Monday, March 3, 2008

Homework 4

Question 1
To be more realistic, people always make a portfolio choice between money and other possible financial assets. For this reason, model PC introduces government bills, interest payments and central bank into model SIM.
Government bills are purchased by households and central bank. The interest payments of each period are generated by stocks of assets in existence at the end of the previous period. Because of this time lag, the rate of the payment is the rate that was set at the end of the previous period. Then, each period, interest payments of households are +r−1 • Bh−1 and payments of central bank are +r−1 • Bcb−1. One thing to be noticed is that interest payments of the bills are not part of national income. Cause national income derived from the sales to households and government. Interest payments are just transfer of money not a creation. What’ more, central bank has two components here: a current account and a capital account. The former one describes the stock flows of current operations while the latter one depicts changes in the balance sheet of the central bank. One thing maybe a little confusing is that why central bank profits is −r−1 • Bcb−1 and meantime the amount becomes assets in government sector. To do so, we can be in line with the current practice of most central banks of the world. Those are the differences from SIM which can be seen from the transactions matrix.
Transactions Matrix
In households sector, a key assumption is that households should make two-stage decision: first, decide how much will save out of their income; second, decide how to allocate their wealth.
YD = Y − T + r−1 • Bh−1 (1)
T = ! • (Y + r−1 • Bh−1) (2)
V = V−1 + (YD − C) (3)
C = "1 • YD + "2 • V−1, 0 < "1 < "2 < 1 (4)
From (3) we can find that the difference between deposable income and consumption is equal to the change of the total wealth (not just the change of money in model SIM).
Hh / V = (1- #0) - #1 • r+#2 •(YD/V) (5)
Bh / V = #0 + #1 • r- #2 •(YD/V) (6)
The two formula above are from the Brainard-Tobin formula with slightly amended. The main point is that households wish to hold a certain proportion #0 of their wealth in the form of bills which is equal to hold (1 - #0) proportion of their wealth with the form of money. It is modulated by two elements: the rate of return of bills and the level of disposable income relative to wealth.
∆Bs = Bs − Bs−1 = (G + r−1 • Bs−1) − (T + r−1 • Bcb−1) (7)
∆Hs = Hs − Hs−1 = ∆Bcb (8)
Bcb = Bs − Bh (9)
r = r (10)
Central bank purchases all the bills issued by the government that households are not willing to hold by the given interest rate. Equations (7) to (9) imply that when the central bank acts as a residual purchaser, it provides cash money on demand. And it’s easy to know that the households wish to hold some more money if they fail to buy the bills. Combine (8) and (9), we can find that the central bank is providing money to those who demand it. The amount of cash money in the system is endogenous and demand-led, while the rate of interest on bills is the exogenous variable, as shown explicitly through equation(10).

Question 2
1. How does Keynes define liquidity preference?
Liquidity-preference is a potentiality or functional tendency, which fixes the quantity of money which the public will hold when the rate of interest is given; so that if r is the rate of interest, M the quantity of money and L the function of liquidity-preference, we have M = L(r). This is where, and how, the quantity of money enters into the economic scheme. The marginal propensity to consume decides how much of your money you spend and how much will be reserved or saved. The schedule of the amount of resources valued in terms of money or of wage units which he/she will wish to retain in the form of money.

The interest rate determines the Liquidity Preference in part as it is the reward for parting with the liquidity. There are three divisions of Liquidity Preference: 1) Transactions motive, this explains that people need cash for day to day spending. 2) Precautionary motive, this refers to money being held in case of emergencies e.g. illnesses. 3) Speculative motive, this means that people may take advantage of profit making opportunities that might arise. If you decrease the interest rate you increase the quantity of money but that’s not always true. E.g. if the Liquidity Preference of the public was increasing faster than the increase in the quantity of money.

In the diagram, we show the quantity of money on the horizontal axis and the interest rate on the vertical axis. For example, if the rate of interest is Ra, people want to hold Ma of money, whereas if the rate of interest were to go down to Rb, people would increase their demand for monetary assets to Mb. (http://william-king.www.drexel.edu/top/prin/txt/money/QT7.html)

2. Is PC a faithful representation of Keynes’ original vision of household decision-making? If so, why? If not, why not?
PC is a faithful representation of Keynes’ original vision of household decision making for the following reasons:
• Both Keynes and the PC model identify a key behavioural assumption – that individuals make a two-stage decision. Firstly they decide how much of their income they will consume and how much they will save “in some form of command over future consumption”. Secondly they decide in what form they will hold the command over future consumption which they have saved.
• In both models interest rates represent a state of equilibrium. In the PC model portfolio decisions are based on the relevant interest rate r, which is defined as the rate of interest which equalizes the supply of and the demand for bills at the end of the current period. In Keynes’ theory the rate of interest is referred to as “price” which equilibrates the desire to hold wealth in the form of cash with the available quantity of cash.
• Both models infer that the share of wealth that people wish to hold in the form of money is negatively related to the interest rate. This is conveyed in the PC model through the following formula:

Hh/V = (1-λ0) – λ1.r + λ2.(YD/V)

Keynes on the other hand, explains this relationship through the transactions-motive. If interest rates fall, national income will increase and the amount of money which it is convenient to keep for transactions will be increased more or less proportionately to the increase in income.
• Both models distinguish between disposable income and consumption. PC recognises these identities respectively through the following equations:
YD = Y – T + r-1.Bh-1
C = α1.YD + α2.V-1
Keynes acknowledges this distinction when he says “…how much of his income he will consume and how much he will reserve in some form of command over future consumption…”
• Finally, another similarity between the models is that the Pc model encompasses Keynes’ ideas of the precautionary, transaction and speculative motives.