Relationship between consumption saving and investment according to keynes
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1. Explain the consumption function according to Keynes! What is the relationship between consumption, saving and investment according to Keynes?
The main hypothesis of Keynes is that the real consumption depends on disposable income. This hypothesis you can express like this:
C = C(Y).
In that case consumption (C) and disposable income (Y) are measured in units. 
Kenynes also said that people enhance consumption when their disposable income increases. But the increase of consumption is smaller than the increase of disposable income. This hypothesis you can describe with the marginal propensity to consume. It describes the increase of consumption when disposable income rises. 
The marginal propensity to consume is defined as
C' = =
Because of the fundamental-psychological law the marginal propensity to consume is between "0" and "1". 
0 < < 1.
When the marginal propensity to consume is 0.8, consumption increases 0.8 units. At once disposable income increases one unit. The fundamental-psychological law is fullfilled because increase of consumption is smaller than increase of disposable income. When 0.8 units are used for consumption the remaining 0.2 units are obviously used for savings. The proof is expressed by following formula:
Y = C + S
1 = 0.8 + S I - 0.8
S = 0.2 .
Therefore we define the marginal propensity to save money as differential quotient dS /dY. The marginal propensity to save shows how much the savings grow due to increasing disposable income. Through differentiation of the budget restriction after Y you can see that after adding the marginal propensity to consume and the marginal propensity to save the result always amounts "1".
1 I = .
That shows you that every additional income has to be used either way. 
The formula below is the consumption function:
C = Caut + C' · Y .
Caut is autonomous consumption. You can interpret the consumption function like a normal formula which you know from mathematics. Therefore Caut is the intersection point with the ordinate and C'·Y the gradient. The last formula which is important to know concerning the consumption function according to Keynes is the consumption rate. It serves to find out which quantity of disposable income is used for consumption. The formula is
c:= . 
So far almost always mathematical background has been mentioned. But it is also necessary and maybe easier to summarize consumption in my own words. This will help to explain the relationship between consumption, savings and investments according to Keynes.
Basically Keynes had three suppositions which result the consumption function is.  The first was mentioned above. He expected that consumption increases when disposable income increases one unit. This mentioned process is the so called marginal propensity to consume. The value of it is between "0" and "1". You see that consumption can not increase just as money. But the main fact is that people lean towards consuming when they have more disposable income. Furthermore the average propensity to consume decline when dispoable income increases and consumption depends on current disposable income. Keynes thought that the rich spend more of their disposable income than the poor and that interest rates are not important.  Therefore a short example should make this facts more familiar.
We expect, that the average consumption and disposable income was 1700â‚¬ and 2000â‚¬. In a formular you find following average propensity to consum:
c0 = = 0.85. The result says that 85% of disposable income is used for consumption. One year later the disposable income increases more than the consumption expenditures:
c1 = = 0.84. The result says that now 84% of disposable income is used for consumption. Now it is possible to calculate the marginal propensity to consum.
C' = = = .
Now you see that consumption increases 0.75 units when disposable income increases about 1 unit.
While explaining the consumption function different facts were metioned. Also consumptions and savings. One word is still missing - investments. In the consumption function according to Keynes investements have not been borne in mind. There is only a relationship between consumption and savings. It was mentioned that you have an disposable income which you use for your consumption expenditures. When consumption does not exceed the disposable income you have a specified amount which you can save. This is expressed by this formular:
Y = C + S
1 = 0.6 + S I - 0.6
S = 0.4 .
Consumption is 0.6 units. Therefore you are able to spend 0.4 units.
The first important formular is Y = C + I + G. This formular describes the equation for equilibrium in the goods market.  Immediately you find consumption, investments and government spending. You see that there is an existing relationship. Now there is the possibility to "play" with the formular what could be interesting to identify the different relationships. Therefore taxes (T) and consumption (C) have to be subtracted from Income (Y). It is also necessary to subtract taxes from government spending. The new formular looks like this:
Y - T - C = I + G - T.
We know that S = Y - T -C takes effect. Therefore the term reads
S = I + G - T .
The term on the left side is private saving. On the right side you find investment (I) and public savings (G - T). The government is able to run a budget surplus. This happens when taxes exceed government spending. But when government spending is higher than taxes there is a budget deficit. That means that the country consumed too much. 
2. Which monetary policy instruments does a Central Bank have to control the money supply? Which of these instruments is the most effective and why?
A Central Bank has three different instruments to control the money supply. It is able to choose between standing facilities, minimum required reserves and open market operations. 
