What an Engel Curve Records
A consumer’s optimal bundle depends on income as well as prices and preferences. If prices and preferences are held constant while income varies, the consumer may choose different quantities of each good. An Engel curve records this relationship for one good: it plots that good’s optimal quantity against income.
This is a useful way to organize the results of consumer choice. The curve does not show how a consumer responds to a price change; that is the role of a demand curve. Instead, it asks how the optimal quantity changes when the consumer has more or less income, with the prices and preferences unchanged.
The horizontal and vertical axes are commonly income \(m\) and quantity \(x^*\), respectively. The same relationship can be drawn with the axes reversed, so check the axis labels before interpreting a graph. The curve is a locus of optimal choices, not a graph of utility or of the budget constraint.
How an Engel Curve Relates to Other Choice Diagrams
An income-consumption curve traces the optimal bundle as income changes at fixed prices. An Engel curve extracts one coordinate of that bundle and plots it against income. Thus, an income-consumption curve can be used to construct an Engel curve, but the two diagrams do not display the same information.
A demand curve holds income and other prices fixed while the price of a good varies. An Engel curve instead holds prices fixed and varies income. This distinction matters because a graph can look similar while answering a different economic question.
| Relationship | What changes? | What is recorded? |
|---|---|---|
| Engel curve for good \(x\) | Income; prices and preferences fixed | Optimal quantity of \(x\) |
| Income-consumption curve | Income; prices and preferences fixed | Optimal bundles of all goods |
| Demand curve for good \(x\) | Price of \(x\); income and other prices fixed | Optimal quantity of \(x\) |
Deriving an Engel Curve from Optimal Choice
The general method is to solve the consumer’s optimization problem as a function of income, then isolate the quantity of interest. For a two-good consumer with budget \(p_xx+p_yy\leq m\), where prices are positive, a change in income shifts the budget constraint without changing its slope. Preferences determine which affordable bundle is selected at each income.
For an interior optimum with differentiable preferences, the familiar tangency condition equates the marginal rate of substitution to the price ratio, alongside the binding budget constraint. But this method does not apply automatically to every income level: a choice may be at a boundary, or preferences may have a kink. In those cases, identify the optimum using the relevant boundary or kink conditions instead.
Worked Example: A Cobb-Douglas Engel Curve
A consumer has utility \(U(x,y)=x^{1/2}y^{1/2}\), faces fixed prices \(p_x=4\) and \(p_y=2\), and has income \(m>0\). Derive the Engel curves for both goods and calculate the optimal bundle at incomes \(m=24\) and \(m=48\).
Method. The utility function is increasing in both goods, so the budget binds. For an interior optimum, marginal utilities are \(MU_x=\tfrac12 x^{-1/2}y^{1/2}\) and \(MU_y=\tfrac12 x^{1/2}y^{-1/2}\). Therefore \(MU_x/MU_y=y/x\). The tangency condition is
Thus \(y=2x\). Substitute into the budget equation:
It follows that \(x^*(m)=m/8\) and \(y^*(m)=m/4\). These are the two Engel curves: each is a straight line through the origin, with income on the horizontal axis and the relevant quantity on the vertical axis.
At \(m=24\), \(x^*=24/8=3\) and \(y^*=24/4=6\). The budget check is \(4(3)+2(6)=12+12=24\).
At \(m=48\), \(x^*=48/8=6\) and \(y^*=48/4=12\). The budget check is \(4(6)+2(12)=24+24=48\). In each case the bundle satisfies the tangency condition \(y/x=2\), as required.
Common mistake avoided: Applying the price ratio in the wrong direction would give \(x/y=2\), inconsistent with the actual marginal-utility ratio. Here \(MU_x/MU_y=y/x\), so the correct tangency equation is \(y/x=p_x/p_y\).
Engel Curves with Kinked Preferences
The Engel curve need not be derived from a smooth interior optimum. With perfect complements, for example, the consumer chooses goods in a fixed proportion. If both prices are positive and more of either good is useful only when the other good is also available in the required proportion, an optimal bundle lies at the kink of the indifference curve. Income then determines how many bundles in that proportion the consumer can afford.
Worked Example: Perfect Complements and the Bundle Count
A consumer has utility \(U(x,y)=\min\{x,2y\}\), faces fixed prices \(p_x=3\) and \(p_y=2\), and has income \(m>0\). Derive the Engel curve for \(x\), and calculate the optimal bundle when \(m=24\).
Method. At an efficient bundle, the two arguments of the minimum are equal: \(x=2y\). Otherwise, the consumer could reduce the good in excess and use the saved expenditure to buy more of the limiting good. Write \(x=2t\) and \(y=t\), where \(t\) is the number of bundles in the \((2,1)\) ratio. The utility of this choice is \(\min\{2t,2t\}=2t\); \(t\) itself is not the utility level.
One such \((2,1)\) bundle costs \(3(2)+2(1)=8\). With income \(m\), the consumer can afford \(t=m/8\) bundles. Consequently,
At \(m=24\), \(t=24/8=3\), so \(x^*=2(3)=6\) and \(y^*=3\). The budget check is \(3(6)+2(3)=18+6=24\), and utility is \(\min\{6,2(3)\}=6\). The Engel curve for \(x\) is a straight line through the origin with slope \(1/4\).
Common mistake avoided: Calling \(t\) “units of utility” would be incorrect: the utility of \((2t,t)\) is \(2t\). Keeping the bundle count and the utility level distinct prevents a factor-of-two error in the Engel curve.
