Model PC (portfolio choice) is a continuation on the Model SIM. Agents must choose between money and financial assets. Their decision is aided through the rate of interest on other assets; hence it is called Porfolio choice.
The Matrices of Model PC
Government bills, interest payment and a central bank are all added to Model SIM to make up Model PC. In the balance sheet of Model PC, the first column is for Households, which contain money H or bills Bh, the sum of these 2 is private wealth (V). There is no production sector as it is assumed to be a pure service economy, which means the value of the household sector is also the value of the private sector. Column 3 consists of public debt – amount of outstanding bills B issued by the Government to households and the central bank together. In column 4 includes the central bank purchasing bills from government, which increases its stock of assets Bch. On the other side the central bank provides money to households.
The two differences from the Model SIM flow matrix are that flow-of-funds accounts include two financial assets and there are interest payments on government debt. National income doesn’t include these interest payments; it consists of income derived from the sales to households and to the government. The new central bank sector consists of a current account (involves the inflows and outflows of daily activities) and the capital account (shows changes in the balance sheet). It is assumed that the central bank has zero net worth and profits are always paid to the government.
The equations of the Model PC
The first model assumes that producers sell exactly what is demanded and that households have correct expectations regarding their incomes. It begins with production is equal to consumption plus government expenditure. Godley follows by defining disposable income (4.2) and taxable income (4.3). Both (4.2) and (4.3) are includes interest payments.
- Y = C + G (4.1)
- YD = Y – T + r-1. Bh-1 (4.2)
- T = Ơ . (Y + r-1 . Bh-1) (4.3)
Keynes two-stage decision means households decide how much they will save and how they will allocate their wealth between money and bills. V is total wealth.
- V = V-1 + (YD – C) (4.4)
Consumption now includes total wealth;
- C = α1 . YD + α2 . V-1 0< α2< α1<1 (4.5)
Wealth is allocated between bonds and money dependent on the interest rates and liquidity preferences. However the allocation will always sum to one to satisfy the following formulas; - Hh/V = (1- λ0) – λ1 . r + λ2 . YD/V (4.6A)
- Bh/V = λ0 + λ1 . r - λ2. YD/V (4.7)
To solve the model, with the demand for cash a residual equation (4.6A) is dropped and (4.6) is used.
- Hh = V – Bh (4.6)
Money holdings is the total wealth and the demand for bills by households.
Equations (4.8)-(4.11) involve the government and central bank.
- ∆Bs = Bs – Bs-1 = (G + r-1 . Bs-1) – (T + r-1 . BCB-1) (4.8)
- ∆Hs = Hs – Hs-1 = ∆Bcb (4.9)
- Bcb = Bs - Bh (4.10)
- R = r (4.11)
Equation (4.8) explains the government budget constraint. Equation (4.9) shows the capital account of the central bank. Equation (4.10) and (4.11) are related to the fact that the central bank purchase all governmental bills that households don’t want. The interest rate will affect this amount.
Given these equations, the cash household’s have is equal to the cash provided by the central bank.
- Hh = Hs (4.12)
Expectations in Model PC
Consumption will depend on expected income and that expectation is usually incorrect. Therefore the Consumption function is now;
- C = α1 . YDe + α2 . V-1 (4.5E)
YDe is the expected disposable income. The rest of the equations must be changed because wealth as well as income is unknown.
Any miscalculation of expected disposable income is balanced by a change in money balances. This means, households actually invest in bills on the basis of their expectations with respect to disposable income from the beginning of the period.
- Therefore Bh = Bd (4.15)
Expected disposable income including the possibility of a random process is defined as;
- YDe = YD . (1 + Ra) (4.16)
Our conclusion from earlier equations is that the difference in the amount of money held and demanded by households is dependent on expected income, which is equal to the difference between realised and expected income.
- Hh – Hd = YD -YDe (4.17)
The Steady State
The effect of rising interest rates on the allocation of wealth between cash and bills is examined, assuming no change in external variables and opening from a full stationary state.
A higher interest rate means that households would hold more interest paying bills as would be expected due to the rate of return being higher at a constant level of risk.
Although it is also found that a higher interest rate encourages an increase in disposable income and consumption both in the short run and in the new stationary state.To explain this unusual result, the steady state solution for national income is established. The government budget must be balanced (government revenue plus central bank profits equal to government expenditures plus cost of servicing government debt)
The steady-state solution for national income is determined:
- Y* = G + r.Bh*.(1 - θ)/θ (4.18)
The steady state solutions for disposable income can also be obtained keeping in mind that in the stationary state consumption equals disposable income:
- C* = YD* = Y* + r.Bh* - T* = Y* + r.Bh* - θ.(Y* + r.Bh*) (4.18)
It can be seen that in the full stationary state the aggregate income flow and disposable income flow are an increasing function of the interest rate. As higher interest payments accumulate on government debt, disposable income increases as does consumption and national income. . As disposable income increases households hold more wealth and a larger absolute amount of bills as well as a larger proportion of their wealth in bills. All of this leads to even larger interest payments on debt. Higher interest rates generate more economic activity.
Fully developed solutions
The solutions need to be examined further because the value of Bh* needs to be found in terms of the different limits and need to put this value into the equations to reach the fully developed stationary solutions of the Model PC.
