Quadratin Utility Function

The core reading says that the general form of the quadratic utility function is

U(w) = a + bw + cw^2.

It then says that, since adding a constant to a utility function, or multiplying it by a constant will not affect the decision making process, we can write the general form as:

U(w) = w+ dw^2.

Why can we write this?

If constants don't change anything, can't we write:

U(w) = w + w^2 ?

 

Adding a constant to a utility function, or multiplying it by a constant will not affect the decision making process:

Therefore add (-a) to U(w) & multiply results by (1/b) & set d=(c/b). The resulting utility function will give the same results.



Step by step:

Add (-a) to U(w):

Then U(w) = a + bw + cw^2 is equivalent to U1(w) = bw + cw^2

(since adding a constant to a utility function will not affect the decision making process)

[Where U1(w)=U(w)-a]


Multiply U1(w) by (1/b):

Then U1(w) = bw + cw^2 is equivalent to U2(w) = w + (c/b)w^2

(since multiplying a utility function by a constant will not affect the decision making process)

[Where U2(w)=(1/b)*U1(w)=(1/b)*U(w)-(a/b)]


Let U2(w) be U(w) & (c/b) be d

Thus U(w) = a + bw + cw^2 can be written as:

U(w) = w + dw^2

 

As you pointed out constants don't change, but writing the function as:

U(w) = w + w^2 would mean setting the constant d=(c/b)=1 in the equation U(w) = w + dw^2

Which may end up changing the non-changing value of the constant d (unless of course d=1 to begin with).

Does this help?

 

In terms of making investment choices, it's the curvature of the utility function (and hence the degree of risk aversion) that's important and not its vertical intercept.

Changing "a" (the vertical intercept) just shifts the utility function up or down on a graph but doesn't affect its curvature, or the degree of risk aversion, and hence won't affect investment choices. This is why "a" doesn't appear in the absolute and relative risk aversion formulae for the quadratic and why you can just delete it without changing anything.

The curvature of the utility function then depends on the ratio of "c" to "b" and not on their individual values. So, you can divide them both by "b", so that "b/b" becomes 1 and write "c/b" as "d<0", giving U = w + dw^2, without affecting the important (ie choice-influencing) properties of the quadratic utility function.

 

When we divide by "b" the curvature of the utility function still changes.

For e.g u(w) = 2x - x^2
Multiplying through by 1/2 gives; u1(w) = x - (x^2)/2 (where d=1/2)

The curvature of u(w) is different from the curvature of u1(w), which then affects the risk aversion hence decision making process?

 

Really? Why do you say that u(w) has a curvature that is different to that of U1(w)?

Using, for example, absolute risk aversion as a measure of curvature you will easily find that the curvature is in fact the same. [Solution being 1/(1-x)]

How are you defining curvature?

 

Ohh thanks I completely misunderstood the curvature concept. I thought its just the shape of the curve and the difference between the curvature of u(w) and u1(w) can be seen by plotting the two curves which is what I did in my graphical calculator.

However, I can see now by using the correct formula for curvature it is indeed the same. Is relative risk aversion a measure of curvature as well? I can't recall coming accross curvature formulae like ARA and RRA before or may be these were named differently in calculus.

 

This thread has gone a long way to helping me understanding it but I still don't understand, if multiplying by a constant doesn't change things, why we need the 'd' in there?

U(w) = w + dw^2 should be the same as U(w) = w + w^2 if you're saying it doesn't matter what 'd' is (yes, I realise d needs to be 1 in algebra for this to match, but you're saying that multiplying by ANY constant doesn't change things, so you're saying d could be 3 and it still wouldn't change things). It clearly does here though.

Confused.

 

The value of "a" in U = a + bw + cw^2 doesn't matter, as all "a" does is shift the utility function vertically upwards and downwards. It doesn't affect the shape or curvature of the function, which is what is important. So, setting a = 0 doesn't affect anything important.

The ratio of "c/b" is important, as that is what determines the shape / curvature of the curve and hence the degree of risk aversion. So, if you divide both "b" and "c" by "b" and call them "1" and "d = c/b" respectively, then you haven't changed anything, other than reducing the apparent number of parameters in the function. This is different to "multiplying by a constant", which will change things.

For example, the absolute risk aversion function is:

ARA = -2c/(b+2cw) = -2d/(1+2dw)

With c = -4, b = 2 this is:

ARA = 8/(2-8w)

and with d = c/b = -2

ARA = 4/(1-4w)

which is exactly the same.

However, multiplying any of b, c and d by a constant will change the function.

 

Are you trolling?