Microeconomics Cobb-Douglas Utility Function

[iheartmicro]

Hello,

I have been learning about Cobb-Douglas Utility functions and Cobb-Douglas Production functions. Some of the reading I have incontured have left me confussed. Does anyone know how the utility function difffers from the production function in terms of explaining the exponents. I know that Cobb-Douglas production function's exponents measure returns to scale, but what I am confussed on is what do the utility function's exponents measure. Is it elasticity or still returns to scale? I know that they must be positive and add up to 1.

If anyone could enlighten me on the differences that would be of great help.
Thank you :)

 

[Pyrrhus]

The difference is that for Production Theory, the magnitude of the output is important. In the case of utility theory, the magnitude of utility is not important (except in some cases such as welfare theory). In other words, cardinality of production matters, but only ordinality of utility matters. Thus, the exponents of utility functions are made to sum to one rather than being different (they do not exhibit the "IRS", or "DRS".. Only "CRS" if you want to use production theory nomenclature).

 

[iheartmicro]

Thank you. I just want to make sure that I understand what you are sying though. To define the exponents of the basic Cobb Douglas Utility Funtion you would not mention anything about returns to scale. I still am confussed at what exactly the exponents purpose are in the basic Cobb Douglas Utility Function. I have looked in textbooks and online and unable to understand what they represent.

 

[Pyrrhus]

It depends on what you want to know.

From a mathematical perspective

The Cobb Douglas function is an homogenous function of degree equal to the sum of the exponents.

When the sum is greater than 1 is IRS, less than 1 DRS, and equal to 1 CRS.

For utility function, we only use equal 1 CRS.

From a microeconomics perspective,

Utility functions are mapping representing preference orderings. In other words, if you have

A, B, and C, and you have some preference order such as A > B > C, then such can be conveyed in a function known as utility functions as

U(A) > U(B) > U(C)

However, for this to work we can use any sort of values where it holds.

For example, U(A) = 5, U(B) = 3, U(C) = 1 or U(A) = 25, U(B) = 15, and U(C) = 5.

So you can see the actual value doesn't matter. We just care about the preservation of the order. This is why, we can solve max utility problems with positive monotonic transformations (such as the natural log of the utility function), because we only care about order.

Now, if you want to know what different values of exponents do to the Cobb Douglas utility function, you could graph it, and see how it affects the preferences.

 

[iheartmicro]

Thank you. That helped :)