Recruiting for Jan-Apr internships is becoming really really difficult! Suppose the full cost (salary + hiring costs) for a 4-month Jan-Apr internship increases by a whopping $101,000.

With this change, how many students should the Port Authority hire for 4-month Jan-Apr internships?

Hint: You should be able to answer this question by inspecting the model solution, or by using the sensitivity report.

Question

Recruiting for Jan-Apr internships is becoming really really difficult! Suppose the full cost (salary + hiring costs) for a 4-month Jan-Apr internship increases by a whopping $101,000.

With this change, how many students should the Port Authority hire for 4-month Jan-Apr internships?

Hint: You should be able to answer this question by inspecting the model solution, or by using the sensitivity report.

BlackTom AI · Accepted Answer

Context and goal:
- The question describes a change in the full cost (salary + hiring costs) for a 4-month Jan-Apr internship, increasing by 101,000, and asks how many students Port Authority should hire for that period under the new cost.
- There are no answer choices provided, so we can’t select from alternatives. Instead, we’ll outline a clear method to determine the new hiring quantity using the model solution or a sensitivity report.

Step-by-step analysis:
1) Identify the baseline decision: Begin by locating the baseline optimal number of interns Q* for the 4-month Jan-Apr window from the model solution or the sensitivity report. This is the amount chosen before the cost increase.
2) Understand what changed: The problem states a fixed change in total cost by 101,000 for the 4-month internship. Determine whether this cost increase is a fixed amount (shifts the total cost curve upward by 101k regardless of Q) or whether it interacts with Q (e.g., a per-intern cost that scales with Q). The wording "full cost ... increases by 101,000" typically implies a fixed shift in total cost, but confirm from the model documentation.
3) Translate cost change into the optimization problem:
   - If the cost increase is a fixed total increment ΔC that applies to any level of Q, the objective typically remains of the form Net Benefit = Revenue/Value of intern output (or a benefit function B(Q)) minus Total Cost C(Q) (including the new fixed ΔC).
   - The new optimal Q is the value that maximizes B(Q) − [C(Q) + ΔC], which is equivalent to maximizing B(Q) − C(Q) with the same ΔC added to cost; in many cases, a fixed ΔC lowers the net benefit uniformly but does not change the marginal condition unless the optimization is constrained by a budget or capacity.
   - If the cost increase effectively raises the marginal cost of hiring each additional intern (e.g., if ΔC is tied to each unit), then you would recalculate the marginal condition where marginal benefit equals marginal cost plus the added ΔC per intern.
4) Use the sensitivity report if available:
   - A sensitivity (or shadow price) report often provides dQ*/dC, i.e., how the optimal quantity Q* changes in response to small changes in total cost C.
   - If the report gives a derivative or a direct table for the effect of a ΔC = 101,000, apply that to the baseline Q*: New Q ≈ Q* + (dQ*/dC) × ΔC. The sign of dQ*/dC will tell you whether hiring should increase or decrease as costs rise.
5) Consider practical constraints:
   - If there are capacity limits, hiring quotas, or minimums, these bounds will cap how much Q can change even if the economic optimum would suggest more or fewer interns.
   - If the model uses discrete units (integer number of interns), round to the nearest feasible integer after applying the sensitivity adjustment.
6) Derive the numeric answer once you have the necessary inputs:
   - Locate Q* from the baseline solution.
   - If ΔC is fixed and the marginal benefit schedule is unchanged, check whether the optimality condition (marginal benefit = marginal cost) remains satisfied; otherwise, recompute using the new cost function.
   - If the sensitivity report provides dQ*/dC, compute New Q = Q* + (dQ*/dC) × 101,000, ensuring you apply the sign correctly and respect any bounds.

What you can do right now with available resources:
- Open the model solution and locate the baseline Q* (the number of interns initially recommended).
- Check the sensitivity analysis for any stated dQ*/dC or explicit impact of a 101,000 change in total cost.
- If the report gives a direct instruction or snapshot (e.g., "a 101k increase reduces interns by 2"), apply it; otherwise, perform the marginal analysis as outlined above.

Common pitfalls and misconceptions to avoid:
- Assuming the number of interns must remain the same simply because a fixed cost increased; depending on the shape of the benefit and cost curves, the optimal Q can shift.
- Treating a fixed cost increase as if it alters per-intern marginal cost unless you’ve confirmed how the cost change enters the optimization.
- Ignoring constraints like minimum/maximum hiring or discrete unit changes when translating the sensitivity result into an actual integer Q.

In summary, without the exact model outputs or a clear specification of how the 101,000 change interacts with Q, you should rely on the baseline Q* from the model solution and use the provided sensitivity information to compute the new optimal Q, applying the change in total cost accordingly and respecting any practical constraints.

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