# Household Expenditure

Chapter
Part of the Mathematics Textbooks for Science and Engineering book series (MTSE, volume 3)

## Abstract

A household allocates its income to variety of goods and services. A collection of goods and services that a household purchases and consumes over a specific time period or horizon (a week, a month, or a year) is called a consumption bundle (bundle for short), or a basket.

## Keywords

Mortgage Loan Budget Equation Household Budget Repeat Part Mortgage Payment
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 1.1 Consumer’s Expenditure and Budget Constraint

A household allocates its income to variety of goods and services. A collection of goods and services that a household purchases and consumes over a specific time period or horizon (a week, a month, or a year) is called a consumption bundle (bundle for short), or a basket. Assume that a household’s annual bundle consists of $$n$$ different goods and services. If we denote the quantity of the $$i$$th item purchased by this household by $$\;Q_i \;\;\,i=1,2,\,...,\,n,\;$$ and the corresponding price of each unit of $$Q_i$$ by $$P_i$$, then we can write the household’s expenditure $$E$$ as
\begin{aligned} E = P_1Q_1+P_2Q_2+P_3Q_3+\cdots + P_nQ_n \end{aligned}
or more compactly, using summation or sigma notation,1 as
\begin{aligned} E = \sum _{i=1}^{n} P_i Q_i \end{aligned}
If we denote the household’s annual income by $$I$$, then
\begin{aligned} E = P_1Q_1+P_2Q_2+P_3Q_3+ \cdots + P_nQ_n \le I \end{aligned}
or, using sigma notation
\begin{aligned} E = \;\sum _{i=1}^{n} P_i Q_i \; \le I \end{aligned}
(1.1)
is the household budget constraint. Strict equality in (1.1) implies that this household spends all its income on its bundle and finishes the year without any savings. In this case future improvement in the standard of living of this household, as partly measured by purchase of additional quantity of current goods and services and/or consumption of new items not currently in the bundle, must be supported by future additional income.
Strict inequality, on the other hand, implies that this household saves part of its income. In that case we can include an item S, dollar amount or quantity of savings with the associated price of 1, to the household’s bundle and rewrite (1.1) as
\begin{aligned} \sum _{i=1}^{n} P_i Q_i + S = I \end{aligned}
(1.2)
If household’s unspent income is invested in interest bearing, income generating, and wealth enhancing asset(s), then future improvement of household’s standard of living could be partially financed by savings. For example, if rate of interest is $$r,$$ then the next period income of this household would increase from $$I$$ to $$I + rS$$.
The third (and not uncommon) possibility for household’s budget equation is
\begin{aligned} \sum _{i=1}^{n} P_i Q_i \; > \; I \end{aligned}
(1.3)
This implies that this household runs a “budget deficit” and its current consumption must be partially supported by “borrowing”. By denoting the dollar amount of this borrowing by B with the associated price of 1, we can rewrite (1.3) as
\begin{aligned} \sum _{i=1}^{n} P_i Q_i = I + B \;\;\; \rightarrow \;\;\;\sum _{i=1}^{n} P_i Q_i - B = I \end{aligned}
(1.4)
If there is no prospect for future higher income, this situation is, of course, not sustainable. $$B$$ in (1.4) does not necessarily represent only the amount of borrowing, but rather the interest payment and partial repayment of borrowed amount. A typical example is home mortgage loan. When a household purchase a house it typically borrows a substantial portion of value of the house from a bank or a mortgage company at certain interest rate, called mortgage rate. In this case household monthly expenditures will include a new item “mortgage payment”, which is a combination of interest and partial repayment of initial amount borrowed, called the principal. In a 25-year fixed mortgage loan, the fixed monthly mortgage payment is structured such that over 300 payments the borrower pays off the loan, principal plus interest (learn more about mortgage loans in Chap.  Mathematics of Interest Rate and Finance).

Tables  1.1 and 1.2 give the results from the Consumer Expenditure Survey released by the Bureau of Labor Statistics (BLS) of the U.S. Department of Labor (USDL), for years 2008, 2009, and 2010.2

Bureau of Labor Statistics, Consumer Expenditure
Table 1.1

Consumer expenditure survey 2008–2010

2008

2009

2010

% change

% change

2008–2009

2009–2010

Average annual expenditures

$50,486$49,067

$48,109 $$-$$2.8 $$-$$1.95 Food 6,443 6,372 6,129 $$-$$1.1 $$-$$3.8 At home 3,744 3,753 3,624 0.2 $$-$$3.4 Away from home 2,698 2,619 2,505 $$-$$2.9 $$-$$4.35 Housing 17,109 16,895 16,557 $$-$$1.3 $$-$$2.0 Apparel and services 1,801 1,725 1,700 $$-$$4.2 $$-$$1.4 Transportation 8,604 7,658 7,677 $$-$$11.0 0.25 Health-care 2,976 3,126 3,157 5.0 1.0 Entertainment 2835 2,693 2,504 $$-$$5.0 $$-$$7.0 Personal insurance and pensions 5,605 5,471 5,373 $$-$$2.4 $$-$$1.8 All other expenditures 5,060 5,113 5,127 0.01 0.27 Number of consumer units (000’s) 120,770 120,847 121,107 Income before taxes$63,563

