Step 1: Restate the consumer's problem in plain terms:
The consumer wants to be as happy as possible (reach the highest indifference curve) but is limited by a fixed budget (must stay on or inside the budget line) — equilibrium is simply the best affordable bundle.
Step 2: Explain why any non-tangency point is NOT equilibrium:
If the budget line crosses (rather than just touches) an indifference curve, the consumer can slide along the budget line to a point on a HIGHER indifference curve while still spending the same total income — so they haven't yet maximised satisfaction; they will keep adjusting until no such improving move is possible, which happens only at a tangency.
Step 3: Explain why convexity/diminishing MRS is also required:
If the indifference curve were concave at the touching point, moving slightly away along the budget line would actually reach a HIGHER indifference curve on either side — so a concave tangency is a point of minimum, not maximum, satisfaction; only a convex (diminishing-MRS) curve guarantees the tangency is truly the best point.
Step 4: Diagram description:
Multiple convex indifference curves nested like contour lines, with a straight budget line touching the outermost affordable one at a single point E — that tangency point E is equilibrium, where the budget line's slope ($-P_x/P_y$) matches the indifference curve's slope ($-MRS_{XY}$).
Final Answer:
Equilibrium requires (i) tangency, $MRS_{XY}=P_x/P_y$, and (ii) the indifference curve being convex (diminishing MRS) at that point, jointly ensuring maximum satisfaction within the budget.