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 The Economic Costs of a Rise in Sea Levels

One major concern about global warming is the impact of a rise in sea levels. The 20th century was a period of increased global temperatures and the estimate of the increase in globally averaged sea levels was 10 to 20 cm. While global temperature will probably continue to rise it is not certain that the major ice caps of Greenland and Antarctica will melt. These ice caps have generally high altitudes and now their interior levels appear to be increasing as a result of higher levels of precipitation. So while there may be some melting on the edges this is being offset by growth in their thickness. Sea levels could in fact go down from global warming but most likely they will stay about the same or increase only slightly. But even if the sea levels were to increase the economic consequences are not likely to be catastrophic. The analysis below shows why the economic adjustments are likely to be marginal.

Conside a low cone-shaped island in which the value of the land is a linear function of its distance from the shore. Suppose the sea level rises taking away a band of land adjacent to the shore. While it might seem that this is the loss of the most valuable land, this is not the case when compensating adjustments take place. What used to be the band of the second-most valuable land now becomes shore-front property, thus cancelling out some of loss of the land in the first band. But there is not quite as much land in this second band as there was in the first band. The third band now is the band of the second-most valuable property. Each band gets upgraded until the upgrading reaches the center of the island. There is no property to take the place of the inner-most band.

Let R be the original radius of the island and the width of the band of land lost to the rising sea be ΔR. Let R/ΔR be an integer n, for illustration purposes, say 5. Let the value of the land be equal to V=v0-v1(R-r). The area of a band is equal to its width times the circumference at its midpoint. For example, the first band's midpoint is (4+0.5)Δr and hence its area is 2π(4.5)(ΔR)². Its value is the value at the midpoint times the area of the band; i.e., 2π(4.5)(ΔR)²(v0-v1(0.5ΔR)).

The losses and gains are then:

LossesGainsNet Loss
Band 1:
2π(4.5)(ΔR)²(v0-0.5v1)
2π(3.5)(ΔR)²(v0-0.5v1)
2π(1.0)(ΔR)²(v0-0.5v1)
Band 2:
2π(3.5)(ΔR)²(v0-1.5v1)
2π(2.5)(ΔR)²(v0-1.5v1)
2π(1.0)(ΔR)²(v0-1.5v1)
Band 3:
2π(2.5)(ΔR)²(v0-2.5v1)
2π(2.5)(ΔR)²(v0-2.5v1)
2π(1.0)(ΔR)²(v0-2.5v1)
Band 4:
2π(1.5)(ΔR)²(v0-3.5v1)
2π(0.5)(ΔR)²(v0-3.5v1)
2π(1.0)(ΔR)²(v0-3.5v1)
Band 5:
2π(0.5)(ΔR)²(v0-4.5v1)
2π(0.5)(ΔR)²(v0-4.5v1)

The total loss is then

#### 2π(ΔR)²(4v0-8.0v1) + 2π(0.5)(ΔR)²(v0-4.5v1) = 2π(ΔR)²(4.5v0-10.25v1) rather than the loss of all of Band 1 which has a value of 2π(ΔR)²(4.5v0-2.251)

The upgrading of the value of the inner bands reduced the loss by 8v1(2π(ΔR)²)

A more general case can be dealt with as follows. The Value of land V is given by:

#### V = ∫0R(v0-v1(R-r))(2πr)dr and therefore ∂V/∂R = v0(2πR) - v1πR²

The first term is the value of the outer band, in this case due to an increase in the radius of the island. The second term is the effect of the revaluation of the interior land. Because the terms depend upon different powers of R there could be a value of R at which the derivative switches sign. This is a flaw in the modeling. The integration should be only over the land of non-negative value. The radius at which the value becomes zero is

#### v0 - v1(R−R0) = 0 hence R0 = R − v0/v1

The lower limit of the integral should then be the maximum of zero and R0. When R0 is positive there is some land in the interior which is valueless. The width of the developed area is given by v0/v1. When the dependence of R0 on R is taken into account the effect of a change in R is given by:

#### ∂V/∂R = v0(2πR) - v1π[R²−R0²] which, noting that R0²=R²-2R(v0/1) + (v0/v1)² reduces to ∂V/∂R = v0(2πR) - v1π[2R(v0/v1)−(v0/v1)²] and still further to ∂V/∂R = v0(2πR) - 2πRv0 + π(v0)²/v1

On the right hand side of the above, the first term is the value of the outer band and it is exactly cancelled by the second term to give the final result:

###### ∂V/∂R = π(v0)²/v1

The picture which is appropriate for this version of the analysis is:

The model may justifiably be modified in one more way. In the previous analysis the limit was over land of non-negative value. It may be more reasonable to have the analysis over land that has a value above its value for subsistence agriculture or nomadic pastoralism. (Think Australia.) Let vg be the minimum value of the land. Then

#### v0 - v1(R−Rg) = vg hence Rg = R − [v0−vg]/v1

Again the width of the developed area appears in the analysis. This time as [v0−vg]/v1. Let this be denoted as w so Rg = R−w and hence Rg² = R²−2Rw+w².

Now the equations are

#### ∂V/∂R = v0(2πR) − vg(2πRg) −v1π[R²−Rg²] or, equivalently ∂V/∂R = v0(2πR) − vg(2π)(R-w) − v1π[2Rw − w²] which reduces to ∂V/∂R = [v0−vg](2πR) + vg(2πw) − 2πR[v0−vg] + π[v0−vg]²/v1and still further to ∂V/∂R = π[v0²−vg²]/v1

Since the width of the developed areas is w=[v0−vg]/v1 the above equation can be expressed as

###### ∂V/∂R = 2πw[v0+vg]/2

It is notable that the effect of a change in R does not depend upon the value of R.

## Conclusions

If sea levels rise as a result of global warming the loss of the value of shore land will be largely offset by the revaluation of the remaining land. Where the land masses contain in the interior land of minimal value the impact of the loss of shore land would be minimal. The loss of shore development can be reduced by the creation of sea walls as in case of such cities as Galveston, Texas. The city of Seattle in the 19th century when it found that it had built at too low a level effectively raised the level of the city streets. The Netherlands has coped with sea levels above the level of much of its land for centuries.

The measures that would need to be taken to deal with rising sea levels would no more thwart economic development than did the devastation of the cities in Europe and Japan as a result of World War II. Both Europe and Japan not only recovered but went on to incomparably greater development.