THE SECOND BEST CITY 1
2.2. Comparison of the Market Benefit and the Market Rent
difficulty, and can estimate, with some more difficulty, the compensated price elasticity,
)
The difference between the shadow rent and the market rent can be calculated by solving the differential equation (2.29) with the boundary condition (2.30). Then the social marginal cost of widening the road is simply the sum of the difference, r(x), and the market rent, R(x). Although it is not extremely difficult to solve the differential equation numerically in simple models like ours, the calculation is likely to be formidable in a more realistic model.
Once the difference between the shadow rent and the market rent is obtained, the social benefit can be easily calculated from (2.23):
∫
′′ ′ ′ ′2.2. Comparison of the Market Benefit and the Market Rent
Having simplified and interpreted first order conditions, we can now proceed to examine the consequence of the benefit-cost analysis based on market prices. Our ultimate goal is to compare the market benefit, B(x), and the market rent, R(x), at the second best optimum. It is convenient to compare the social rent, µ(x) , with each of these first.
In this subsection we consider the case where compensated demand for land is not completely price inelastic: e>0.
The social rent is equal to the market rent in the optimum city with optimal congestion tolls. If, however, congestion t tolls are not levied, the market rent diverges from the social rent. Since transportation costs are lower than they should be,
Households tend to locate too far from the CBD. People seeking land farther from the center bid up the rent at larger radii, and the market rent tends to be flatter than the social rent. The following Theorem shows that the market rent crosses the social rent at some intermediate radius, and that the social rent must be higher than the market rent inside the radius and lower outside the radius. This is illustrated in Figure 1.
Theorem 1: If e>0 for any radius, then there exists an xˆ strictly between 0 and x )
ˆ 0
( <x <x such that µ(xˆ) =R(xˆ), and )
( ) (x >R x
µ for 0≤ x< xˆ, )
( ) (x <R x
µ for xˆ<x≤x.
Proof:
From (2.26) and e>0, it is impossible to have µ(x)>R(x) for all x or )
( ) (x <R x
µ for all x. Since both µ(x) and R(x) are continuous, they must cross somewhere: there exists an xˆ , 0<xˆ<x , where µ(xˆ) =R(xˆ). From (2.20), at this point λ(x) satisfies
y u xˆ)= −
λ( .
From (2.6), (2.9), (2.10), (IV.1.3), and (IV.1.4), we obtain 0 /
)
( =− >
′ x µgT gL
λ .
This inequality is strict at xˆ since µ(xˆ)=R(xˆ)>Ra >0. Hence we obtain the following inequalities:
y u x)< −
λ( x<xˆ
y u x)> −
λ( xˆ<x.
From (2.20), these inequalities imply ) ( ) (x >R x
µ x<xˆ
) ( ) (x <R x
µ xˆ<x,
which in turn implies that xˆ must be strictly between 0 and xˆ to satisfy (2.26).
Q.E.D.
We next compare the market benefit and the social rent. The next Theorem shows that they are equal at x=0 and that the market benefit is greater than the social rent in the rest of the city. Thus the market benefit overestimates the true social benefit. This is illustrated in Figure 2.
The result can be understood intuitively as follows. Recall that the difference between the market benefit and the social rent is the social value of the adjustment of consumptions in response to a decrease in transportation costs. First, consider the social value of the adjustment caused by a transportation improvement at x=0. The improvement reduces commuting costs for all households by the same amount, which is equivalent to
an increase in the income, y, of every household in the city. Since y is optimally chosen, the change in the utility level caused by an infinitesimal increase in y is zero.
The social value of the consumption adjustment is, therefore, zero for an improvement at x=0.
Next, consider an improvement at any radius x beyond xˆ in Theorem 1. This decreases commuting costs of households living farther than x and raises the market rent there. Since the social rent is lower than the market rent beyond xˆ , this works in a
socially undesirable direction and causes a social loss. Thus the social benefit (and hence the social rent) is less than the market benefit at any radius beyond xˆ .
An improvement inside xˆ benefits both households living outside xˆ and inside xˆ . The consumption adjustments of households outside xˆ cause social losses for the same reason as above, but those of households inside xˆ are socially beneficial since the social rent is higher than the market rent there. The next Theorem shows, however, that the former is always greater than the latter except for an improvement at
=0
x in which case the two are equal.
Theorem 2: If e>0 for any x, then we obtain )
0 ( ) 0 ( =B
µ
and
) ( ) (x <B x
µ , for 0<x<x. Proof:
We first show that for any x strictly between 0 and x ,
∫
xxµ−RReNdx′<0.For x greater than or equal to xˆ , this can be immediately obtained since )
( ) (x <R x
µ from Theorem 1. For x less than xˆ this is obtained from
∫
∫
xxµ−RReNdx′=− 0xµR−ReNdx′<0.Hence (2.23) yields
) ( ) (x <B x
µ 0<x<x. At x=0, the following equality is obtained:
∫
−−
= L T x eNdx
R L R
T g
B(0) ( (0), (0)) 0
) 0
( µ
µ
) 0 (
=B ,
where the second equality is obtained from (2.26), since gL can be seen to be finite at
=0 x .
At x = x, however gL becomes infinite and we must use L’Hôpital’s Rule to obtain
[ ]
where the first equality is obtained from (2.9), the second equality by L’Hôpital’s Rule, the third equality from (2.17), and the inequality from R(x)>µ(x)=Ra and the elementary result that the limit must be nonpositive when it is approached through nonpositive values. From (2.23) this implies
)
Combining Theorems 1 and 2, we can immediately see that the market benefit is greater than the market rent near the center. However, it is not clear whether or not this remains to be true when we move farther from the center. The next proposition throws a light on this question.
