The market demand for gadgets is given byQD=100-3p , where QD is the quantity of
ID: 2496506 • Letter: T
Question
The market demand for gadgets is given byQD=100-3p , where QD is the quantity of gadgets demanded and p is the price of a gadget in dollars. The market supply is given byQ S=p , where QS is the quantity of gadgets supplied. The production of gadgets releases a toxic effluent into the water supply. The marginal external cost can be described by MEC Q=2Q. a) Compute consumer surplus, producer surplus and the cost pf pollution in the market equilibrium. b) Compute consumer surplus, producer surplus and the cost pf pollution in the social optimum. c) Suppose the government is considering imposing a tax of $T per unit on the producers. Find the level of the tax that ensures the socially optimal amount of gadgets will be produced in a competitive equilibrium.
Explanation / Answer
Demand: Q = 100 - 3P, or P = (100 - Q) / 3
Market supply: Q = P
(a) In market equilibrium, Demand = Market supply
(100 - Q) / 3 = Q
100 - Q = 3Q
4Q = 100
Q = 25
P = (100 - 25) / 3 = 75 / 3 = 25
From demand curve, When Q = 0, P = 100/3 = 33.33 [Reservation price]
Consumer surplus (CS) = Area between demand curve & equilibrium price
= (1/2) x $(33.33 - 25) x 25 = (1/2) x 8.33 x 25 = $104.125
Producer surplus (PS) = Area between supply curve & equilibrium price
= (1/2) x $(25 - 0) x 25 = (1/2) x $25 x 25 = $312.5
Cost of pollution = MEC = 2Q = 2 x 25 = 50
(b) In social optimum, demand = Soially oprimal supply = Q + 2Q
(100 - Q) / 3 = 3Q
100 - Q = 9Q
10Q = 100
Q = 10
P = (100 - 10) / 3 = 90 / 3 = 30
CS = (1/2) x $(33.33 - 30) x 10 = (1/2) x $3.33 x 10 = $16.65
PS = (1/2) x $30 x 10 = $150
Cost of pollution = 2Q = 2 x 10 = 20
(c) With the tax, effective supply function becomes
P = Q - T
P = 25 - T [Since Q = 25 from part (a)]
Demand: P = (100 - Q) / 3
Total revenue, R = P x Q = (100Q - Q2) / 3
Marginal revenue, MR = dR / dQ = (100 - 2Q) / 3
In competitive equilibrium, MR = MC = Revised supply curve
Since we obtained Q = 25 [Part (a)],
(100 - 2 x 25) / 3 = 25 - T
(100 - 50) / 3 = 25 - T
50 = 75 - T
T = 75 - 50 = $25
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