Derivation of compensated demand curve:
Hicksian compensated demand function for x_{1} is given by x_{1}=x_{1}(p_{1}, p_{2}, U), where Hicksian compensated demand curve for a good represent the relationship between price of that good with its own demand quantity for given prices of other goods and real income in terms of utility.
We now derive this graphically. Suppose, initial equilibrium is attained at e0 in Figure A where price of good on is p_{1}^{0} and price of good two is p20 respectively and utility is fixed at U0. Corresponding indifference curve is IC0. Compensated Hicksian demand for x_{1} is at x10. Expenditure line is AB at initial equilibrium with absolute slope p_{1}^{0}/p_{2}^{0}. Plot this x_{1}^{0} and p_{1}^{0} in Figure B. Suppose, for given utility and p_{2}, p_{1} decreases to p_{1}^{1}. Therefore, absolute slope of the budget line decreases, i.e., expenditure line become flatter. Since utility is constant, the indifference curve remains the same as before. Therefore, expenditure is minimised for given utility at point e1 in Figure A, as indifference curve is downward sloping strictly convex to the origin. So compensated Hicksian demand for good I increases to x_{1}^{1} plot p_{1}^{1} and x_{1}^{1} in Figure B. By joining all such pair of p1 and x1 in Figure B, we have a downward sloping curve in p_{1}-x_{1} plane, for given p_{2} and utility. This downward sloping demand curve is the Hicksian compensated demand curve. This is shown in the above Figure B.