Homework 5

Homework 5
Modify the PC model to allow for a simulation of a stagflation type episode in the economy

Y = G + C
Government expenditure should be held constant. It has been demonstrated that efforts of fiscal policy, i.e. increasing taxes and/or reducing expenditure only exacerbates inflationary problems. Substitute G for G*
Because of increased inflation, an individuals’ purchasing power will be reduced meaning that while his/her income has not changed he/she will not be able to purchase the same amount as previous time periods. As a result their consumption patterns and expenditure decisions will become more disciplined and will depend on the prices of goods i.e. the inflation rate π. Substitute C for C(π)
Adjusted equation: Y = G* + C(π) (1)

YD = Y – T + r-1.Bh-1

An individuals’ disposable income will remain dependent on their income adjusted for taxes and on the interest received from government debts that they hold. However, this amount should be subjected to a MPS parameter, i.e. (1-MPC) or (1- α1) because during times of recession, people are uncertain about the future and therefore, should consider saving portions of their present income.
It is important to note that while their income will not change during periods of stagflation, the yield that they receive on their investments will be below their expectations since it is usual for interest rates to fall during recessions.
Adjusted equation: YD = (1 - α1).(Y – T + r-1.Bh-1) (2)

T = θ.(Y + r-1.Bh-1)
Taxable income will remain as before – tax will be paid on income and interest received from government bills. As mentioned before, efforts of fiscal policy do not work during a recession and therefore, tax rates should remain constant. Replace θ with θ*.
Adjusted equation: T = θ*.(Y + r-1.Bh-1) (3)

V = V-1 + [YD – C(π)] (4)

An individuals’ stock of wealth will stay as per the PC model, with consumption adjusted for inflation rates.

C = α1.YD + α2.V-1
An individuals’ behavioural decisions to consume should be modified to consider savings as per the equation for disposable income.
Adjusted equation: C = (1- α1).(α1.YD + α2.V-1) (5)

Hh = (1 – λ0) – λ1.r + λ2.(YD)
V (V)
Bh = λ0 + λ1.r - λ2.(YD)
V (V)
The portfolio allocation decision of households depends on the level of interest on bills and the level of disposable income relative to wealth. During a recessionary period, interest rates are inclined to decline (see below that interest rates are not fixed). While yields will be lower than usual, individuals should realise that bonds are a good bet for investors seeking safety, flexibility and inflation protection. In an uncertain environment investors need to make careful investment decisions. Therefore, the extent to which individuals should allocate their wealth between bills and cash should take investors’ tolerance for risk into account. The inclusion of a risk tolerance parameter, R, is seen in the adjusted equations:

Hh = R.(1 – λ0) – λ1.r + λ2.(YD) (6)
V (V)
And, Bh = R.λ0 + λ1.r - λ2.(YD) (7)
V (V)

The money held by the household still equals to the wealth of the household minus the money held in bills.
Hh = V – Bh (8)

Balancing equations are also as per PC model:
ΔBs = Bs – Bs-1 = (G + r-1.Bs-1) – (T + r-1.Bcb-1) (9)
ΔHs = Hs – Hs-1 = ΔBcb (10)
Bcb = Bs – Bh (11)

r = r*
In this model, interest rates should not be held constant. As demonstrated by Paul Volcker, Federal Reserve chairman during the 1970s period of stagflation, increased interest rates will reduce money supply and incur “disinflationary” measures. In addition, theory goes that higher interest rates should depress growth in demand which should lead to lower prices. Adjust r* for rnf, indicating non fixed interest rates.
Adjusted equation: r = rnf (12)

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.

Homework 3

Question 1

1.1 Explain the differences between SIM and SIMEX when both models are in their steady state

The SIM model assumes that consumers have perfect foresight with regard to money and that consumers’ disposable income is constant and actual. The consumption function for the model is Cd = α1.YD + α2.H-1

The SIMEX model on the other hand considers the fact that consumers do not know precisely what their disposable income is going to be, thus assumes that households make some estimate of the income they will receive within the given period. The consumption function is adjusted to reflect expected disposable income as opposed to actual known income; Cd = α1.YDe + α2.H-1, where YDe denotes an expected value.

