Derivation of compensated demand curve:
Hicksian compensated demand function for x1 is given by x1=x1(p1, p2, 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 p10 and price of good two is p20 respectively and utility is fixed at U0. Corresponding indifference curve is IC0. Compensated Hicksian demand for x1 is at x10. Expenditure line is AB at initial equilibrium with absolute slope p10/p20. Plot this x10 and p10 in Figure B. Suppose, for given utility and p2, p1 decreases to p11. 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 x11 plot p11 and x11 in Figure B. By joining all such pair of p1 and x1 in Figure B, we have a downward sloping curve in p1-x1 plane, for given p2 and utility. This downward sloping demand curve is the Hicksian compensated demand curve. This is shown in the above Figure B.