## Objectives

At the end of this section you should be able to:

• Identify and sketch linear consumption functions.

• Identify and sketch linear savings functions.

• Set up simple macroeconomic models.

• Calculate equilibrium national income.

• Analyse IS and LM schedules.

Macroeconomics is concerned with the analysis of economic theory and policy at a national level. In this section we focus on one particular aspect known as national income determination. We describe how to set up simple models of the national economy which enable equilibrium levels of income to be calculated. Initially we assume that the economy is divided into two sectors, households and firms. Firms use resources such as land, capital, labour and raw materials to produce goods and services. These resources are known as factors of production and are taken to belong to households. National income represents the flow of income from firms to households given as payment for these factors. Households can then spend this money in one of two ways. Income can be used for the consumption of goods produced by firms or it can be put into savings. Consumption, C, and savings, S, are therefore functions of income, Y: that is,

for some appropriate consumption function, f, and savings function, g. Moreover, C and S are normally expected to increase as income rises, so f and g are both increasing functions.

We begin by analysing the consumption function. As usual we need to quantify the precise relationship between C and Y. If this relationship is linear then a graph of a typical consumption function is shown in Figure 1.23. It is clear from this graph that if then a > 0 and b > 0. The intercept b is the level of consumption when there is no income (that is, when Y = 0) and is known as autonomous consumption. The slope, a, is the change in C brought about by a 1 unit increase in Y and is known as the marginal propensity to consume (MPC). As previously noted, income is used up in consumption and savings so that

It follows that only a proportion of the 1 unit increase in income is consumed; the rest goes into savings. Hence the slope, a, is generally smaller than 1: that is, a < 1. It is standard practice in mathematics to collapse the two separate inequalities a > 0 and a < 1 into the single inequality

The relation

enables the precise form of the savings function to be determined from any given consumption function. This is illustrated in the following example.

Sketch a graph of the consumption function C = 0.6Y + 10

Determine the corresponding savings function and sketch its graph.

Solution

The graph of the consumption function C = 0.6Y + 10

Example

has intercept 10 and slope 0.6. It passes through (0, 10). For a second point, let us choose Y = 40, which gives C = 34. Hence the line also passes through (40, 34). The consumption function is sketched in Figure 1.24. To find the savings function we use the relation

which gives

(subtract C from both sides) (substitute C) (multiply out the brackets) (collect terms)

The savings function is also linear. Its graph has intercept -10 and slope 0.4. This is sketched in Figure 1.25 using the fact that it passes through (0, -10) and (25, 0).

Practice Problem

1 Determine the savings function that corresponds to the consumption function C = 0.8Y + 25

For the general consumption function

The slope of the savings function is called the marginal propensity to save (MPS) and is given by 1 - a: that is,

Moreover, since a < 1 we see that the slope, 1 - a, is positive. Figure 1.26 shows the graph of this savings function. One interesting feature, which contrasts with other economic functions considered so far, is that it is allowed to take negative values. In particular, note that autonomous savings (that is, the value of S when Y = 0) are equal to -b, which is negative because b > 0. This is to be expected because whenever consumption exceeds income, households must finance the excess expenditure by withdrawing savings.

The result, MPC + MPS = 1, is always true, even if the consumption function is non-linear. A proof of this generalization can be found on page 274.

The simplest model of the national economy is illustrated in Figure 1.27, which shows the circular flow of income and expenditure. This is fairly crude, since it fails to take into account government activity or foreign trade. In this diagram investment, I, is an injection into the circular flow in the form of spending on capital goods.

Advice

Let us examine this more closely and represent the diagrammatic information in symbols. Consider first the box labelled 'Households'. The flow of money entering this box is Y and the flow leaving it is C + S. Hence we have the familiar relation

For the box labelled 'Firms' the flow entering it is C + I and the flow leaving it is Y, so

Suppose that the level of investment that firms plan to inject into the economy is known to be some fixed value, I *. If the economy is in equilibrium, the flow of income and expenditure balance so that

From the assumption that the consumption function is C = aY + b for given values of a and b these two equations represent a pair of simultaneous equations for the two unknowns Y and C. In these circumstances C and Y can be regarded as endogenous variables, since their precise values are determined within the model, whereas I * is fixed outside the model and is exogenous.

