Public Goods, Externalities, Development, and Regulations

A developer can earn more by adding another floor while making neighbouring homes darker. A neighbourhood can value a park more than its purchase price while no resident wants to pay for it alone. A driver can save time by entering a road while slowing everyone already on it. These situations share a problem: the decision-maker’s private calculation omits some benefits or costs experienced by others.

Learning goals

Distinguish public goods from externalities and congestible resources, explain free-riding, calculate uncoordinated and cooperative density choices, and derive a congestion toll. Compare private planning, regulation, and bargaining while identifying whose benefits each calculation includes.

1. Three related problems with different definitions

A pure public good is non-excludable and non-rival. Non-excludable means it is difficult to prevent someone from benefiting; non-rival means one person’s benefit does not diminish another person’s benefit. A landscape view can have public-good characteristics.

An externality is an effect on others that is not fully accounted for through the decision-maker’s prices, payments, or compensation. A building casting an unwanted shadow is a negative externality; preserving a pleasant view can create a positive one.

A congestible shared resource becomes rival when additional users reduce the quality available to others. A crowded park or road is therefore not a pure public good at that level of use. Parks can also restrict entry, and roads can be tolled. Classify the relevant service and access conditions rather than labelling an entire asset once and for all.

The supplied lecture distinguishes broad shared amenities from identifiable parcel-to-parcel effects. That is useful for organising examples, but whether the source can be traced to one parcel is not the defining test for a public good. Public-good benefits and externalities can overlap.

2. Why a valued open space may not be purchased

Ten neighbours each value preserving a vacant lot at 12,000. The lot costs 100,000. Their combined willingness to pay is 120,000, so preservation creates 20,000 of net benefit before other costs. An equal contribution would be 10,000 each, below each neighbour’s benefit.

Yet no neighbour wants to pay 100,000 alone for a benefit worth only 12,000 to that person. Each may hope others will fund it while everyone enjoys the result. That is free-riding.

For identical neighbours, the supplied condition is p/n < MV < p, where p is the purchase cost and MV is each household’s willingness to pay for this discrete preservation choice. Here MV is a total benefit for the decision, not automatically a marginal benefit curve for a continuous amount of parkland.

With varying preferences, compare the sum of individual benefits with the cost. Collective contributions, a funded neighbourhood association, or public acquisition can overcome the funding problem, but require agreement about rights, payments, and maintenance. A benefit exceeding equal cost share does not guarantee voluntary coordination.

The supplied estimate of a 20-25% home-price premium within 100 feet of a park is a lecture-specific empirical claim without its study details. It is not a universal premium. Park quality, nuisances, neighbourhood selection, and the statistical method affect measured associations. Capitalised property value also does not automatically measure all social benefits or establish causality.

Teach it back: How can all ten neighbours prefer a funded park while none wants to buy the lot individually?

3. Free entry can overcrowd a shared space

Suppose n identical visits to a park each yield net benefit h(n), which declines as crowding increases. An individual visitor ignores the reduction in other visitors’ enjoyment. With free entry and an interior equilibrium, use can expand until h(n0) = 0.

For this particular symmetric model, total net benefit is n h(n). A coordinated optimum satisfies:

h(n) + n h’(n*) = 0**, or h(n) = -n h’(n*)**.

The lecture calls h marginal valuation MV, but here it must be interpreted as each user’s net benefit at crowding n for this total-benefit expression to apply. If MV instead denotes an inverse demand curve for heterogeneous visitors, total benefit must be integrated and the derivation changes.

Example: Let h(n) = 100-n. Free entry gives n0 = 100 visits. Total benefit is 100n-n squared, maximised at n = 50*, yielding 2,500 benefit units. At free entry, each user’s net benefit and aggregate net benefit in this stylised model are zero. The extra 50 visits erase the value other visitors would have retained.

This is an illustration, not a claim that every freely accessible park becomes worthless. Capacity, alternatives, access costs, and heterogeneous preferences matter.

Teach-back self-check: what does the extra visitor ignore?

The visitor considers their own enjoyment and access costs but not the reduction in enjoyment imposed on everyone else. The gap between private and collective calculations creates overuse under the assumptions.

4. Adding a neighbour to the density model

Return to a developer choosing Floor Area Ratio F. Let neighbouring density be f. Sale price per unit of floor area is:

P = alpha - beta F - gamma f,

while average construction cost per floor area is C = mu + tau F. Positive gamma represents a negative neighbouring-density effect, such as lost light or crowding, holding other inputs fixed. This assumes an adverse spillover; density can also create positive access and service benefits in other settings.

Residual land value per land unit is:

p(F,f) = [(alpha-mu) - (tau+beta)F - gamma f]F.

Write a = alpha-mu and b = tau+beta to simplify the algebra. Assume a > 0, b > 0, gamma > 0, symmetric parcels, and interior solutions. Values below are development residuals before land acquisition, with other relevant costs treated consistently.

5. Nash density: optimise my parcel, hold yours fixed

A developer treats f as given. Differentiating their own residual gives:

dp/dF = a - 2bF - gamma f = 0.

