Do Cobb Douglas Utility Exponents HAVE TO Sum to One? 

De nada, Amigo. Thank me by liking, subscribing, and telling all of your friends! Glad to help.

You are welcome! Thanks for the GREAT question!

Let's solve for one of the point on the original graph where U=10 for U=x^.7* y^.1. Looking at the graph, it looks like when x=19, y= around 11. Solving 19^.7*y^.1=10 for y gives y=11.187. So, with the old utility function x=19, y=11.187 gives 10 utils. Now plug the same values into the other utility function, and we get 19^(1/8)*11.187^(7/8)=17.78 (which rounds to 17.8 utils)

Caveat: I certainly don't know everything there is to know about this... I am a "Jack of all trades, master of none." But, from what I can tell, There aren't really restrictions on the values that would matter for utility analysis, since "a positive linear transformation" of a utility function leaves underlying preferences unchanged. The only, small picky thing that one could quibble about is, if you had say, U=x^3*y^3, the the marginal utility of x = 3x^2y^3, which gets larger as x increases. So, this would seem to violate our standard rule that "marginal utility should be decreasing" as we consumer more of it. However, I guess the important thing isn't so much that it decreases, but that it decreases RELATIVE to the MU of other goods as you consume more x, ceteris paribus. So, as you consume more x, other goods look comparatively more and more attractive.

@@BurkeyAcademy Hi Mark, Would this Utility Function, U=x^3*y^3, be valid for an addictive substance/good (e.g. heroin, opioids, Pringles potato chips, etc.), i.e. once you consume the first little bit, you demand/crave more (not less)? Thanks, Lenny P.S. I enjoy your videos/explanations very much! Great job!!

@@BurkeyAcademy Incase of production function (Suppose we have K and L as our input) and the marginal return of K or L should be diminishing as we use more and more of the input of our interest by keeping fixed the other input constant. For example the ratio of K/L decreases as we increase L by keeping K constant. Therefore any new coming labor will have fewer and fewer capital to work with and hence its marginal return decreases. This diminishing marginal rate of return can be expressed mathematically by a a power function where its exponent is between 0 and 1.

You are right, you cannot change a production function because it measures a real thing-- units of output. That is why at 3:20 I stress that Utility is a made up concept, so the units do not matter. But units of production do matter, and so cannot be "normalized" in this way.

Great questions! It makes some of the common solutions a little simpler when when they add to 1. For example, Marshallian demand for x= Ba/[(a+b)px], where B= budget and a,b asre exponents for x and y. If (a+b) is set to be =1, the it simplifies this to x=Ba/px. Similar simplifications can be seen when you solve for Indirect utility, Hicksians, etc. So, it help write these generalized solutions in a much simpler way, which makes life easier when you really start taking a deep dive through the mathematics of consumer theory.

@@BurkeyAcademy that helps, ty!

1) This is only a useful concept in production functions, not utility. 2) In production functions you have to be careful not to confuse marginal returns with returns to scale. Marginal returns asks what happens to output when we add more of ONE input. Returns to scale is when we are scaling up all inputs by a certain proportion. In short, diminishing marginal returns would require each individual exponent to be less than one, but the total of them relates to what is going on with returns to scale.

@@BurkeyAcademy thanks, awesome response I understand it now

@@jadonschroeder1583 Very nice- That's what I love to hear!

I don't understand the question. Are you asking "if the exponents are the same, do they buy the same amount of the two goods?" If so, no- it depends on the prices.

@@BurkeyAcademy Sorry, let me try to clarify: If the exponents are the same, does x1 times price of good 1 equal x2 times price of good 2 in the optimal bundle

@@bakteriebakterie6096 Yes... If the exponents are the same then the consumer spends exactly half of their income on each good. So PxX =PyY =B/2

@@BurkeyAcademy Thank you! Struggled to find proof so i wanted to make sure:D