Proposition 1: If the compensated demand for land is not completely price inelastic )
0
(e> , then the market benefit is always greater than the market rent near the CBD.
Near the edge of the city, however, the market benefit is smaller than the market rent if the price elasticity is less than one, and is greater than the market rent if the elasticity is greater than one.
Proof:
The first half is immediately obtained from Theorem 1 and 2.
From the proof of Theorem 2, we obtain
)
Noting that the denominator and the square bracket of the numerator are both positive, we get
Figure 3 illustrates the relationship between the market benefit and the market rent in the case of price inelastic demand for land: the market benefit is greater than the market rent near the center of the city, but drops below it near the edge. As a result, the naive benefit-cost criterion has a tendency to overinvest in roads near the center and to underinvest near the edge. When demand for land is price elastic as in Figure 4, the benefit-cost criterion tends to overinvest in roads both near the center and near the edge of the city4.
Since the Cobb-Douglas type utility function (IV.2.27) has the elasticity 1−α, which is always less than 1, there is a tendency in that case to overinvest in roads near the center and to underinvest near the edge.
The conclusion depends on the elasticity of demand for land since difference between the market benefit and the social rent reflects the side effect due to the change of housing consumption, and the change of housing consumption is greater when the elasticity is bigger.
Notice that since these results are valid only in the neighbourhood of the second best solution, we do not have a definite answer as to whether the second best solution has a wider road than the market solution.
When the naive benefit-cost analysis based on market prices is adopted only in a small ring at x, and roads are built in other parts of the city to achieve the second best allocation, the above comparison between two equilibria is valid. If, for example, the market benefit is greater than the market rent in the ring between x and x+dx, the naive criterion calls for the road to be widened until the marginal market benefit of further widening falls to the market rent. When the ring is very narrow the market rent is not significantly affected by a change in road width there, and the preceding conclusions hold.
If, however, the naive benefit-cost criterion is adopted in the entire city, this
4 Note that the case where B(x) is lower than R(x) somewhere in the middle of the city is not excluded.
argument cannot be applied because the market rent curve changes. Widening of the road in the rest of the city might cause such a rise in market rent at some locations that, even though the market rent at the second-best allocation was below the market benefit, the road might become narrower as a result of changes elsewhere.
Furthermore, since the market rent is higher than the rural rent at the edge of the city, the city tends to expand. This causes another tendency toward overinvestment in roads. The reader may think that this effect would not appear if the second best problem were solved with the additional constraint that the market rent equal the rural rent at the edge of the city. In our model, however, under the reasonable assumption that transportation costs per mile, g(T,LT), are infinite when the width of the road is zero, the constraint is superfluous and the effect does not disappear.
The constraint on the market rent at the boundary, Ra
u x t y
R( − ( ), )= , (2.28)
would restrict y, t(x), and u to a hypersurface. The optimum allocation for the problem with this additional constraint is essentially the same as that for the problem without the constraint: the allocation is exactly the same within the boundary x, and
) , (T LT
g is made infinite at x causing a jump in t(x) of an appropriate size to satisfy the constraint (2.28). Since the jump which occurs in an infinitesimally small interval does not involve a finite social cost, the same maximum without the constraint is attained.5
Now, we briefly consider the possibility that t(x) has jumps even without the constraint (2.28). In such a case the usua l maximum principle like the Theorem of Hestenes in Appendix IV cannot be applied since it assumes that state variables are continuous. Kanemoto (1977b) analyzed the case by considering the problem with an upper bound on g(T,LT) and letting the upper bound tend to infinity.
The following argument shows that a jump in t(x) is indeed possible. Equation (2.21) suggests that η(x) must be nonpositive, since µ(x) is nonnegative. There is no guarantee, however, that η(x) is nonpositive since η(x) must also satisfy (2.19).
If compensated demand for housing is sufficiently price elastic, the indirect benefit from increasing transportation costs (the second term on the right side of (2.19)) may overwhelm the direct cost (-T), in which case η(x) becomes positive. Then the necessary conditions for the optimum involve contradiction, which suggests that the maximum does not exist within the range of functions assumed by the maximum principle.
In order to show that such a case can occur, we rewrite (2.19) as
∫
+ − ′−
= x
x e Ndx
R
x µ R
η( ) 1 .
5 It can be shown that, if g is infinite when LT is zero, then a jump in t(x) may occurs at x. See Kanemoto (1977b). Although the proof there has a minor error, the conclusion can be easily seen to be correct.
This equation shows that, if e(x)>R(x)/
[
R(x)−Ra]
, η(x) is positive near x. In particular, if Ra =0 and e(x) >1, then η(x) is positive. There certainly exists a well-behaved utility function whose compensated demand function is price elastic.In Kanemoto (1977b) it was shown that, if g(T,LT) tends to infinity as traffic density, T LT , approaches infinity, a jump in t(x) occurs at a point where η(x) is positive. Theorem 1 remains valid even when a jump occurs. Theorem 2 and Proposition 1 are also valid if Ra and R(x) are replaced by the left side limits,
) lim ( )
( x
x
x x µ
µ = ↑ ,
) lim ( )
( R x
x x x
R = ↑ .
If g remains finite even when T LT approaches infinity, LT becomes zero for a finite length. It can be easily seen that if the upper bound for g is sufficiently large, the same results are obtained.