- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Functions of two variables and partial derivatives practice question 15
A consumer’s utility from units of one good and units of another is . A student finds the slope of the indifference curve through the bundle , the level curve of through that point, as shown. Find the first line that is wrong.
Check each line against what an indifference curve is: a curve along which stays the same. Does each formula the student uses follow from that?
On an indifference curve is constant: here . Differentiate that equation in , treating as a function of .
Implicit differentiation gives . Solve for and watch the sign.
Theory UBC Math 100 §15.1Key results
Corrected working
Strategy
An indifference curve is a level curve, U = constant. Differentiate along it implicitly and solve for dy/dx.
Working
The level of U at (3, 2): 9 × 2 = 18.
Implicit differentiation in x, with the product rule on x²y.
The curve falls: less of one good must be made up with more of the other. As a check, y = 18/x² has dy/dx = −36/x³ = −4/3 at x = 3.
Answer
The answer is The slope is negative 4/3; line 3 drops the minus sign..
Check
Along x²y = 18 the curve falls: at x = 3.01, y = 18/9.0601 ≈ 1.9867, a drop of about 0.0133 over a step of 0.01, a slope of about −4/3.
Where this goes wrong
Writing the slope of a level curve as U_x/U_y. Differentiating U = constant gives U_x + U_y dy/dx = 0, so the slope is −U_x/U_y, and an indifference curve slopes downward.
Leans on
Implicit differentiation, from Implicit differentiation and inverse trigonometric functions
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