Engel Curve Explained: Shapes, Engel's Law and a Worked Example
Jude Wallis
Founder of EconLearn · 2nd place internationally at the Economics Olympiad (econolympiad.org)
The Engel curve is a graph of the quantity of one good a consumer buys at each level of income, with all prices and tastes held constant. It slopes upward for a normal good and downward for an inferior good. Whether it rises more or less than in proportion to income separates necessities, whose quantity grows by a smaller percentage than income, from luxuries, whose quantity grows by a larger percentage. Graphically, a necessity's Engel curve is flatter than a straight line from the origin to that point, and a luxury's is steeper. It is named after the statistician Ernst Engel.
The Engel curve itself goes beyond the AP Microeconomics course. The number it is built from, income elasticity of demand, is tested in AP Micro Topic 2.5, Other Elasticities. The glossary entry on the Engel curve has the short definition.
What an Engel curve shows
An Engel curve answers one thing: how much of this good a person would buy with more money and nothing else changed.
- Horizontal axis: income per month or per year.
- Vertical axis: quantity of one good bought per period.
- Held constant: the price of this good, the prices of all other goods, and the consumer's tastes.
Fixed prices keep the curve about income alone; a price change would mix in the substitution effect that the income effect vs substitution effect guide separates out. Some textbooks put income on the vertical axis instead, so read the labels before you read the shape.
How to draw an Engel curve from the income-consumption path
The Engel curve comes out of the consumer choice graph, with good X (groceries) on the horizontal axis and all other spending on the vertical axis.
1. Draw the budget line for the first income and find where it touches the highest indifference curve. That tangency is the consumer's choice.
2. Raise income. Prices are fixed, so the budget line shifts out in parallel. Find the new tangency.
3. Connect the tangencies. This line is the income-consumption curve, also called the income expansion path.
4. Plot the quantity of X at each tangency against the income that produced it. Those points form the Engel curve.
A hypothetical case: groceries cost $40 a basket. At $2,000 a month the budget line reaches 2,000 ÷ 40 = 50 baskets on the groceries axis and the household picks 20. At $3,000 it reaches 75 and the household picks 24; at $4,000 it reaches 100 and the household picks 27. The Engel curve passes through (2,000, 20), (3,000, 24) and (4,000, 27).
If the income-consumption curve bends back toward the vertical axis, X is inferior over that range. The budget constraint guide covers the shifting line and the indifference curves guide covers the tangency. Shift income in the consumer choice sandbox to watch the chosen bundle move.
Consumer choice: the optimum where the budget line touches the highest indifference curve. Tap it to drag the curves, open the full version, or put it on your own site free, or turn it into a five-minute class activity.
Engel curve shapes for normal, inferior, necessity and luxury goods
The slope and curvature of an Engel curve match the sign and size of income elasticity of demand.
| Type of good | Income elasticity | Engel curve shape | Budget share as income rises | Hypothetical example |
|---|---|---|---|---|
| Luxury (normal) | Greater than 1 | Rises, typically getting steeper | Rises | Weekend trips |
| Unit elastic (normal) | Exactly 1 | Straight line through the origin | Stays the same | A good bought in fixed proportion to income |
| Necessity (normal) | Between 0 and 1 | Rises, typically flattening | Falls | Groceries |
| Income neutral | Exactly 0 | Horizontal | Falls | Table salt |
| Inferior | Less than 0 | Slopes downward | Falls | Discount frozen dinners |
A normal good has a rising Engel curve and an inferior good a falling one. Necessities and luxuries are both normal goods; they differ in how fast quantity rises compared with income.
Slope is not the same as elasticity
Income elasticity equals the Engel curve's slope scaled by income over quantity:
Income elasticity = (change in Q ÷ change in I) × (I ÷ Q)
Take the groceries curve from $2,000 to $3,000. The slope is 4 baskets ÷ $1,000 = 0.004 baskets per dollar. At the midpoint, I = 2,500 and Q = 22, so the elasticity is 0.004 × (2,500 ÷ 22) = 10 ÷ 22 = 0.45. The slope carries units; the elasticity is a pure number. For upward-sloping straight-line Engel curves there is a shortcut: a line through the origin has elasticity 1 everywhere, a line that meets the quantity axis above zero is a necessity throughout, and a line that starts at a positive income is a luxury throughout.
