# Homework Help: Montonic Functions (Econ)

1. Jun 23, 2010

### djalap

1. The problem statement, all variables and given/known data
Suppose that the utility functions u(x,y) are related to v(x,y) = f(u(x,y)). In each case below, select "Yes" if the function f is an increasing, monotonic transformation and "No" if it is not.

2. Relevant equations

A differentiable equation f(u) is an increasing function of u if its derivative is positive.

3. The attempt at a solution

f(u) = 3.141592u (YES)
f(u) = 5000-23u (YES)
f(u) = u-100000 (YES)
f(u) = log(base 10)u (NO)
f(u) = -e^-u (NO)
f(u) = 1/u (NO)
f(u) = -1/u (YES)

My answers are in parenthesis. Are they right? If not, can you explain briefly why?

2. Jun 23, 2010

### Staff: Mentor

Above, check the derivative. Is it positive?
Above, Why do you think this?
Above, check the derivative. Is it negative?
The two above are a little tricky. For the first, the derivative is not defined for u = 0, so, for example, even though 1/u is decreasing on each half of its domain it is not decreasing overall. -1 < 1, but 1/-1 < 1/1, which should not be the case for a decreasing function.

3. Jun 23, 2010

### djalap

Yes
No
Yes
Yes
IDK? (NO?)
No
Yes

4. Jun 23, 2010

### Staff: Mentor

d/du(log u) is not a constant.
Yes, but that's not the function - it is -e-u. What is d/du(eu)? What is d/du(e-u)? Take another look at the differentation rules for log x, ln x, ex.

5. Jun 23, 2010

### djalap

so d/du (log u) = 1/(u LN 10) = no?

d/du (-e^-u) = e^-u = no

6. Jun 23, 2010

### Staff: Mentor

You have the derivative right, but isn't 1/(u ln 10) > 0?
Again, the derivative is right, but isn't e-u > 0?

7. Jun 24, 2010

### djalap

With U in the function, can't U be negative, making the answer negative?

8. Jun 24, 2010

### Staff: Mentor

The domain for log u is {u | u > 0}. This has an effect on the values of 1/(u ln 10).
The domain for e-u is all real numbers, and the range is the same as for eu. (The graphs are different, of course.)

9. Jun 24, 2010

### djalap

So they are both positive.

Because e^-u is always positive.

Right?

10. Jun 24, 2010

### Staff: Mentor

"They" being the derivatives - yes.