Separate feasibility from preferences
Before choosing a bundle, establish which bundles are affordable. With non-negative quantities x and y, prices pₓ and pᵧ and budget m, feasibility means pₓx+pᵧy≤m. Equality describes the budget line; the inequality describes the budget set.
Preferences determine which affordable bundle is preferred. Do not assume an interior tangency or full expenditure without checking the assumptions. MIT OpenCourseWare provides a primary learning reference for this distinction.
Work an original two-good example
Take a fictional budget of 24 units and prices of 3 for x and 6 for y. The line 3x+6y=24 meets the axes at (8,0) and (0,4). These are hypothetical teaching quantities, not college prices.
With y vertical, y=4−0.5x. Moving right by one unit requires giving up half a unit of y along the line. The bundle (4,2) exhausts the budget; (2,2) costs 18 and is affordable but below the line.
Distinguish a rotation from a parallel shift
Reduce only the price of x to 2. The new line is 2x+6y=24, with intercepts 12 and 4 and slope −1/3. The unchanged vertical intercept shows the line rotating outward around that intercept.
Instead, keep both original prices and increase the budget to 30. The intercepts become 10 and 5 while the slope stays −0.5. This is a parallel shift. A larger feasible set alone does not reveal the chosen bundle.
Check reasoning before a tutorial
At the reduced x price, (6,2) costs 24 and lies on the line. Can you call it optimal? No: preferences have not been specified. This distinction is the central self-check.
Bring your own diagram and identify whether the problem concerns units, algebra, the graph or the economic interpretation. Practise on permitted teaching questions and produce assessed work independently.
References
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