Difficulty: Intermediate | Prerequisites: Marginal analysis (PS2 Q1), basic calculus
When you have a fixed budget to split between two activities, the optimal split equalises the marginal benefit per dollar across both. If one activity's MB per dollar is always higher than the other's, you spend the entire budget on that activity (a corner solution). The budget constraint replaces the simple MB = MC rule with an "equal bang for your buck" rule.
Constrained optimisation
Choosing the best allocation of a scarce resource (money, time, capacity) across competing uses, subject to a constraint that limits total spending. Think of it as: you cannot have everything, so you must decide how to divide what you have.
Budget constraint
The equation that says total spending cannot exceed the budget. Here: P × H_E + P × H_B ≤ M, where P = price per hour and M = total budget. In simple terms, the hours of engine work plus body work, at $100 each, cannot cost more than $2,000.
Marginal benefit per dollar (MB/P)
The additional benefit you get for each additional dollar spent on an activity. Calculated as MB divided by the price of that activity. This is the measure you equalise across activities at the optimum.
Equal marginal principle ("bang for your buck")
At the optimal allocation, the last dollar spent on each activity should yield the same additional benefit: MB_E / P_E = MB_B / P_B. If one side is higher, you should shift spending toward it.
Corner solution
An optimum where you spend the entire budget on one activity and nothing on the other. This happens when one activity's marginal benefit per dollar remains higher than the other's across the entire feasible range.
You have a $2,000 budget. Each hour of repair costs $100. Total hours available: 2000/100 = 20 hours to split between engine work (H_E) and body work (H_B).
Benefit functions:
Engine: B_E(H_E) = 250H_E − 25(H_E)²
Body: B_B(H_B) = 250H_B − 15(H_B)²
Constraint: 100H_E + 100H_B = 2000, which simplifies to H_E + H_B = 20.
MB_E(H_E) = dB_E/dH_E = 250 − 50H_E
MB_B(H_B) = dB_B/dH_B = 250 − 30H_B
Maximise B_E(H_E) + B_B(H_B) subject to H_E + H_B = 20, with H_E ≥ 0 and H_B ≥ 0.
Since both activities have the same price ($100), equalising MB per dollar reduces to equalising MB directly:
MB_E(H_E) = MB_B(H_B)
250 − 50H_E = 250 − 30H_B
−50H_E = −30H_B
50H_E = 30H_B → H_B = (5/3)H_E
Substitute into the constraint: H_E + (5/3)H_E = 20 → (8/3)H_E = 20 → H_E = 7.5 hours.
Then H_B = 20 − 7.5 = 12.5 hours.
The intuition: body work has diminishing returns that kick in more slowly (the coefficient on H² is 15, not 25), so you allocate more hours to body work.
Now suppose B_B(H_B) = 900H_B − 15(H_B)². The new marginal benefit is:
MB_B(H_B) = 900 − 30H_B
Set MB_E = MB_B: 250 − 50H_E = 900 − 30H_B.
Using the constraint H_B = 20 − H_E:
250 − 50H_E = 900 − 30(20 − H_E)
250 − 50H_E = 900 − 600 + 30H_E
250 − 50H_E = 300 + 30H_E
−50 = 80H_E
H_E = −0.625
A negative number of hours is not feasible. This means the interior solution does not exist. Body work's marginal benefit per dollar is always higher than engine work's over the entire feasible range.
The corner solution: spend the entire budget on body work. H_E = 0, H_B = 20.
To verify: at the corner, MB_B(20) = 900 − 600 = 300, and MB_E(0) = 250. Since MB_B is still higher than MB_E even after spending the full budget on body work, shifting any money to engine work would reduce total benefit.
Equal marginal principle (same price for both activities):
MB_E(H_E) = MB_B(H_B)General form (different prices):
\frac{MB_E}{P_E} = \frac{MB_B}{P_B}Budget constraint:
P_E \cdot H_E + P_B \cdot H_B = MCorner solution test: if the interior solution gives a negative quantity for one activity, set that activity to zero and allocate the full budget to the other.
