ian preston, university college london econ b003 lecture notes …uctp100/b003/b003lect.pdf · econ...

Ian Preston, University College London Spring Term 2005

Econ B003 Lecture Notes Applied Economics

Lecture times: Mon 1 and Fri 12 in the Darwin Lecture Theatre

Practical lecture times:

Fri 1 in the Darwin Lecture Theatre Weeks beginning 24/1, 7/2, 21/2, 7/3, 14/3

Tutorials:

Weeks beginning 24/1, 7/2, 21/2, 7/3, 14/3

Lecturer's Office Hour: Mon 10-11

Course outline

Introduction to statistical methods

Wage equations Consumer demand

Labour Supply Consumption Investment

©Ian Preston 2003 2

Introduction

Purpose of the course • To provide an introduction to the application of economic theory to data • To develop an understanding of simple and commonly used econometric techniques • To impart an ability to understand and interpret results both statistically and economically Economic theory suggests relationships but rarely specifies magnitude. • We want to test these relationships by investigating their compatibility with data • We want to estimate the empirical magnitude of effects For example, consumer theory suggests quantities purchased should depend on prices and income. We wish to test whether such relationships exist. If so, we want to know whether demands are income elastic or inelastic, whether different goods are substitutes or complements and so on.

Types of data • Cross section data (different units observed at same point in time) eg household surveys,

firm censuses etc Such data are good for investigating relationships between variables which vary widely

across individuals at any point in time such as incomes and commodity demands but may lack variation in other variables of interest such as prices, interest rates etc.

• Time series data (same unit observed at different points in time) Such data are good for investigating effects of variables which vary mainly over time.

However data may often be aggregate data on units (such as whole countries) which correspond poorly with the units to which the theory applies.

• Repeated cross sections Such data combines cross sectional and time series variation. When the cross sections

cover the same units at different points in time this is referred to as panel data and allows particularly powerful econometric techniques to be used (though these are beyond the scope of the current course).

Tools for analysis

The tools used in the course are those of econometrics (mainly the technique of linear regression). Econometrics is "the application of statistical and mathematical methods to the analysis of economic data, with a purpose of giving empirical context to economic theories and verifying them or refuting them." (Maddala, Introduction to Econometrics, 1992, p.1) The purpose of these notes is an informal presentation of the main properties of common estimation techniques rather than the sort of detail and rigour appropriate to more advanced courses. Data description and economic modelling We should draw a distinction between simply describing data and the more ambitious task of empirically modelling the social or economic processes involved. Let us illustrate this with a simple example of the relationship between two discrete variables – newspaper readership and attitudes to immigration. Table 1 below is based on cross sectional survey data on 3800 newspaper readers in the UK over the years 1983-1991. [The data can be found in a file


called tabloid.dta] It shows clearly that surveyed individuals who read tabloid newspapers regularly are more hostile to immigration than are those who read broadsheets. Furthermore this relationship is statistically strong. The probability of observing an association as strong as this in a data set this size if there were none in the population would be negligible. Without some theory about the causal processes relating these two characteristics of respondents’ lives this is just data description however – a possibly interesting but as yet uninterpreted fact. One obvious question to which these data are relevant is that of how newspaper readership affects social attitudes but to make inferences about this we need a model. It is tempting to conclude from Table 1a that reading tabloid newspapers, in which coverage of news in this area tends to a more hostile perspective, leads a reader to greater hostility. One might then want to interpret the difference between the percentages in the two columns as an estimate of the effect. However we should not do this without being clear about the assumptions needed to justify such an interpretation. The sort of model that might lead this way would have the form:

In order to draw conclusions about the magnitude of the effect of interest from data of this sort we need firstly to consider issues about the origin of the sample.

• We need to assume the sample of observations has been drawn in a way that doesn’ t induce any unwanted correlation between the variables. This is probably not of great concern in this instance but could be in other cases.

Secondly we need to make assumptions about reliability of the data in the context for which we wish to use them

• Variables in the data need to match up to the theoretical concepts in the model. If the correspondence is poor then we would expect the relationship in the data to be an attenuated version of that we are trying to identify. In particular, if the supposed explanatory variable (newspaper readership) is imprecisely measured then the difference between the column percentages will tend to understate the magnitude of the true influence. In this instance neither of the characteristics involved are without ambiguity so this is something we might worry about.

Thirdly we need to consider assumptions about the process generating the data. In particular, we need to be aware that newspaper readership was not controlled by the survey investigators, it was chosen by the subjects of the inquiry and the implications of this need to be recognised.

• We need to be confident about our identification of the direction of causation. If it is attitudes that affect newspaper readership rather than or as well as the reverse then we may be seriously misinterpreting the association we are observing. For example, if people choose their newspaper to confirm their preexisting prejudices then it may be that all we are picking up is this fact and what is printed in the newspaper may have no impact on opinions at all.


• We need to assume that the other influences on attitudes are not associated with newspaper readership or else we may mistakenly be attributing their influence to that of newspaper readership. For example, education may well influence attitudes and may also affect choice of newspaper. Broadsheet readers may be less hostile to immigration not because of what they read in newspapers but because they are typically more highly educated. In this data we have information on education and the evidence of Table 2 bears out associations compatible with this – the more highly educated do read more broadsheet newspapers and do have more liberal attitudes. Furthermore, given this information there are things we can do. We can control for education by checking to see whether correlation between readership and attitudes persists within education groups. Table 1b suggests that this is so. Not all other influences are observed however and in these cases we need to rely on assumption if we are to justify a simple estimate of the effect of newspaper readership based on this sort of cross tabulation.

