Project: Estimate a consumption function for the UK economy explaining the statistical techniques you have used.

### Abstract

The purpose of doing this project is to estimate different consumption functions and to observe the relationships between consumption and a set of variables, such as household disposable income, house price inflation and inflation. I have extended my data to include the periods 1999 to 2001. I will construct and apply a model to such data and apply an appropriate set of tests to it. This project will be achieved using PC GIVE, which is a computerised statistical package. This will enable me to produce graphical analysis and present results in an appropriate manner.

### Introduction

To address economic problem in context we separate an economy’s aggregate expenditure (Y) into four categories: consumption (C), investment (I), government expenditure (G), and net foreign expenditure, or exports less imports (X-M). The aggregate demand identity

Y=C + I + G + (X – M)

Is used to represent these four elements. It is useful to make this kind of categorization, because different agents are responsible for each type of expenditure and therefore they must have different determinants. (William E.Griffiths, R, Carer Hill, George G. Judge, (1993) Learning and Practicing Econometrics, 1993 by John Wiley ; Sons, Inc, page 261)

Consumption is the most important element in aggregate demand, it accounted for almost 70 per cent of GDP in 1989. It is thus very important to forecasters for them to be able to predict consumption correctly. Even a small percentage of error can result in a very large absolute error. For example, forecasters make an error of 1 per cent in predicting consumption, which seems to be insignificant. This will account for an error of 0.7 per cent of GDP. Therefore its accurate prediction is essential to the management of the economy. If we can model the aggregate consumption function, then we can go along predicting future consumption level and use fiscal and monetary policy to manage the economy efficiently. Another reason that consumption is so important is that the marginal propensity to consume is one of the factors determining the size of the multiplier. This is important in determining the effects of changes in investment and government spending. According to J.M Keynes, consumption and disposable income are related and this was accepted for many years.

However, during the 1950s there appeared to be a discrepancy between the consumption function estimated from long run, short run and cross-section series of data. It also failed to explain some of the more interesting features of aggregate consumer behaviour and failed to predict certain periods of sharp fall in the proportion of personally disposable income consumed, such as in the early 1970s and early 1980s. Evidence suggests that the Keynesian consumption function could not resolve these problems and there was need for a more accurate consumption function. This led to many attempts to estimate an equation, which can predict consumer expenditure, such as the development of the Permanent Income and Life-Cycle Hypotheses, which, it was claimed, fitted the facts better than the simple Keynesian view of consumption.

A consumption function describes the relationship between consumer expenditure and income. This research was given by Keynes’s initial conceptual break through in The General Theory of Employment, Interest and Money (Keynes 1936).

Main body of the project

The consumption function was introduced in Keynes (1936)

” We shall therefore define what we shall call the propensity of consume as the functional relationship, between Y, a given level of income and C the expenditure on consumption out of that level of income…

The amount that the community spends on consumption depends (i) partly on the amount of their income, (ii) partly on other objective attendant circumstances, and (iii) partly on the objective needs and the psychological propensities and habits of the individuals comparing it…

The fundamental psychological law upon which we are entitled to depend with great confidence both a priori from our knowledge of human nature and from the detailed facts of experience, is that men are disposed, as a rule and on the average, to increase their consumption as their income increases, but not by as much as the increase in their income. That is …Dc/Dy is positive and less than unity.” (Keynes, 1936, p96)

A simple ‘Keynesian’ Consumption Function:

Keynes argued that on average men increase their consumption as their income increases, but not by as much as their income. This means the marginal propensity to consume is less than 1. The proportion of income consumed tends to fall as income rises. He also suggested that a higher absolute level of income would widen the gap between income and consumption. From this statement the Keynesian consumption function has been derived:

C = c0 + c1 Y

C is consumption, or in other words consumer expenditure, c0 is a constant and represents autonomous consumption which is the sum of expenditure that is not influenced by real GDP; c1 is the marginal propensity to consume (m.p.c), which is a fraction of a change in disposable income that is consumed. (Michael Parkin Melanie Powell Kent Matthews, Economics Fourth Edition 1998, Addison – Wesley), Y is national income. To find the estimates for c0 and c1 we can fit a regression line, or line of ‘best fit’, to the data. Using linear regression I obtain the equation below:

C = 13064.8 + 0.925Y

The marginal propensity to consume is 0.925. This supported the Keynesian’s view that the m.p.c is less than 1. The autonomous consumption is 13064.8. This is positive and very large. Real interest rate can influence autonomous consumption. Normally the lower the real interest rate the greater is autonomous consumption.

