Fast way to write out the utility optimization problem for a Cobb-Douglas function?

In my last problem set, I had to solve both the Utility Maximization Problem (UMP) and Expenditure Minimization Problem (EMP) for a Cobb Douglas utility function. Recall, Cobb Douglas is defined as $$U(x,y) = x^\alpha y^\beta$$

I can compute the EMP or UMP. I can then find the other by "duality". But writing the problem was very tedious and messy. I ended up with an indirect utility function like this

$$v(p_x,p_y,m) = \left(\frac{\alpha m}{p_x}\right)^\alpha \left( \frac{\beta m }{p_y}\right)^\beta$$

and doing the Lagrangian was kind of clunky with all this multiplication going on.

Is there a simpler way to write the problem that can save me time / effort?

• This question could be improved by spelling out what a UMP and EMP are. While it is often the case that Cobb-Douglas production functions have exponents that sum to 1 it is not required. – BKay May 11 '15 at 23:37
• Take the natural logarithm. – Alecos Papadopoulos May 12 '15 at 0:30
• That's a good idea. That's a monotonic transform, right? But doesn't that change the utility function? – Stan Shunpike May 12 '15 at 0:33
• It changes the cardinality of the utility function but not the preferences over bundles of x and y (preserves ordinality). – BKay May 12 '15 at 12:43

Well all Cobb-Douglas have an structure, it is the utility that makes shares constants as budget shares independent of prices $p_{l}x^{CD}_{l}(p,w)/w=\alpha_{l}$ with $u^{CD}(x)=\sum_{l}\alpha_{l}log(x_{l})$ and $\sum_{l}\alpha_{l}=1$. then you can go back and compute $x_l(p,w)$, with this the indirect utility, then invert to obtain the expenditure and then derive to get the hicksian. Not sure if this is what you want but some utility functions have a defining property, the same with the leontief or CES.