These concerns – measurement error, omitted influences, simultaneous causation – are exactly the same sort of concerns that we worry about repeatedly throughout the course in the context of more sophisticated models.


Table 1a Attitudes to immigration and newspaper readership

Newspaper readers, UK

Newspaper readership

Tabloid Broadsheet

All Same or more 931

(29.74) 297

(44.33) 1228

(32.32) Preference for immigration

Less 2199 (70.26)

373 (55.67)

2572 (67.68)

All 3130 670 3800 Figures in parentheses are column percentages

Table 1b Attitudes to immigration and newspaper readership by education

A level holders

Newspaper readership

Tabloid Broadsheet

All Same or more 106

(32.42) 136

(54.18) 242

(41.87) Preference for immigration

Less 221 (67.58)

115 (45.82)

336 (58.13)


Non A level holders

Newspaper readership Tabloid Broadsheet

All

Same or more 397 (31.09)

49 (41.18)

446 (31.95)

Preference for immigration

Less 880 (68.91)

70 (58.82)

950 (68.05)



An Introduction to OLS Regression

Wage equations OLS regression is the main econometric technique to be used this term. We choose to introduce it in the context of a specific economic issue – the effect of education upon wages. Economic theory suggests that more educated individuals should earn higher wages. Firms will be willing to pay higher wages if education either raises productivity or identifies to employers the more productive in the population. We can investigate whether education does raise wages and, if so, by how much by using cross sectional data on education and wages. It is unlikely that individuals would be prepared to forgo the earnings or pay the fees necessary to acquire education without some return in increased future earnings (albeit that there are some consumption benefits). We investigate this issue using data on the wages of married women in the US. [The data can be found in a file called work.dta] A model is a simplified representation of the real world process involved. When discussing theory we can often make do with a precise deterministic relationship - for instance

ln wi = α + β ei where ln wi and ei are the values of ln hourly wage and years of education for the ith individual. Here α is to be interpreted as the ln hourly wage of someone with no years of education and β is to be interpreted as the effect of an extra year of education on the ln hourly wage, often referred to as the rate of return to education. Note that for this example we choose a model where the effect on the ln hourly wage is constant rather than the effect on hourly wage so that an additional year of education raises any individual’s wage by the same proportionate, rather than by the same absolute, amount. For applied purposes, however, we need to recognise that no deterministic linear relationship can ever fit the data. We modify the relationship by adding a further term ui to reflect the discrepancy. This has to capture all other influences (experience, innate ability, social background etc)

ln wi = α + β ei + ui

Estimation The model has the form

yi = α + β xi + ui Here yi is called the dependent variable and xi is the explanatory variable or regressor, α is referred to as the constant or intercept and β is referred to as the coefficient of xi. Our aim is to estimate α and β and to test hypotheses about them (eg that β = 0). The term ui is referred to as the disturbance or error term. The assumptions made about ui are crucial in determining the properties of the techniques to be employed.


Ordinary least squares (OLS) Suppose that �α and �β are possible estimates for α and β. Then we can define the corresponding residual �ui for the ith observation as the difference between the actual and predicted value of the dependent variable

� � �u y xi i i= − −α β Since large residuals imply a failure to fit the data, it is sensible to expect good estimates �α and �β to make the residuals small in some sense.

The OLS regression estimates �α and �β are those that minimise the sum of squared residuals

( )Q u y xii

i ii

= = − −� �� 22

α β

First order conditions for minimisation require

0ˆ2ˆ

0ˆ2ˆ

==−

==−

�

�

iii

ii

xuQ

uQ

β∂∂

α∂∂

which are solved by

( )( )

�

� �

β

α β

=−

−

= −

�

�

x x y

x x x

y x

i ii

i ii

where y and x are sample means of y and x.

Note that �β is the ratio of the sample covariance of y and x to the sample variance of x. It is positive if y and x are positively correlated in the data and negative if y and x are negatively

correlated. Note also that �β is a linear estimator ie it can be expressed as a weighted sum of

the observations on yi. Given �β , �α is set so that the fitted relationship passes through the means of y and x (and the mean residual is therefore zero).

Applying this technique to data on US women, gives us the estimates in Table 2a. Here �β is 0.109 which means that each extra year of education is estimated to add about 11 per cent to the wage.

Properties of OLS estimates The properties of the OLS estimate �β depend on the properties of the error term ui. Assumption 1: The average of ui is zero across all individuals in the population with any value of xi ie E u x xi i i( | ) = 0 for all . This assumption has a number of implications. Firstly it implies that the average of ui is zero across all individuals in the population as a whole. Secondly, and crucially, it implies that ui and xi are uncorrelated. Another way of seeing the assumption is that it says that the average of yi across all individuals in the population with xi is equal to α+βxi , ie the structural component of the model, for any value of xi.

The implications for �β are very desirable.

Property 1: �β is unbiased ie it delivers the true value β on average across different samples.


Property 2: �β is consistent ie the probability of any deviation from the true value diminishes towards zero as sample size gets very large. Assumption 1 is reasonable in experimental contexts where the values of the explanatory variable are fixed by the investigator. This is almost never possible in economic applications where the reasonableness of the assumption needs careful consideration. Assumption 1 may fail to hold for any of a number of reasons: • Omitted influences on yi may be correlated with xi. If an omitted positive influence on yi

is positively correlated with xi then OLS will tend to mistakenly attribute the omitted

influence to xi and �β will tend to be an overestimate. Conversely, if an omitted positive

influence is negatively correlated with xi then �β will tend to be an underestimate. In the wage equation example, experience could be an omitted positive influence on the

wage, which is typically negatively correlated with education (because the two are substitutes). People with high education tend to have lower experience, this pulls down their wages and the correlation between wages and education is reduced. Omitting experience from the regression leads us to underestimate the return to education.

