The dual problems of

utility maximization & expenditure minimization

Using the interactive graphs linked below, try changing any of the parameters to see the ordinary (Marshallian) and compensated (Hicksian) demand response.

Cobb-Douglas example

Try increasing the price of x, and notice how both the ordinary and the compensated quantity demanded for x decrease, but x^* decreases faster than x^c (ordinary demand is more elastic).

Notice also that (when Px increases) the compensated response for y is positive, but the ordinary response is zero: this is because the income effect exactly cancels out the substitution effect, thus the cross-price elasticity of demand for Cobb-Douglas is zero: x and y are neither (gross) complements nor substitutes.

Quasilinear utility

Because x enters non-linearly and y enters linearly in the utility function, MRS will only depend on x. This means that the ratio of prices is sufficient to determine the quantity of good x consumed. Then x^c=x^*, and the income effect on x is zero.