Best Choice on a Budget

You want the most satisfaction you can afford. The best choice is where a level curve of satisfaction just touches your budget line, so their gradients point the same way.

How it works

Lagrange multipliers

In one sentence

At the best affordable bundle, the gradient of utility is parallel to the gradient of the budget: $\nabla U = \lambda\nabla g$ with $g = p_x x + p_y y = M$ .

General formula

{\htmlData{sym=0}{\nabla U}} = {\htmlData{sym=2}{\lambda}}\,{\htmlData{sym=1}{\nabla g}}, \qquad {\htmlData{sym=3}{g(x, y)}} = {\htmlData{sym=4}{p_x}} x + {\htmlData{sym=5}{p_y}} y = {\htmlData{sym=6}{M}}

At the optimum the indifference curve just touches the budget line, so their normal vectors line up; $\lambda$ is how much utility one more dollar buys.

What the symbols mean

$\nabla U$
Utility gradient
The direction in which satisfaction rises fastest.
$\nabla g$
Budget gradient
Points straight across the budget line.
$\lambda$
Lagrange multiplier
Extra satisfaction per extra dollar. It measures how much the budget is holding you back.
$g(x, y)$
Spending
Total cost of buying x units of one good and y of the other.
$p_x$
Price of good x
Dollars per unit of the first good.
$p_y$
Price of good y
Dollars per unit of the second good.
$M$
Budget
The total money available to spend.

Level curves of $U(x, y) = x^{a}y^{1-a}$ are indifference curves: combinations you like equally.
The budget line $p_x x + p_y y = M$ is the constraint.
At the optimum the curves are tangent, $\nabla U \parallel \nabla g$ . For this utility you spend a share $a$ of income on $x$ .

Learn the math

OptimizationCalc I · Lesson 33, Applications of Derivatives Lagrange MultipliersCalc III · Lesson 29, Applications of Partial Derivatives