• There may be simultaneous influence of yi on xi. If this is so then since ui affects yi and yi affects xi there will be correlation between ui and xi. In the example, this is not usually considered a worry since it would require current wage to affect past education.

• There may be errors in measurement of xi. This can be shown to cause correlation between the mismeasured explanatory variable xi and the error in the mismeasured relationship ui.

If Assumption 1 fails to hold then there exist econometric techniques to deal with the problem. Inclusion of omitted variables in the estimation is one response dealt with below. Other techniques are the subject matter of more advanced courses. Much of practical applied work is concerned with recognising and dealing with problems of this sort. For the purpose of this course you need only be aware of the existence and nature of these problems and their relevance to the applications dealt with.

It is undoubtedly desirable to have an estimate �β , which is correct on average, but we are also concerned with the variability of the estimate around its average. Assumption 2: The variance of the error is the same on all observations. Assumption 3: Errors on different observations are uncorrelated.

If we add Assumptions 2 and 3, �β has even more desirable properties.

Property 3: Among all linear and unbiased estimators, �β is best or efficient ie it is the estimate with the lowest variance around the true value β. Assumption 4: Errors have a normal distribution. If we add Assumption 4, another desirable property holds.

Property 4: Among all estimators whatever, �β has the lowest variance around the true value β in large samples.

Standard errors We know that under the right assumptions �β has lowest variance of all estimates within a certain class (linear and unbiased estimates). However, for practical purposes we would like to be able to estimate the magnitude of that variance in order to judge the precision of our estimates.


If Assumptions 1 to 3 hold, then it is possible to be precise about the variance of �β . Two things influence the precision of our estimates. • The higher the variance of the errors, the less closely the data is likely to be concentrated

close to the line α+βxi and the more difficult it is to estimate β precisely. • The higher the variance of the explanatory variable xi, the greater is likely to be the range

of values in the data and the more informative the data is likely to be about β. The variability of xi is easily estimated from the values in the data. The variability of the

errors can be estimated from the variability of the residuals �ui implied by �β . The estimated standard deviation of the errors ui is called the standard error of the equation and is usually reported in regression output. This can be combined with the sample variance of xi to give an

estimate of the standard deviation of the coefficient �β known as the standard error (of the coefficient). This standard error can be interpreted as a measure of the imprecision in our

estimate �β .

Interval estimation and hypothesis testing A confidence interval is a band of values around �β based on the standard error and intended to capture the imprecision in the estimation. A 95% confidence interval is a band, which has a 95% probability of containing the true value of β. The usual rule of thumb is to construct

such an interval by taking (roughly) 2 standard errors either side of �β . If we wish to test a hypothesis that β takes a specific value then it is natural to check whether this value lies in such a confidence interval. If it does not then, given the assumptions above,

we would have been very unlikely to get an estimate such as �β if β did take that value. On this basis we would reject the hypothesis. If it does lie in the confidence interval then we are unable to reject - this could either be because the hypothesis is true or because the information in the data is weak. The most usual hypothesis of interest is β=0. In the wage equation example this is the

hypothesis that education does not affect wages. Zero lies in the confidence interval for β if �β

is less than two standard errors from zero ie the ratio of �β to its standard error is less than 2 in absolute value. This ratio is known as the t value (or t statistic or t ratio) and is a useful means for testing hypotheses of this form. If the absolute value of the t statistic is greater

than (roughly) 2 we say that �β is statistically significantly different from zero at the 5% level or, more often, simply significant.

In the example of US married women the standard error on �β is 0.104. Since the estimated

coefficient is over 7 times the standard error we would firmly reject �β =0 at any conventional significance level.

Goodness of fit It is important to assess how well the model fits the data. One standard measure is called R2, and is based on a decomposition of the variance of the dependent variable into an explained part and an unexplained part. R2 is the proportion of the variance of yi, which is explained by the estimated model and varies from 0 if there is no fit at all to 1 if the fit is perfect. A low value of R2 need not mean that the model is wrong but does imply that it fails to capture the main influences. In the example we explain about 11 per cent of the variation in wages.


Multiple regression

The OLS analysis can be extended straightforwardly to allow for more than one explanatory variable. Suppose we add an extra explanatory variable zi to the model with a coefficient γ.

yi = α + β xi + γ zi + ui The sum of squared residuals can still be minimised, fitting a line through higher dimensions. The estimated coefficients are to be interpreted as partial effects, ie effects conditional on the value of other variables in the equation. The properties of the OLS estimates as outlined above continue to hold. Standard errors, t statistics, R2 and so on bear the same interpretation. In the wage equation example, the additional variable could be, say, experience if that is also in the dataset. Experience may typically be expected to raise wages. It was argued above that

omission of a positive influence on yi would lead to an upward bias to �β if positively correlated with xi and to a downward bias if negatively correlated. Incorporating the omitted variable in the regression eliminates the bias. Thus adding experience to the regression will typically reduce the estimated return to education if experience and education are positively correlated and increase it if, as is more likely, they are negatively correlated. For our data on married women we do not have experience but we do have age. Its estimated effect proves however to be statistically insignificant at the 5% (or even 10%) level and adding it to the regression leaves the estimated effect of education almost unaffected.

