Consumer preference and price in the Cobb-Douglas function

Question

I believe I’m using the most basic version of Cobb-Douglas: $U(x,y)=x^\beta * y ^{(1-\beta)}$. The question I have is: in this example would a consumer’s preference ($\beta$) change if the price of either X or Y changes?

I.e., when presented with several options of a good at different prices the budget line will change for each. Given this change the indifference curve will at least shift, if the entire function doesn’t change altogether (different total Utility due to a different basket of goods).

This question is came up when solving for beta and utility using actual data where 1 good is X and the other is all goods not X (income less spend on X).

Got started solving for beta after seeing how they solved for the total derivative here.

Answer 1

No, preferences are stable. That is not to say that the quantity demanded or marginal utility obtained at the new price level is the same though.

If we'd allow the exponent of the utility function to vary for different price levels we'd let utility depend on prices, which it does not. Of course the actual utility that can be obtained depends on prices, but not the utility itself.

Back to your problem. You probably obtained a $\beta$ by observing one price and X quantity, with Y the remaining budget, filled that in in the formula:

$\frac{dY}{dX}=-\frac{\beta Y}{(1-\beta) X}$

and solved for $\beta$, and then did that again for a different price and found a different $\beta$. Given that you use real world data, that in itself is not surprising.

Unfortunately, the real world does not always match our models, so you would have to estimate $\beta$ using econometrics. Given that in CD models expenditures are fixed proportions of the budget you should be able to simply estimate:

$\beta=\frac{p_x X}{M} + \epsilon$

That is $\beta$ is the average proportion of the budget spent on X.