Question

Consider a market where demand is P = 10 - 2Q and supply is P = Q/2. There is a consumption positive externality of $2.50/unit of consumption

a. Calculate the market equilibrium.

b. What is the social optimum quantity and price?

c. If the government uses a tax to get producers to internalize their externality, what is the net price received by producers?

d. Calculate the total surplus in the market equilibrium, at the social optimum and with the tax.

e. How can government increase the total surplus?

f. Impose the program you indicated in part e and calculate the total surplus.

Answer 1

(a) In market equilibrium, quantity demanded equals quantity supplied at a co

10 - 2Q = Q/2

20 - 4Q = Q

5Q = 20

Q = 4

P = 4/2 = 2

(b) Marginal social benefit (MSB) = Private demand + Positive externality = 10 - 2Q + 2.5 = 12.5 - 2Q

Efficient outcome is achieved when MSB equals supply.

12.5 - 2Q = Q/2

25 - 4Q = Q

5Q = 25

Q = 5

P = 5/2 = 2.5

(c) The tax will be equal to the externality cost per unit. When tax is imposed, new supply function becomes

P = (Q/2) + 2.5

Equating with MSC,

12.5 - 2Q = (Q/2) + 2.5

25 - 4Q = Q + 5

5Q = 20

Q = 4

P = 12.5 - (2 x 4) = 12.5 - 8 = 4.5 (Price paid by buyers)

Price received by sellers = 4.5 - 2.5 = 2

(d)

Consumer surplus (CS) = Area between demand curve and price

Producer surplus (PS) = Area between supply curve and price

Total surplus (TS) = CS + PS

From demand function, when Q = 0, P = 10 (Vertical intercept)

From MSB function, when Q = 0, P = 12.5 (Vertical intercept)

From supply function, when Q = 0, P = 0 (Vertical intercept)

(i) In market equilibrium

CS = (1/2) x (10 - 2) x 4 = 2 x 8 = 16

PS = (1/2) x (2 - 0) x 4 = 2 x 2 = 4

TS = 16 + 4 = 20

(ii) In efficient outcome

CS = (1/2) x (12.5 - 2.5) x 5 = 2.5 x 10 = 25

PS = (1/2) x (2.5 - 0) x 5 = 2.5 x 2.5 = 6.25

TS = 25 + 6.25 = 31.25

(iii) With tax,

CS = (1/2) x (12.5 - 4.5) x 4 = 2 x 8 = 16

PS = (1/2) x (2.5 - 0) x 4 = 2 x 2.5 = 5

TS = 16 + 5 = 21

(e) Government can increase total surplus by providing a unit subsidy to consumers, equal to unit externality of $2.5 per unit.

(f) The subsidy will increase demand by $2.5 at every output level. New demand function becomes

P = 10 - 2Q + 2.5 = 12.5 - 2Q = MSB

Equating MSB and supply, we obtain the efficient outcome where Q = 5 (as in part b).

Price paid by buyers = $2.5

Price received by sellers = $2.5 + $2.5 = $5

CS = (1/2) x $(12.5 - 2.5) x 5 = $25

PS = (1/2) x $(5 - 0) x 5 = $2.5 x 5 = $12.5

TS = $(25 + 12.5) = $37.5