Deposit facilities comply with deposits of merchant banks. Business partners of the european system of central banks (ESCB) are able to invest redundant central bank money until the next business day. There are no restrictions for using this facility and no corresponding amount limits for deposits. The interest rate is the bottom line for overnight money.  Business partner of the ESCB are able to use marginal lending facilities to provide overnight liquidities by national banks. There is a fixed interest rate for these liquidities. But the business partnes will not get the liquidities if they are not able to deposit refinancable securities. There are also no existing credit lines or restrictions which could make barriers. The interest rate for marginal lending facilities is the maximum limit for overnight money. 
The second monetary instrument of the ECB to control the money supply are reserve requirements. Due to competition international financial centers to decrease the reserve debt. It was introduced in the beginning of the 20th century. The idea was to secure the bank's solvency. Nowadays this task is done by banking supervisory standards.  Banks always hold reserves. The reason is very simple. They want to satisfy the depositors. But that money reserves which are required are much bigger. Therefore the aim to satisfy the depositors is not the only one. To prohibit disadvantages in competition between different currency areas where banks do not have to hold reserves interests are payed on reserves of European banks by the ECB.  The bank has to have reservers which are equal to 10% of their checkable deposits.  But it is to mention that the monthly average percentage has to be 10%. If there are liquidity bottlenecks the bank is able to go back to their reserves. When all banks average reserves at the beginning of the month are below 10% they can feel certain that the ECB provides liquidities at the end of the month. Otherwise banks would not be able to fullfil their reserve requirements. 
The most important instrument are the open market operations. When a central bank buys bonds they pay for them with "creating" money. This activity is used to increase the amount of money. Vice versa the central bank is able to sell bonds. It has to remove the money which is in circulation because the central bank get it for the selling of the bonds. The amount of money will decline. In modern economies these procedures are the most popular to increase or decline the money stock. Open markt operations are distinguishable into expansionary and contractionary open market operation. To understand that operation an example with a short balance sheet would be helpful.  Below you find the situation before buying bonds (a) and after buying bonds (b):
Assets Liabilities Assets Liabilities
1 Mio. â‚¬ 1 Mio. â‚¬ 1 Mio. â‚¬ 1 Mio. â‚¬
+ 1 Mio. â‚¬ + 1 Mio. â‚¬
1 Mio. â‚¬ 1 Mio. â‚¬ 2 Mio. â‚¬ 2 Mio. â‚¬
This is an example for an expansionary open market operation. In (b) the amount of bonds is 2 Mio. â‚¬ that means 1 Mio. â‚¬ higher than before. But therefore the amount of money in the economy is also 1 Mio. â‚¬ higher.
Assets Liabilities Assets Liabilities
1 Mio. â‚¬ 1 Mio. â‚¬ 1 Mio. â‚¬ 1 Mio. â‚¬
- 1 Mio. â‚¬ - 1 Mio. â‚¬
1 Mio. â‚¬ 1 Mio. â‚¬ 0 â‚¬ 0 â‚¬
In the example above you see an example for contractionary open market operation. You see that the central bank decided to decline the supply of money. The procedure is the same like for the open market operation only vice versa. The amount of bonds decline and the money in the economy, too. Both decline about 1 Mio. â‚¬.
Now it is interesting to know what kind of effects are resulting by open market operations. Therefore it is necessary to take a look at Treasury bills which are one-year bonds. The main fact of Treasury bills is that you get a fixed amount of dollars after one year. An example will explain to you the interest rate which you get after one year. At first we fix that you get 200 â‚¬ after one year. â‚¬PB is the price of the bond after one year. In our example we take 150 â‚¬ . Below you will find the formular for the interest rate:
This example shows that in that case the interest rate is 33% per year. When you the interest rate for the bond you can "play" with the formula and change it so that you are able to find the future price of the bond after one year. For that operation the formula below is an example:
Because of this mathematical facts you can say that the price for bonds is less than the final price when the interest rate is positive. Sometimes you read in newspapers that bonds went up. This means that the interest rates for bonds decrease and the prices for bonds increase. 
3. Which monetary policy strategies are the most widely practised? Which monetary policy strategy is currently used by the European Central Bank?
The most widley practised monetary policies strategies are the nominal exchange rate peg, inflation targeting and monetary targeting.
1974 the Deutsche Bundesbank was the first one which used monetary targeting. Monetary targeting is based on the quantity theory of money. Therefore it is necessary to take a look at the quantity formula:
P are prices, V is velocity, Y is real GNP and M money supply. If you change the formula you are able to see which amount of money has to be choosen to achieve a certain price level:
For practical use the stability of velocity is very important. If you take a look at formula (1) you are able to see that an increase of V increases P. That means that less money is needed to achieve an intended price level. V is determined by demand for money. If the demand for money decrease money supply (M) is converted faster.  A country has to do some tests which proof if monetary targeting can be practiced. 
