Utility Function

[aolattus]

Hi Pascal,

I was wondering where the separable utility function you used in the videos comes from? I am trying to understand how it connects to Cobb-Douglas, CES, and the log transform of Cobb-Douglas because it seems very similar but not quite the same. Do you have any resources for understanding the theory behind this function? Thanks!

[aolattus]

As a follow up question -- why does epsilon have to be greater than 1? My elasticity of substitution is 0.16 and I do get a solution in the model, so I'm not sure why this matters.

[pmichaillat]

You need $\epsilon > 1$ so the AD curve is decreasing in tightness $x$—as you can see in this lecture video. This property will guarantee that the model always has a solution and the solution is unique, as you can see in this video. Notice that if $\epsilon<1$, then $[1+\tau(x)]^{1-\epsilon}\to+\infty$ (instead of $\to 0$) when $x\to x^m$ so there is no guarantee that the model has a solution or the solution is unique.

The restriction that $\epsilon>1$ is quite common in macro models with consumption and wealth in the utility function. Blanchard & Kiyotaki (1987) also make this assumption for instance.

[aolattus]

This makes sense, thanks! I suppose a follow-up question would then be what should I think about given that my elasticity (which comes from the data) is less than 1?

My thought is that I need to find a way to say something about the solution: is it unique, etc. After solving the model, there does appear to be a unique solution in the range of values from 0 to the bound on supply, $k$ (or in my project I denote it $F$). Also, $[1+\tau(x)]^{1-\varepsilon}$ doesn't show up in this project, so how would I determine what I can and can't say about the solution?

Sorry for the questions, I am obviously now in the mathematical weeds of the project. :)

[pmichaillat]

Yes, you should be able to say whether the solution always exists and is unique. To do that, try to follow the steps from the course. In particular, if all the variables can be determined from tightness, then you only need to show that there is a unique tightness that solves the model. You can do that by studying the equation that implicitly gives tightness and showing that it always admits a solution and the solution is unique—as we do in this lecture video.

[aolattus]

Awesome, thank you!

[pmichaillat]

The separable utility function is just a concave transformation of the CES utility function. It is obtained by taking the CES utility function to the power of $(\epsilon-1)/\epsilon < 1$. Unlike the CES utility function this utility function is not homothetic (homogeneous of degree 1), but that does not matter since there is no growth in the model. Having a separable utility function greatly simplifies the analysis.

You can also see the utility function as the sum of two isoelastic utility functions: one over consumption plus one over relative real wealth.

[aolattus]

Ok yes, thank you, that clarifies this for me.