Given fixed expectations and perfect foresight both models eventually converge to the steady state level of national income. Convergence in the SIMEX model takes longer however, due to the fact that people act on wrong expectations.

1.2 Second, what does it mean for the stability of the model when the presence of mistakes allows household’s incomes to suffer? Can you draw any general conclusions about the real world from this model?

The fact that households can incorrectly estimate disposable income does not affect the stability of the model. When people act on wrong expectations, for example underestimating disposable income, saving is higher than expected and hence the stock of wealth grows faster than in the perfect foresight state. Consumption will eventually reach the same steady state value that it would have reached in a perfect foresight model, and the model will solve for the same steady state level of national income.

From this one can conclude that people will often underestimate their disposable income and that they will revise their behavioural consumption patterns as their wealth stocks deplete and their expectations about future income are adjusted.


1.3 Solve SIMEX for the following values for 3 periods: G=30, α1=0.6, α2=0.6, θ=0.2. Follow the format of table 3.6 on page 81 of GL in presenting your results.

Calculations are as follows:

Period 1: This period assumes no economic activity, thus reflected by the zero values.

Period 2: Expected disposable income, YDe, is derived by calculating the individual’s tax liability, θ*G = 0.2 * 30 = 6, and then subtracting this figure from government expenditure, 30 – 6 = 24. This figure is then used to solve for consumption, Cd = α1.YDe + α2.H-1 = 0.6*24 + 0.4*0 = 14.4. The figure for income is solved by adding government expenditure and consumption, 30+14.4=44.4. The tax rate is 20% of income, 44.4*0.2=8.88. These figures are then plugged into the remaining calculations to solve for ΔHs, ΔHh, H, ΔHd and Hd.

Period 3: Expected disposable income, YDe, is the same figure for period 2’s actual disposable income, 35.52. All other figures are derived as per period 2’s calculations. One difference however, is that there is now a value for H-1 which is equal to period 2’s H figure.

Period ∞: Y* is calculated using the fiscal stance formula, i.e. G/θ. This gives a value of 150.

Question 2
2.1 Is it possible to specify a version of SIM that replicates the ISLM model?

Yes it is possible to specify a version of the SIM that replicates the ISLM model. This is because their consumption functions are similar.

The ISLM model’s consumption is as follows: C= α0 + α1YD, where the intercept α0 represents autonomous consumer expenditure and the coefficient α1 indicates the consumer’s MPC.

The SIM model’s consumption function is Cd = α1.YD+ α2.H-1, where α1 represents the propensity to consume out of regular (present income) and α2 represents the propensity to consume out of past wealth.

2.2 Write one down and comment on the stability of this model

One of the main properties of the stationary state, i.e. SIM equilibrium, is that consumption must equal disposable income. In other words the average propensity to consume must equal one. This implies that YD=C, which can be replicated in the ISLM model by assuming an MPC of 1.

Equilibrium Shocks to Interest Rate
Show a change in interest rate on bonds from 0.07 to 0.1. What effect does this have on the u-vh space?

The second diagram illustrates the change in the value of the capital output ratio that has occurred as a result of increasing the interest rate on bonds. The diagram illustrates an initial increased dip in the curve but then proceeds to increase at a steady rate to finish at a much higher value than before the interest rate shock.

Before shock

After shock

Similarly the value of the household decreases slightly but then increases at a rapid pace with the final value at 11, in comparison to the pre shock value of less than 4.5. This shock has had a very positive impact on the value of the household.

Before shock

After shock

Reset the model. Show a change in the value of α from 0.3 to 0.7

The reaction to the value of the capital output ratio to the change in alpha is initially good as it jumps to 1. However, it then falls and steadies out just above 0.9 which is less than the value prior to the shock.

