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Diplomarbeit, 2003
56 Seiten, Note: 2.3 (B)
1. Introduction
2. The data and some key facts and gures
2.1. Data
2.2. Key economic indicators of the U.S. economy
3. Estimating the households' demand system
3.1. Deriving the estimation equation
3.2. The estimation of the households' preferences using the method of ordinary least squares
3.2.1. A general view on linear models
3.2.2. The OLS-estimation
4. Discrepancies in demand and exact de ators
4.1. Di erences between actual and notional demand
4.2. An outline of index number theory
4.3. Exact de ators
5. Conclusion
A. Personal Consumption Expenditures by type of expendi- ture
B. Chained-type price indices
C. Disposable personal income
D. U.S. population
E. The categories of expenditure
F. Analysis of residuals
G. Observed and estimated demands
2.1. The 12 categories of expenditure
3.1. Estimated coe cients (estimated standard errors in paran- theses)
3.2. Estimated and observed budget shares (1945)
4.1. Consumer Prices (P, 1996=100), Quantities (Q) and Expenditures (X), observed and estimated values of 1945
4.2. Estimated De ators referred to 1946
2.1. Growth rate of annual per capita GDP, 1996 chained Dollars
2.2. Growth rate of CPI, 1996 chained Dollars
2.3. Disposable personal income (1996 chained USD)
2.4. Disposable personal income, annual growth rate
2.5. Mid-year population (thousands)
During World War II, the private customers in the United States were confronted with conditions of rationing concerning several consumer goods. The government regulated the allocation of those goods, for example oil, leather, and grain, particularly with regard to their tness to ght. The U.S. army was preferred to receive such goods instead of giving them to the citizens. Therefore, people had to optimize their consumption decisions with respect to both their budget constraint and rationing of goods.
Under the assumption that the households' preferences didn't change during World War II, one major trouble exists from the point of view of index number theory. Because of rationing, the prices of such goods usually change in contrast to a market economy, where the allocation of goods is presumed to be e cient, given prices which cannot be a ected by the households consumption. Therefore the actual consumption of any given customer under conditions of quantity constraints need not be optimum with respect to his money outlay and the existing price system. In spite of rationing, the households optimize their utility from consumption given their money expenditures and the price sys- tem. Based on this relative strong assumption, one has to estimate a price system, which helps to measure changes in real income under such conditions of rationing. The central idea to solve that problem was in- troduced by Erwin Rothbarth (1941), who assumed that the quantities consumed under rationing were the same quantities consumed by house- holds in an open market economy to receive their optimum. Rothbarth called such an estimated price system a virtual price system , which allowed him to work with the common index number theory. Such a virtual price system makes the quantities consumed under rationing a consumption optimum, i.e. the maximum level of utility, of the private customers. Rothbarth's method will be used in this article to estimate de ators, which will help to unlock changes in real income conditional on quantity constraints.
First, I will give an overview of the U.S. economy in chapter 2. After having done that, I will present several aspects concerning the calcula- tion of a Rothbarian virtual price system . Thereby, it will be impor- tant to build an opinion about the preferences of the private households, which will be expressed as a generalized Cobb-Douglas demand system. Based upon that demand system an estimation equation for the house- holds' preferences will be derived in section 3, followed by an ordinary least squares regression of the preference parameters.
The second important step will be a presentation of the basic infor- mation we can win from index number theory. It will be especially interesting to say a few words about economic index number theory (see chapter 4.2).
The combination of the outcomes from estimation with index number theory makes up the most important step to unlock hidden di erences between wartime-prices and postwar prices, i.e. I will compare wartime demand to postwar demand and the impact of rationing on changes in income. This procedure will be done by calculating de ators using both the estimated demand behavior and virtual prices.
In every section, I will present both the basic theoretical aspects, which will help the reader understand the steps to de ators, and applied theory, i.e. estimations using empirical data in most cases.
In this rst chapter some key economic facts based on the data I used are presented. This might help the reader get a feeling for the economic background in the last century in the United States of America.
The data I use in this article are available at the Bureau of Economic Analysis (BEA)-website (cf.). The BEA delivers, among other things, time series of di erent price indices, Gross Domestic Product (GDP), and personal consumption expenditures. Of course not all of the data is necessary in this article, therefore it follows a list of used time series, which are annual data from 1929 to 2001:^{1}
- Personal Consumption Expenditures by Type of Expenditure (cf.[2], shortform cf. appendix A)
- Chain-Type Quantity and Price indices for Personal Consump- tion Expenditures by Type of Expenditure (cf. [3], shortform cf. appendix B)
- Selected Per Capita Product and Income Series in Current and Chained Dollars (cf. [4], shortform cf. app. C and appendix D)
All other relevant numbers are calculated by using entries in these tables, especially the variables
- Per Capita Personal Consumption Expenditures by Type of Ex- penditure, and
- Per Capita Real Expenditure (1996 prices).