Before we can assume that MPC is 0.925, we need to decide whether the predicted equation is acceptable or not. To do this we can test whether the equation fits the actual data correctly. Another way is to check how big the residuals, in other words standard errors are. The smaller the residual, the better the estimation. Graph 1.1 shows that the predicted line does fit the actual data quite well, except it ignored the fluctuation, which took place during the1970s to 1990s. Indeed it failed to predict the rise in consumption during the 1970s and the sharp fall during the 1990s. This failure can be explained by the rises or falls in national income. As Friedman once suggested, men base their consumption on permanent income, they do not increase their consumption due to short-term changes in income.

The second graph on graph 1.1 shows consumption against time, it is clear to see that consumption tends to rise through time, but at a slower rate.

Figure 1.1

By a closer examination of the data, it indicates that using changes in household disposable income to explain changes in consumption is not enough. It may help to look at the ‘errors’ more carefully. The third graph on figure 1.1 shows the remained residuals or error. Between the periods of the 1950s and early 1970s residuals seem to be small, however, very large positive errors occurred during the 1980s. This means that actual consumption was very much more than predicted. It appears that consumers were optimistic towards their future incomes. Again during the 1990s very large negative errors occurred, so actual consumption was much lower than predicted. It appears consumers were pessimistic about their future incomes during that period. The figure for DW (Durbin Watson) should be equal to 2. So it is uncorrelated, which means residuals are randomly distributed and it would be easy to model. However the DW figure for this consumption function is 0.426, which is much more below 2.

Coefficient Std.Error t-value t-prob Part.R^2

Constant 13064.8 2801. 4.66 0.000 0.2950

HDY 0.924901 0.008219 113. 0.000 0.9959

sigma 8375.07 RSS 3.64737352e+009

R^2 0.99591 F(1,52) = 1.266e+004 [0.000]**

log-likelihood -563.386 DW 0.426

no. of observations 54 no. of parameters 2

mean(CON) 300958 var(CON) 1.6516e+010

R2 is called the coefficient of determination. It indicates how well the predicted line fits the actual data. The closer it is to 1 the greater the predictive ability of our model over the sample observations. The R2 for this consumption function is 0.996. This is roughly equal to one. This means that the estimation fits the line quite well.

Before we go on consider other consumption functions we will make a small change to this simple consumption function. This is to use logarithms of consumption and income to estimate this function in the form:

ct = c0 + c1yt or Ln(Ct) = c0 +c1Ln(Yt)

The reason to use logarithms is because it makes the residuals smaller, thus a better estimation. C = log(C) and y = log(Y). The relationship between C and Y is:

C = c0Yc1

c1 is the elasticity of consumption with respect to income. Estimating this log consumption function we obtain:

c = 0.773 + 0.937y

The elasticity with respect to income is 0.937 and the level of consumption predicted is very similar to the simple Keynesian consumption function.

R2 is 0.997. This is closer to 1 than the previous value. This means the estimation fits the actual line even better.

Permanent Income / Life cycle Theories

Friedman’s Permanent Income Hypothesis (PIH) and Modigliani’s Life Cycle Hypothesis (LCH) state that consumers base their consumption on expected life- time income, not on current income.

Permanent income is defined as the constant income stream, which has the same present value as an individual’s expected lifetime income, and can thus be taken as a simple measure of lifetime income. (Backhouse, R (1991) Applied UK Macroeconomics, Blackwell)

The problem with the Permanent Income consumption function is that the measurement of Permanent income depends on consumers’ expectations. There are 2 solutions to this problem: one is the assumption that Permanent Income responds with a lag to actual income and the other is the assumption of rational expectation.