In symmetric equilibrium f = F_m, so:

F_m = a/(2b+gamma),

p_m = a^2 b/(2b+gamma)^2.

This is a Nash equilibrium: given the neighbour’s choice, no developer wants to change their own choice alone. It need not maximise the sum of their values, because each ignores the harm their density imposes on other parcels.

6. Cooperative density: include both sides of the spillover

With coordinated equal densities, f = F across parcels. Per-parcel residual becomes p(F,F) = aF-(b+gamma)F^2. The symmetric cooperative maximum is:

*F = a/[2(b+gamma)]**,

*p = a^2/[4(b+gamma)]**.

For gamma > 0, F_m > F* and p_m < p*. At gamma = 0, the outcomes coincide, so the strict inequalities require an actual negative spillover.

A two-parcel example

Let a = 600, b = 100, and gamma = 100. These are hypothetical model coefficients in consistent monetary and FAR units.

Outcome	Density per parcel	Residual per land unit	Combined residual for two unit-area parcels
Symmetric Nash	2	400	800
Symmetric cooperative	1.5	450	900

Together, the owners gain 100 by coordinating lower density. But if the neighbour stays at 1.5, an individual owner’s best response is F = (600-100 x 1.5)/200 = 2.25. That owner’s residual becomes 506.25, above 450, while the neighbour’s falls to 337.50. The private temptation to deviate explains why an agreement needs enforceable rights or another coordinating mechanism.

The symmetry assumption is important. For a literal two-parcel problem, the equal-density cooperative solution is a unique joint interior maximum when b > gamma/2. With sufficiently strong cross-effects, asymmetric or boundary solutions can dominate. The worked example meets this condition.

Also, maximising combined land residuals is not automatically a full social optimum. Tenants, excluded households, public infrastructure, environmental effects beyond the parcels, and distributional consequences may be missing. A model showing excessive density under one spillover does not establish that lower density is always better.

Teach it back: Why can both owners benefit from restraint while each still wants to add floor area individually?

7. Incompatible uses and historical lock-in

Industry and housing can both value access while imposing costs on one another. The supplied illustrative bid-rent forms are:

r_I(d) = r_I0 - k_I d,

r_H(d) = r_H0 - k_H d + gamma |m-d|.

Here d measures position along a line, m locates a nuisance source or relevant interface, and the residential distance term represents a benefit from separation. Intercepts, distances, and gamma must have consistent units. This is a stylised specification, not a general pollution-dispersion equation; the meaning of m must be fixed before interpreting a diagram.

Different historical arrangements can persist because buildings, infrastructure, and businesses are costly to move. A better coordinated arrangement on an empty map may not be worth implementing once demolition, lost output, relocation, and compensation are counted. Existing assets’ future services are opportunity costs even when their original construction spending is sunk.

Multiple spatial equilibria are possible under suitable assumptions, but these two rent expressions alone do not prove that an industrial core and an industrial fringe are both equilibria. Market-clearing, land allocation, transport, and boundary conditions must also be specified. No owner being willing to move first can be a coordination problem rather than proof that relocation has no collective value.

8. Three ways to coordinate decisions

Mechanism	How it can help	Important limit
Large-scale private master planning	One owner captures effects between the parcels it controls	Effects outside its boundary and on non-owners remain
Public planning and regulation	Rules or charges can address spillovers across ownership boundaries	Information, enforcement, political incentives, and access costs matter
Bargaining over defined rights	Parties negotiate compensation for mutually beneficial changes	Holdouts, many parties, uncertainty, and transaction costs can block agreement

A master developer may leave a parcel open if the lost development residual is smaller than the resulting gain across other lots. Covenants, conditions, and restrictions (CC&Rs) can preserve agreed design features, though their enforceability and effects depend on their terms and jurisdiction. Common ownership does not internalise every regional spillover.

Public tools include performance standards, development rules, and charges related to harm. A uniform FAR cap might help in the symmetric adverse-density example but fit poorly when parcels differ or access benefits are omitted. Maximising incumbent land value is not the only public objective: housing access, health, environment, and distribution also matter.

The Coase bargaining benchmark says clearly defined transferable rights and frictionless bargaining can support efficient agreements when all affected parties and relevant effects are included. Standard claims that the efficient allocation is independent of initial rights also require conditions such as the absence of relevant wealth effects. Initial rights still affect who pays and who receives compensation.

Example: A firm can spend 60,000 to eliminate annual harm worth 100,000 to neighbours on the same present-value basis. There is a potential 40,000 gain from abatement. If the firm has the relevant right and bargaining is feasible, neighbours could jointly pay between 60,000 and 100,000. If neighbours hold an enforceable right against the harm, the firm’s obligations and payments change. Uncertain damage, costly organisation, or holdouts can prevent the efficient result.

Compensation is a transfer; pollution reduction or relocation changes real resource costs and harms. Count those separately. Negotiating a payment does not by itself make an activity socially beneficial, and an imposed fee does not automatically equal marginal damage.

9. Growth boundaries have two distinct channels

Protecting open space can affect property markets through:

1. Supply: A binding restriction reduces feasible development, potentially raising housing costs relative to an otherwise comparable unconstrained outcome.
2. Amenity: Preserved landscape, recreation, or environmental quality can increase willingness to pay for nearby or regional housing.