Before shock

After shock


The after shock diagram illustrates how value of the household experiences a massive decline and is deemed almost worthless after the change in alpha.

Before shock

After shock

Homework 2

MODEL SIM

Question 1
1.1 The vertical columns must necessarily sum to zero, because the change in the amount of money held must always be equal to the difference between households’ receipts and payments, however these are determined. Similarly the change in the amount of money created must always, by the laws of logic, be equal to the difference between the government receipts and outlays. And, as they are assumed to hold no cash, producers’ receipts from sales must equal their outlays on wages.

1.2 Unlike the vertical columns, it may not be assumed in advance that the horizontal entries sum to zero. Thus, we have to specify the mechanisms by which these equalities come about. The most realistic mechanism is related to the existence of inventories. Firms hold a buffer of finished goods, which can be called upon whenever demand exceeds production. Sales are always equal to demand because it is assumed that inventories are always large enough to absorb any discrepancy between production and demand.
This approach generates the assumption that whatever is demanded (services, taxes and labour) is always supplied within the period i.e.
Cs = Cd
Gs = Gd
Ts = Td
Ns = Nd
These equalities thus give rise to a zero sum across the horizontal rows.

Question 2
Consumption
Households open each period with a stock of wealth rolled over from the previous period (H-1) and during the period they receive post tax income equal to YD. Households will consume either of these sources of funds. Consumption is therefore determined as some proportion, α1, of the flow of disposable income and some smaller portion, α2, of the opening stock of money.

Government Expenditure
This refers to the amount of money that the government spend on goods and services produced by the production sector.
Gd represents the purchases of government services. This denotes an outflow of funds and thus has a negative sign. On the otherhand Gs represents the sale of government services to the production sector. This results in an inflow of funds to the government which is illustrated by the positive sign.

Output
Total production, [Y], is not a transaction between two sectors and hence only appears once, in the production column 2. Total production is defined as the sum of all expenditures on goods and services or as the sum of all payments of factor income.
Y = C + G = WB

Factor income
This is the returns earned on labour, i.e. wages. W is the wage rate and N is the employment. Then, W*N is the ‘wage bill’. W*Ns is the total wage that earned by the households. This is a source of funds so it has a positive sign. On the other hand, W*Nd is the total wage that paid to employees by the production sector. This represents an outflow of funds so it has a negative sign.

Taxes
Taxes are the amount that households pay to the government. Td is the amount that the government receives in a period, hence the positive sign. Ts is the amount that the households pay in the same period, hence the negative sign.

Change in Money Stock
The change in the stock of money issued by the government in each period, ∆Hs, is given by the difference between government receipts (Td) and outlays in that period (Gd), i.e. ∆Hs = Hs - Hs-1 = Gd - Td. This also represents the budget constraint of the government.

The change in the accumulation of household’s wealth in each period, ∆Hh, is determined by their financial balance - the excess of disposable income over expenditure, i.e. ∆Hh = Hh – Hh-1 = YD-Cd. This also represents the budget constraint of the household.

Homework 1

DEFINITION TERMS

Aggregate Demand Relation[1]
The aggregate demand relation captures the effects of the price level on output. It is derived from the equilibrium conditions in the goods and financial market. This relation implies that the level of output is a decreasing function of the price level. It is represented by a downward sloping curve called the aggregate demand curve.

Animal Spirits[2]
This is a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities.


Bank Run[3]
This is a kind of crisis where a large portion of a bank’s customers withdraw their money in a short time because they feel that there is a problem with the bank’s solvency.

In January 2007, Zhonghua Bank in Taiwan suffered one of the most serious bank runs in years. In one day, about 50 billion Yuan was withdrawn by customers.


Bond[4]
A bond is a debt security, in which the authorized issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon) at a later date, termed maturity.