Functional form issues We have already seen one functional form issue in our example when we decided to take ln hourly wage rather than simply hourly wage as the dependent variable. This was because we could then interpret β as a constant proportional effect since

β ∂∂

∂∂

= =lnw

e w

w

ei

i i

i

i

1

and this seemed more reasonable than a constant absolute effect on the wage. The assumption of a constant proportional effect can also be questioned. We might wish to allow for the possibility that the marginal rate of return decreases or increases with education. One way of doing this is to adopt a more general functional form, such as a quadratic relationship between ln wage and education

lnw e e ui i i i= + + +α β δ 2 . With this functional form

∂∂

β δlnw

eei

ii= + 2 .

The rate of return is increasing if δ>0, decreasing if δ<0 and constant if δ=0, so all cases of interest are covered. Such an equation is easily estimated by linear regression if we simply add education squared as an additional explanatory variable. The estimated coefficient on education squared is an estimate of δ and if this is significantly different from zero then we can reject the hypothesis of constant returns. Note that β is now the rate of return only if ei=0 ie it has now to be interpreted as the return to early years of education. Supposing δ<0, then a useful way to judge the extent to which returns diminish is to find the number of years of education at which the marginal return falls to zero, which is easily shown to be -β/2δ.


In the example we can include age and age squared in the regression. Neither turn out to be significant considered in isolation. However the estimated shape of the relationship is concave and ln wage reaches a peak, given education at 0.054/2×0.533=50.7 years.


Table 2a: Wage Equation Married women, US

Number of obs = 428 R2=0.118

Dependent variable: ln wage

St error=0.680

Coefficient Standard error t value Years of education 0.109 0.014 7.545

Constant -0.185 0.185 -1.000

Table 2b: Wage Equation Married women, US



St error=0.679


Age/10 0.068 0.043 1.606 Constant -0.488 0.264 -1.848

Table 2c: Wage Equation Married women, US



St error=0.679


Age/10 0.533 0.467 1.142 Age squared/100 -0.054 0.054 -1.000

Constant -0.185 0.185 -1.451


lnw

educ5 10 15 20

-2

0

2

4

lnw

age30 40 50 60

-2

0

2

4


Consumer Demand Consumer theory

A consumer's demand for goods depends on their budget constraint and on their preferences. The position of the budget constraint is determined by their income or total budget and by the prices of the goods. Their preferences are affected by many factors including the age and demographic composition of the household. These are the variables typically used as explanatory variables in studies of demand.

Total budget Consumer theory does not constrain the direction of the income effect for any one good. A good for which demand increases as total budget rises is called a normal good. If demand falls as total budget rises then the good is called inferior - this might happen because better quality substitutes for the good or more variety in expenditure become affordable at higher incomes. Normal goods can be divided into those for which budget share rises and those for which it falls as total budget rises. Budget share rises with total budget if the total budget elasticity for the good exceeds one and such a good is called a luxury. Budget share falls with total budget if the total budget elasticity for the good is below one and such a good is called a necessity. If consumers are on their budget constraints then spending on all goods must add up to the total budget - this requirement is known as adding up. It implies that there must be some goods which are normal - not all goods can be inferior (and indeed none may be). It also implies that it cannot be the case that all goods are luxuries or that all are necessities - we may expect to find some goods in each category. Income elasticities are interesting if we wish • to predict changes in demand as incomes rise or fall • to analyse the distributional incidence of government policies affecting the good, such as

taxes

Own price A rise in a good's own price has two effects. Firstly it makes the good more expensive relative to other goods - the effect of this is called the substitution effect and, under usual assumptions, must decrease demand. Secondly, it reduces the consumer's total spending power for a fixed total budget - the effect of this is called the income effect and its direction depends upon whether the good is normal or inferior. The magnitude of the income effect depends upon the good's share in the total budget. If the good is normal then the income effect is negative and the total effect is therefore also negative. This is an important testable assumption of consumer theory - normal goods must have negative own price effects. If the good is inferior then the income effect is positive and it is theoretically possible that the total effect also be positive - this is the case of a so-called Giffen good. Such a possibility would require the good to take a very big share of the total budget and can be practically discounted in usual empirical contexts. (The requirement that the substitution effect be negative still, of course, holds for these goods). All things considered, it would be cause for considerable concern if empirical estimation suggested a positive own price effect. Goods can be divided into those for which total spending (and therefore budget share) rises and those for which it falls as price rises. Total spending rises with price if the own price elasticity for the good is less than one in magnitude (ie less negative than -1) and such a good


is called price inelastic. Total spending falls with price if the own price elasticity for the good is more than one in magnitude (ie more negative than -1) and such a good is called price elastic. Price elasticities are interesting if we wish • to predict the effect of price changes on demand • to predict the effect of changes in taxes on the good • to assess efficiency costs of tax distortions

Other prices The effects of prices of other goods - cross price effects - can be in any direction. If an increase in the price of another good raises demand then the other good is regarded as a substitute. If an increase in the price of another good decreases demand then the other good is regarded as a complement. If demands for several goods are being estimated then the cross price effects should be consistent between equations. One should not, for instance, find two goods to be substitutes when estimating demand for one of them but complements when estimating demand for the other. These sort of requirements are referred to as symmetry restrictions. (Strictly speaking, these restrictions apply to substitution effects ie effects of price changes net of income effects).