$62,857$62,481

$$-$$1.1

$$-$$0.6

Average number in consumer unit

Persons

2.5

2.5

2.5

Earners

1.3

1.3

1.3

Vehicles

2.0

2.0

1.9

Percent homeowner

67

66

66

Consumer Expenditure Survey data records how consumers allocate their spending to different consumer goods and services. According to Table 1.1 the number of households (in BLS’s terminology number of consumer units) in the US in 2010 was 121,107,000 with average size of 2.5 persons. Average income before tax (average gross income) of the households is reported as $62,481 and the average expenditure as$48,109. As a clear evidence of the “great recession”, for the first time since 1984 consumer spending registered a drop of 2.8 % in 2009 from 2008 and 1.95 % in 2010 from 2009. Loss of income have forced consumers to further cut back in their consumption expenditure in 2010, specially for entertainment and food. With the average saving rate of the American households negligibly small, it is safe to say that a major portion of the difference between average gross income and expenditure is the average annual amount of taxes households paid to various levels of government—local, state, and federal.
Table 1.2

Consumer expenditure survey 2010

2010

Percentage

Average annual expenditures

$48,109 100.0 Food 6,129 12.7 At home 3,624 7.5 Away from home 2,505 5.2 Housing 16,557 34.4 Apparel and services 1,700 3.5 Transportation 7,677 15.96 Vehicle purchase 2,588 5.4 Other vehicle expenses 2,464 5.1 Gasoline and motor oil 2,132 4.4 Public transportation 493 1.0 Health care 3,157 6.6 Entertainment 2,504 5.2 Personal insurance and pensions 5,373 11.2 Other expenditures 5,012 10.4 BLS classifies household expenditure on consumer goods and services in eight broad categories, like Food, Housing, and Transportation. From Table 1.2 it is clear that the largest household expenditure item is Housing; 34.4 % of the total annual expenditure in 2010. Housing expenditure generally consists of mortgage payment for homeowners, rental payment for households that do not own their homes, maintenance and repair costs, fuel and utility costs, and homeowner insurance cost. Transportation and food are the next two big items. While the share of Transportation from the household’s total expenditure is 15.96 % (an 11 % decline in 2009 from 2008), gasoline and public transportation costs constitute only 4.4 and 1.0 % of expenses, respectively. A combination of deep recession and historically high gas price has lead, for the first time, to a decline in gasoline consumption in 2009 compared to 2008. The data, however, indicates that while other industrially advanced nations have cut their oil consumption since 1980 (Sweden and Denmark by as much as 33%), U.S. oil consumption has increased by more that 21 %. The United States still has the lowest gasoline price in the industrial world. 6.7 % of household expenditure is related to Health Care and 11.2 % to personal insurance and pension. As Table 1.1 indicates health care is still, and almost the only, growing expenditure item for the household. An increase of 5 % from 2008 to 2009 comes on the heels of 4.3 % increase from 2007 to 2008, 7.9 % increase from 2004 to 2005, and 18.9 % increase from 2003 to 2004. ### 1.1.1 A Simple Two-Commodity Model Assume a household consumes two goods. Let $$X$$ denote the units of good 1 and $$Y$$ the units of good 2 consumed by this household in a year. Assume $$P_1$$ and $$P_2$$ are the market prices of good 1 and 2, respectively. Also assume that this household earns$ $$I$$ as annual income. With these assumptions we can write this household’s budget equation as
\begin{aligned} P_1X+P_2Y = I \end{aligned}
Solving for $$Y$$, we have
\begin{aligned} Y = \frac{1}{P_2}I- \frac{P_1}{P_2}X \end{aligned}
