ECON7870 Chap.4 Budget Constraints and Consumer Choice
Budget Constraints and Consumer Choice
Combine feasibility with preference
Prices and income define the budget set, while preferences rank its bundles. Under monotonicity the budget binds.
A smooth interior optimum equates the consumer's MRS with the market price ratio, preventing any affordable marginal reallocation from improving utility.
Use the Lagrangian as a method
The multiplier system joins the objective and constraint, then uses first-order conditions to recover a bundle ratio and scale. The candidate still needs feasibility, non-negativity and a boundary comparison.
The multiplier measures the local value of relaxing the budget.
Read movements before solving again
An income change shifts both intercepts proportionally and preserves the slope. A change in one price rotates the budget around the intercept for the other good. Mark those intercepts before predicting the new optimum.
The graph supplies a direction check that can expose a swapped price or a demand response inconsistent with the expanded feasible set.
Audit the candidate
After solving first-order conditions, multiply quantities by prices and confirm total spending. Check non-negativity and compare relevant corners. A negative quantity means the interior calculation has left the feasible set; it is not a meaningful demand.
With indivisible goods, evaluate nearby whole-number bundles rather than rounding each quantity independently and breaking the constraint.
Interpret the multiplier locally
The Lagrange multiplier measures how much maximised utility rises from a small additional unit of income around the current solution. It does not remain constant across a large budget expansion. State its units and local nature.
In a managerial application, that shadow value can help compare the benefit of relaxing competing capacity constraints.
Separate prediction from verification
Before solving a changed budget, predict whether the line shifts or rotates and which intercept stays fixed. After solving, restore the budget equation and compare the candidate with relevant corners.
For smooth monotone preferences, unexplained unspent income is a warning; for a proposed interior solution, a negative quantity is decisive evidence that the maintained case is wrong. If goods are indivisible, calculate utility at nearby feasible integer bundles instead of rounding coordinates separately.
This two-stage routine uses geometry to predict direction and arithmetic to verify feasibility, making a transposed price or sign error easier to locate.
Handle non-interior cases explicitly
Tangency is a candidate rule, not a guarantee. Linear preferences can place the optimum at an intercept when the consumer's constant trade-off differs from the price ratio.
Fixed-proportion preferences direct spending to a kink because extra units of only one component add no utility. Quantity restrictions can create a new boundary even for a smooth utility function. In each case, list the feasible boundary candidates, calculate utility at each and compare them with any valid interior solution.
If a capacity cap binds, include its shadow value only as a local measure of relaxing that specific constraint. This boundary audit keeps a mathematically neat first-order solution from replacing the actual feasible choice. Report which boundary binds and why the rejected alternatives deliver lower utility, not merely that the numerical optimiser selected a corner.
What this chapter covers
- 01
Budget sets and lines
- 02
Price and income changes
- 03
Interior tangency
- 04
Lagrange multipliers
- 05
Marginal utility of income
- 06
Corner solutions
- 07
Kinks and boundaries
Worked example · free
Allocate a fixed budget
- 3Set MRS equal to the price ratio.
- 3Combine the ratio with the binding budget.
- 2Check spending and interpret the bundle.
Key terms
- Budget Line
- The boundary containing bundles that spend all available income at stated prices.
- Interior Optimum
- A utility-maximising bundle with positive quantities of every good and a valid tangency condition.
- Lagrangian
- An expression that combines an objective with a constraint using a multiplier.
- Shadow Value
- The local improvement in the optimised objective from a small relaxation of a constraint.
- Corner Solution
- An optimum at the boundary where at least one chosen quantity is zero.
Budget Constraints and Consumer Choice FAQ
What rotates a budget line?
A change in one price rotates the line around the intercept associated with the unchanged price. A change in income shifts both intercepts proportionally and leaves the slope unchanged. Label intercepts before describing the movement so a rotation is not mistakenly called a parallel shift.
What does the Lagrange multiplier mean?
At an interior optimum, the multiplier is the marginal utility of income: the local increase in maximised utility from one additional unit of budget. Its units are utility per dollar and its interpretation is local, so it should not be extrapolated across a large income change.
Why can first-order conditions produce a wrong answer?
They identify a candidate under smooth interior assumptions. A candidate may be negative, unaffordable, a local minimum or dominated by a corner. Perfect substitutes, perfect complements and quasi-linear preferences often make boundary checks decisive. Feasibility and shape complete the solution.
Exam move
Sketch the budget line with labelled intercepts before solving. Derive the MRS condition, use the budget to obtain scale, then test both corners even when an interior answer looks plausible. Recompute nearby feasible integer bundles when the application involves indivisible quantities.
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