Question

Please answer the following question. Please show all your working/solutions.

Optimality conditions in general frequently require us to relate slopes of different curves (for example, the slope of an indifference curve and the slope of a budget constraint). For this reason, we deal with a lot of derivatives. The logarithm has a particularly convenient form, so we will use it extensively in this course. Consider the function f(X) = a log(X) where a is a parameter with a > 0. You may assume that log(X) is the same as the natural logarithm, ln(X).

(a) What is the first derivative of f?

(b) What is the second derivative of f?

(c) Is f convex or concave (and how do you know)? (Hint: Recall that the log function is only well defined for X > 0.)

Answer 1

y = f(X) = a log X

(a)

First derivative = f'(X) = dy/dX = d/dX(a log X)

= a / X

(b)

Second derivative = f''(X) = d²y/dX² = d/dX(a / X) = d/dX(aX^-1) = -1 x a x X^-2 = -a / X²

(c)

Since X > 0, X² > 0, therefore (-a / X²) < 0.

Since second derivative is negative, the function is concave.