To do this the wealth accumulation function is obtained
- ∆V = α2 . (α3 . YD – V – 1) (4.20)
Noting that in the stationary steady state households accumulate no additional wealth and so ∆V equals zero. Thus wealth and disposable income are in a constant ratio, combining this ratio with the portfolio decisions of households eventually highlights that the steady state level of GDP is dependant on the fiscal stance which, whatever the target wealth to disposable income ratio and the average interest rate payable on overall government liabilities depends on the G/ θ ratio.
- Y* = (G/ θ ) { 1 + α3. (1 – θ) . ѓ (4.26)
[θ/(1- θ)] - α3 . ѓ }
Implications of changes in parameter values for temporary and steady state income
Puzzling results
An increase in the permanent level of government expenditure boosts the stationary level of income and disposable income; dY*/dG >0.
Any permanent decrease in the on the whole tax rate θ has the same effect; dY*/d θ <0.
Alternatively an increase in the interest rate or in the rate of interest payable on government liabilities leads to an increase in the steady state level of income and disposable income as the higher rate increases payments arising from the government sector thus leading to more consumers spending from households. Also the higher rate of return on bills means households will hold more interest paying government debt.
An increase in the desire to hold bills increases the proportion of government debt taking the form of bills, thus raising ѓ leading to an increase in the steady state value of national income. So holding more bills leads to an increase in national income.
The model also shows that any increase the wealth to disposable income ratio (a3), leads to an increase in stationary income or disposable income. This is because dY*/a3 >0 where a3 = (1-a1)/ a2. (Where a1 and a2 are the propensities to consume out of current income and out of accumulated wealth.) Thus if households decide to save larger amounts of both their income and wealth then the level of steady state income will be higher.
This result challenges a ‘paradox of thrift’ that has been stated by Keynes. If people decide to save more at each level of income one might expect that this would increase the total amount of savings, but Keynesian multiplier model predicts a paradox of thrift, that total savings will remain the same and income will decline. The reason that higher thrift leads to higher stationary level of income is because a larger a3 parameter suggests households aim for a higher wealth level, but a higher wealth objective implies higher interest payments on government debt held by households and in due course higher absolute consumption and income levels once the steady state is reached.
Graphical Analysis
- Expectations about current disposable income are based on actual disposable income in the previous period and
- An increase in the propensity to consume implied a decrease in the target wealth to disposable income ratio
An increase in the propensity to consume out of disposable income causes GDP to rise in the short run, but in the long run a lower steady state value is achieved which is lower than the original steady state National Income operates as follows.
National income first rises in the short run because for a given level of accumulated wealth when households increase consumption out of current income there is an increase in aggregate demand leading to an increase in national income. However, the propensity to save decreases, reducing the stock of wealth as consumption exceeds disposable income. Ultimately the reduced consumption out of the wealth compensates for higher consumption out of current income. Wealth and consumption keep falling and national income falls to reach steady state level which is lower than the initial steady state.
Why then with a higher propensity to consume does both household wealth and government debt fall? This is due to fact the starting point is a stationary state which indicates the private sector is neither saving /dissaving and the government budget is balanced. As household consume more, their wealth is falling. There is increased revenue for government which means the budget is in surplus and the debt level can be reduced. So both household wealth and government debt decrease by the same amount.
The consequences of an increase in the rate of interest set by the central bank is also shown assuming we begin from a stationary position where accumulated wealth is the targeted level of wealth. Any increase in the rate of interest or the propensity to consume leads to a brief increase in disposable and national incomes. The target level of wealth also increases, suggesting that households will be increasing savings and their demand for bills. In the next period, consumption and income rise, leading to a higher stationary level of income.
The effect of a decrease in the propensity to consume out of current income is examined. The target wealth to income ratio would rise as a result. There would be an initial fall in national income but there would also be a difference between current wealth and target levels of wealth, making it appealing for households to buy bills.
The puzzling impact of interest rates reconsidered
Assuming the propensity to save is not constant, but is a value which depends negatively on the interest rate on bills.
The first result of the increased interest rate is that it causes a fall in consumption, disposable income and national income.
However this model takes stocks into account. A reduction in the propensity to consume increases the target wealth to income ratio, leading to an increase in the stock of wealth. If households’ wealth keeps increasing consumption demand is lower than disposable income once household wealth is increasing.
There is also an increase in government debt. The short term recession caused by the higher interest rates on consumption drives tax revenues down. This negative effect is heightened by an increase in government expenditures due to higher cost of debt. This causes a budget deficit. This deficit gets smaller as government expenditures and taxes continue to increase up to stationary level where the government budget is balanced, and eventually the deficit flow is wiped out by rising consumption expenditures encouraged by increasing household wealth.
A Government target for the debt to income ratio
Godley asks can the government do anything about the debt to income ratio in the long run.
In this model the ratio is defined as V/ Y where V is the wealth of households which corresponds to government debt and Y is national income or GDP This ratio is determined by the behaviour of households.
If households are trying to achieve a high wealth ratio government could attempt to achieve a budget surplus (e.g. by reducing expenditure) but this would be unsuccessful as there would be a reduction in disposable income, but no change in the ratio unless the reduction in income led to households revising their saving propensities.
Government and securities rating agencies are more concerned with public debt as a ratio of GDP, which is easier to adjust. If households are targeting too high a wealth ratio, the government can reduce its stationary debt to GDP ratio. If government wishes to decrease this, they need to increase tax rates or reduce interest rate on bills, resulting in a reduced level of stationary income. Thus, the debt to income ratio must be disregarded to sustain full employment income. This is irreconcilable with maintaining full employment in the long run.
