A: Part of an exam problem that had surprisingly many strange answers ...
Consider a consumption-investment trade-off problem in T periods, where your running utility is \(((1+c)x)^{1?1/m}\), where the constant m is >1.
Here c is the fraction of current stock x that you consume. (That is, you get utility both from owning the stock and from consuming from it.) Being a fraction, so it should be in the interval [0,1]. If you choose \(c_t\) in period t, next period's stock will be \(x_{t+1}=(1?c_t)\cdot x_t\) .
The question:
If you own x at the start of the last period T, which c do you choose and what is then your indirect utility \(J_T(x)\) at that time?
(Do you think this was a trick question? You could be right.)
B: Dot product drill
- Let w be a given vector of length 1. Solve the problem \(\max \text{ or } \min \{\mathbf w'\mathbf x\}\ \text{subject to }||\mathbf x||=1\).
- Use the previous item to prove that \(-||\mathbf u||\ ||\mathbf v|| \leq \mathbf u\pmb\:\cdot\: \mathbf v\leq ||\mathbf u||\ ||\mathbf v||\).
C: Dot products: the "physics" definition.
In math courses that are too much hijacked by physicists (like when yours truly went to high school ...), one will sometimes define the dot product to be \(\mathbf u\pmb\:\cdot\: \mathbf v := ||\mathbf u||\ ||\mathbf v||\ \cos t\) where t is the angle in between them. That definition coincides with ours when we are in dimension 2 or 3; the following exercise shows a little bit of it:
- Exercise:
Recall the "unit circle" definition of cos and sin: (x,y) = (cos t, sin t). Let u be this vector (cos t, sin t), and dot it with something that has angle t with it.
Mathematicians generally agree with economists that the "Math 2" definition is better. In fact, we can define the angle between two nonzero vectors as \(\cos^{-1}\left(\frac{\mathbf u\pmb\:\cdot\: \mathbf v}{\sqrt{\mathbf u\pmb\:\cdot\: \mathbf u}\ \:\sqrt{\mathbf v\pmb\:\cdot\: \mathbf v}}\right)\), where "\(\cos^{-1}\)" is the inverse of the restriction of cos to [0,pi].
- Exercise:
Explain why the formula \(\cos^{-1}\left(\frac{\mathbf u\pmb\:\cdot\: \mathbf v}{\sqrt{\mathbf u\pmb\:\cdot\: \mathbf u}\ \:\sqrt{\mathbf v\pmb\:\cdot\: \mathbf v}}\right)\)is well-defined. - Exercise (this is outside Math 3, so skip it if you are not interested).
If we let X and Y be random variables with zero mean, and in place of the dot product, use E[XY], and define the "angle" between X and Y as \(\cos^{-1}\left(\frac{\mathsf E[XY]}{\sqrt{\mathsf E[X^2]\:\sqrt{\mathsf E[Y^2]}}}\right)\)
- what does it then mean (in stats/probability terms) that the angle is- 0?
- \(\pi/2\)?
- \(\pi\)?
D: Which direction does a function grow the fastest?
Let f be a (edit: continuously!) differentiable function of n variables. Fix a reference point z that is not a stationary point. The question to solve is: in what direction (from z) does f grow fastest?
To make the question more precise: Fix some h>0.
- Give the Lagrange conditions for the problem \(\quad \max_{\mathbf x}f(\mathbf z+\mathbf x)\quad\text{subject to}\quad \mathbf x'\mathbf x=h^2.\)
- What direction does x have as h tends to 0?