A Piecewise Engel Curve from a Boundary Choice
An income increase does not always change every quantity. A consumer may first be unable to afford a preferred interior bundle because a good is constrained to be nonnegative. Once income is high enough to reach that bundle, additional income can be spent on another good while the quantity of the first good stays fixed. The result can be a piecewise Engel curve.
Worked Example: A Quasilinear Engel Curve with a Corner
A consumer has utility \(U(x,y)=\ln x+y\), with \(x>0\) and \(y\geq 0\). The fixed prices are \(p_x=2\) and \(p_y=1\), and income is \(m>0\). Derive the Engel curve for \(x\), then find the optimal bundles at incomes \(m=0.5\), \(m=1\), and \(m=3\).
Method. The budget binds because utility increases with both goods. If the optimum is interior in \(y\), the marginal utilities are \(MU_x=1/x\) and \(MU_y=1\). The tangency condition gives
The budget then implies \(y=m-2x=m-1\). This interior choice is feasible only when \(m\geq1\), because \(y\) cannot be negative. If \(0<m<1\), the consumer is at the boundary \(y=0\); the binding budget gives \(2x=m\), or \(x=m/2\). Therefore the Engel curve is
At \(m=0.5\), \(x^*=0.5/2=0.25\), \(y=0\), and expenditure is \(2(0.25)+0=0.5\). At \(m=1\), \(x^*=0.5\), \(y=0\), and expenditure is \(2(0.5)=1\). At \(m=3\), \(x^*=0.5\), \(y=3-1=2\), and expenditure is \(2(0.5)+2=3\). Each bundle is affordable, uses the full budget, and satisfies the appropriate boundary or interior condition.
Common mistake avoided: Extending the interior solution \(x=1/2,\ y=m-1\) to incomes below 1 would imply \(y<0\), which is infeasible. The boundary solution must be used there.
Reading and Checking an Engel Curve
A curve can be checked both algebraically and graphically. For a proposed income level, substitute it into the derived quantity function, recover the rest of the optimal bundle when needed, and verify the budget and choice conditions. On the graph, the point should have income as its horizontal coordinate and the optimal quantity of the specified good as its vertical coordinate.
The curve’s shape summarizes how the chosen quantity varies over the income range being considered. It may be linear, piecewise, or more complicated, depending on preferences and constraints. A kink can mark a transition between a boundary and an interior choice, as in the quasilinear example. A flat section means the quantity plotted is unchanged over that income range; it does not mean the consumer’s entire bundle is unchanged.
This construction is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. The central exam skill is to keep the experiment clear: income varies, but prices and preferences do not. If prices change between observations, the points do not trace the Engel curve defined here.
Common Mistakes and a Reliable Workflow
- Changing prices along with income. That combines income and price effects, so the resulting points are not an Engel curve at fixed prices.
- Plotting the whole bundle as one quantity. An Engel curve is for a specified good; record its optimal quantity against income.
- Assuming every optimum is interior. Check nonnegativity and other constraints at each income level. If a candidate solution violates a constraint, solve the boundary case.
- Confusing a bundle parameter with utility. In the perfect-complements example, \(t\) counted \((2,1)\) bundles, while utility was \(2t\).
- Skipping the budget check. Substituting the proposed bundle into \(p_xx+p_yy\leq m\) is a quick way to expose arithmetic or parameter errors.
A dependable solution states what is fixed, solves the choice problem as a function of income, checks any boundary conditions, and only then draws the good’s quantity against income. This separates the economic reasoning from the final graph and makes the result easier to verify.
Check Your Understanding
For each problem, hold the stated prices and preferences fixed as income varies. Derive or use the relevant optimal-choice relationship, then check the reported bundle against the budget.
- A consumer has \(U(x,y)=x^{0.4}y^{0.6}\), prices \(p_x=5\) and \(p_y=2\), and income \(m\). Find the Engel curve for \(x\), then find \(x^*\) and \(y^*\) at \(m=50\). (Answer: \(x^*(m)=0.08m\); at \(m=50\), \((x^*,y^*)=(4,15)\).)
- A consumer has \(U(x,y)=\min\{3x,y\}\), prices \(p_x=2\) and \(p_y=1\), and income \(m=30\). Find the optimal bundle and utility, and state the Engel curve for \(x\). (Answer: \(x^*(m)=m/5\); at \(m=30\), \((x^*,y^*)=(6,18)\) and utility is \(18\).)
- A consumer has \(U(x,y)=2x+y\), prices \(p_x=4\) and \(p_y=1\), and income \(m=12\). Compare utility per dollar and give an optimal bundle and the Engel curve for \(x\). (Answer: \(x\) yields \(2/4=0.5\) utility per dollar and \(y\) yields \(1/1=1\); an optimum is \((0,12)\), and \(x^*(m)=0\).)
- A consumer has \(U(x,y)=\ln x+y\), prices \(p_x=3\) and \(p_y=1\), and income \(m>0\), with \(x>0\) and \(y\geq0\). Find the Engel curve for \(x\), including its transition income, and calculate the bundle at \(m=6\). (Answer: \(x^*(m)=m/3\) for \(0<m\leq3\) and \(x^*(m)=1/3\) for \(m\geq3\); at \(m=6\), \((x^*,y^*)=(1/3,5)\).)
- For \(U(x,y)=\min\{x,2y\}\), prices \(p_x=3\) and \(p_y=2\), and income \(m=40\), calculate the optimal bundle, verify its cost and utility, and explain the meaning of the parameter \(t\) if the bundle is written as \((2t,t)\). (Answer: \(t=40/8=5\), so \((x^*,y^*)=(10,5)\), cost is \(3(10)+2(5)=40\), utility is \(10\), and \(t\) counts \((2,1)\) bundles, not utility units.)