There are different kinds of bonds, such as: fixed rate bonds which have a coupon that remains constant throughout the life of the bond and floating rate notes which have a coupon that is linked to a money market index, such as LIBOR, for example three months USD LIBOR + 0.20%.

The interest earned on a bond investment is known as the bond yield.

Yield on a 30-Year Treasury Bond 1994-2005

Capital Account[5]
The capital account records international transactions involving foreign assets and liabilities. For example, US residents purchase foreign securities to earn a higher rate of return and to diversify their portfolios. US capital flows out when Americans buy foreign assets. Foreign capital flows in when foreigners buy US assets.

In 1995 America developed a capital account problem as a result of over-investment by foreigners.


Debt to GDP Ratio[6]
A measure of a country's federal debt in relation to its gross domestic product (GDP). By comparing what a country owes and what it produces, the debt-to-GDP ratio indicates the country's ability to pay back its debt. The higher the debt-to-GDP ratio, the less likely the country will pay its debt back, and the higher its risk of default.

America’s debt to GDP ratio from 1940 - 2004




Effective demand[7]
This is when the consumers desire to buy something is backed up by a willingness and an ability to pay for it.

The notion that the actual demand for aggregate output in the macroeconomy is based on the actual income or other existing economic conditions and not on income and conditions existing in equilibrium.


Deflation[8]
The price level of a market decreases under the general level during a period of time.

During The Great Depression 1930-1933, the rate of deflation was approximately 10 percent per year.


Consumption function[9]
It is the function that used to calculate the consumption in an economy. It determines how much of people’s disposable income they will spend at different levels of income.
The simple consumption function is shown as the linear function:

C = c0 + c1Yd

where C = total consumption, c0 = autonomous consumption (c0 > 0), c1 = the marginal propensity to consume (0 < yd =" disposable">

Consumer Price Index[10]
The Consumer Price Index (CPI) is designed to measure the change in the average level of prices paid for consumer goods and services by all private households and foreign visitors to Ireland. The CPI is the official measure of inflation in Ireland.



Investment Function[11]
It is a function that used to calculate the investment in an economy.
Such as I=I（r）which represent the relation between interest rate and the total investment.


Fiscal Expansion[12]
This is when the economy is stimulated towards achieving macroeconomic objectives such as full employment, sustained economic growth and price level stability by setting and changing taxes, making transfer payments, and purchasing goods and services.

For example, by increasing government spending or by reducing taxes the government can shift the aggregate demand curve rightward in order to expand real output.


GDP Deflator[13]
The GDP deflator measures changes in the average price of an economy’s output. This is a broader measure than the consumer price index and covers goods and services sold to businesses as well as households. The formula used to calculate the deflator is:
Nominal GDP * 100
Real GDP

Economy-wide price levels have increased at a higher rate in Ireland than in both the EU and the OECD since 1997.



Imports[14]
Products are brought from one country to another so that they can be sold there.

China imported lots of goods from the EU in 2006 –
Germany €27 billion (43% of total imports)
France €8 billion (13% of total imports）
Italy €6 billion (9% of total imports）


Monetary Contraction[15]
Monetary contraction is the reduction in the quantity of money and or volume of spending in the economic system.

For example, when the Irish government increases income tax it is taking part in monetary contraction.


Nominal GDP[16]
Nominal GDP is a gross domestic product figure that has not been adjusted for inflation.

For example, in 2006 Ireland had a nominal GDP per capita of $44500.


Propensity to Consume[17]
This is the proportion of total income or of an increase in income that consumers tend to spend on goods and services rather than to save.
For example if I earn €100 in a week and I spend €80 of it then my marginal propensity to consume would 0.8. The €20 I didn’t spend is my marginal propensity to save i.e. 0.2 or 1-MPC.


Short Run[18]
In macroeconomics, the short run is a period of time in which wages and prices are inflexible and resource markets are not in equilibrium.

