Hong Kong Island features steep, hilly terrain, as well as hot and humid

weather. Travelling up and down the slopes therefore causes problems; this has

led the city authorities to imagine rather unusual methods of transport. One

famous example can be found in the Western District, where one of the busiest

commercial area of Hong Kong can be found. This area stretches from Des

Voeux Road in Central (which is at sea level) up to Conduit Road in the MidLevels (which is the mid section of the hill of Hong Kong Island). Because the

street is so steep, sidewalks are made of stairs. To make travelling up the slope

easier for pedestrians, the Mid-Levels escalators were opened to the public in October 1993. (See http://www.12hk.com/area/Central/MidLevelEscalators.shtml

for some pictures of the escalators and the stairs of this area).

For the sake of this problem set, imagine the following story. Suppose that

the street is one kilometre long (kilometre 0 is down at the crossroad with Des

Voeux Road and kilometre one is up at the crossroad with Conduit Road).

Suppose that 100,000 inhabitants are uniformly distributed along the street.

Without loss of generality, we can approximate the consumer distribution by a

continuum on [0; 1] with a mass set equal to 1 (i.e., we rede...ne all quantities by

dividing them by 100,000).

There are only two shops selling egg drop soup in this area. For simplicity,

we set their marginal cost of production to zero. As it happens, one shop (named

'Won-Ton'and indexed by 1) is located at point 0, while the other shop (named

'Too-Chow'and indexed by 2) is located at point 1. Everyday, each inhabitant

of the street may consume at most one bowl of egg drop soup, bought either

from Won-Ton or from Too-Chow. The price per bowl of the two shops are

respectively denoted by p1 and p2. The net utility for a consumer located at x

on the interval [0; 1] is given by

8<:

r 1 (x) p1 if consumer buys at Won-Ton,

r 2 (1 x) p2 if consumer buys at Too-Chow

0 if consumer does not buy.

where it is assumed that r is large enough so that every consumer buys one

bowl of soup.

1. Before 1993 and the installation of the Mid-Levels escalators, walking up

the street was much more painful than walking down. This is translated

by the following assumptions: 1 (x) = tx and 2 (1 x) = (t + ) (1 x),

with t; > 0.

1

(a) Derive the identity of the consumer who is indierent between the

two shops.

(b) Compute the equilibrium prices and pro...ts of the two shops.

(c) Show that Two-Chow's pro...ts increase if walking up the street becomes more costly for consumers, that is if increases (e.g., because

the temperature has risen). Explain the intuition behind this result.