Minimizing Spending for a Target Utility

Turn the consumer's problem around. Instead of "the most utility for income $m$," ask "the least money for utility $\bar u$." This is the problem behind price indexes and money measures of welfare, and it uses the same tangency as Cobb-Douglas Demand and its relatives.

Key idea

Minimize $p_1x + p_2y$ subject to $U(x, y) = \bar u$. At an interior solution $$MRS = \frac{p_1}{p_2} \quad\text{and}\quad U(x, y) = \bar u.$$ The solution is the compensated (Hicksian) demand $h(p_1, p_2, \bar u)$, and its cost is the expenditure function $E(p_1, p_2, \bar u)$.

Why it works

Picture a family of parallel budget lines, $p_1x + p_2y = E$, one for each spending level. Start with a large $E$ and slide the line toward the origin. The last line that still touches the target indifference curve touches it at a tangency, where its slope $-p_1/p_2$ equals the curve's slope $-MRS$. Same condition, different constraint. The two problems are linked (duality): if income $m$ buys at most utility $u^*$, then the least money that buys $u^*$ is exactly $m$, and the two problems pick the same bundle.

The target curve $\sqrt{xy} = \sqrt{200}$. At prices (3, 6), \$120 is the least spending that reaches it, at the bundle (20, 10).

Worked example: Hicksian demand and expenditure

$U = xy$, $p_1 = 4$, $p_2 = 1$. What is the cheapest way to reach $U = 100$?

Tangency: $\dfrac{y}{x} = \dfrac{4}{1}$, so $y = 4x$.
Target: $x(4x) = 100$, so $x^2 = 25$ and $x = 5$; $y = 20$.
Cost: $4(5) + 1(20) = 40$.
In general $E = 2\sqrt{p_1p_2\bar u} = 2\sqrt{4 \cdot 1 \cdot 100} = 40$.

Answer: $(5, 20)$, costing \$40.

Worked example: Duality check

With $U = xy$, $p_1 = 4$, $p_2 = 1$ and income \$40, what utility can the consumer reach? Compare with the example above.

Cobb-Douglas demand: $x^* = \dfrac{40}{2 \cdot 4} = 5$ and $y^* = \dfrac{40}{2 \cdot 1} = 20$.
$U^* = 100$, and the least money that reaches 100 was \$40: the two problems give the same bundle.

Answer: $U^* = 100$ at the bundle (5, 20).

Three properties of the expenditure function follow from the picture. It rises with $\bar u$ (a higher curve needs a line farther out). It rises with each price, but less than in proportion to one price alone, because the consumer substitutes away from the good that got dearer. And doubling every price exactly doubles $E$, since the cheapest bundle doesn't change. The second fact is why a fixed-basket price index overstates the rise in the cost of living: it prices the old bundle and ignores substitution, which the U.S. Bureau of Labor Statistics calls substitution bias.

Common mistake

Expecting a different tangency rule when minimizing spending. The condition is still $MRS = p_1/p_2$; what changes is the second equation: the utility target replaces the budget. Check: your answer must sit on the target curve ($U = \bar u$ exactly), and its MRS must equal the price ratio.

Hicksian demand holds utility fixed, so it responds to a price change only through substitution. That makes it the right tool for the Substitution and Income Effects split and for Compensating and Equivalent Variation.

Minimizing Spending for a Target Utility

Practice

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