One good can be normal and then inferior
Engel curves often bend. A very low-income household may buy more frozen dinners as income rises, then fewer once it can afford fresh food. The curve rises, peaks and turns down. Normal and inferior describe a stretch of the curve, not the product. The normal vs inferior goods guide has more examples of goods that switch.
A downward-sloping Engel curve does not make a good a Giffen good. Every Giffen good is inferior, but Giffen also requires an upward-sloping demand curve, a statement about price rather than income.
Engel's law
Engel's law says that as household income rises, the share of the budget spent on food falls, even though dollar spending on food usually rises. Ernst Engel, a 19th-century German statistician, drew it from studies of family budgets. In elasticity terms, it says food is a necessity, with income elasticity above 0 but below 1.
Budget share = (P × Q) ÷ I
With price fixed, the share rises only if Q grows by a bigger percentage than I (elasticity above 1) and falls if Q grows by a smaller percentage (elasticity below 1).
Worked example: an Engel curve table with budget shares
A hypothetical household's monthly income rises in steps from $2,000 to $5,000. Prices stay fixed: groceries at $40 a basket, weekend trips at $150 each, and discount frozen dinners at $4 each.
Step 1: budget shares for groceries
- $2,000: 20 baskets × $40 = $800. Share = 800 ÷ 2,000 = 40%.
- $3,000: 24 × $40 = $960. Share = 960 ÷ 3,000 = 32%.
- $4,000: 27 × $40 = $1,080. Share = 1,080 ÷ 4,000 = 27%.
- $5,000: 29 × $40 = $1,160. Share = 1,160 ÷ 5,000 = 23.2%.
Grocery spending rises from $800 to $1,160 while the share falls from 40% to 23.2%. That is Engel's law in one column.
Step 2: midpoint percentage change in income
- $2,000 to $3,000: 1,000 ÷ [(2,000 + 3,000) ÷ 2] = 1,000 ÷ 2,500 = 40%.
- $3,000 to $4,000: 1,000 ÷ 3,500 = 28.57%.
- $4,000 to $5,000: 1,000 ÷ 4,500 = 22.22%.
Step 3: midpoint income elasticity for groceries
- 20 to 24: 4 ÷ [(20 + 24) ÷ 2] = 4 ÷ 22 = 18.18%. Elasticity = 18.18 ÷ 40 = 0.45.
- 24 to 27: 3 ÷ 25.5 = 11.76%. Elasticity = 11.76 ÷ 28.57 = 0.41.
- 27 to 29: 2 ÷ 28 = 7.14%. Elasticity = 7.14 ÷ 22.22 = 0.32.
| Monthly income | Baskets | Grocery spending | Budget share | Midpoint elasticity from previous row |
|---|---|---|---|---|
| $2,000 | 20 | $800 | 40% | (start) |
| $3,000 | 24 | $960 | 32% | 0.45 |
| $4,000 | 27 | $1,080 | 27% | 0.41 |
| $5,000 | 29 | $1,160 | 23.2% | 0.32 |
Every elasticity is between 0 and 1, so groceries are a necessity, not an inferior good. The curve gains 4, then 3, then 2 baskets per extra $1,000: it rises and flattens.
Step 4: the other two goods
Weekend trips go 1, 2, 4, 6. The midpoint changes are 1 ÷ 1.5 = 66.67%, then 2 ÷ 3 = 66.67%, then 2 ÷ 5 = 40%. Dividing by the income changes gives 66.67 ÷ 40 = 1.67, then 66.67 ÷ 28.57 = 2.33, then 40 ÷ 22.22 = 1.80. The budget share climbs from 150 ÷ 2,000 = 7.5% to 900 ÷ 5,000 = 18%.
