Principles of Mathematical Economics pp 1-16 | Cite as

# Household Expenditure

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## 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## 1.1 Consumer’s Expenditure and Budget Constraint

*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

*summation*or

*sigma notation*,

^{1}as

*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.

*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. 12

*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**

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 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 |

| 2,132 | 4.4 |

| 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 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.

*ceteris paribus*, the new budget equation will be

^{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:

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.

## 1.2 Exercises

- 1.A household allocates its $2,000 monthly income to the purchase of three goods. Prices of these goods are $30, $40, and $20 per unit.
- (a)
Write the household monthly budget constraint.

- (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?

- (a)
- 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
- (a)
write the households budget equation.

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

- (c)
Assume that the price of nonfood products increases to $160. Write the new budget equation.

- (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)?

- (a)
- 3.
Assume the household’s income in Problem 2 increases by 5 %. Repeat parts (a) through (d) of problem (2).

- 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.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.
- (a)
Write the household budget constraint.

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

- (c)
What is the equation of household income-consumption path?

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

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

- (f)
Compare your answers in part (b) and (c).

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

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

- (i)
Compare your answers in parts (g) and (h)

- (j)
Compare your answers to parts (f) and (i). Are you surprised?

- (a)

## Footnotes

- 1.
If you are not familiar with sigma notation you should read the Appendix to this chapter first.

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

- 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.