These channels can reinforce prices of some existing homes while making entry more expensive. They do not imply every land parcel gains value. Land denied development rights can lose option value; allowed development sites and homes benefiting from amenities can gain. Substitution, density, infrastructure, population movement, and spillovers to other areas affect the result.

There is a modelling distinction from the fixed-lot urban land lecture: if N and q are fixed, an imposed boundary smaller than sqrt[Nq/(pi V)] leaves insufficient land. A new equilibrium requires an adjustment in density, household numbers, land availability, or another assumption. Raising rents alone cannot fit unchanged fixed lots into too little space.

The supplied London, Seattle-area, and California examples illustrate different types of regional land policy, not interchangeable rules or current legal summaries. Evaluating a specific policy requires its actual scope and counterfactual. A price increase alone does not establish whether total benefits exceed costs, who gains, or who is displaced.

10. Roads: the driver does not pay for all the delay

Let V be trips per period, W(V) the willingness to pay of the marginal trip, and C(V) each trip’s private travel cost in money-equivalent units. C includes the traveller’s own time cost and rises with traffic. Assume trips can adjust, W decreases, C increases, and the relevant social costs are otherwise captured.

Without a toll, user equilibrium satisfies W(V0) = C(V0). Each driver considers their own trip cost, but adding one trip also increases cost for the V existing trips. Total travel cost is V C(V), so marginal social cost is:

MSC(V) = C(V) + V C’(V).

Social net benefit is the integral of W(x) from 0 to V, minus V C(V). The integral matters because W is the marginal trip’s willingness to pay, not a benefit shared equally by every trip.

The efficient interior flow therefore satisfies:

W(V) = C(V) + V* C’(V*)**.

A congestion toll equal to V C’(V)** adds the external delay cost to the private decision. It is the external component, not the entire marginal social cost. With suitable decreasing demand and increasing congestion, V0 exceeds V*.

A numerical road example

Let W(V) = 100-V and C(V) = 10+V, using normalised trip-flow and monetary units. Without a toll, 100-V = 10+V, giving V0 = 45. Social marginal cost is 10+2V, so V = 30*. The congestion toll is 30 x 1 = 30 per trip. At that flow, private cost is 40 and cost including the toll is 70, matching W(30).

Net benefit is 90V-1.5V^2: it equals 1,012.5 at V = 45 and 1,350 at V = 30, a gain of 337.5. Toll revenue of 900 is a transfer in this calculation, not an extra resource cost to subtract a second time. Administration, distribution, revenue use, alternative routes, and other externalities matter for actual policy design.

Teach it back: Why does making each driver pay only for their own travel time leave too much traffic in this model?

Feynman study lab

Explain it simply

Some benefits are shared, so everyone may wait for someone else to pay. Some private decisions impose costs on neighbours, so individually attractive choices can make the group worse off. Coordination can come through common ownership, public rules, or negotiated rights, but each has limits. Congestion pricing illustrates the principle: charge for the delay a trip imposes on others, not only the delay experienced by its driver.

Teach it back without looking

Explain the ten-neighbour park problem, the FAR 2 versus FAR 1.5 comparison, and the road toll using everyday language. Then answer:

Why is the appropriate congestion toll 30 in the road example even though total social marginal cost is 70?

Check your explanation after you try

The driver already experiences private travel cost of 40. The additional 30 is the delay cost imposed on others. Adding that external cost makes the driver’s decision reflect total marginal cost of 70; charging 70 on top would double-count the private component.

Transfer questions

1. Eight households each value a preserved lot at 15,000. The lot costs 100,000. Is combined benefit greater than cost, and would one household want to fund it alone?
2. In h(n) = 80-2n, find free-entry use and use maximising n h(n).
3. In the symmetric density model, let a = 400, b = 100, and gamma = 100. Find F_m, F*, p_m, and p*.
4. If gamma = 0, does coordination lower density in this model?
5. With W(V) = 80-V and C(V) = 20+V, find V0, V*, and the efficient congestion toll.
6. Does a growth boundary necessarily raise the value of land outside it that loses development rights?

Check the transfer questions

1. Combined benefit is 120,000, exceeding cost by 20,000. No individual values preservation enough to pay 100,000 alone; equal contributions would be 12,500 each.
2. Free entry gives 40; total benefit 80n-2n squared is maximised at 20.
3. F_m = 4/3, F = 1*, p_m approximately 177.78, and p = 200* per land unit. The joint interior condition b > gamma/2 holds.
4. No. The outcomes coincide when there is no cross-parcel effect.
5. V0 = 30, V = 20*, and the toll is 20 per trip.
6. No. Losing development rights can reduce that parcel’s option value. Amenity and supply effects must be assessed for specific assets and people.

Find the gap and try again

Record the step you could not explain: non-rival versus congestible, individual versus collective benefit, land value versus social welfare, or private versus social marginal cost. Rework the example, then close the notes and explain it again. Change one assumption and predict the effect before calculating. Tomorrow, reconstruct the congestion toll argument and explain one reason a theoretically efficient bargain might fail in practice.