The first one is found by dividing the entries of tableA by those of table D; the second series is calculated from the nominal values of table A divided by 100, then multiplied by the entries of table B to be divided by table D to obtain per capita values. This calculation of per capita variables is required, because both microeconomic demand theory and index number theory assume single households and not an aggregated point of view.
The data are ordered by N = 12 categories of consumption, which are shown in table 2.1. Every single category is divided into several sub- categories which are listed in appendix E for interested readers. The households' consumption expenditures are assigned to these categories to point out an as exact as possible view of the demand behavior of the private sector.
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Table 2.1.: The 12 categories of expenditure
We can extract some key indicators from the data to describe the development of U.S. economy in the past decades. The most important are shown in gures 2.1 to 2.4.
Figure 2.1 shows the annual growth rates of per capita Gross Do- mestic Product (GDP) during the timeframe 1929 to 2001. Especially in the middle of the 1930s until the beginning of Second World War in 1939, GDP grew with annual rates of about 10%. This must be seen in respect to the 1929-Wall-Street-crash and the following worldwide economic crisis ( Great Depression ), which determined a large world- wide unemployment, crisis in the banking sector, high de ation in most industrial countries all over the world, etc. From that recession (with annual shrinking rates of GDP of about 10% to 14%), the U.S. econ- omy came back on track to prosperity pretty rapidly - till the Japanese
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Figure 2.1.: Growth rate of annual per capita GDP, 1996 chained Dol- lars
attack at Pearl Harbor in 1941 and the resulting United States' o cial entry into the war. Beginning in the late 1940s, we can identify sev- eral peaks, and long-run limits of variation of about − 3% and +7%. The average long-run annual growth rate amounts to approximately 2%. In absolute terms the per capita GDP nearly quintupled starting by approximately 6,700 USD in 1929 and grewing up to approximately 32,500 USD in 2001.^{2}
Another important economic indicator is illustrated in gure2.2: the annual growth rate of the (aggregated) consumer price index (CPI), also known as in ation rate. The BEA publishes the chained-type price in- dex3 of personal consumption expenditures (cf.[3]), which has been used to plot this gure.
Abbildung in dieser Leseprobe nicht enthalten
Figure 2.2.: Growth rate of CPI, 1996 chained Dollars
We can identify negative growth rates of the CPI especially in the early 1930s. This de ation of course is a result of the wordwide eco- nomic crises mentioned above. During World War II high in ation took place with growth rates of up to 12% in 1942, followed by decreasing rates until 1973, when the rst shock of a rising oil price let the prices increase faster than in the decades before.4 The long-run annual U.S. in ation rate averages about 3%.
The trend and the annual growth rate of disposable personal income are presented in the following gures 2.3 and 2.4.
The disposable personal income rose in a nearly linear trend from about 5,000 USD in 1930 up to 24,000 USD in 2001 (measured in 1996 chained dollars), which was conducted by annual growth rates in a range between -15% and +14%. These extremely volatile growth rates were observed especially in the rst half of the twentieth century. Be- ginning in the 1950s, the annual growth rate stabilized at around 2%
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Figure 2.3.: Disposable personal income (1996 chained USD)
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Figure 2.4.: Disposable personal income, annual growth rate
until the present. Also in this plot the relative high negative growth rates in the early 1930s show the e ects of the Great Depression .
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Figure 2.5.: Mid-year population (thousands)
Finally, gure 2.5 shows the development of the U.S. population. The economic process described earlier was achieved by a mid-year population of about 120,000,000 in 1929 and 280,000,000 in 2001, respectively, which was more than doubled within 73 years with averaged annual growth rates of about 1%.
After giving an overview of the development of the U.S. economy, I will discuss the households' preferences in the next chapter.
In this chapter the description and estimation of the households' de- mand system is presented to illustrate the preferences of the private customers.
A formal speci cation of a Cobb-Douglas demand system is as follows. In a market economy, the utility that a household can achieve is usually written as
Abbildung in dieser Leseprobe nicht enthalten
The utility is expressed as the product of consumed quantities, weighted by preference parameters ai. Maximizing this utility subject to the households' bugdet constraint ^{1} [Abbildung in dieser Leseprobe nicht enthalten]leads to Marshallian de- mand functions qi (p , y). If we replace these demands in the Cobb- Douglas utility function (3.1), the resulting expression is called indi- rect utility , which depends on a vector of prices of goods p t and the household's budget constraint yt at any point of time, t = 1 , . . . , T. From the available data, the series Disposable personal income (1996 chained Dollars) is equivalent to the (exogenous) budget constraint.
Then the indirect utility function can be written in a formal general expression:
Abbildung in dieser Leseprobe nicht enthalten
The quotient yt/pt n represents the quantities qi ofgoodsandservices consumed in any period as mentioned in equation (^{3}.^{1} ). In empirical works, researchers are often confronted with the problem that those quantities cannot be observed directly, but there are many income se- ries and price data measured, so that these are used to calculate im- plicit quantities. The budget shares β t n for all n =[1] ,..., [12]categories of expenditure must of course sum to unity in any period t, that is [Abbildung in dieser Leseprobe nicht enthalten] 1. Using this and the exogenous disposable income, the indirect utility function (3.2) should be transformed into
Abbildung in dieser Leseprobe nicht enthalten
This relation will be used to calculate the households' indirect utility ^{3}
in section 3.2.2.
