Housing versus greenway on a fixed council budget
Mayo County Council has a three-year capital budget of €60 million and 900 construction worker-years. It is choosing between social housing and a greenway. Each social home costs €150,000 and requires 2 worker-years; each kilometre of greenway costs €1.2 million and requires 20 worker-years. Land is available, but electricians are scarce and are needed mainly for housing. The council can also apply for an EU grant covering 40% of greenway construction costs; no grant is available for housing. Officials claim the greenway is 'free' because the grant reduces the bill. Local housing groups argue that every kilometre built delays homes for waiting lists. The council must publish a three-year investment plan and justify any point inside, on or outside its production possibility frontier. The finance officer notes that the budget is cash-limited, while the construction labour force is fixed by the training pipeline.
1. Construct the council's production possibility frontier before the grant. Identify the maximum number of homes and maximum greenway kilometres, and explain why the frontier is not a straight line throughout.
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Let H be homes and G be greenway kilometres.
Budget constraint: 0.15H + 1.2G ≤ 60 (millions), which simplifies to H + 8G ≤ 400. Labour constraint: 2H + 20G ≤ 900, which simplifies to H + 10G ≤ 450.
Maximum homes: set G = 0. Budget gives H = 400; labour gives H = 450. The budget binds, so maximum homes = 400. Maximum greenway: set H = 0. Budget gives G = 50; labour gives G = 45. Labour binds, so maximum greenway = 45 km.
The two constraints intersect where H + 8G = 400 and H + 10G = 450. Subtracting gives 2G = 50, so G = 25 and H = 200.
For G between 0 and 25, the budget is the binding constraint and the frontier has slope −8 homes per km. For G between 25 and 45, labour is binding and the slope is −10 homes per km. The frontier is therefore concave to the origin: opportunity cost rises as greenway expands because increasingly scarce construction labour must be diverted from housing. Points on the frontier are productively efficient; points inside are inefficient; points outside are currently unattainable.
2. Calculate the opportunity cost of one kilometre of greenway at G = 10 and at G = 35 before the grant. Recalculate after the 40% grant. Does the grant make greenway 'free'?
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Before the grant:
- At G = 10, the budget binds. One extra km costs 8 homes because H = 400 − 8G.
- At G = 35, labour binds. One extra km costs 10 homes because H = 450 − 10G.
After the grant, the council pays only 60% of greenway cost: €0.72 million per km. The budget becomes 0.15H + 0.72G ≤ 60, or H + 4.8G ≤ 400. Labour is unchanged: H + 10G ≤ 450.
New intersection: H + 4.8G = 400 and H + 10G = 450, so 5.2G = 50 and G ≈ 9.62, H ≈ 353.85.
- At G = 10, labour now binds, so the opportunity cost is 10 homes per km.
- At G = 35, labour binds, so the opportunity cost is still 10 homes per km.
The grant lowers the opportunity cost only while the budget is the binding constraint, at low greenway output. Once labour becomes scarce, the true cost is measured in homes forgone, not euro. Greenway is not 'free': the grant changes who pays, not the resources used. The opportunity cost includes scarce labour and the housing not built.
3. Advise the council on whether to accept the grant and choose a point beyond 25 km. Discuss two factors not captured by the PPF.
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Accepting the grant shifts the budget segment outward and enlarges the feasible set, so it is rational to accept if the council values greenway. But beyond 25 km the labour constraint binds: each extra km costs 10 homes, not 8. The council should compare the marginal social benefit of greenway with the marginal cost in homes, especially waiting-list need. A point at 25 km uses the two resources exactly (H = 200, G = 25) and avoids the steeper labour trade-off. Beyond 25 km, housing losses accelerate.
The PPF omits at least two relevant factors. First, electricians are a specialised bottleneck: the model treats all construction labour as homogeneous, but housing may need skills that greenway does not. Second, the grant may carry conditions, matching funding or time limits, and greenway may bring tourism or health benefits not measured in homes. Distributional effects also matter: the gainers and losers are not the same households. A judgement should therefore use the PPF as a feasibility check, then weigh evidence on waiting lists, grant conditions, long-run tourism and environmental impact.