Econ 385

Problem Set 2 –

1.  Consider the following situation.  The gas distributer Dot, is the only gas

distributer in a certain town.  This town is constituted by a road of 1 mile.

The distributer is located exactly in the middle of the road.  The consumers

are uniformly distributed along the road and face a transportation cost t

per unit of distance.  The consumers have a valuation of V  for buying gas.

(a)  Consider that the gas distributer is deciding how much to charge for

gas.  What price per gallon should he charge if he would like to sell

to all the consumers?

(b)  If the firm considers opening a new gas distributer (and relocate his

old  one),  where should  he  locate  the  distributers?   How  does  the

optimal price change when there are 2 distributers?

(c)  Consider  that opening a  new  distributer  has  a cost  of  F  =  1.  If

t = 20, how many gas distributers should the firm open?

(d)  Assume that some road work will be done in the middle of the road

such that the left consumers cannot travel to the right and the right

consumers  cannot travel  to the left.  Would  this affect the optimal

location of the two distributers?  (use only your intuition to answer

this question).

2.  Consider the model of competition with differentiated products à la Hotelling.

A population of consumers is distributed uniformly in a line of length 1.

A seller is placed at 0.25 of one end of the line and the other at 0.25 of the

other end.  Transportation cost of each consumer is t per unit  of length

traveled.

(a)  Determine the demand of seller 1, D₁, given the price fixed by seller

2.

(b)  Compute the  price elasticity  of  the  demand.  (hint:  elasticity ε  is

(c)  How does the elasticity change as a function of the transport cost?

Justify economically.

3.  Consider a market in which  two firms compete in prices, and they offer

a vertically differentiated product (i.e.  differentiated by quality) v_i(i =

1, 2).  In particular assume v₁= 1 and v₂= 2 and call u_i(x) = V + xv_i−p_i

the utility of consumer x ∈ [0, 1] if he/she buys variety v_iat price p_i.

(a)  Offer a graphical representation  of the above setting when V  = 10

and p₁= 0.25 and p₂= 0.5.