Demographic variables An increase in household size increases a household's needs and, for a given total budget, leaves the household's economic standard of living lower. It may therefore have an effect similar to a fall in income. If the new household members are children, this will also affect the composition of spending, causing greater consumption of children's goods, for instance. Effects of other demographic variables such as age are theoretically unclear but it is important to allow for these sorts of effects because age, for instance, is strongly correlated with economic variables like income.

Estimation The richest sources of data on household budgets, income and demographic variables are household budget surveys. Data on regional or even household-level variation in prices are sometimes available in such surveys but often not. Price effects often need to be estimated from several years of data, merging in information on prices from other sources. Our empirical example uses cross sectional data on the demand for eggs in the Czech Republic. [The data can be found in a file called czegg.dta] Estimation requires a choice for the functional form linking quantities demanded to the explanatory variables. In choosing the functional form we want to maintain ease of interpretation, to keep flexibility in describing the data and to allow restrictions suggested by consumer theory to be tested. We consider two examples.

Double Log Demand Equation A double log (or loglinear or constant elasticity) demand equation has the form


ln ln lnq y p Z uij i k ikk i i= + + + +�α β γ δ

where qij is the quantity of the jth good demanded by the ith household, xi is total expenditure on all goods by the ith household, pik is the price of the kth good paid by the ith household, Zi denotes the demographic characteristics of the ith household and ui is a disturbance term. The interpretative advantage of such a specification is that coefficients on the ln total budget and ln price terms are elasticities - β is the total budget elasticity, γj is the own price elasticity and γk is the cross price elasticity with the kth good if k≠j.

y

q

q

y

q

y

p

q

q

p

q

p

i

ij

ij

i

ij

i

ik

ij

ij

ik

ij

ikk

∂∂

∂∂

β

∂∂

∂∂

γ

= =

= =

ln

ln

ln

ln

.

Since these are constant values this specification necessarily implies constant elasticities. The good is inferior if β<0, a normal good and a necessity if 0<β<1 and a normal good and a luxury if 1<β. The good is a Giffen good if 0<γj, price inelastic if -1<γj<0 and price elastic if γj<-1. The good is a substitute for the kth good if 0<γk and a complement for the kth good if γk<0 where k≠j. Estimates using the Czech eggs data are presented in Table 3a. The estimated total budget elasticity is 0.102 which indicates that eggs are a normnal good and a necessity. It is possible to reject the good being a luxury but not being inferior since a 5% confidence interval for the elasticity goes from -0.058 to 0.262. A strong negative own price effect indicates that eggs are price elastic and the standard error is small enough to allow one to reject a price elasticity of -1 at the 5% level. A rise in the bread price raises demand for eggs suggesting that the two are substitutes and the estimated cross-price elasticity is statistically significant. Estimates of a double log specification for the demand for bread also point towards bread and eggs being substitutes although the cross price term in this equation is poorly determined. Symmetry can not therefore be rejected. There are several strong demographic influences and it is important to include these in the estimation since the demographic variables are strongly associated with economic variables such as income. To motivate an example of an omitted variables argument that might be made in this example, note that the price variables used here are household specific and not corrected for differences in quality of the goods purchased. If eggs differ in quality and better quality eggs are more expensive then one has to worry whether prices might be correlated with the error term ui on the equation because, say, both are correlated with taste for good eggs. For example, households with a liking for eggs may spend a lot and pay high prices because they choose to buy expensive high quality eggs. If so this would mean the estimated price elasticity was biased towards zero since this would be misinterpreted as evidence that high eggs prices do not depress demand.


The double log specification has several disadvantages. • It cannot be used for households purchasing none of the good since the dependent variable

would be undefined. (It may be inappropriate to model such households similarly to other households under any specification but that is a different issue).

• Constancy of elasticities may be felt to be an unacceptable restriction. • It is not possible for all goods to have double log demands since such demands cannot add

up to the total budget.

Budget share specification An alternative specification with none of these particular problems is one that relates the budget share linearly to the same explanatory variables. Such a specification has the form

wp q

yy p Z uij

ij ij

ii k ikk i i= = + + + +�α β γ δln ln

where wij is the budget share for the jth good. Given the difference in functional form, the interpretation of the coefficients is different. For instance, the total budget elasticity is no longer β but

y

q

q

y

q

y wi

ij

ij

i

ij

i ij

∂∂

∂∂

β= = +ln

ln1 .

The elasticity is derived as follows:

p q y w

pq

yw y

w

yw

y

q

q

y wp

q

y w

ij ij i ij

ijij

iij i

ij

iij

i

ij

ij

i ijij

ij

i ij

=

� = + = +

� = = +

∂∂

∂∂

β

∂∂

∂∂

β11

The budget share rises as total budget rises and the good is therefore a luxury if β>0. Notice from the formula for the elasticity that this guarantees an elasticity greater than one. The good is a necessity if β<0. Note carefully how these rules differ from those under the double log specification. Table 4 reports estimates of such a specification for eggs. The coefficient on ln total spending is negative indicating eggs to be a necessity. The average budget share for eggs is 0.00607 in this data and for a household with this budget share the total budget elasticity is 1-0.00545/0.00607=0.102. This is actually exactly the total budget elasticity found with the double log specification.


The coefficients on the ln price terms are not to be interpreted as elasticities either. However there are still simple rules of interpretation. The budget share rises with own price and the good is therefore price inelastic if γj>0. It is price elastic if γj<0. The good is a substitute for the kth good if 0<γk and a complement for the kth good if γk<0 where k≠j. As found previously, eggs appear to be price elastic and substitutes for bread in the Czech Republic.


