# Consumer preference and price in the Cobb-Douglas function

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.

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.

• Is there anything wrong with adding additional factors to the preference calculation (if I find the variables to be significant in a regression)? In this situation I’m fairly certain that the consumer’s beta is influenced by income and household size. I’m imagining the process would be: estimate beta and error using the ratio and then try a regression that includes these additional factors as well. – vizie Jul 20 '18 at 13:19
• @vizie I'm far from an econometrician, so not sure. In fact I'm not even sure the technique proposed above would work. You can certainly ask that question here or at cross validated, though. Before you do, however ask yourself what the added value is. Are you really interested in having different betas for different groups? – Maarten Punt Jul 20 '18 at 13:28
• Will do. Thanks for the answer! The results of the real data are being incorporated into an agent based model (resulting in a version of the agents’ utility function) so if other features in the data have a significant impact on preference I’d like to be able to include them in the model as characteristics of the agents. – vizie Jul 20 '18 at 13:34