It is generally true that the costs of market distortions increase at an increasing rate. You can prove this robust conclusion yourself. Consider the effects of zoning rules or of a greenbelt which restrict how land is used. In this social cost-benefit calculation, ignore the effects of externalities and any shift in the WTP functions associated with the government policy. (If you knew them also then you could determine whether the greenbelt is optimally located; that is an interesting-but-different question with slightly more complicated arithmetic.)

a) Suppose that the willingness to pay for land is $1000 per hectare (ha.) everywhere if used for farming. The willingness to pay for land used for residential purposes depends on the distance (d, in km.) from the CBD (i.e. d= 0):

WTPR= 2000- 20 d.

i)Find d which would be the boundary of the suburban fringe of the city if land use was determined in a free market.

ii) If the side of the greenbelt closest to the CBD was inside that boundary by 1 km, estimate the deadweight loss (i.e. cost of inefficiency) due to the fact that the land is not used to its highest and best use.

iii) If the boundary of the greenbelt closest to the CBD was inside that boundary by 10 km, estimate the deadweight loss due to the fact that the land is not used to its highest and best use. What is the ratio of your answers to parts ii) and iii)?

iv) If the WTPR changed to WTPR= 1200- 20 d, estimate the deadweight loss if the greenbelt were located at the suburban boundary found in part i).

b) Consider zoning which prevents conversion and redevelopment of residential properties to office property.

WTPR= 1000- 20 d

WTPO= 1200- 40 d

i) If the government imposes no zoning rules, where is the boundary between the residential parts of the city and the part with offices.

ii) If the mayor imposes zoning rules which increase the residential area by 0.1 km. and shrinks the office area by 0.1 km. (Ignore any effects on the willingness to pay.) What is the cost of the deadweight loss?

iii) If the mayor imposes zoning rules which increase the residential area by 1 km. and shrinks the office area by 1 km. (Ignore any effects on the willingness to pay.) What is the cost of the deadweight loss?

iv) If the mayor imposes zoning rules which increase the residential area by t km. and shrink the office area by t km. (Ignore any effects on the willingness to pay.) What is the cost of the deadweight loss?

(Hint: the standard formula for calculating the area of a triangle does not apply in part b) because measuring the "base" is a bit awkward. If you compute the relative premium by office land users that would be paid at any location then you could find a way to apply the standard formula.

Alternatively, you could search for "Heron's formula" which calculates the area of any triangle based on the length of the sides. Using some high school geometry and straight lines, you should be able to prove that the cost of a zoning distortion of distance t is proportional to t²: doubling the size of the distortion increases its social cost by a factor of 4. If you think that the WTP functions are not straight lines then you should be able to convince yourself that the simple formula is approximately correct if t is small and a bit of economic logic would reveal whether the computation over under- or over-estimates the cost if t is large.)