Why Trace Choices as Income Changes?
A consumer’s chosen bundle depends on both prices and income. To study the role of income alone, hold all prices fixed and vary the consumer’s budget. The resulting path shows how the quantities chosen change as the consumer can afford more. It may be a straight ray, a curved path, a segment along an axis, or even a set of bundles rather than a single path.
The construction is a direct application of utility maximization: solve for an optimal bundle at each income, then place those bundles in commodity space. The shape should be read with care. It records the chosen quantities, not the consumer’s indifference curves, and it need not rise in both goods.
With positive prices and locally nonsatiated preferences, an optimal bundle uses the entire budget: otherwise, the consumer could afford a small additional amount of a good and obtain a preferred bundle. Thus, in the examples below, the budget constraint binds. This conclusion requires local nonsatiation; without it, a consumer might prefer to leave some income unspent.
An ICC keeps prices fixed and varies income. This is different from a price-consumption curve, which keeps income fixed and varies a price. To read an ICC, note which coordinate changes as income rises and whether the locus is a single-valued path or a collection of tied optima.
Constructing an ICC from Optimal Bundles
A useful method is to solve the consumer’s problem as a function of income, rather than solving it separately for just one budget. The resulting demand formulas can then be plotted or described as a locus. The following steps organize the calculation.
Write the budget constraint with \(p_x\) and \(p_y\) held constant. Make clear which values of income are being considered.
Use the relevant interior conditions or examine boundary choices where an interior solution is infeasible.
Substitute the bundle into the budget and verify that it is feasible and optimal. If several bundles tie, include all of them.
Worked Examples
Worked Example: A Cobb-Douglas ICC Is a Ray
A consumer has utility \(U(x,y)=x^{1/3}y^{2/3}\), faces prices \(p_x=3\) and \(p_y=2\), and has income \(m>0\). Find the optimal bundle as a function of income, then give the ICC and the chosen bundles at incomes \(18\) and \(36\).
Method and assumptions. Both prices and income are positive, and the consumer chooses nonnegative quantities. Utility is strictly increasing in each good and strictly quasi-concave on positive bundles, so the budget binds and the optimal bundle is unique and interior. We can maximize the increasing transformation \(\ln U=\frac13\ln x+\frac23\ln y\).
Let \(\lambda\) be the multiplier on \(3x+2y\leq m\). The interior first-order conditions are \(\frac{1}{3x}=3\lambda\) and \(\frac{2}{3y}=2\lambda\). Thus \(\lambda=\frac{1}{9x}=\frac{1}{3y}\), which gives \(y=3x\). Substituting into the binding budget gives \(3x+2(3x)=9x=m\). Therefore \(x^*(m)=m/9\) and \(y^*(m)=m/3\).
Why this formula holds. The first-order conditions equate each good’s marginal contribution to log utility per dollar: \(\frac{1/(3x)}{3}=\frac{2/(3y)}{2}\). Rearranging gives the fixed quantity ratio \(y=3x\). As income rises, the consumer can scale both quantities up in that same ratio; the budget then determines the scale.
At \(m=18\), \(x=18/9=2\) and \(y=18/3=6\). The cost is \(3(2)+2(6)=6+12=18\). At \(m=36\), \(x=36/9=4\) and \(y=36/3=12\); the cost is \(3(4)+2(12)=12+24=36\). The ICC is the ray \(y=3x\) in the nonnegative quadrant, excluding the origin when only positive incomes are included.
Common mistake avoided: Do not infer that each good receives one-third or two-thirds of the physical units. The exponents give expenditure shares: \(x\) receives one-third of income and \(y\) receives two-thirds. Here those shares produce \(x=m/(3p_x)=m/9\) and \(y=2m/(3p_y)=m/3\).
Worked Example: Perfect Complements Produce a Proportional ICC
Suppose \(U(x,y)=\min\{x,2y\}\), with prices \(p_x=4\) and \(p_y=3\). Find the optimal bundle at incomes \(22\) and \(55\), and describe the ICC for all positive incomes.
Method and assumptions. The consumer needs the goods in the proportion \(x=2y\) to avoid paying for units that do not raise utility. For any utility level \(q\), a bundle must have \(x\geq q\) and \(y\geq q/2\). The least-cost bundle achieving \(q\) is therefore \((q,q/2)\), at cost \(4q+3(q/2)=\frac{11}{2}q\). At a binding budget, the largest attainable utility is \(q=2m/11\), so the unique least-cost optimal bundle is \(x=2m/11\), \(y=m/11\).
At \(m=22\), \(x=44/11=4\) and \(y=22/11=2\). Its cost is \(4(4)+3(2)=16+6=22\), and utility is \(\min\{4,4\}=4\). At \(m=55\), \(x=110/11=10\) and \(y=55/11=5\). Its cost is \(4(10)+3(5)=40+15=55\), and utility is \(\min\{10,10\}=10\).