Table 3a: The demand for eggs: double log specification Czech Republic 1990


Dependent variable: ln quantity eggs

St error=1.070

Coefficient t value ln expenditure 0.102 1.246

ln price eggs -1.848 -10.875 ln price bread 0.623 2.651

age 0.009 3.017 no of children -0.0001 -0.263

no of adults 0.003 3.449 Constant -0.302 -0.243

Table 3b: The demand for bread: double log specification Czech Republic 1990


Dependent variable: ln quantity bread

St error=0.463

Coefficient t value ln expenditure 0.164 4.645

ln price eggs 0.014 0.185 ln price bread -0.938 -9.222

age 0.008 6.601 no of children 0.002 13.943

no of adults 0.005 13.297 Constant 0.856 1.589


Table 4: The demand for eggs: budget share specification Czech Republic 1990


Dependent variable: eggs share

St error=0.0043

Coefficient t value ln expenditure -.00545 -16.612

ln price eggs -.00093 -1.366 ln price bread .00073 0.772

age .00007 6.536 no of children 0.000004 2.332

no of adults .00002 5.145 Constant .04574 9.140


ln

egg

s

ln exp7 8 9 10

4

6

8

10

eggs

sh

ln exp7 8 9 10

0

.005

.01

.015

.02


Labour Supply

Labour supply theory The economic model of labour supply applies the ideas of consumer theory to a two good choice problem where the two goods are consumption and non-labour time (leisure). The budget constraint depends on the individual's income from other sources m and the real wage rate w. Consumption c is equal to total income c=m+wh where h denotes hours worked. Non-labour time l is the difference between total available time T and hours worked ie l=T-h. Thus the budget constraint can be written c+wl=m+wT. Chosen hours of work will depend on the determinants of the budget constraint and on any influences on preferences between consumption and non-labour time, such as children, health, age and so on.

Income from other sources The role of income from other sources is straightforward. If non-labour time is a normal good then increases in income should decrease hours worked.

Real wage rate An increase in the real wage rate, like a change in any price, has two effects - a substitution effect and an income effect. Firstly it makes non-labour time more expensive relative to consumption - under usual assumptions, this substitution effect must decrease demand for leisure ie encourage hours of work. Secondly, it raises the value of the individual's hours of work sold - the income effect of this is called the income effect and should discourage hours of work if non-labour time is a normal good and hours of work are positive. Note that the two effects are opposing and the total effect is therefore theoretically ambiguous. The ambiguity arises because of the nature of the income effect when the individual is a net seller of the good concerned – in this case, of their own time. The direction of the total effect depends upon whether the income or substitution effect dominates. It may be reasonable for different effects to dominate for different people – for instance, those on high and low wages. This is the explanation for the possibility of a backward bending labour supply curve ie one which slopes up at low wages and down at high wages - this would happen if the substitution effect dominated at low wages but the income effect dominated at high wages. (Forward bending labour supply curves are also possible). A formula known as the Slutsky equation allows income and substitution effects to be separated:

dm

dhh

dw

dh

dw

dh

um

+=

Here mdw

dh is the total effect (ie the effect at fixed income),

udw

dh is the substitution effect (ie

the effect at fixed utility) and hdh

dm is the income effect. The intuitive rationale for calculating

the income effect in this way is that the effect of a wage change on income depends on the

number of hours sold h and the effect of that on labour supply is given by dh

dm. Note that this


gives us a means to calculate the substitution effect dm

dhh

dw

dh

dw

dh

mu

−= since all terms on

the right hand side of this expression are calculable. This is important since it allows us to check for the negativity of the substitution effect as required by theory.

Demographic influences

The presence of children increases household needs which should encourage longer hours of work but may also increase the value of non-labour time discouraging hours of work.

Estimation

Quadratic labour supply We consider a single commonly estimated specification - a quadratic labour supply function

h w w m Z ui i i i i i= + + + + +α β γ δ φ2 where hi is hours of work of the ith household, wi is the real wage rate of the ith household, mi is the income of the ith household from other sources, Zi denotes other observed influences and ui is a disturbance term. Our example in this instance comes from a published paper and appears in Table 5.

For such a specification i

mi

i wdw

dh γβ 2+= . Thus the labour supply curve slopes upward at

low wages if β>0 and is backward bending if γ<0. In the example of Table 5 this is indeed so and both wage terms are stroingly significant. Labour supply is rising in the wage for wi<$1.14 and decreasing in the wage thereafter.

The substitution effect of a wage change is readily calculated as ii

ui

i hwdw

dh δγβ −+= 2 .

And this calculation is demonstrated for someone on the mean wage in the panel below the Table. Note that the substitution effect is positive, in accordance with theory, even though the total effect is negative. Empirical work on labour supply is complicated by a number of factors including nonworking households and the effect of taxation and social security.

Non working households Hours worked cannot fall below zero. Functional forms which may well describe the behaviour of working individuals well can predict that individuals with low wages and high unearned incomes would want to work negative hours. This needs to be accommodated somehow in estimation. A second problem arises because wages will typically not be observed for nonworking households.

• One response could be to try and predict their wages using other information (such as education).


• Another response might be to drop non-working households from the estimation. This though is usually a bad idea since the remaining sample could be very unrepresentative with regard to the factors not in the model, ui. In particular, it would lead to underrepresention of individuals with a high taste for leisure but not only this – it would do so most sharply for those with low observed incentives to work. If we

suppose for simplicity that 0>mi

i

dw

dh then it could be that most high waged people

would work but the only low waged people who would work and therefore end up in the sample would be those with high ui so our selected sample would be one in which we had induced a correlation between wi and ui.

