An Engel curve describes how household expenditure on a particular good or service varies with household income. There are two varieties of Engel curves. Budget share Engel curves describe how the proportion of household income spent on a good varies with income. Alternatively, Engel curves can also describe how real expenditure varies with household income. They are named after the German statistician Ernst Engel (1821–1896), who was the first to investigate this relationship between goods expenditure and income systematically in 1857. The best-known single result from the article is Engel's law which states that the poorer a family is, the larger the budget share it spends on nourishment.
The shape of Engel curves depend on many demographic variables and other consumer characteristics. A good's Engel curve reflects its income elasticity and indicates whether the good is an inferior, normal, or luxury good. Empirical Engel curves are close to linear for some goods, and highly nonlinear for others.
Graphically, the Engel curve is represented in the first-quadrant of the Cartesian coordinate system. Income is shown on the Y-axis and the quantity demanded for the selected good or service is shown on the X-axis.
For normal goods, the Engel curve has a positive gradient. That is, as income increases, the quantity demanded increases. Amongst normal goods, there are two possibilities. Although the Engel curve remains upward sloping in both cases, it bends toward the y-axis for necessities and towards the x-axis for luxury goods.
For inferior goods, the Engel curve has a negative gradient. That means that as the consumer has more income, they will buy less of the inferior good because they are able to purchase better goods.
Many Engel curves feature saturation properties in that their slope tends to diminish at high income levels, which suggests that there exists an absolute limit on how much expenditure on a good will rise as household income increases. This saturation property has been linked to slowdowns in the growth of demand for some sectors in the economy, causing major changes in an economy's sectoral composition to take place.
In microeconomics, an Engel curve shows how the quantity demanded of a good or service changes as the consumer's income level changes. In order to be consistent with the standard model of utility-maximization, Engel curves must possess certain properties. For example, Gorman (1981) proved that a system of Engel curves must have a matrix of coefficients with rank three (or less) in order to be consistent with utility maximization.
When considering a system of Engel curves, the adding-up theorem also dictates that the sum of all total expenditure elasticities, when weighted by the corresponding budget share, must add up to unity. This rules out the possibility of saturation being a general property of Engel curves across all goods as this would imply that the income elasticity of all goods approaches zero starting from a certain level of income. The adding-up restriction stems from the assumption that consumption always takes place at the upper boundary of the household's opportunity set, which is only fulfilled if the household cannot completely satisfy all its wants within the boundaries of the opportunity set.
In microeconomics Engel curves are used for equivalence scale calculations and related welfare comparisons, and determine properties of demand systems such as aggregability and rank.
Engel curves have also been used to study how the changing industrial composition of growing economies are linked to the changes in the composition of household demand.
In trade theory, one explanation of inter-industry trade has been the hypothesis that countries with similar income levels possess similar preferences for goods and services (the Lindner hypothesis), which suggests that understanding how the composition of household demand changes with income may play an important role in determining global trade patterns.
Low explanatory power
Heteroscedasticity is a well known problem in the Estimation of Engel curves: as income rises the difference between actual observation and the estimated expenditure level tends to increase dramatically. Engel curve and other demand function models still fail to explain most of the observed variation in individual consumption behavior.
As result, many scholars acknowledge that influences other than current prices and current total expenditure must be systematically modeled if even the broad pattern of demand is to be explained in a theoretically coherent and empirically robust way.
Accounting for their shape
No established theory exists that can explain the observed shape of Engel curves and their associated income elasticity values. Ernst Engel himself argued that households possessed a hierarchy of wants that determined the shape of Engel curves. As household income rises some motivations become more prominent in household expenditure as the more basic wants that dominate consumption patterns at low-income levels, such as hunger, eventually become satiated at higher income levels.
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