Question

The demand for T-shirts is Q=20 – (P/2), where Q is the number of t-shirts and P is the price. The private cost of producing t-shirts is C(Q) = Q^2, with the private marginal cost being MC(P) = 2Q. Washing and dyeing t-shirts causes water pollution with marginal external costs of MC(E) = Q.

A) Based on the above information, net welfare: consumer surplus + producer surplus - damages of consumption at market equilibrium is ______(no decimal point)...(numeric answer). Hint: Consumer Surplus and Producer Surplus are calculated as usual on the graph. You can also plot the damage curve and find the area UNDER the damage curve to understand total costs to society

B) Based on the above information, net welfare: consumer surplus + producer surplus at socially efficient equilibrium is _____(no decimal point)...(numeric answer). Hint: There are no damages incurred by society that are not already taken into consideration.

Answer 1

Solution:

Please note that graph is not necessary to be drawn for this question, it is just helpful to calculate the required dead weight loss and thus, attached along (at the end of the question). Consumer surplus and producer surplus are calculated in usual manner, and are fairly easy areas to mark (thus, excluded from the graph to avoid confusion).

At market equilibrium, the price pertaining to demand equals the private marginal cost, PMC (private supply curve). With MC(private) = 2Q, this supply curve becomes: Q = P/2, so we have

20 - P*= P*/2

20 = P*/2 + P*/2

P* = $20 per t-shirt

And Q* = 20/2 = 10 t-shirts

At social optimal level, price pertaining to demand equals the social marginal cost (calculates social supply curve). Social marginal cost, SMC = PMC + MC(E)

SMC = 2Q + Q = 3Q

So, social supply curve is: Q = P/3

Then, social optimal level occurs where: 20 - P'/2 = P'/3

20 = P'/2 + P'/3

5P'/6 = 20

So, P' = 20*6/5 = $24

And, Q' = 24/3 = 8 shirts

Dead weight loss incurred due to market equilibrium = a + b (refer to the graph)

This area can be calculated as: a = (1/2)*difference in quantity*(30 - social optimal price)

b = (1/2)*difference in quantity*(social optimal price - market price)

So, DWL = a + b

DWL = (1/2)*(10 - 8)*(30 - 24) + (1/2)*(10 - 8)*(24 - 20)

DWL = (1/2)*2*(6 + 4) = $10

Finally, consumer surplus = area above price and below demand curve

So, CS = (1/2)*(quantity chosen)*(maximum payable price - price chosen)

Maximum payable price (or price intercept of demand curve is: Q = 20 - P/2, so, P = 40 - 2Q, thus, price intercept = 40)

Producer surplus = area below the price and above supply curve

So, PS = (1/2)*(quantity chosen)*(price chosen - minimum acceptable price).

Minimum acceptable price is the price intercept of supply curve. In this case, it is 0, for both SMC and PMC.

Then,

a). Under market equilibrium,

CS = (1/2)*10*(40 - 20) = $100

PS = (1/2)*10*(20 - 0) = $100

DWL = $10 (already calculated)

So, net welfare = 100 + 100 - 10 = $190

b). Under social optimal level,

CS = (1/2)*8*(40 - 24) = $64

PS = (1/2)*8*(24- 0) = $96

So, net welfare = 64 + 96 = $160