Example

Find the equilibrium level of income and consumption if the consumption function is

Solution

We know that

C = 0.6Y + 10 (given in problem) I = 12 (given in problem)

If the value of I is substituted into the first equation then

The expression for C can also be substituted to give

0.4Y = 22 (subtract 0.67 from both sides)

The corresponding value of C can be deduced by putting this level of income into the consumption function to get

The equilibrium income can also be found graphically by plotting expenditure against income. In this example the aggregate expenditure, C +1, is given by 0.67 + 22. This is sketched in Figure 1.28 using the fact that it passes through (0, 22) and (80, 70). Also sketched is the '45° line', so called because it makes an angle of 45° with the horizontal. This line passes through the points (0, 0), (1, 1),..., (50, 50) and so on. In other words, at any point on this line expenditure and income are in balance. The equilibrium income can therefore be found by inspecting the point of intersection of this line and the aggregate expenditure line, C + I. From Figure 1.28 this occurs when 7 = 55, which is in agreement with the calculated value.

2 Find the equilibrium level of income if the consumption function is C = 0.8Y + 25

and planned investment I = 17. Calculate the new equilibrium income if planned investment rises by

Practice Problem

Practice Problem

### 1 unit.

To make the model more realistic let us now include government expenditure, G, and taxation, T, in the model. The injections box in Figure 1.27 now includes government expenditure in addition to investment, so

We assume that planned government expenditure and planned investment are autonomous with fixed values G* and I * respectively, so that in equilibrium

The withdrawals box in Figure 1.27 now includes taxation. This means that the income that households have to spend on consumer goods is no longer Ybut rather Y- T (income less tax), which is called disposable income, Yd. Hence

In practice, the tax will either be autonomous (T = T* for some lump sum T*) or be a proportion of national income (T = tYfor some proportion t), or a combination of both (T = tY + T*).

Example

C = 0.9 Yd + 70 T = 0.2Y + 25 calculate the equilibrium level of national income.

### Solution

At first sight this problem looks rather forbidding, particularly since there are so many variables. However, all we have to do is to write down the relevant equations and to substitute systematically one equation into another until only Y is left. We know that

Y = C +1 + G (from theory) (1) G = 20 (given in problem) (2) I = 35 (given in problem) (3) C = 0.9Yd + 70 (given in problem) (4) T = 0.2Y + 25 (given in problem) (5) Yd = Y- T (from theory) (6)

This represents a system of six equations in six unknowns. The obvious thing to do is to put the fixed values of G and I into equation (1) to get

This has at least removed G and I, so there are only three more variables (C, Yd and T) left to eliminate. We can remove T by substituting equation (5) into (6) to get

and then remove Yd by substituting equation (8) into (4) to get C = 0.9(0.8Y- 25) + 70 = 0.72Y- 22.5 + 70

We can eliminate C by substituting equation (9) into (7) to get

= 0.72Y + 47.5 + 55 = 0.72Y + 102.5 Finally, solving for Y gives

0.28 Y = 102.5 (subtract 0.72Y from both sides) Y = 366 (divide both sides by 0.28)

Practice Problem

3 Given that

T = 0.1 Y + 10 calculate the equilibrium level of national income.

To conclude this section we return to the simple two-sector model:

Previously, the investment, I, was taken to be constant. It is more realistic to assume that planned investment depends on the rate of interest, r. As the interest rate rises, so investment falls and we have a relationship

I = cr + d where c < 0 and d > 0. Unfortunately, this model consists of three equations in the four unknowns Y, C, I and r, so we cannot expect it to determine national income uniquely. The best we can do is to eliminate C and I, say, and to set up an equation relating Y and r. This is most easily understood by an example. Suppose that

We know that the commodity market is in equilibrium when

Substitution of the given expressions for C and I into this equation gives

Y = (0.8 Y + 100) + (-20r + 1000) = 0.8Y- 20r + 1100

which rearranges as

This equation, relating national income, Y, and interest rate, r, is called the IS schedule.

We obviously need some additional information before we can pin down the values of Y and r. This can be done by investigating the equilibrium of the money market. The money market is said to be in equilibrium when the supply of money, MS, matches the demand for money, MD: that is, when ms = md

There are many ways of measuring the money supply. In simple terms it can be thought of as consisting of the notes and coins in circulation, together with money held in bank deposits. The level of MS is assumed to be controlled by the central bank and is taken to be autonomous, so that

for some fixed value M*.

The demand for money comes from three sources: transactions, precautions and speculations. The transactions demand is used for the daily exchange of goods and

services, whereas the precautionary demand is used to fund any emergencies requiring unforeseen expenditure. Both are assumed to be proportional to national income. Consequently, we lump these together and write

where Ll denotes the aggregate transaction-precautionary demand and kl is a positive constant. The speculative demand for money is used as a reserve fund in case individuals or firms decide to invest in alternative assets such as government bonds. In Chapter 3 we show that, as interest rates rise, speculative demand falls. We model this by writing

where L2 denotes speculative demand, k2 is a negative constant and k3 is a positive constant. The total demand, MD, is the sum of the transaction-precautionary demand and speculative demand: that is,

= kJY + k2 r + k3 If the money market is in equilibrium then

This equation, relating national income, Y, and interest rate, r, is called the LM schedule. If we assume that equilibrium exists in both the commodity and money markets then the IS and LM schedules provide a system of two equations in two unknowns, Y and r. These can easily be solved either by elimination or by graphical methods.

