Question

Consider a Hotelling line of length 1. A grocery store is located at each endpoint. The grocery store at the west end point is owned and operated by Jack Donaghy and offers customers the use of a personal “grocery concierge”, free of charge, who assists the customer in shopping. The grocery store at the east endpoint is owned and operated by Milton Greene and does not offer its customers a grocery concierge. The utility of a consumer at location x who visits Jack’s store is given by u_x^J = v_J − tx − p_J and utility from visiting Milton’s store is given by u_x^M = v_M − t(1 − x)− p_M, where v_J > v_M because of the grocery concierge and the p terms represent prices and t is the transportation cost per unit of distance. Assume for simplicity that the grocery stores have no marginal costs of production.

[20 points] Given some prices, p_J and p_M, what is the location of the indifferent consumer? If those prices were the same, would the location of the indifferent consumer be to the left or right of 0.5, or exactly at 0.5?

[20 points] Solve for the equilibrium prices charged by each grocery store and call them p^∗_J and p^∗_M. Are they equal?

[10 points] Where is the location of the indifferent consumer in equilibrium?

Answer 1

Answer 1. Assuming utility of a consumer at location x who visits Jack’s store: uxJ = vJ − tx − pJ

              ;And utility from visiting Milton’s store is given by uxM = vM − t(1 − x)−Pm; where vJ > vM because of the grocery concierge and the p terms represent prices and t is the transportation cost per unit of distance.

For a given prices: pJ and Pm, the location of indifferent Consumer (x1) who is indifferent buying from either Jack's or Miller's store will be, when both utilities are same for him:

                               vJ – tx1 − pJ = vM − t(1 – x1)−pM; x1 is villing to pay either of the price at the store, hence, vj=vm.;

                        à pJ + tx1 = pM + t(1-x1) à pJ +tx1 = pM + t1-tx1 à 2tx1 = pM-pJ +t1

                        à x1 = (pM-pJ +t1)/2t    ; this is location of indifferent consumers for a given price pM & pJ.

If prices are same in both stores (i.e. pM = pJ), then x1 (indifferent consumer) will be at half a distance exactly (0.5) as he's ready to pay any price for the goods as well as has same utility. If prices in Jack's store (pJ) are higher than pM, then x1 will move toward right whereas if pM>pJ then x1 will move towards left.

Answer 2. Solve for equilibrium prices: p^∗_J andp^∗_M:

As indifferent consumer is the one farthest from either of the store, v(utility) has to be high enough for him to go and buy the product:

As, vJ> vM; pJ + tx > pM +t(1-x) or for an indifferent consumer: x1 = (pM-pJ +t1)/2t

The demand function of J-firm(DJ) & M-firm (DM) are :

DJ (P*j, P*M ) = x1 = (pM-pJ +t1)/2t à ((pM – pJ)/2t) + 1/2

DM (P*j, P*M) = 1-x1 = 1- [((pM – pJ)/2t) + 1/2] = > ((pJ-pM)+t)/2         ;Demand of firm J depends positively on the difference (pM -pJ ) and negatively on the transportation costs. If firms set the same prices pB=pA then transportation costs do not matter as long as the market is covered, firms split the market equally (and the indifferent consumer is located in the middle of the interval 0.5).

The maximization profit of Jack's firm is:

max p*jx1 = maxp*j [(p*M-p*J +t1)/2t]

à F.O.C = ((p*m-2p*j)/2t) +1/2 = 0 à p*M – 2p*J +t = 0

à p*M = 2p*J – t ------ (i) ---- equilibrium price at Jack's store

Simimarly, Miller's firm maximises profit by:

max p*m * x1 = maxp*m [(p*j-p*m +t1)/2t]

à F.O.C = ((p*J-p*m +t)/2t) = p*M/2t à p*J + t = 2p*M

Now applying equation (i): à p*J + t = 2(2p*J -t) à p*j = p*m = t – equilibrium price at Miller's store which is same as at Jack;s store.

3)the location of indifferent consumer will be at the middle of line (length 1) at equilibrium.