Tax and social security Taxes reduce the effective after tax real wage ie the return to hours of work. Withdrawal of social security benefits as earned income rises will have a similar effect. This is easily handled if marginal tax rates do not vary with hours of work. However actual tax and social security systems are highly nonlinear with multiple rates of tax and complex social security rules.


Table 5: Labour supply Poor urban white males, US

Dependent variable: Annual hours of work

Coefficient t value Wage ($ per hour) 494.71 4.18 Wage2 -217.17 5.83 Other income ($ per year) -0.134 8.32 Children 94.65 2.84 Own health 295.65 4.23 Spouse's health -150.73 2.61 Constant 1172.64

[Source: C. R. Hill "The determinants of labour supply for the working urban poor" in: Cain and Watts, 1973, Income Maintenance and Labour Supply]

Calculating the substitution effect of a wage change Consider, for example, someone with average wage, w=$1.40, no children, in good health and with a spouse in good health. Hours of work are:

h = 1172.64 + 494.71×1.4 - 217.17×1.42 + 295.65 - 150.73 = 1584.5

dh/dm = -0.134

dh/dw|m = 494.71 - 2×217.17 w

= 494.71 - 2×217.17×1.4 = -113.36

Thus the substitution effect is

dh/dw|m - h dh/dm = -113.36 - 1584.5 × (-0.134) = 98.96


Consumption and Saving

Life cycle model The main points in the life cycle model of saving decisions can be understood in a simple two period context. Suppose individuals are choosing how to divide consumption between two periods, earning incomes of y0 in the first (current) period, y1 in the second (future) period and starting with assets of A (which would be negative if carrying debts). Consumptions are c0 in the current period and c1 in the future. If the (real) interest rate between the two periods is r then the budget constraint is 1010 )1)(()1( yrAycrc +++=++ . Consumption decisions will

be influenced by the determinants of the budget constraint – income, assets and the interest rate - and any variables affecting intertemporal preferences, such as impatience, anticipation of demographic change and so on.

Incomes Both current income y0 and future income y1 affect current consumption through lifetime resources. An increase in income which is expected to endure (ie which affects y0 and y1) will have greater effects than one which is not (ie which affects y0 only).

Real interest rate Changes in the real interest rate have both income and substitution effects which need to be considered. The substitution effect of an interest rate rise inhibits current consumption in favour of future consumption. The income effect of an interest rate rise depends upon whether the consumer is a saver or borrower - a saver gains from an interest rate rise whereas a borrower loses out.

Permanent income model Suppose individuals divide total income yt into uncorrelated permanent and transitory components, yt

P and ytT so that y y yt t

PtT= + . Consumption is related to the permanent part

only:

Ttt

Ptt

yya

yac

κκκ

−+=

+=

A simple regression of consumption ct on current income yt will underestimate the marginal propensity to consume out of permanent income κ because the transitory part of income is uncorrelated with consumption. In the usual regression terminology, the error term here is

Ttt yu κ−= which is clearly correlated (negatively) with the regressor yt .

Suppose assessments of permanent income are updated from period to period according to a rule t

Ptt

Pt yyy νλλ +−+= −1)1( where νt is a term capturing news not in current income. The

coefficient λ reflects the proportion of income changes, which are permanent. Then


ttt

tPtt

Ptt

cya

yya

yac

κνλκλλκνλκκλ

κ

+−++=

+−++=

+=

−

−

1

1

)1(

)1(

If consumption is regressed on previous period's consumption and current income then the coefficient on lagged consumption indicates λ and this and the coefficient on income can be used to infer κ . Our empirical application uses UK data for the period 1974-1997 on average consumption and income of households grouped into cohorts by ten year bands for date of birth of head of household. Table 6 reports a regression of consumption on current income and on current income and lagged consumption. From the final column we derive an estimate of 1-

� from the coefficient on lagged

consumption. Since 1-�=0.890 we infer an estimate

� of 0.110 suggesting a low proportion of

income changes are permanent. Turning to the coefficient on income of 0.099 we infer an estimate for of 0.900=0.099/0.110. This is considerably higher than biased estimate of the marginal propensity to consume of 0.517 which would be drawn from the first column.

Interest rates If consumers are saving optimally then the marginal rate of substitution between current and future consumption equals the slope of the budget constraint ie

MU

MUr0

1

1= +

where MU 0 and MU1 are marginal utilities today and in the future. This defines a relationship between consumption in the two periods and the real interest rate r .

Suppose period-specific utilities are β

βδ ttt cU1

)1(

1

+= . (Here is a subjective discount rate

capturing impatience to consume). Then 1

)1(

1 −

+= β

δ ttt cMU and therefore the equality

between the MRS and the slope of the budget constraint gives

δβ

β

β

++=+= −

−

−

1

1)1( 1

11

10 r

gc

c

where gc

c= −1

0

1 is the growth rate of consumption. The growth rate of consumption is high

when the interest rate is high because of the incentive this gives to delay consumption.

This relationship is well approximated by )(1

1ln δ

β−

−≈∆≈ rcg t and suggests regressing

the growth rate of consumption on the real interest rate. The coefficient β−1

1 reflects the

willingness of consumers to substitute consumption in one period for that in another and is


referred to as the intertemporal elasticity of substitution. The intercept β

δ−

−1

reflects

consumers’ patience. Using the UK cohort data we see from Table 7 that the real interest rate is indeed strongly associated with the growth of consumption. Such simple models might be extended to allow for the effect of household demographic change on consumption decisions and for the role of borrowing difficulties in limiting intertemporal substitution.