For example, if a government or central bank increases the interest rate in a country then the short run will last until the effects are evident in wages and prices especially housing prices i.e. when they began to decrease. When the ECB increased the interest rate there was a period of time before the effects were visible.


Real Exchange Rate[19]
A real exchange rate between two countries is calculated as the product of the nominal exchange rate and relative price levels in each country, i.e. it is the nominal exchange rate that takes the inflation differentials among the countries into account

For example, the real exchange rate between the Dollar and the Brazilian Real at the moment is 1USD to 1.7790 BRL, this has taken into account the inflations levels in the respective countries.


Trade Surplus[20]
A positive balance of trade, i.e. exports exceed imports.

Or example, for the month of February China had a trade surplus of $23.76billion.


STEADY-STATE SOLUTION QUESTION

The stationary state flow of aggregate income is:

Y* = G
θ

If for example, Government expenditure (G) is €20 million and the tax rate (θ) is equal to 20%, the aggregate income (Y*) is equal to:

20
0.20

=

€100 million

However, if the tax rate increases to 30%, the steady state value of Y* will decrease to:

20
0.30

=

€66.7 million

This is because:

1. An increase in the tax rate means that consumers will have to pay higher taxes on their income.

For example if consumers are earning €1000 at a 20% rate of tax, the tax they pay is equal to:
T = θ * Y
200 = 0.20 * 1000

However, if the tax rate increases to 30%, they will have to pay taxes of:

300 = 0.30 * 1000

2. As a result of increased taxes consumers will have less disposable income

Disposable income: YD = Y – T

At a 20% tax rate consumers’ disposable income is equal to:

800 = 1000 – 200

However, at a 30% rate their disposable income is equal to:

700 = 1000 – 300

3. A reduction in disposable income will inevitably lead to a decline in consumption of consumers and thus the national income will also decline since
Y = C + I + G

4. The national income will decline until it reaches the steady-state solution of €66.7 million as per this example.





[1]http://www2.widener.edu/SBA/econdept/ch07-part%202-4e-sp07-agg%20dem.ppt#1
[2] http://distance-ed.bcc.ctc.edu/econ100/ksttext/keynes/keynes.htm
[3]http://www.gusu.org/catalog.asp?tags=%E9%93%B6%E8%A1%8C%E6%8C%A4%E5%85%91
[4] http://en.wikipedia.org/wiki/Bond_(finance)
http://www.marketoracle.co.uk/Article1384.html
[5] McEachern W. V (2006) Macroeconomics: A Contemporary Introduction, Thompson South-Western: UK
http://www.jubileeinitiative.org/BePrepared-CapitalAccountProblem.htm
[6] http://www.investopedia.com/terms/d/debtgdpratio.asp
[7] http://en.wikipedia.org/wiki/Effective_demand
[8] http://en.wikipedia.org/wiki/Deflation_(economics)
[9] http://en.wikipedia.org/wiki/Consumption_function
http://tutor2u.net/economics/content/topics/consumption/consumption_theory.htm
[10] http://www.cso.ie/releasespublications/documents/prices/current/cpi.pdf
[11] http://en.wikipedia.org/wiki/Investment
[12] Parkin, M. (2006) Economics, Pearson Education: US
[13] http://www.forfas.ie/ncc/reports/ncc_annual_05/ch03/ch03_03.html
[14] http://finance.sina.com.cn/stock/t/20071128/02401819156.shtml
[15] http://www.gold-eagle.com/gold_digest_03/
[16] http://www.investopedia.com/terms/n/nominalgdp.asp
[17] http://www.britannica.com/eb/article-9061553/propensity-to-consume
[18]http://www.amosweb.com/cgi-bin/awb_nav.pl?s=wpd&c=dsp&k=short%20run,%20macroeconomics
[19]http://www.rba.gov.au/PublicationsAndResearch/Bulletin/bu_nov01/bu_1101_2.pdf
[20] http://www.investorwords.com/5028/trade_surplus.html