The conventional way to measure permanent income is to take a weighted average of past incomes. The formula for yp is yp = (yt +yt-1 + yt-2) / 3. By using this formula I obtained:

Ct = 0.6064 + 0.952ytp

This look very similar to the one predicted before. The elasticity with respect to permanent income is 0.95 compared to the previous value of 0.94. However, the short – run consumption and income will be very different between the two consumption functions. This is because y p is only based on the average of the current and the past two years income. This may not look very representative. Observing the value of R2, it is 0.992; this is lower than the simple Keynes consumption function and the log consumption function. This means its predictive ability as a model is lower than the first two functions. The remained residuals varied in a great deal, between the period of the 1970s to mid 1980s, there were large negative residuals, this means people were pessimistic about the future income. However, between 1986 to 1990 and early 2000, there were large positive errors, this means people were optimistic about their future income.

Another common approach to modelling permanent income is to assume that permanent income is a weighted average of all past incomes, with geometrically declining weights. Using a more complicated lag structure does this. The function obtained:

Ct = 0.3175 + 0.5437yt +0.4309ct-1

The short run elasticity of consumption with respect to income is 0.5437. Observing the R2, it is 0.997; this is higher than all other consumption functions I predicted before. This indicates the predicted value of this log permanent income function fits the line better than all the other ones. The remained residuals fluctuate a lot throughout the period. It was quite small between the periods 1950 to 1974. However, the size of the error became larger between the periods of 1975 to 2002.

### The Life – Cycle Hypothesis (LCH)

The permanent income hypothesis discussed above is clearly better than the simple Keynesian model. However, it is still not very satisfactory and additional factors or variables need to be taken into account, such as inflation and house price inflation. Inflation should affect consumption because it reduces the real value of any debts. This means there is a transfer of resources from creditor to debtors. Government is the largest debtor and personal sector is a large net creditor. This means inflation reduces real income.

The life – Cycle Hypothesis (LCH) developed by Modigliani is very similar to the Permanent Income Hypothesis. It is also seen to be proportional to Yp, however, he stresses the age of the consumer also plays a part in determining their consumption over their lifetime. Such as young and very old households, tend to have low income and both have high propensity to consume. On the other hand, the high – income groups containing a higher proportion of middle – aged household tend to have a low propensity to consume. We could add inflation to the life – cycle consumption function and we obtain an equation:

ct = c0 + c1yt + c2ct-1 + c3?t

? is the inflation rate. However, one of the problems with this function is that standard consumer theory suggests that in the long run permanent income will be proportional to actual income, so consumption should be proportional to income. But in the long run we cannot expect consumption to be exactly proportion to income. This has led economists to use what are known as error correction mechanisms. This involves two components: in the long run we assume consumption is proportional to income, however in the short run consumers adjust their consumption towards their target level and this adjustment spread out over time. (Backhouse, R(1991) Applied UK Macroeconomics, Blackwell.)

I obtained a consumption function:

?Ct = 0.0075 + 0.788?yt + 0.122st-1 – 0.0009?t

s is the saving ration and it is obtained by y (income) minus c (consumption).

Coefficient Std.Error t-value t-prob Part.R^2

Constant 0.00746478 0.004738 1.58 0.122 0.0492

DLHDY 0.787978 0.07590 10.4 0.000 0.6919

St-1 0.121970 0.05434 2.24 0.029 0.0950

INF -0.000931982 0.0004286 -2.17 0.035 0.0897

sigma 0.0138087 RSS 0.00915268714

R^2 0.754598 F(3,48) = 49.2 [0.000]**

log-likelihood 150.984 DW 1.24

no. of observations 52 no. of parameters 4

mean(DLCON) 0.0288347 var(DLCON) 0.000717243

The predicted value does not always fit the actual data. The R2 is 0.75; it is not very high compared to other consumption functions. However, its DW value is 1.24, this is closer to 2. This means that residuals are random distributed, which means it would not be easy to model.

Conclusions

Consumption is the most important element over aggregate demand in the economy.

The simple Keynesian theory suggested as income increases people only spend part of their increased income. The permanent income hypothesis and the Life – Cycle hypothesis suggested that people base their spending on their future income. However, Life – Cycle hypothesis stressed that the age of household also plays a part in determining their consumption.

In this project I started by estimating the simple Keynesian consumption function using the data collected from 1948 to 2002. Then I introduced a number of additional factors, which should influence consumption. I used the log to minimise errors and lag structure to compare the difference between last years consumption to the current consumption.

There are also other important factors, which influence consumption, such as house prices and uncertainties. If I were to do this project again I would include these factors in my estimated consumption function.