Consequently, the ICC is \(x=2y\) for positive quantities, excluding the origin for positive incomes. The kinked preferences make the chosen bundle proportional as income rises, even though the indifference curves are not smooth.
Common mistake avoided: Treating the two goods as interchangeable misses the fixed proportion. A bundle such as \((8,2)\) yields utility only \(4\), the same as \((4,2)\), while costing more. At the stated prices and income \(22\), it is unaffordable anyway: its cost would be \(4(8)+3(2)=38\).
Worked Example: A Quasilinear ICC Has a Boundary Segment
A consumer has \(U(x,y)=\ln x+y\), where \(x>0\) and \(y\geq0\), and faces \(p_x=2\), \(p_y=1\). Find the optimal bundles at incomes \(m=0.6\) and \(m=1.6\), and describe the ICC for all \(m>0\).
Method and assumptions. Utility increases in both goods, so the budget binds. For any fixed income, the consumer can be described as choosing \(x\) and then spending the remainder on \(y\), if any. When \(y\geq0\), the budget gives \(y=m-2x\), so the relevant objective is \(\ln x+m-2x\), with \(0<x\leq m/2\).
The derivative with respect to \(x\) is \(1/x-2\), which is zero at \(x=1/2\). If \(m<1\), then \(x\leq m/2<1/2\), and the derivative is positive throughout the feasible range. The optimum is therefore the boundary choice \(x=m/2,\ y=0\). If \(m\geq1\), \(x=1/2\) is affordable and maximizes the objective; the remaining income gives \(y=m-1\). At \(m=1\), these formulas agree at \((1/2,0)\).
At \(m=0.6\), \(x=0.6/2=0.3\) and \(y=0\). The cost is \(2(0.3)+0=0.6\). At \(m=1.6\), \(x=0.5\) and \(y=1.6-1=0.6\). The cost is \(2(0.5)+0.6=1.6\). Thus the ICC first follows the \(x\)-axis from positive quantities below \(x=1/2\), then continues vertically at \(x=1/2\) as \(y\) increases.
Common mistake avoided: Applying the interior first-order condition \(x=1/2\) at every income ignores affordability. For \(m=0.6\), that quantity alone costs \(2(1/2)=1\), more than the available income. The boundary solution must be used below \(m=1\).
Interpreting Shape and Multiple Optima
The examples show why an ICC is best understood as a record of optimal choices, not as a curve with one universal shape. Under Cobb-Douglas preferences, both quantities rise in a fixed ratio. With perfect complements, the consumer also scales a fixed-proportion bundle. In the quasilinear example, one quantity rises initially, while the other remains at zero; above the threshold, \(x\) is fixed and \(y\) rises.
An ICC can also contain multiple optimal bundles at the same income. For example, let \(U(x,y)=x+y\), with \(p_x=p_y=2\). At any positive income \(m\), every nonnegative bundle satisfying \(2x+2y=m\) is optimal: each dollar spent buys the same amount of utility. As income varies over all positive values, the full ICC is \(\{(x,y):x\geq0,\ y\geq0,\ x+y>0\}\). The origin is excluded because it costs zero and is not optimal at any positive income.
Check Your Understanding
For each problem, hold the stated prices fixed as income changes. Check both the chosen quantities and the cost of each reported bundle.
- A consumer has \(U(x,y)=x^{1/2}y^{1/2}\), prices \(p_x=5\), \(p_y=10\), and incomes \(60\) and \(120\). Find both optimal bundles and the ICC. (Answer: \((6,3)\) at income \(60\), \((12,6)\) at income \(120\); the ICC is \(y=x/2\), excluding the origin for positive incomes.)
- A consumer has \(U(x,y)=\min\{x,3y\}\), prices \(p_x=2\), \(p_y=4\), and incomes \(40\) and \(80\). Find both optimal bundles and the ICC. (Answer: \((12,4)\) at income \(40\), \((24,8)\) at income \(80\); the ICC is \(x=3y\), excluding the origin for positive incomes.)
- A consumer has \(U(x,y)=3x+2y\), prices \(p_x=6\), \(p_y=4\), and income \(24\). State all optimal bundles at that income and determine whether the origin belongs to the ICC over all positive incomes. (Answer: every \((x,y)\) with \(x,y\geq0\) and \(6x+4y=24\) is optimal; the origin does not belong to the positive-income ICC.)
- A consumer has \(U(x,y)=\ln x+y\), prices \(p_x=3\), \(p_y=1\), and income \(0.6\). Find the optimal bundle and explain why the interior quantity \(x=1/3\) is infeasible. (Answer: \((x,y)=(0.2,0)\); \(x=1/3\) alone costs \(3(1/3)=1>0.6\).)
- For the same quasilinear consumer as in question 4, find the optimal bundle at income \(6\), and state the direction of the ICC above the income threshold where the interior solution first becomes affordable. (Answer: \((1/3,5)\); above the threshold \(m=1\), the ICC is vertical at \(x=1/3\) as \(y\) rises.)