Lind on Measuring Benefits

© Allen C. Goodman, 2009

Suppose

• We have 2 types of land, half polluted (p), half not (u).

• Pp = 100; Pu = 200• If a program of air pollution

control equalizes productivity everywhere, rent may end up at 150.

• If so total property values remain same.

• Does this mean there were no benefits?

Diagram

200

100

Pp Pu

100 100 200

150

Analysis

• Assume class of programs w/ output specific to particular locations.

• How do we measure benefits?

• Two major findings– Benefits of a project can be approximating by measuring net

increase in profits of activities located on land directly affected by project.

– W/ zero profits, total change in productivity equals change in land values.

Analysis

aij = maximum rent individual i would pay for parcel j.

Alternatively:

aij = Net earnings of firm i at parcel j exclusive of land cost.

n activities, n parcels

pj = rental value of parcel j.

Activity locates at parcel j ONLY IF:

(2.3) aij pj AND

(2.4) aij - pj aik - pk pj,aij

aij

pj

AnalysisDefine consumer surplus (or profit) as sij.

sij aij – pj.

SO, the total benefit from land is

(2.5) aij = sij + pj

Land is ENHANCED if there is some aij > aij for some parcels, and aij = ij for the rest.

Let:

A = ajj

(2.7) A a j(j)

(2.8) A a j(j)- a jj pj,aij

sij

Analysis(2.5) aij = sij + pj

Let:

A = ajj (2.7) A a j(j)

(2.8) A a j(j)- a jj From (2.5)

(2.9) A s j(j) - s jj ) + p j - pj)What does this mean?Change in total land values = change in productivity ONLY if

surplus terms = 0, or surplus terms don’t change!Essentially an open city argument.

Benefit MeasurementLind shows that any cycle of relocation NOT involving a

parcel of land affected by the project will leave net productivity unchanged. We need to consider ONLY parcels that are directly affected, where land is improved.

Benefits can be measured by considering in surplus of those activities alone that move onto improved land.

Suppose we have 3 unimproved parcels

in productivity = [a12 – a11] + [a23 – a22] + [a31 – a33]

Since there is no improvement,

in productivity = [a12 – a11] + [a23 – a22] + [a31 – a33]

From the equilibrium conditions:

(2.3) aij pj AND

(2.4) aij - pj aik - pk

[ajj+1 – ajj] p j+1 – p j

AND

[ag1 – p 1] [agg – p g]

New rents

Suppose we have 3 unimproved parcels

[ajj+1 – ajj] p j+1 – p j

AND

[ag1 – p 1] [agg – p g]

ALSO

[ajj+1 – ajj] p j+1 – p j

AND

[ag1 – p 1] [agg – p g]

New rents

Old rents

Suppose we have 3 unimproved parcels

a12 – a11 p2 – p1

a23 – a22 p3 – p2

a31 – a33 p1 – p’3

Sum 0

New rents

Old rents

a12 – a11 p2 – p1

a23 – a22 p3 – p2

a31 – a33 p1 – p3

Sum 0Benefits must equal 0!

Let’s improve a parcel (#1)

D = Benefits. Assume activity 1 moves to parcel 2, activity 2to parcel 3, etc.

1

111

)'()'(g

gggjjjj aaaaD

With no improvement on parcels 2 through g:

1

11

1

11

)()'(g

jjjj

g

jjjj aaaa

pj,aij

“Winning” aij changes;

Eq’m rent pj changes.

From eq’m for jth firm

jjjjjjjj ppaapp 111 ''

Why they’re here now Why they weren’t before

So, for g-1 firms:

jj

g

jjjj

g

jj

g

ppaapp

1

1

11

1

11

1

1

''

We know:

1

111

1

111 ''''

g

gjj

g

gjj

pppp

pppp

From eq’m for jth firm

)()'()''()'(

)'('')'(

)'(''

1111

1111

111

gggggggg

gggggggg

ggggg

ppaaDppaa

aappDppaa

ppaaDpp

Increased profit at new prices Increased profit at old prices

jjjjjjjj ppaapp 111 ''If we’re operating on the margin, we get equalities above:

)()'()''()'( 1111 gggggggg ppaaDppaa

1 1( ' )D p p

Earlier Problem

Activity is on polluted land p rather than unpolluted land u.

aip – pp aiu – pu

pu – pp aiu – aip

If pollution is eliminated, all land is same and net benefits are:

(for half of parcels) (aiu – aip)From above,

(for half of parcels) (aiu – aip) (n/2)(p2 – p1)Here, with 200 parcels, 100 polluted, 100 not

(for half of parcels) (aiu – aip) 100 (200 – 100) = 10000

Another exampleTwo types of land

Demand for unpolluted land: P = 250 – 0.5Q

Demand for polluted land: P = 110 –0.1Q

100 acres of each. Pu = 200; Pp = 100. Su = 2500; Sp = 500.

Total surplus = 3000.

If all land is now unpolluted, bidders will use first demand curve Q = 200; P = 150. Property values unchanged. New surplus = 10000

From (2.5)

(2.9) A s j(j) - s jj ) + p j - pj)

A + From above, (for half of parcels) (aiu – aip) 10000

You shouldgraph thisand do the

calculations

Polinsky and Shavell

© Allen C. Goodman 2009

Polinsky and Shavell

Examine the distinction between closed and open cities when looking at the measurement of benefits.

Take a little different tack in the modeling but with similar results.

They use an indirect utility function:

V = V (y - T(k), p(k), a(k))

where k = distance, y = income, T = transportation costs, a(k) is an amenity.

We have V1 > 0, V2 < 0, V3 > 0. Why?

What can we do with this?

V = V (y - T(k), p(k), a(k))

Within a city dV = 0.

dV/dk = -V1T´ + V2p´ +V3 a´ = 0.

Leads to p´ = (V1/V2)T´- (V3/V2)a´.

(V1/V2) = [Utility/$]/[Utility/(acres/$)] (1/Land).

We have negative price-distance function.

With an open city V is fixed at V*, so suppose there is an increase in a(k).

For V to stay equal to V*, p(k) must rise.

What can we do with this?

Suppose we have a closed city.

V** = V (y - T(k), p(k), a(k))

V starts at V**. Suppose amenities increase everywhere but k. V** must rise.

Since amenities haven’t improved at k, p(k) must fall relative to elsewhere. This is an indirect effect.

Now increase a(k). p(k) must rise to maintain V**. This is the direct effect.

Regression analysis w/ PS

Cobb-Douglas Example

U = Axqa(k); + = 1.

x(k) = (y – T(k))

q(k) = (y – T(k))/p(k)

Putting x and q into U V(k) = C[y-T(k)]p(k)-a(k)C is a constant

Solve for p(k) as:

log p(k) = (1/) log (C/V*) + (1/) log [Y – T(k)] + () log a(k)

= b0 + b1 log [Y – T(k)] + b2 log a(k)

Regression analysis w/ PS

log p(k) = (1/) log (C/V*) + (1/) log [Y – T(k)] + () log a(k)

= b0 + b1 log [Y – T(k)] + b2 log a(k)In an open city, since V* is fixed, a change in a(k) will predict change in

log p(k).In closed city V* V**. Must know what happens to a(k) all over city.

Gen’l eq’m model is necessary.SO:

Changes in aggregate land values correspond to WTP only with an open city model.Eq’m rent schedule will give enough information to identify demand for a(k), all else equal; in “closed city” all else may not be equal.