their own income and start to bid up asset prices.  But, assuming they eventually cotton on to what is happening, this will eventually reverse.

Having said all this, I would still say that increased wealth concentration has contributed to falling yields.  The reasons for saying so are as follows.

The argument above is based on what happens in a monetary economy.  In looking at how things pan out in a monetary economy, we can't get very far without asking how monetary policy is framed.  For example, if monetary policy takes the form of simply fixing or targeting a particular interest rate, then almost by definition changes in savings propensity are not going to change that rate.

However, for our purposes here we need to recognise that current monetary policy takes the form of inflation targeting.  This means that the central bank responds to perceived deflationary pressures by reducing the policy rate.  So an increased propensity to save does indeed lead to lower interest rates, but it does so because it depresses demand and because the central bank reacts to that.

One interpretation here is to suppose that implementation of monetary policy brings the economy back to its original level of demand, at whatever interest rate that requires.  Thus we can go back to our loanable funds diagram and say that the Sd curve has indeed shifted to the right, once we have taken into account the full operation of monetary policy.

This is one way of looking at it and I'm all in favour of looking at things in different ways, for the additional insight it provides.  However, I find it a little problematic.  We can easily conceive of a situation where there is an increased propensity to save but where reductions in interest rates are ineffective in restoring demand.  In this scenario, there is no solution compatible with the unchanged output interpretation and it fails as an explanation of interest rates.

Even if we think that interest rate manipulation can bring output back to its original level, there is no general reason to suppose that the terminal interest rate of this process is independent of the path taken to get there.  (In most models it is independent, but that doesn't mean it is in reality.)  There would then be no meaningful ceteris paribus solution to the loanable funds model. 

There is another important way in which wealth concentration has contributed to falling yields, one that was particularly important in the run up to the crisis.  This is not to do with increased propensity to save, but rather with portfolio preference.  (We might think of portfolio preference as a more general form of liquidity preference, when we are considering a range of assets rather than simply bonds and money.)

Wealth concentration makes investors more concerned about large exposures to single name risk.  Pre-crisis, managers of large cash pools already holding sizeable unsecured bank deposits increasingly sought alternative low risk short term investments.  This created a strong demand for traded high quality assets which could be used as collateral for short term secured instruments.  This in turn led to increased demand for the assets that could used to create such collateral, such as securitisable mortgages.  The effect of this demand pressure was to drive down the yields on such assets relative to policy rates and rates on unsecured bank deposits.

In short, wealth concentration is certainly an important part of the picture of what has happened to financial asset yields.  But whilst the loanable funds model might provide some kind of insight it is, in my view, an insufficient framework for understanding the mechanisms at play.

Fiscal Policy in Neo-Fisherite Versions of New Keynesian Models

This post is intended to enable me to explain in a bit more detail something I have been describing in an exchange with Stephen Williamson on his blog.

The context is how fiscal policy might impact on the results of models of interest rate changes, particularly in the impulse response on output.  I have in mind here a model based on John Cochrane's, which I've looked at before.  The two main equations are a New Keynesian IS curve and Phillips curve.  There are also assumed to be lump sum taxes and single period government bonds, so there needs to be a government budget constraint which determines the evolution of the stock of bonds.  The three equations in y (output), π (inflation) and b (the real end period value of the stock of bonds) are:

(1)          y_t = E[y_t+1] . ( β . i_t+1 / E[π_t+1] )^-σ

(2)          π_t = E[π_t+1]^β . y_t^κ

(3)          b_t = b_{t-1 .} i_t / π_t - τ_t

where i is the interest rate and τ is the level of real lump sum taxes.

Here, monetary policy is an interest rate peg with i as the policy tool - we have no central bank reaction function.  We want to consider the impact of a 1% cut in this rate, announced five period in advance.  In the first instance, we will assumes that fiscal policy involves keeping τ constant.

The results of this are shown below (showing deviations from starting values).

Inflation starts to decline immediately and eventually reflects the drop in nominal interest rates.  This means that, prior to the actual rate cut, real interest rates are higher, which implies that output rises.  Once actual nominal rates are cut, real rates are lower and output declines again.

This change results in a temporary increase in the real stock of bonds.  This happens because real rates have risen and real rates are a key factor in determining how the stock of bonds changes over time.

So I want to consider an alternative fiscal policy rule; one where the level of taxes, τ_t, is adjusted to prevent b_t ever rising above b₀.  We therefore need a fourth equation which is:

(4)          τ_t = GREATER OF { τ₀ } OR { b_{t-1 .} i_t / π_t - b₀ }

Adding in this equation gives us the results shown below (b_t = b₀ for all t, so I have not included it in the chart).

Under this fiscal policy rule, both output and inflation drop sharply on the announcement, and then both rise until the actual rate change takes place.  These paths clearly satisfy equations (1) and (2) for every period where there is no surprise.  Output is rising when inflation is lower than nominal interest rates; inflation is rising when output is below its equilibrium level.  Equations (3) and (4) selects this equilibrium path from multiple others that would also satisfy (1) and (2) (although I cannot be sure that there are no other alternative paths that would satisfy all four equations).

Note on parameters.  I have used Cochrane's values of β = 0.97 and κ = 0.2, (as well as assuming a τ₀ equal to 20% of y₀).  These parameters suggest time periods in the order of a year, so we are talking about changes announced five years in advance.  If we adjust the parameters to reflect shorter periods (like a month), then announcing five periods in advance has a more limited impact (although still the same pattern).

A Portfolio Balance Model of Exchange Rates

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 non-functional 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 = f₁ ( 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 = f₂ ( 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 micro-founded 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 Mundell-Fleming 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 long-term steady-state 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 = α₁ . YD  + α₂ . ( 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 = x₀ . ( 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 . λ₁ . 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 = λ₂ . 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 re-arranged 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.