Frozen dinners go 30, 32, 26, 18. The midpoint changes are 2 ÷ 31 = 6.45%, then -6 ÷ 29 = -20.69%, then -8 ÷ 22 = -36.36%. The elasticities are 6.45 ÷ 40 = 0.16, then -20.69 ÷ 28.57 = -0.72, then -36.36 ÷ 22.22 = -1.64.
| Income step | Groceries | Weekend trips | Frozen dinners |
|---|---|---|---|
| $2,000 to $3,000 | 0.45 (necessity) | 1.67 (luxury) | 0.16 (necessity) |
| $3,000 to $4,000 | 0.41 (necessity) | 2.33 (luxury) | -0.72 (inferior) |
| $4,000 to $5,000 | 0.32 (necessity) | 1.80 (luxury) | -1.64 (inferior) |
The frozen dinner curve rises, peaks near $3,000 and then falls: a bending Engel curve. Check any row with the income elasticity of demand calculator or the midpoint method calculator. The income elasticity of demand guide covers the formula in depth.
Engel curve vs demand curve
| Feature | Engel curve | Demand curve |
|---|---|---|
| Variables | Income and quantity | Price and quantity |
| Held constant | All prices and tastes | Income, other prices and tastes |
| A rise in income | Moves you along the curve | Shifts the curve (right if normal, left if inferior) |
| A change in the good's price | Shifts the whole curve | Moves you along the curve |
| What the slope sign tells you | Normal (up) or inferior (down) | Downward for almost every good (law of demand) |
When income rises from $3,000 to $4,000, the household slides along its grocery Engel curve from 24 to 27 baskets. On its demand curve, the same event is a shift right: at the fixed $40 price, it now wants 27 baskets instead of 24. If the grocery price rose to $50, the demand curve would not move, but the whole Engel curve would be redrawn, usually lower at every income.
See income elasticity vs the Engel curve for the number against the full curve, budget constraint vs Engel curve for how many budget lines make one Engel curve, and the consumer choice module for all of these graphs together.
Frequently asked questions
What is an Engel curve in economics?
An Engel curve is a graph of how much of one good a consumer buys at each income level, with every price and the consumer's tastes held constant. An upward-sloping Engel curve marks a normal good and a downward-sloping one marks an inferior good. A curve along which quantity rises by a smaller percentage than income (income elasticity between 0 and 1) marks a necessity, and one along which quantity rises by a larger percentage than income (income elasticity above 1) marks a luxury.
How do you draw an Engel curve from indifference curves?
To draw an Engel curve, start with a budget line and find its tangency with the highest indifference curve. Raise income with prices fixed, so the budget line shifts out in parallel, and find each new tangency. Connecting the tangencies gives the income-consumption curve. Plotting the quantity of the good at each tangency against the income that produced it gives the Engel curve.
What is Engel's law?
Engel's law says that as household income rises, the share of the budget spent on food falls, even though dollar spending on food usually rises. It is named after the 19th-century German statistician Ernst Engel. In elasticity terms, Engel's law says food is a necessity, with an income elasticity of demand above 0 but below 1, so food's Engel curve rises more slowly than income.
What is the difference between an Engel curve and a demand curve?
An Engel curve relates quantity to income with all prices fixed, while a demand curve relates quantity to the good's own price with income fixed. A rise in income moves a consumer along the Engel curve but shifts the demand curve, right for a normal good and left for an inferior good. A change in the good's price does the reverse: it moves along the demand curve and shifts the Engel curve.
What does an Engel curve look like for a luxury good?
The Engel curve for a luxury good rises faster than income, so with income on the horizontal axis it typically gets steeper as income grows. A luxury has an income elasticity of demand above 1, which means its share of the budget rises as income rises. A straight-line Engel curve that starts at a positive income, where the consumer first begins buying, also describes a luxury.
Is the Engel curve on the AP Microeconomics exam?
The Engel curve itself goes beyond the AP Microeconomics course. The idea behind it is tested, though: AP Micro Topic 2.5, Other Elasticities, covers income elasticity of demand, where a positive value means a normal good and a negative value means an inferior good. Topic 2.1, Demand, covers how a change in income shifts the demand curve for normal and inferior goods.
More on Engel curve
- DefinitionEngel Curve, AP Economics
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