A further monetary policy strategy is inflation targeting. The short-term or middle-term aim of a central bank which is using inflation targeting is to achieve low inflation rates or their inflation targets. The expected inflation rate play an important role. Central banks are able to influence the inflation target by using interest rates. It is a very simple example which should explain this instrument: If inflation is higher than the target, the bank can raise interest rates. The effect is that the inflation will decrease. If inflation is lower than the target, the bank can decrease interest rates. The result is that inflation will increase. 
Concerning inflation targeting it is necessary to take a look at Taylor's Rule which determines the interest rates. The central banks may follow the rule. Therefore you have to take a look a the following formula:
is the inflation rate, is the inflation target, is the nominal interest rate, is the target interest rate, is the unemployment rate and is the natural unemployment rate. and are positive coefficients. If and the nominal interest rate should be equal to , the inflation target. This procedure has to be done by the central bank. When >, the inflation rate is higher than the inflation target. In that case the central bank has to intervene. It has to increase above . The effect is that umeployment will increase because of the higher interest rate and the inflation will decrease. The coefficient has to be larger than one. The higher it is the better the inflation target will be achieved. 
2. The more central bank increase interest rate
3. The more economy slow down
4. The more unemploy-ment
5. Inflation target achieved faster
1. The higher
The central bank has to increase the real interest rate when inflation increases. But therefore the aim of the central bank should be decreasing spending and ouput. When > the nominal interest rate should be decreased by the central bank. The result is that the output will increase and the unemployment will decrease. has the the same aim like . It shows the care about unemployment in relation to inflation by the central bank. If Î² is higher, the c.bank is more concerned to bring the unemployment close to natural level (to increase employment) and to increase output than to keep the inflation close to its target. 
2. Central bank bring unemployment close to natural level
3. Central bank increase output
4. Central bank achieve inflation target
The nominal exchange rate is also a monetary policy strategy. Through pegging the domestic currency to a very stable currency it is possible to control the exchange rate and the inflation.  But there are some requirements which have to be fulfilled. At first there has to be a country which has a currency that is powerful enough to be an anchor currency. The issue bank of the "anchor currency country" needs a huge reliability because of well developed stability in the past. Furthermore a large part of the foreign trade has to be done with the "anchor currency country" or in its currency. That means that for example Poland has to trade with Germany or in â‚¬. It is also possible to trade in US-$ but therefore the anchor currency country has to be the USA.  Furthermore an aim is to attract capital flows and to stabilize the movement of interest rates. When there is a danger that the domestic currency will decrease the central bank will increase central banks. If there is an danger that the domestic currency will increase the central bank will decrease interest rates. 
The European Central Bank (ECB) is using the two-pillar strategy. The first pillar is a wide economic analysis to identify short- and middle-term risks for keeping price stability. For the analysis the ECB has a lot of indicators which are used to identify these risks. For example the economic growth, wage development, exchange rate development and the balance of payment situation, consumer price and producer price index development, asset price and real-estate price development and the inflation forecast.  The content of the second pillar is based on the quantity theory of money and is similar to the strategy of the Deutsche Bundesbank. The formular below expresses the annual increase of M3:
M3 · V = P · Y
V is velocity and it is between 0,5% to 1% per year. P is price increase and it is up to 2%. Y is the growth of the production potential and it is between 2% to 2,5%. Because of these factors the annual increase of M3 is 4,5%. With the help of these two-strategies the ECB is able to achieve their main aim, price stability. 
4. List of sources
Felderer, B. / Homburg,S., Makroökonomik und neue Makroökonomik, 9.Auflage, Berlin 2005
Borchert, M., Geld und Kredit, 7.Auflage, München 2001
Gischer, H. /Herz, B. /Menkhoff, L., Geld, Kredit und Banken, 2.Auflage, Berlin-Heidelberg 2005
Kazandziska, M., Macroeconomics I, Session 13
Bofinger, P. / Reischle, J. / Schächter, A., Geldpolitik, 1.Auflage, München 1996
Blanchard, O., Macroeconomics, 3.Auflage, Upper Saddle River 2003
Heine, M. / Herr, H., Volkswirtschaftslehrer, 3.Auflage, München 2003
Mankiw, N. Gregory, Mikroökonomik, 5.Auflage, Stuttgart 2003
Burda, M. / Wyplosz, C., Macroeconomics, 3.Auflage, New York 2001
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