Table 6: Consumption function UK cohort data 1974-1997

Dependent variable: mean consumption

Coefficient St error Coefficient St error mean income 0.517 0.058 0.099 0.025

lagged mean consumption 0.890 0.034 Constant 19.564 1.857 0.902 1.026

Number of obs 120 115

R2 0.479 0.925 St error 4.310 1.654

Table 7: Consumption growth

UK cohort data 1974-1997 Dependent variable: � ln mean consumption

Coefficient St error

real interest rate 0.386 0.068 Constant -0.005 0.004

Number of obs 115

R2 0.224 St error 0.036


Investment The final topic is concerned with behaviour of firms. Investment I t includes replacement investment RI t , which replaces capital stock Kt lost through depreciation, and net investment NI t , which adjusts the capital stock towards a new desired level Kt

* . In a simple model the depreciation in need of replacement is a constant proportion δ of the previous period's capital stock and net investment is a constant proportion λ of the difference between desired and actual stock so that

RI K

NI K K

t t

t t t

=

= −−

−

δλ

1

1( )*.

Thus I RI NI

K K

t t t

t t

= +

= + − −λ δ λ* ( ) 1

.

Expressing investment as a proportion of existing capital gives another form which can be useful

δλλδλ

+−≈

+−=

−

−−

1*

1*

1

lnln

)1/(/

tt

tttt

KK

KKKI

where the approximation should be good so long as 1

* / −tt KK is close to 1.

To derive an expression for desired capital stock we can consider input choice as the outcome of profit maximisation decisions given technology. For instance, suppose that technology is such that output ]/,/min[ γβ ttt LKY = where Lt is

labour used. This is a model in which there is no possibility to substitute between capital and labour in production and the two are used in fixed proportions. Then desired capital stock is proportional to output, ie tt YK β=* . Note that, in this case, cost of capital ρt does not affect

desired capital stock given output. Investment is given by

1)( −−+= ttt KYI λδλβ

or

11 lnln)ln(/ −− −++≈ tttt KYKI λλδβλ

Alternatively, suppose that technology is Cobb-Douglas so that αα −= 1

ttt LAKY . This

is a model which allows much greater scope for capital-labour substitution. Firms choose capital stock to maximise profit

ttttttt KLwYp ρ−−=Π

where pt is output price and wt is wage. At the profit-maximising choice

0=−=Π

tt

tt

t

t

dK

dYp

dK

d ρ


which implicitly defines an optimum stock of capital. Since, in this case,

t

ttt

t

t

K

YLAK

dK

dY αα αα == −− 11 this equation implies t

ttt

YpK

ρα

=* . Desired capital stock can be

seen to depend on expected output and on the cost of capital which will both therefore affect investment.

)/ln(lnln)ln(

)ln()ln()ln(

)/ln(/

1

1

11

tttt

tttt

tttttt

pKY

KYp

KYpKI

ρλλλδαλρλλδαλ

ραλδ

−−++=−++=

+≈

−

−

−−

Note that there are several ways of writing the same model. The last line differs from the earlier model of investment in the presence of the term involving )/ln( tt pρλ and the two

models are therefore empirically distinguishable throught the effect played by the real cost of capital. We apply these ideas empirically to a sample of UK textile firms observed over the 1990s. Table 8 shows that estimated effects of the logarithm of current sales and lagged capital stock are both strongly significant, of the right sign and close to each other in magnitude. Adding the real interest rate to the regression does not affect these estimates and its estimated effect is not significant.

Notice that firms should invest if the earnings of capital at the margin t

tt dK

dYp exceed the

rental cost of capital tρ . For the technologies chosen here, and for others, this is so if

average earnings of capital exceed the average rental value of capital. But future average earnings of capital should be reflected in the market worth of the firm and the firm should therefore invest if the ratio of its market value to the replacement value of its capital exceeds one. This ratio is known as Q and this leads to an alternative approach explaining investment by Q. To develop this idea for the Cobb Douglas case above note that optimal choice of labour is

t

ttt w

YpL

)1(* α−= so that earnings on capital are tttttt YpLwYp α=− . Thus the ratio 1

* / −tt KK

is equal to the ratio of earnings on capital to the rental value of current capital stock. In a stable environment this should equal the ratio of the market worth of the firm (as measured by the market vale of its equity and debt) to the replacement value of its capital. This ratio is known as Q. Table 9 gives an example of an empirical application to the same data showing that Q is indeed strongly associated with investment intensity.


Table 8: Investment UK textile firms 1990-2000

Dependent variable: investment to lagged capital ratio

Coefficient St error Coefficient St error ln sales 0.073 0.018 0.073 0.018

ln lagged capital stock -0.055 0.017 -0.055 0.017 ln real interest rate 0.045 0.037

Constant -0.146 0.088 -0.194 0.096

Number of obs 162 162 R2 0.098 0.106

St error 1.095 0.095

Table 9: Q model of investment

UK textile firms 1990-2000 Dependent variable: investment to lagged capital ratio

Coefficient St error

ln Q 0.039 0.007 Constant 0.095 0.007

Number of obs 162

R2 0.138 St error 0.092

ian preston, university college london econ b003 lecture notes …uctp100/b003/b003lect.pdf · econ...

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