If price of good 1 is $100, price of good 2 is$200, and household annual income is 50,000, then the household budget equation is \begin{aligned} 100X + 200Y = 50000 \end{aligned} leading to \begin{aligned} Y = \frac{50000}{200} - \frac{100}{200} X = 250- 0.5 X \end{aligned} (1.5) Figure 1.1 represents the graph of this linear budget equation. Points on the budget line signify combinations of units of good 1 and 2 that the household can buy, spending its entire annual income. Points $$A$$ (250 units of $$Y$$ and 0 unit of $$X$$) and $$B$$ (500 units of $$X$$ and 0 unit of $$Y$$) illustrate the cases when the household spends its income entirely on good 2 or on good 1. Fig. 1.1 Budget line Assume this household picks the combination of 200 units of good 1 and 150 units of good 2. This is the point $$C$$ on the line. This household can choose a combination of $$X$$ and $$Y$$ at a point like $$D$$ below the line. In this case the household is not spending its entire income and has some savings. All the points on the sides and inside of the triangle AOB constitute the feasible consumption set, i.e. all combinations of $$X$$ and $$Y$$ that the household can buy. Given the current income of this household, it cannot buy any combinations of $$X$$ and $$Y$$ above the budget line unless it borrows additional fund. Points above the budget line are non-feasible consumption set for this household. $$P_1, P_2$$, and $$I$$ are parameters of the budget constraint. Changes in any of these parameters lead to a new budget equation. The slope of the budget line is $$\;-\dfrac{P_1}{P_2}$$. Changes in $$P_1$$ and $$P_2$$, individually or simultaneously, will impact the household budget and change its budget equation. For example, if the price of good $$X$$ increases from100 to 110 ceteris paribus, the new budget equation will be \begin{aligned} 110X + 200Y = 50000 \;\;\; \longrightarrow \;\;\; Y = 250 - 0.55X \end{aligned} Here the slope of the budget line changes from 0.5 to $$-$$0.55, rotating inward and reducing the size of the feasible consumption set. This situation is depicted in Fig. 1.2. Fig. 1.2 Change in budget line due to increase in price of $$X$$ To examine the impact of change in income, assume the household income increases from50,000 to 60,000. In this case $$100X + 200 Y = 60000$$ and the new budget equation is \begin{aligned} Y = 300 - 0.5 X \end{aligned} The X-and Y-intercepts of the new budget line are now 600 and 300 while the slope of the line remains the same at $$-0.5$$. An increase in income generates an outward parallel shift in the budget line and leads to a new and expanded feasible consumption set. The reverse is also true; a loss of income shrinks the feasible set. If the household income declines by 20 % to40,000 a year, then the new budget equation is
\begin{aligned} Y = 200 - 0.5 X \end{aligned}
(1.6)
with the new budget line $$\;A'B'\;$$ in Fig. 1.3 Fig. 1.3 Shift in budget line due to loss of income
While a change in income leads to a parallel shift in the budget line, a proportional change in prices has the same effect. Assume that the inflation rate (rate of increase in the general level of prices) is 20 %. Because of 20 % inflation prices of good 1 and 2 rise to $120 and$240, respectively. The new budget equation is
\begin{aligned} Y = \frac{50000}{240}-\frac{120}{240}X \;\;\; = \; 208.33 - 0.5 X \end{aligned}
(1.7)
Notice that due to the proportional changes in price, the slope of this line ($$-0.5$$) is the same as line in (1.5) and (1.6), i.e. the new budget line is parallel to $$A'B'$$ and $$AB$$. This creates an impression that the impact of inflation must be similar to the loss of income—a parallel downward shift in the budget line. But does inflation have the same impacts on household’s consumption as the loss of income? In order to examine this question we must make certain assumptions about the household preference. Here we assume that the household consumes $$X$$ and $$Y$$ in the same proportions as the combination of 150 units of $$Y$$ and 200 units of $$X$$, i.e. in the ratio of 150/200 $$=$$  0.75.3 This means that the number of units of $$Y$$ consumed is always equal to 0.75 units of $$X$$. This can be expressed as:
\begin{aligned} Y = 150/200 X \;\;\;\; \text {or} \;\;\; Y = 0.75 X \end{aligned}
(1.8)

The graph of this equation is a ray from the origin to point $$C$$ on the budget line and is called the income-consumption path (see Fig. 1.4). Any change in the household income or in the prices of $$X$$ and $$Y$$ would lead to parallel shifts or rotations of the budget line. In response to the new budgetary realities, the household must adjust its bundle and pick a new combination of $$X$$ and $$Y$$. The assumption of an equal proportion of consumption of $$X$$ and $$Y$$ dictates that these new combinations must be points on this line. This assumption adds a new condition or constraint; not only must a bundle be on the budget line, it must also be on this line. So the new combination must be at the intersection of these two lines.

