Two kilns, one emissions cap: where should the retrofit money go?
A cement producer operates two kilns and is the only domestic supplier. Market demand is P = 120 − Q, where Q is thousands of tonnes per year and P is euros per tonne. Kiln 1 has marginal cost MC₁ = 20 + 2q₁; Kiln 2 has MC₂ = 20 + q₂, with q measured in thousands of tonnes and cost in euros per tonne. Fixed costs are sunk and there are no capacity limits on either kiln. A binding emissions rule caps the firm's total output at 30 thousand tonnes per year; allowances cannot be bought or sold, and the cap applies to total tonnage. The firm currently splits output evenly, 15 thousand tonnes per kiln, because that is how the plant managers have always scheduled production. The operations director proposes a €240,000 retrofit of Kiln 2 that would change its marginal cost to MC₂ = 20 + 0.5q₂, arguing that 'cheaper abatement at Kiln 2 is the obvious place to spend'. The board must decide within a month, and wants the production plan justified rather than asserted.
1. Ignoring the emissions cap for a moment, derive the profit-maximising output levels for each kiln, the market price, and the firm's profit. Explain the condition that determines the split between kilns.
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Step 1 - one price means one marginal revenue. The firm sells a homogeneous good at a single price, so MR = 120 − 2Q is common to both kilns. Splitting production is optimal only when MC₁ = MC₂; otherwise shifting one tonne from the high-cost to the low-cost kiln reduces cost with no change in revenue.
Step 2 - solve the system. Setting MC₁ = MC₂: 20 + 2q₁ = 20 + q₂, so q₂ = 2q₁ and Q = 3q₁. Setting MR = MC: 120 − 6q₁ = 20 + 2q₁, so 100 = 8q₁, giving q₁ = 12.5 and q₂ = 25. Then Q = 37.5 thousand tonnes, P = 120 − 37.5 = 82.5, and MC = 45 = MR at the optimum, as required.
Step 3 - verify profit. Revenue = 82.5 × 37.5 = 3,093.75 thousand euros. Variable cost = ∫₀^12.5(20 + 2q)dq + ∫₀^25(20 + q)dq = (250 + 156.25) + (500 + 312.5) = 1,218.75 thousand euros. Profit = 1,875 thousand euros.
Step 4 - interpret the equal split. At 15/15, MC₁ = 50 and MC₂ = 35: the firm is producing too much in the expensive kiln. Reallocating towards Kiln 1 until the marginal costs meet is a pure cost saving, which is why the 15/15 rule is a costly habit rather than a neutral convention.
2. Now impose the 30 thousand tonne cap. Find the cost-minimising allocation, the price, and the shadow price of the cap. Quantify what the 15/15 rule costs under the cap.
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Step 1 - the cap binds. Unconstrained output is 37.5 > 30, so the cap is binding and the firm produces Q = 30. Price follows from demand: P = 90. Revenue = 2,700 thousand euros.
Step 2 - minimise cost subject to Q = 30. The only decision is the split. Equalising marginal costs: 20 + 2q₁ = 20 + q₂ with q₁ + q₂ = 30 gives q₂ = 2q₁ and 3q₁ = 30, so q₁ = 10, q₂ = 20, MC = 40. Cost = ∫₀^10(20 + 2q)dq + ∫₀^20(20 + q)dq = (200 + 100) + (400 + 200) = 900. Profit = 1,800 thousand euros.
Step 3 - shadow price. At Q = 30, MR = 120 − 60 = 60 exceeds MC = 40. Marginal profit from one extra tonne is 60 − 40 = 20, so the shadow price of the cap is €20 per tonne (€20,000 per thousand tonnes). This is the maximum the firm should pay for a relaxation of the cap, and it is the number to use in any negotiation over allowances.
Step 4 - cost of the 15/15 habit. At 15/15, cost = (300 + 225) + (300 + 112.5) = 937.5, which is 37.5 thousand euros above the minimum. The habit is not a rounding error: it is 4% of profit under the cap.
3. Evaluate the €240,000 retrofit using the capped production plan. Should the board approve it? Under what conditions would your answer change?
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Step 1 - re-optimise with the new cost function. With MC₂ = 20 + 0.5q₂, equalising marginal costs under Q = 30 gives 20 + 2q₁ = 20 + 0.5q₂, so q₂ = 4q₁ and 5q₁ = 30, hence q₁ = 6 and q₂ = 24. MC = 32.
Step 2 - compute the profit change. Cost = ∫₀^6(20 + 2q)dq + ∫₀^24(20 + 0.5q)dq = (120 + 36) + (480 + 144) = 780. Revenue is unchanged at 2,700 because the cap fixes output and price, so profit rises from 1,800 to 1,920: a gain of 120 thousand euros.
Step 3 - compare with the cost. €120,000 < €240,000, so the retrofit does not pay back. Reject it on these numbers. The breakeven retrofit cost is €120,000.
Step 4 - conditions that flip the answer. (i) If the cap were relaxed or tradeable, the retrofit would be worth more: it lowers marginal cost at the margin, and the shadow price rises from 20 to 60 − 32 = 28, so the value of extra allowances rises by €8 per tonne. If the firm could buy allowances at a price between 20 and 28, the retrofit plus extra allowances could dominate. (ii) If the regulation were expected to tighten further, saving abatement cost would matter less, not more, because output would fall. (iii) If a €120,000-plus subsidy were available, the arithmetic changes. The board should ask for the allowance price and the regulatory trajectory before signing, not just the engineering quote.