For the arguement of "Compansated variation is always bigger than consumer surplus under all price changes", could you please discuss whether it is true or not by drawing the necessary graphs? And i would be pleased if you can give a numerical example to support the arguement (utility function is a Cobb-Douglas utility function). thank you in advance.
This is discussed in the video, CV EV and Change in CS. The graph below is from the spreadsheet used to make that video. Let's review the definitions of CV and CS and then consider the determinants of which is bigger.
CV - this is the area to the left of the compensated (Hicksian) demand curve for the original optimum between the original price and the new price.
Decrease in CS - this is area to the left of the ordinary demand curve between the original price and the new price.
Remember that the compensated demand measures the substitution effect only, but that the ordinary demand measures the substitution effect and the income effect in combination. For a good where there is no income effect, CV = Decrease in CS.
More generally, what matters are:
(1) the direction of the price change, and
(2) whether the good is normal or inferior.
In the graph above the original price is given by the height of the dashed horizontal line. Then the price rises and the new price is indicated by the height of the dotted horizontal line. The blue curve is the ordinary demand curve. The red curve is the compensated demand curve for the original optimum. In this diagram, the blue curve is more elastic at the original price than the red curve. That will be the case for a normal good. The area to the left of the red curve between the two prices is greater than the area to the left of the blue curve between the two prices. Thus in this case CV > Decrease in CS.
I leave it to you to consider the case of a price decrease and/or the case where the good is inferior. By the way, if the utility function is Cobb-Douglas, then the good is normal.