The new bundle that the household purchases after a 20 % loss of income, would consist of 160 units of $$X$$ and 120 units of $$Y$$ (point $$C'$$ on $$A'B'$$ and OC.) We obtain this result by solving (1.6) and (1.8) as a set of simultaneous equations.
\begin{aligned} {\left\{ \begin{array}{ll} Y = 200-0.5X\\ Y=0.75X \nonumber \end{array}\right. } \end{aligned}
\begin{aligned}&200-0.5 X = 0.75 X \;\;\longrightarrow \;\; 1.25 X = 200 \;\; \text {and} \;\; X = \frac{200}{1.25} = 160 \;\; \text {units}\\&Y = 200 - 0.5 (160) = 120 \;\;\; \text {units} \end{aligned}
The income-consumption path and the result of the loss of income are depicted in Fig. 1.4
The new bundle resulting from 20 % inflation would consist of 166.67 units of $$X$$ and 125 units of $$Y$$, obtained by solving (1.7) and (1.8) simultaneously in the same manner.
\begin{aligned} {\left\{ \begin{array}{ll} Y = 208.33-0.5X\\ Y=0.75X \nonumber \end{array}\right. } \end{aligned}
It is clear that inflation and loss of income both adversely affect the standard of living of this household, measured in real terms in the number of units of good 1 and good 2 purchased. With the assumption of a linear income-consumption path, it is also clear that the impact of a loss of income is greater than that of inflation. Unemployment and underemployment may have a much larger negative impact on welfare and standard of living of households than a comparable decline in purchasing power due to inflation.
Next let’s examine the effect of a non-proportional change in the prices. Let assume the price of $$X$$ increases from $100 to$110 and the price of $$Y$$ increases from $200 to$210 per unit, while the household income remains at 50,000. The new budget equation is \begin{aligned} 110 X + 210 Y = 50000 \longrightarrow Y = 238.1- 0.524 X \end{aligned} (1.9) Compare the new budget equation (1.9) with the original equation (1.5). There are changes in both the intercept and the slope. To determine the new consumer’s equilibrium, we must find the intersection of this budget line with the income-consumption path by solving (1.9) and (1.8) simultaneously. \begin{aligned} {\left\{ \begin{array}{ll} Y = 238.1-0.524 X\\ Y=0.75X \nonumber \end{array}\right. } \end{aligned} We obtain $$X = 186.9$$ units and $$Y = 140.2$$ units ## 1.2 Exercises 1. 1. A household allocates its2,000 monthly income to the purchase of three goods. Prices of these goods are $30,$40, and $20 per unit. 1. (a) Write the household monthly budget constraint. 2. (b) If this household purchases 40 units of good three each month, write its budget equation and graph it. What is the slope of the budget line? 2. 2. Assume a household with a monthly income of$5,000. This household allocates its income to the purchasing of food and nonfood products. If the average price of food products is $20 per unit and nonfood items costs$150 per unit
1. (a)

write the households budget equation.

2. (b)

If this household consumes 100 units of food products, how many units of nonfood items it can buy?

3. (c)

Assume that the price of nonfood products increases to $160. Write the new budget equation. 4. (d) If this household wants to purchase food and nonfood items in the same proportion as in part (b), what is the household’s new bundle in part (c)? 3. 3. Assume the household’s income in Problem 2 increases by 5 %. Repeat parts (a) through (d) of problem (2). 4. 4. Assume that due to competition prices of food and nonfood products in Problem 2 decline by 5 % while the household’s income remains the same. Repeat parts (a) and (b) of problem (2). Compare your result with problem (3). 5. 5. A household splits its$4,000 monthly income between necessity and luxury goods. The average price of necessities is $30 per unit and that of luxuries is$100 per unit.
1. (a)

Write the household budget constraint.

2. (b)

Determine the household equilibrium bundle if its proportion of necessity and luxury goods purchases is 10 to 1.

3. (c)

What is the equation of household income-consumption path?

4. (d)

Assume the household income declines by 10 %. What is the household’s new bundle?

5. (e)

Assume no loss of income but an inflation rate of 10 %. What is the household’s new bundle?

6. (f)

7. (g)

Assume the original household income increases by 10 % to \$4,400. What is the household’s new bundle?

8. (h)

Assume no change in income but the price level declines by 10 %. What is the household’s new bundle?

9. (i)

10. (j)

## Footnotes

1. 1.

If you are not familiar with sigma notation you should read the Appendix to this chapter first.

2. 2.

The most recent data available are for 2011, released by BLS on February 2013. I am using 2008, 2009, and 2010 to highlight the impact of the great recession on consumers.

3. 3.

This is a rather restrictive assumption. We must consider this problem in more detail in the context of consumers’ welfare maximization strategy.

4. 4.

Note that in $$x_1 + x_2 + \cdots + x_n,$$ the use of ellipses ‘$$\cdots$$’, representing the omitted terms, in itself is one step in the direction of more economical and time- and space-saving expression of long sums.