A hiring mandate that reduces hiring: constrained cost minimisation
A regional manufacturer produces a single good using labour L and capital K according to Q = L^0.5 K^0.5, with Q in thousands of units per month. The wage is €16 per hour and the rental rate of capital is €4 per hour; both inputs are variable over the planning horizon. The firm has signed a contract to deliver 100 thousand units this month and cannot store output, so it must produce exactly Q = 100. A condition attached to the industrial park's subsidised land requires the firm to employ at least 30 workers (L ≥ 30) at all times. Finance treats cost as linear in hours, but the operations team reports that they have always simply set L = K. The plant manager wants to know the cheapest compliant input mix, how much the employment condition actually costs, and whether to ask the park authority for a waiver that would be priced at €150 per month.
1. Set up and solve the cost-minimisation problem without the employment condition. Interpret the Lagrange multiplier.
Read the worked answer
Set-up. Minimise C = 16L + 4K subject to L^0.5 K^0.5 = 100. The Lagrangian is ℒ = 16L + 4K + λ(100 − L^0.5 K^0.5).
First-order conditions. 16 = λ(0.5)K^0.5/L^0.5 and 4 = λ(0.5)L^0.5/K^0.5. Dividing the first by the second gives 4 = K/L, so K = 4L. This is the tangency condition: the marginal rate of technical substitution equals the input price ratio.
Substitute into the constraint. L^0.5(4L)^0.5 = 2L = 100, so L = 50 and K = 200. Cost = 16(50) + 4(200) = 800 + 800 = €1,600 per month.
Interpret λ. Substituting back, 16 = λ(0.5)(200/50)^0.5 = λ(0.5)(2) = λ, so λ = 16. The multiplier is the marginal cost of an extra unit of output, consistent with the closed form C = 2Q√(wr) = 2(100)√64 = 1,600, whose derivative is 2√64 = 16. The equal-cost outcome is not a coincidence: at the optimum the firm spends equal amounts on the two inputs because the production function is Cobb-Douglas with equal exponents.
Check on the operations rule. Setting L = K would require L = K = 100 and cost 16(100) + 4(100) = €2,000, so the habit wastes €400 per month.
2. Now impose L ≥ 30. Find the compliant optimum, the extra cost, and the shadow price of the constraint.
Read the worked answer
Step 1 - does it bind? The unconstrained optimum has L = 50, which already satisfies L ≥ 30, so at first sight the constraint is slack and the cost should remain €1,600. That is the trap: the constraint is a minimum, and the unconstrained labour demand of 50 exceeds 30. The constraint binds only if the unconstrained L were below 30. Re-examining the numbers, the constraint is therefore non-binding at Q = 100 and L = 50.
Step 2 - what does bind is the opposite margin. If the park authority raises the minimum above 50, or if the firm's contract required a smaller output, the minimum would bite. To see the cost of the mandate as written, compare L = 50 with the minimum: it does not force substitution at all.
Step 3 - compute the case where it does bind. Suppose the requirement were L ≥ 60 (for example after an expansion of the subsidised space). Then K = 100²/L = 10,000/60 = 166.67, cost = 960 + 666.67 = €1,626.67, an extra €26.67. The multiplier is |dC/dL| = w − r(Q/L)² = 16 − 4(100/60)² = 16 − 11.11 = €4.89 per additional worker: each worker added beyond the optimum raises cost by €4.89 at that point.
Step 4 - the general method. Write C(L) = 16L + 4(10,000)/L, differentiate, and set the derivative to zero: 16 − 40,000/L² = 0 gives L = 50. For any mandated L ≠ 50 the excess cost is C(L) − 1,600, and the shadow price on the mandate is |C′(L)| at the mandated level. Check the borderline case: with L = 30 forced (if output were lower, say Q = 60, giving unconstrained L = 30), the condition is exactly satisfied and costs nothing.
3. The plant manager asks whether to request the €150 waiver. Using the structure of the cost function, advise her and explain the long-run policy implication of an employment mandate with substitutable capital.
Read the worked answer
Framing the decision. The waiver is worth paying only if the mandated input mix costs more than €150 per month relative to the unconstrained optimum. The excess cost is C(L*) − C(50), where C(L) = 16L + 40,000/L and L* is the mandated level. Because C is convex with a minimum at L = 50, the penalty is small for mandates near 50 and rises roughly quadratically as the mandate moves away.
Illustration. At L = 30, C = 480 + 1,333.33 = €1,813.33, so the excess is €213.33 and the waiver at €150 is worth buying. At L = 40, C = 640 + 1,000 = €1,640, an excess of €40, so the waiver is not worth it. The answer therefore depends on the exact mandated level, not on the principle.
Insight the manager should report. A minimum-labour rule can reduce employment. At L = 30 the firm hires fewer workers (30) than it would freely choose (50) and substitutes towards capital, raising the capital-labour ratio from 4 to 11.1. The mandate is only effective if the firm's unconstrained labour demand is below the floor; if the floor is set below the market outcome it is irrelevant, and if set above it may push capital substitution that offsets the intended effect, especially in the long run when both inputs are flexible.
Recommendation. Compute the excess cost at the actual mandated level before paying; request the waiver only if the excess exceeds €150; and in the negotiation, present the substitution arithmetic to argue for a floor linked to output rather than a headcount fixed in advance.