The '45 Degree' Diagram

The '45 Degree' Diagram

Many of you will have spent quite a lot of time looking at '45 degree' diagrams, or 'Keynesian cross' diagrams. Until a few years ago, they were the main way in that the expenditure and income aggregates where analysed. Nowadays, aggregate demand and supply diagrams are preferred, although many teachers still like to explain the situation using the 'good old' 45-degree diagram. It should be noted, though, that examiners prefer students to use aggregate demand and supply analysis (see the next three Learn-Its).

On those grounds, I do not intend to spend too long on this. I do feel, though, that these diagrams are useful for explaining the concept of marginal propensity to consume (MPC). This, in turn, is important for the understanding of the multiplier.

Look at the diagram below.

The consumption function

You can see why this is called the 45-degree diagram. There is a line that comes diagonally out of the origin at an angle of 45 degrees. The reason why these diagrams have this 45-degree line is that for every point on the line, the value of whatever is being measured on the x-axis is equal to the value of whatever is being measured on the y-axis. In this case, that means that Y = C.

Actually, this fact it not so important when dealing with consumption functions, but it is very important if the y-axis is measuring planned expenditure. Equilibrium national income occurs where Y = E, and this would be every point on the 45 degree line.

Anyway, the other line in the diagram is the one that represents consumption in the economy at each level of income. For simplicity, we shall assume that we are dealing with a 2-sector model of the economy. Households either consume or save their income. The line starts at point 'a'. This means that even when income is zero, consumption is 'a'. This is called autonomous consumption. How can households consume if they have no income? We have to assume that 'saving' is negative, or that 'dissaving' is going on. In other words, households have to borrow money to consume, or dip into previously saved money, while they temporarily (one hopes!) have no income.

As income rises, consumption also rises. The gradient of the line represents something called the marginal propensity to consume. This is the amount of every extra pound earned that is spent. If you earned an extra £1, and spent 60p of it, then the value of the MPC would be 0.6. Notice that if you spend 60p of an extra £1, you must, by definition, be saving the other 40p. Hence, the marginal propensity to save (MPS) will be 0.4. It should also make sense that MPC + MPS = 1.

Here are the formulas:


Notice that the consumption line has been labelled C = a + BY. You may remember the equation of a line in GCSE maths (Y = MX + C). The consumption equation is exactly the same structure. 'a' is the intercept point on the y-axis (autonomous consumption) and 'b' is the gradient (the MPC).

This next diagram shows the relationship between the consumption function and the savings function.

The relationship between the consumption function and the savings function

Where the consumption line crosses the 45 degree line, Y = C. Hence savings must be zero. You can see that this is the case in the bottom diagram. To the left of this point (Y1), C > Y, so saving must be negative. Dissaving is occurring. To the right of Y1, C < Y, so saving is positive. Notice that since MPC + MPS =1, the gradient of the saving line is '1-b' where 'b' is the gradient of the consumption line.

The multiplier in the 2-sector economy

The multiplier is a very important concept in macroeconomics. The best way to explain the multiplier is to use a circular flow diagram.

Circular flow diagram

Assume an initial £100 million of autonomous investment. This money will return to the households in terms of income earned from the factors of production (land, labour and capital) hired to the firms (the black line). It is assumed that the MPC = 0.8, and so the MPS = 0.2. Hence, households save 20% of this income. £20 million is saved and the rest is spent on the goods and services produced by the firms. This £80 million again returns to the households in terms of factor incomes. 20% of this is saved (£16 million) leaving £64 million to be spent on goods and services. This process keeps going. The initial £100 million will multiply to give a final increase in total national income of much more than £100 million. In fact, there is a formula that you can use to find the multiplier, and then another formula that can be used to find the final increase in national income.

The multiplier in the 2-sector economy

So, using the formulae, the value of the multiplier in the example above is 5 (1 divided by 0.2). The final increase in national income is £500 million (£100 million times 5).

The multiplier in the more realistic 4-sector model

If we now bring in the government and the foreign sectors, the multiplier works in the same way, but there are more withdrawals from the economy.

The multiplier in the more realistic 4-sector model

In this, more realistic, example, the MPC is still 0.8, giving a MPS of 0.2. We now assume that money is withdrawn from the economy via taxation and the purchase of imports. The marginal propensity to tax (MPT) is assumed to be 0.4 (the actual average tax paid by an individual is just under 40%). The marginal propensity to import (MPM) is assumed to be 0.2. You can see from the diagram that with so many withdrawals, the amount of money that subsequently flows around the economy after the initial injection is much smaller. The higher the withdrawals, the lower the value of the multiplier. In the 2-sector formula, we divided one by the MPS. In a sense, the MPS represented the marginal propensity to withdraw (MPW), because saving was the only withdrawal. The formula is the same for the 4-sector model, except we now have three withdrawals.

The multiplier in the more realistic 4-sector model

So, in the example above, adding the three marginal propensities to withdrawal gives 0.2 + 0.4 + 0.2 = 0.8. Hence the multiplier is one divided by 0.8, which equals 1.25. Finally, the total increase in national income is £125 million (£100 million times 1.25).

This example is much more realistic. The value of multipliers is rarely above 2 in the real world. Even though the savings ratio is quite low in the UK (less than 10%), the tax burden is about 38%, and around 30% of GDP is spent on imports. So you see, the MPW is likely to be well over 0.5, giving a value of the multiplier of less than two.

The concept of the multiplier is very powerful. For those who believe in the Keynesian 'demand management' way of running an economy, the idea that a given injection of money by the government can lead to a multiple increase in the final national income (and so the creation of more jobs) is very persuasive. The important final point to note is that the process is a dynamic one. In the last example, the £100 million does not turn into £125 million overnight. There will be a time lag between when the households receive their factor incomes and when it all gets spent. And then the money has to go round the circular flow again and again until the full £125 million is finally spent.

Remember that the multiplier process can work in reverse as well. When the German owned Rover decided to cut back its production in the UK, the resulting loss in jobs meant a reduction in consumer spending in the area, which will create further job losses in local shops that depended on the Rover employees' spending. Again this process can keep going.