Engel's law ^{4} says, that the share of expenditure for a good (espe- cially for food) declines with rising income. In other words: poorer families will spend a larger share of their total expenditures on food than wealthier families will. We have to let the budget shares β n vary systematically with indirect utility itselft to achieve consistency with Engel's law and obtain an equation for the budget shares:^{5}
Abbildung in dieser Leseprobe nicht enthalten
In this equation, the α n are percentages of goods based on disposable
income, and would be proportional to the budget shares β n, if γ n would be equal to γ m. The parameters γ j , j = m, n are the responses of β j to a change in real income (which itself is hidden in indirect utility). An interpretation of those parameters γ j in equation (3.4) directs to the concept of income elasticity considering the fact that disposable income is part of indirect utility: if income raises by 1% the demand for consumer good n will raise by γ n %.
Now we can transform equation (3.4) to eliminate the denominator, so as reference category I arbitrarily choose food (index f), which is the rst mentioned category in table 2.1 on page 10,6 and obtain
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In this step we have changed the point of view from a single budget share to the relative budget share of good n to a reference f.
To obtain a more useful expression, log-form of (3.5) provides an estimation equation
Abbildung in dieser Leseprobe nicht enthalten
By using natural logarithm of indirect utility, the estimated coe - cients γ n can be interpreted as income elasticities again.
Later on in the estimation procedure one will has to substitute the ln ν -term in this equation, which is the result of an equivalent trans- formation (using ln-operation) of the indirect utility expressed in equa- tion (3.3), and is set by the observed values of income, observed budget shares, and observed prices.
In the previous section I derived the estimation equation, which will be used to estimate a demand system. We can simply use the ordinary least squares regression method, which will be rst modelled in general, followed by a second step in which I will present the detailed regression results.
Linear models are often used to solve economic questions. The idea of linear regression models is to examine by which factors (independent variables) anotherone is determined (dependent variable) in a linear way. Such a linear regression equation is usually written as
Abbildung in dieser Leseprobe nicht enthalten
We can see that one economic factor yn is determined by a vector of independent variables xi, i = 1 , . . . , K. The coe cients β 1 , . . . , β K 8
tell us the way in which the dependent variable is determined by independent variables xni:
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Unfortunately, estimations are not exact, but it remains a vector
of errors un, so we have to make some assumptions concerning these residuals:
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In words these assumptions mainly mean that the expected value of all residuals is equal to zero (3.7), the variance of all residuals is constant (3.8), and there is no autocorrelation between two residuals (3.10). The residuals follow a normal distribution (3.9).
To estimate the impact of the independent variables to a dependent one we have to minimize the sum of residuals squared subject to β:
Abbildung in dieser Leseprobe nicht enthalten
The resulting estimators β n are BLUE , i.e. best linear unbiased estimators.
Further on in our analysis we can use di erent statistical test proce- dures to examine the statistical evidence of the estimated parameters and the respective assumptions. To evaluate the estimated parameters β n, we use the simple Student's t-test, which indicates whether an esti- mated coe cient is equal to zero or not. Another usually declared size is the stability index R, which tells us how much of the whole variance is determined by the estimated model. To form an opinion about the evidence of the assumptions 3.7 to 3.10, a popular test procedure is the so-called Jarque-Bera test for goodness-of- t to a normal distribution (in residuals). Well-known test procedures concerning heteroskedastic- ity are either White's Test or ARCH-LM. Autocorrelation will be tested by Durbin-Watson-test. In a nal step, Ramsey's RESET-test is used
[...]
^{1} The delivered data sets contain the timeframe 1929 to 2001, but for most sub- sequent estimations and calculations the timeframes 1940 to 2001 and 1946 to 2001 will be used.
^{2} Both numbers are valued at 1996 Dollars.
^{3} Besides price indices, quantity indices are also calculated. Chained-type indices are calculated to show the development of prices of goods relative to the previous period; another calculation method is using a xed period. Chained-type indices are used by the U.S. and France for example, a representative of xed period indices is Germany.
^{4} The countries of the OPEC reduced their delivery of oil by approximately 25%.
^{1} A households' consumption opportunities are restricted by their income, in other words: a consumer can only buy as much as he/she can a ord.
^{2} I will use data which are ordered from 1946 to 2001, i.e. t = 1 , . . . , 56.
^{3} To avoid another subscript I renounce the index t in the following calculations.
^{4} Ernst Engel, 1821-1896, German statistician.
^{5} Of course the budget share β n may be interpreted as demand for good n when it is multiplied by the income y t.
^{6} Of course any other category is allowed to choose. By choosing food, I consider the fact that expenditures on food are one of the most important for the private households.
^{7} The matrix notation y = X β + u is often used instead of such a system of equations.
^{8} β 0 is a constant term.
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