Example

Determine the equilibrium income and interest rate given the following information about the commodity market

and the money market

What effect would a decrease in the money supply have on the equilibrium levels of Y and r?

### Solution

The IS schedule for these particular consumption and investment functions has already been obtained in the preceding text. It was shown that the commodity market is in equilibrium when

For the money market we see that the money supply is MS = 2375

and that the total demand for money (that is, the sum of the transaction-precautionary demand, L1, and the speculative demand, L2) is

The money market is in equilibrium when

that is,

The LM schedule is therefore given by

Equations (1) and (2) constitute a system of two equations for the two unknowns Y and r. The steps described in Section 1.2 can be used to solve this system:

Step 1

Double equation (2) and subtract from equation (1) to get

Step 2

Divide both sides of equation (3) by 70 to get r = 5

Step 3

Substitute r = 5 into equation (1) to get

0.27 = 1000 (subtract 100 from both sides) 7 = 5000 (divide both sides by 0.2)

Step 4

As a check, equation (2) gives

The equilibrium levels of Y and r are therefore 5000 and 5 respectively.

To investigate what happens to Y and r as the money supply falls, we could just take a smaller value of MS such as 2300 and repeat the calculations. However, it is more instructive to perform the investigation graphically. Figure 1.29 shows the IS and LM curves plotted on the same diagram with r on the horizontal axis and Y on the vertical axis. These lines intersect at (5, 5000), confirming the equilibrium levels of interest rate and income obtained by calculation. Any change in the money supply will obviously have no effect on the IS curve. On the other hand, a change in the money supply does affect the LM curve. To see this, let us return to the general LM schedule klY + k2r + k3 = M* and transpose it to express Y in terms of r:

klY = -k2r — k3 + M* (subtract k2r + k3 from both sides)

Expressed in this form, we see that the LM schedule has slope —k2/k1 and intercept (—k3 + M*)/k1.

Any decrease in M* therefore decreases the intercept (but not the slope) and the LM curve shifts downwards. This is indicated by the dashed line in Figure 1.29. The point of intersection shifts both downwards and to the right. We deduce that, as the money supply falls, interest rates rise and national income decreases (assuming that both the commodity and money markets remain in equilibrium).

It is possible to produce general formulae for the equilibrium level of income in terms of various parameters used to specify the model. As you might expect, the algebra is a little harder but it does allow for a more general investigation into the effects of varying these parameters. We will return to this in Section 5.3.

Practice Problem

4 Determine the equilibrium income, Y, and interest rate, r, given the following information about the commodity market

and the money market

Sketch the IS and LM curves on the same diagram. What effect would an increase in the value of autonomous investment have on the equilibrium values of Y and r?

Example EXCEL

(a) Given the consumption function

C = 800 + 0.9 Y and the investment function

I = 8000 - 800r find an equation for the IS schedule.

(b) Given the money supply

MS = 28 500 and the demand for money

Md = 0.75 Y - 1500r find an equation for the LM schedule.

(c) By plotting the IS-LM diagram, find the equilibrium values of national income, Y, and interest rate, r. If the autonomous investment increases by 1000, what effect will this have on the equilibrium position?

Solution

(a) The IS schedule is given by an equation relating national income, Y, and interest rate, r.

In equilibrium, Y = C + I. By substituting the equations given in (a) into this equilibrium equation, we eliminate C and I, giving

0.1 Y = 8800 - 800r (subtract 0.9Y from both sides)

(b) The LM schedule is also given by an equation relating Y and r, but this time, it is derived from the equilibrium of the money markets: that is, when MS = MD. Substituting the equations given in (b) into this equilibrium equation gives

0.75Y = 28 500 + 1500r (add 1500r to both sides) Y = 38 000 + 2000r (divide both sides by 0.75)

(c) To find the equilibrium position, we plot these two lines on a graph using Excel in the usual way. We need to choose values for r and then work out corresponding values for Y. It is most likely that r will lie somewhere between 0 and 10, so values of r are tabulated between 0 and 10, going up in steps of 2. We type the formula

=88000-8000*A2

in cell B2 and type

=38000+2000*A2

in cell C2. The values of Y are then generated by clicking and dragging down the columns. Figure 1.30 shows the completed Excel screen.

Placing the cursor at the point of intersection tells us that the lines cross when r = 5% and Y = 48 000

If the autonomous investment increases by 1000, the equation for the IS schedule will change, as the equation for investment now becomes

The new IS schedule can be plotted on the same graph by adding a column of figures into the spreadsheet, as shown in Figure 1.31.

