Notesale: Turn your study into money

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Annotated worked example for solving rent gradient (the height and slope)
Question: from Chen, 2016
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People live along a line (in a “half line city”) from 0 to xmax, commuting
to work at the city centre at 0
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There are N = 100 people in the city and each consumes a lot of
size (length on the line) of 1
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Commuting reduces work time at the rate t = 0
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(a)

Solve for the rent gradient: height and slope
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It is the slope of the function p in terms of x (which is a straight line in this example)
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Using the Budget constraint, w(1-tx)=z+ph,
(where w is the wage, t is the time it takes to commute per x units of distance travelled,
z is quantity of ‘other goods’ which the consumers consume and h is quantity of housing
which the consumers can consume) you can obtain:
𝑤(1 − 𝑡𝑥) = 𝑧 + 𝑝ℎ
∴ 𝑤 = 𝑧 + 𝑝ℎ + 𝑤𝑡𝑥
𝑝ℎ = 𝑤 − 𝑧 − 𝑤𝑡𝑥 (By algebraic manipulations)
𝑝=

𝑤−𝑧−𝑤𝑡𝑥
ℎ

∴

𝜕𝑝
𝜕ℎ

=−

𝑤𝑡
ℎ

(Partial differentiation
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*)

𝑤 = 5, ℎ = 1, 𝑡 = 0
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002)𝑥 (Inserting numbers given in
question)
𝑝 = 5 − 𝑧 − 0
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01
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The slope can be easily observed as -0
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01
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01(100)

𝑧 = 5−1= 4
This implies p=5-4, p=1, therefore the height is 1
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01
The rent gradient equation is p=1-0
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, 2016
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London: London School of Economics

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