Students often equalise total benefits (B_E = B_B) rather than marginal benefits. The total levels do not need to be equal; it is the marginal benefit per dollar that must be equalised.
Forgetting the constraint when solving. After setting MB_E = MB_B, you still need to substitute into the budget constraint to find the actual values. Two equations, two unknowns.
Assuming a negative solution means "the answer is zero for both." A negative interior solution means one activity dominates; you spend the full budget on the dominant one, not nothing on both.
Confusing MB = MC (unconstrained) with MB_E/P_E = MB_B/P_B (constrained). In constrained problems, cost is already embedded in the budget constraint, so you compare marginal benefits per dollar, not marginal benefit to marginal cost.
⚠️ Expect a problem that requires you to identify whether the solution is interior or corner. The negative-quantity check is the signal.
⚠️ When prices differ between activities, you must divide MB by price before equalising. The equal-price case in this problem set is the simple version.
⚠️ This constrained framework is the stepping stone to consumer choice theory: replace "activities" with "goods," "marginal benefit" with "marginal utility," and "budget" with "income," and you have the standard consumer optimisation problem.
True or False: In a constrained optimisation problem, you set MB equal to MC for each activity separately. _______
If MB_A/P_A > MB_B/P_B, you should shift spending toward activity _______.
True or False: A corner solution means the budget constraint is not binding. _______
The equal marginal principle says: at the optimum, the last _______ spent on each activity yields the same additional benefit.
True or False: If the interior solution gives H_E = −3, you should set H_E = 3 and adjust H_B accordingly. _______
Answers: 1. False (you equalise MB per dollar across activities, subject to the budget constraint). 2. Activity A. 3. False (the budget is still fully spent; you just spend it all on one thing). 4. Dollar. 5. False (you set H_E = 0 and give the full budget to H_B).
Q: Given B_E(H_E) = 250H_E − 25(H_E)² and B_B(H_B) = 250H_B − 15(H_B)², with a budget of $2,000 and a price of $100/hour, find the optimal allocation.
A: MB_E = 250 − 50H_E, MB_B = 250 − 30H_B. Set MB_E = MB_B (same price): 250 − 50H_E = 250 − 30H_B → H_B = (5/3)H_E. Constraint: H_E + H_B = 20 → H_E + (5/3)H_E = 20 → H_E = 7.5, H_B = 12.5.
Q: If B_B changes to 900H_B − 15(H_B)², what happens to the optimal allocation?
A: MB_B = 900 − 30H_B. Setting MB_E = MB_B and substituting the constraint gives H_E = −0.625, which is infeasible. Corner solution: H_E = 0, H_B = 20. At the corner, MB_B(20) = 300 > MB_E(0) = 250, confirming body work is always more valuable per dollar.
Q: Suppose two goods cost $5 and $10 respectively. You have $100. MB_A = 40 − 2Q_A and MB_B = 60 − 4Q_B. Find the optimal bundle.
A: Equalise MB/P: (40 − 2Q_A)/5 = (60 − 4Q_B)/10 → 2(40 − 2Q_A) = 60 − 4Q_B → 80 − 4Q_A = 60 − 4Q_B → 4Q_B − 4Q_A = −20 → Q_B = Q_A − 5. Constraint: 5Q_A + 10Q_B = 100 → 5Q_A + 10(Q_A − 5) = 100 → 15Q_A = 150 → Q_A = 10, Q_B = 5.
This framework leads directly into consumer theory (ECON300 Q3 and Q4), where "maximise utility subject to a budget" is the same structure with utility replacing benefit. The equal marginal principle becomes MU_X/P_X = MU_Y/P_Y. It also connects to firm input choice: a firm choosing between labour and capital subject to a cost constraint uses the identical logic with marginal products.
constrained optimisation, budget constraint, marginal benefit per dollar, equal marginal principle, bang for buck, corner solution, interior solution, resource allocation, two-good optimisation, constrained maximisation, Lagrangian (related method), ECON300, intermediate microeconomics, optimal allocation