Notice that the point of intersection has shifted both upwards and to the right. The equilibrium position has now changed, resulting in a rise in interest rates to 6% and an increase in income to 50 000.

A |
B |
c |
D |
E |
F |
6 H |
i | |||||||||

1 |
r |
IS Schedule |
LM Schedule | |||||||||||||

2 |
0 |
88000 |
38000 | |||||||||||||

3 |
3 |
73000 |
43000 | |||||||||||||

4 |
4 |
58000 |
46000 | |||||||||||||

5 |
6 |
40000 |
50000 | |||||||||||||

6 |
8 |
24000 |
54000 | |||||||||||||

7 |
10 |
8000 |
5S000 | |||||||||||||

8 | ||||||||||||||||

9 | ||||||||||||||||

10 |
National income, |
80000 ■ | ||||||||||||||

11 |
\ | |||||||||||||||

12 |
60000 - |
—X |
e'* |
IS Schedule LM Schedule | ||||||||||||

13 |
40000 -30000 - |
\ | ||||||||||||||

14 | ||||||||||||||||

15 |
\ | |||||||||||||||

16 | ||||||||||||||||

17 |
u |
a iu ID | ||||||||||||||

18 |
Interest rate, r | |||||||||||||||

19 | ||||||||||||||||

20 | ||||||||||||||||

31 |

Autonomous consumption The level of consumption when there is no income. Autonomous savings The withdrawals from savings when there is no income. Consumption The flow of money from households to firms as payment for goods and services.

Consumption function The relationship between national income and consumption. Disposable income Household income after the deduction of taxes and the addition of benefits.

Factors of production The inputs to the production of goods and services: land, capital, labour and raw materials.

Government expenditure The total amount of money spent by government on defence, education, health, police, etc.

Investment The creation of output not for immediate consumption. IS schedule The equation relating national income and interest rate based on the assumption of equilibrium in the goods market.

LM schedule The equation relating national income and interest rate based on the assumption of equilibrium in the money market.

Marginal propensity to consume The fraction of a rise in national income which goes on consumption. It is the slope of the consumption function.

Marginal propensity to save The fraction of a rise in national income which goes into savings. It is the slope of the savings function.

Money supply The notes and coins in circulation together with money held in bank deposits. National income The flow of money from firms to households.

Precautionary demand for money Money held in reserve by individuals or firms to fund unforeseen future expenditure.

Speculative demand for money Money held back by firms or individuals for the purpose of investing in alternative assets, such as government bonds, at some future date.

Taxation Money paid to government based on an individual's income and wealth (direct taxation) together with money paid by suppliers of goods or services based on expenditure (indirect taxation).

Transactions demand for money Money used for everyday transactions of goods and services.

Practice Problems

5 If the consumption function is given by

(a) autonomous consumption

(b) marginal propensity to consume

Transpose this formula to express Y in terms of C and hence find the value of Y when C = 110.

6 Write down expressions for the savings function given that the consumption function is

7 For a closed economy with no government intervention the consumption function is

Calculate the equilibrium level of

(a) national income

(b) consumption

(c) savings

show that

1 - a and obtain a similar expression for C in terms of a, b and I*. 9 An open economy is in equilibrium when

where

Y = national income C = consumption I = investment G = government expenditure X = exports M = imports

Determine the equilibrium level of income given that C = 0.8Y + 80 I = 70 G = 130 X = 100 M = 0.2Y + 50

money supply, MS = 4000

transaction-precautionary demand for money, MS = 0.15 Y speculative demand for money, L2 =-20r + 3825

determine the values of national income, Y, and interest rate, r, on the assumption that both the commodity and the money markets are in equilibrium.

11 (Excel) Consider the consumption function

C = 120 + 0.8 Yd where Yd is disposable income.

Write down expressions for C, in terms of national income, Y, when there is

(c) a proportional tax in which the proportion is 0.25

Sketch all three functions on the same diagram, over the range 0 < Y < 800, and briefly describe any differences or similarities between them.

Sketch the 45 degree line, C = Y, on the same diagram, and hence estimate equilibrium levels of national income in each case.

12 (Excel) If the consumption function is C = 0.9Y + 20

and planned investment I = 10, write down an expression for the aggregate expenditure, C + I, in terms of Y.

Draw graphs of aggregate expenditure, and the 45 degree line, on the same diagram, over the range 0 < Y < 500. Deduce the equilibrium level of national income.

Describe what happens to the aggregate expenditure line in the case when

(a) the marginal propensity to consume falls to 0.8

(b) planned investment rises to 15 and find the new equilibrium income in each case.

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