Before the final, flavour C

Why is the best point on a budget line where a level curve touches it?

Slide along a budget line and watch the level curve through you: while it crosses the line you can still do better, and where it only touches, the Lagrange conditions hold.

Builds on functions of two variables and partial derivatives and optimization.

Bundle, (x, y): (6.00, 2.00)
Utility, U = xy²: 24.00

Step 1 of 5: Spend the budget

Your utility from $x$ units of one good and $y$ of another is $U = xy^2$ . They cost $2 and $3 a unit and you have $18, so you choose a point on the budget line $2x + 3y = 18$ . Each curve joins bundles of equal utility, and one of them passes through your point.

Predict first. At $(6, 2)$ , $U = 24$ , and the level curve through the point crosses the line. Which way along the line raises $U$ ?

1. Spend the budget
Your utility from $x$ units of one good and $y$ of another is $U = xy^2$ . They cost $2 and $3 a unit and you have $18, so you choose a point on the budget line $2x + 3y = 18$ . Each curve joins bundles of equal utility, and one of them passes through your point.
Question: At $(6, 2)$ , $U = 24$ , and the level curve through the point crosses the line. Which way along the line raises $U$ ?
Towards more of $y$ . Where a level curve crosses the line, part of the line lies beyond it, on higher curves. Up the line at $(4.5, 3)$ , $U = 4.5 \cdot 3^2 = 40.5$ . Below, $U$ all along the line.
The figure: Level curves of U = xy² for U = 12, 48 and 96, with the budget line 2x + 3y = 18. A point on the line at (4.50, 3.00), where U = 40.50; the level curve through it crosses the line. U along the budget line against x, from 0 to 9.0: it rises to 48.00 at x = 3.00 and falls to 0. The point is at x = 4.50, U = 40.50.
2. Touch, do not cross
Keep sliding. While the level curve through the point crosses the line, one direction along it still does better. The best point is where the curve stops crossing.
Slide the point until its level curve touches the budget line without crossing it.
At $(3, 4)$ the curve $U = 48$ just touches the line: every other point of the line lies on a lower curve, and below, $U$ along the line peaks at $48$ . Touching means the curve and the line have the same slope there.
The figure: Level curves of U = xy² for U = 12, 48 and 96, with the budget line 2x + 3y = 18. A point on the line at (3.00, 4.00), where U = 48.00; the level curve through it touches the line without crossing it. U along the budget line against x, from 0 to 9.0: it rises to 48.00 at x = 3.00 and falls to 0. The point is at x = 3.00, U = 48.00.
3. Same slope, same value per dollar
A level curve has slope $-U_x/U_y$ , and the line has slope $-2/3$ . They match when $U_x/2 = U_y/3$ : a dollar buys the same utility at the margin on either good. At $(6, 2)$ it is $2$ on the first and $8$ on the second, so money should move to the second.
Question: At the best point $(3, 4)$ , $U_x = 16$ and $U_y = 24$ . What is the common value $\lambda$ of $U_x/2$ and $U_y/3$ ?
$\lambda = 8$ : $16 = 8 \cdot 2$ and $24 = 8 \cdot 3$ . These are the Lagrange conditions: each partial derivative of $U$ is the same multiple $\lambda$ of the matching partial derivative of the budget $2x + 3y$ .
$U_x = 2\lambda, \qquad U_y = 3\lambda, \qquad 2x + 3y = 18$
The figure: Level curves of U = xy² for U = 12, 48 and 96, with the budget line 2x + 3y = 18. A point on the line at (3.00, 4.00), where U = 48.00; the level curve through it touches the line without crossing it. U along the budget line against x, from 0 to 9.0: it rises to 48.00 at x = 3.00 and falls to 0. The point is at x = 3.00, U = 48.00.
4. Where, and what
A question asks for the largest utility the budget allows. On the way the work finds $x = 3$ , $y = 4$ and $\lambda = 8$ , and each is a number that could be reported by mistake.
Question: Which number answers the question?
$48 = 3 \cdot 4^2$ , the height of the peak below. The amber line drops to where the peak is, $x = 3$ : that answers where, not what. $\lambda = 8$ answers yet another question, the next step's.
The figure: Level curves of U = xy² for U = 12, 48 and 96, with the budget line 2x + 3y = 18. A point on the line at (3.00, 4.00), where U = 48.00; the level curve through it touches the line without crossing it. U along the budget line against x, from 0 to 9.0: it rises to 48.00 at x = 3.00 and falls to 0. The point is at x = 3.00, U = 48.00. An amber dashed line drops from the peak to x = 3.00 on the axis; a dashed line runs across to U = 48.00.
5. One more dollar
Now the point stays at the best bundle while the budget changes. Raise the budget and the line moves out, and the best point moves with it, onto a higher level curve.
Question: Raise the budget from $18 to $19. By about how much does the largest utility grow?
About $8$ : the best bundle moves to $\left(\tfrac{19}{6}, \tfrac{38}{9}\right)$ , where $U = 56.45$ , up $8.45$ from $48$ . That is what $\lambda$ measures: the utility one more dollar buys, spent the best way.
The figure: Level curves of U = xy² for U = 12, 48 and 96, with the budget line 2x + 3y = 19. A point on the line at (3.17, 4.22), where U = 56.45; the level curve through it touches the line without crossing it. U along the budget line against x, from 0 to 9.5: it rises to 56.45 at x = 3.17 and falls to 0. The point is at x = 3.17, U = 56.45.

Step 1 of 5: Spend the budget

The lesson as text

1. Spend the budget

2. Touch, do not cross

3. Same slope, same value per dollar

4. Where, and what

5. One more dollar