Question

Consider a Stackelberg price-leader duopoly. There are two firms: A leader and a follower. Assume marginal cost to be zero. The market demand is given as: p = a-bq: Show that:

(a) The leaders profit-maximizing output q is the same as a monopolist in this market. But, the leaders profit and the market price are lower compared to monopoly. The followers output is one-half the output of the leader.

(b)Leaders output is lower than when two firms behave as Cournot oligopolists

(c) Suppose the market demand is given by Q = 70-P. There are 1000 small (fringe) suppliers and the supply curve of the fringe firms is given by QSF = 5 + 4P: There is a price-leader whose marginal cost is constant and equal to $3 = MC = AC: Find, the total market output and market price and the total profits of the price-leader.

(d) Compare the price and output with monopoly.

(e) Why do the optimality conditions MU1/p1 = MU2/p2 in the case of consumer, and MPL/w = MPK/r in the case of producer holds only in the long-run but do NOT hold in the short-run.

Answer 1

a. Market demand is P= a - bQ, Marginal cost (MC)= 0

For a monopolist, profit maximizing output will be determined as follows:

TR = P * Q = (a - bQ)Q = aQ - bQ²

TC = 0

MR= a - 2bQ

MR=MC

a - 2bQ = 0

Q_monopolist = a/2b

Profit of monopolist , $\pi$ = TR - TC = aQ - bQ² - 0 = a(a/2b) - b (a/2b)²

$\pi$_monopolist= a²/4b

Market price = a- b(a/2b) = a/2

For the stackelberg model,

P= a - b( Q_L+ Q_F), where Q_L is the output for leader and Q_F is the output for the follower.

$\pi$_L= TR_L- TC = aQ_L - bQ_L² - bQ_LQ_F

Partial derivative of $\pi$_L w.r.t. Q_L is, d$\pi$_L/ Q_L= a-2bQ_L-bQ_F=0

Q_L= (a - bQ_F)/2b Reaction function of leader

$\pi$_F = TR_F - TC = aQ_F - bQ_F² - bQ_LQ_F

partial derivative of $\pi$_F w.r.t. Q_F is, d$\pi$_F/dQ_F = a-2bQ_F-bQ_L=0

Q_F = (a - bQ_L)/2b Reaction function of follower

Leader will use follower's reaction function in its own profit function to determine its profit maximizing output;

$\pi$_L = aQ_L - bQ_L² - bQ_L(a- bQ_L)/2b = 1/2(aQ_L - bQ_L²)

for maximizing the profit, first order derivative is d$\pi$_L/ Q_L =0

1/2(a- 2bQ_L)= 0

Q_L = a/2b

thus, leader's output is same as that of monopolist.

$\pi$_L = a²/8b , which is lower than the monopolist profit.

follower's output will be, Q_F = [a - b(a/2b)]/2b = a/4b = 1/2 Q_L

Market price = a- b( a/2b + a/4b)= a/4, which is lower than the monopolist price

b. When firms behave as cournot oligopolists:

reaction functions of both the firms are as follows:

Q_L= (a - bQ_F)/2b

Q_F = (a - bQ_L)/2b

In cournot model, profit maximizing outputs are found where both the reaction curves intersect.

Therefore using Q_F in Q_L

Q_L= [a - b (a-bQ_L)/2b]/2b

Q_L= a/3b

Subsitituting Q_L= a/3b into reaction function of follower

Q_F= [a - b (a/3b)]/2b = a/3b

Total output Q= Q_L+ Q_F= a/3b + a/3b = 2a/3b

Therefore, leaders output Q_L= a/2b, is lower than output when both firms behave as cournot oligopolist.

c. Market demand curve is Q= 70 - P

supply function for fringe firms is, Q_SF = 5 + 4P

Q = Q_DF + Q_SF , where Q_DF is the output produced by dominant firm(price leader)

70 - P = Q_DF + Q_SF

Q_DF= 70 - P - (5+4P) = 65 - 5P

The dominant firm acts as a monopolist by setting the price for the industry at the profit maximizing condition, MR=MC

TR_DF= P*Q_DF= 65P - 5P²

MR_DF= 65 - 10P

MC_DF=3

MR_DF= MC_DF

65 - 10P = 3

P= 6.2 (Market price)

Q = 70 - 6.2 = 63.8 (Market total output)

Q_DF= 65 - 5(6.2) = 34

profit of price leader, $\pi$_DF= TR_DF - TC_DF = 6.2* 34 - 3* 34 = 108.8

d. if the market is monopoly,then

Q = 70 - P is monopolist's demand curve

TR = P*Q = 70P - P²

MR = 70 - 2P

MC = 3

profit maximizing pricing condition is,

MR=MC

70 - 2P = 3

P= 33.5

Q = 70 - 33.5 = 36.5

The monopolist output is less than the total market output with the price leader and small fringe firms, whereas the monopolist price higher than the price leader's price.

e. The optimality conditions MU₁/P₁ = MU₂/P₂ for consumers and MP_L/w = MP_K/r for producers hold only in the short run and not in long run. These are the pareto optimality conditions as deviation from these conditions does not make anyone better off. But this is true only in the long run with fixed bundle of goods and fixed factor endowments. In the long run, the bundle of goods available to consumers changes and also their prices and consumer preferences. the factor endowments can also change in long run with improvement in technology and human capital. also in the long run, the industry tries to operate at the full employment level.