Question

I Page 1 Assignment 1 Due: Beginning of class, February 13, 2020 1. Deadweight loss of monopoly Suppose that electricity consumers have an inverse demand curve given by: P(e) = 1-e, There is one supplier of electricity, which has a total cost function given by: cle) = ? You can assume that there are no externalities associated with electricity consumption or production. 1.1. What is the marginal cost function? Draw the marginal cost function and the inverse demand curve on the same diagram, labeling axes and intercepts. 1.2. What is the socially optimal level of production? Show this point on your diagram from 1.1. 1.3. How much electricity does the monopolist produce, assuming the monopolist maximizes profits? Show this point on your diagram from 1.1. 1.4. Calculate social welfare in 1.1 and 1.2. What is deadweight loss of monopoly?

Answer 1

(1.1)

MC = dc(e)/de = 2e/4 = 0.5e

When e = 0, MC = 0 (vertical intercept)

From inverse demand, when e = 0, P = 1 (vertical intercept) and when P = 0, e = 1/(1/2) = 2 (horizontal intercept).

In following graph, D and MC are inverse demand and MC curves respectively.

Price /мс PO 10.3-- MRI / 0.67 0 Q1 QO Quantity

(1.2)

Socially optimal production is where P = MC

1 - 0.5e = 0.5e

e = 1

P = 0.5 x 1 = 0.5

In above graph, socially optimal outcome is at point A where D intersects MC with price P0 (= 0.5) and output Q0 (= 1).

(1.3)

Monopolist equates MR and MC.

TR = P x e = e - 0.5e²

MR = dTR/de = 1 - e

1 - e = 0.5e

1.5e = 1

e = 0.67

P = 1 - 0.5 x 0.67 = 1 - 0.33 = 0.67

In above graph, socially optimal outcome is at point B where MR intersects MC with price P1 (= 0.67) and output Q1 (= 0.67).

(1.4)

Social welfare (TS) = CS + PS

In efficient outcome,

CS = (1/2) x (1 - 0.5) x 1 = 0.5 x 0.5 = 0.25

PS = (1/2) x (0.5 - 0) x 1 = 0.5 x 0.5 = 0.25

TS = 0.25 + 0.25 = 0.5

In monopoly,

CS = (1/2) x (1 - 0.67) x 0.67 = 0.33 x 0.33 = 0.11

When e = 0.67, MC = 0.5 x 0.67 = 0.33

PS = (1/2) x [(0.67 - 0) + (0.67 - 0.33) x 0.67 = 0.33 x (0.67 + 0.33) = 0.33 x 1 = 0.33

TS = 0.11 + 0.33 = 0.44

Deadweight loss = Change in TS = 0.5 - 0.44 = 0.06