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Markets and the Environment Exercises 2 Instruments for correcting market failure Academic Year 2014/2015 This material is NOT for distribution 1.

a) P = MC(X ∗ ) ⇔ X ∗ = 10. Benefit function: B(X) = 10X − 0.5X 2 and external cost function: EC(X) = 0.25X 2 Hence B(10) = 100 − 0.5 · 100 = 50 and EC (10) = 0.25 · 100 = 25. b) At the social optimum P = MC(X o ) + MEC(X o ) ⇔ 10 = 1.5X o ⇔ X o =

20 3

20 10 )= 3 3    2 20 20 20 20 10 − 0.5 · = B( ) = 3 3 3 3  2  2  2 20 20 20 20 SW F ( ) = − 0.25 = 0.75 · 3 3 3 3 MEC(

Comparison: X o < X ∗ , B o < B ∗ , EC o < EC ∗ . c) Pigouvian tax t = MEC(X o ) =

10 3

d) Social optimum is characterized by P = SMC(X o ). Firms equate P − t = MC(X) ⇔ X = 10 −

1

20 10 30 − 10 = = = Xo 3 3 3

2.

a) competitive equilibrium: P = MC(X ∗ ) ⇔ 8 = 2X ∗ − 2 ⇔ X ∗ = 5 Net social welfare: W (X ) = pX − C(X ) − CE(X) = 8X − X 2 + 2X − 1, 5X 2 = X (10 − 2, 5X ) W (5) = 5(10 − 12, 5) = −5 · 2, 5 = −12, 5 b) social optimum : P = MC(X o )+ MEC(X o ) ⇔ 8 = 2X o − 2+3X o ⇔ X o = 2. W (2) = 2(10 − 5) = 10. Social welfare increases by 22,5 compared to a) c) Pigouvian subsidy s = MEC(X o ) = 3X o = 6

3.

a) Benefits of firm 1: B1 (N1 ) = P1 X1 (N1 ) − wN1 = 100N1 − N12 − 10N1 = N1 (90 − N1 ) Benefits of firm 2: B2 (N1 , N2 ) = P2 X2 (N1 , N2 ) − wN2 = 150N2 − N22 − 0, 8X1 In competitive equilibrium firms maximize profits. First-order conditions are: dB1 (N1 ) = 0 ⇔ 90 − 2N 1∗ = 0 ⇒ N1∗ = 45 dN1 dB2 (N1 , N2 ) = 0 ⇔ 150 − 2N 2∗ = 0 ⇒ N2∗ = 75 dN2 Levels of production: X1∗ = X1 (45) = 55 · 45 = 2475 and X2∗ = X2 (75) = 75(80 − 0, 5 · 75) − 0, 4 · 2475 = 3187, 5 − 990 = 2197, 5 Social welfare: W (N1 , N2 ) = X1 (N1 ) − 10N1 + N2 (150 − N2 ) − 0, 8X1 (N1 ) = 0.2X1 (N1 ) − 10N1 + N2 (150 − N2 ) W ∗ = W (45, 75) = 0, 2 · 2475 − 450 + 75(150 − 75) = 495 − 450 + 5625 = 5670 b) maxN1 ,N2 W (N1 , N2 ) = 10N1 − 0, 2N12 + 150N2 − N22 first-order conditions: dW = 0 ⇔ 10 − 0, 4N1o = 0 ⇔ N1o = 25 dN 1 dW = 0 ⇔ 150 − 2N2o = 0 ⇔ N2o = 75 dN 2 W o = W (25, 75) = 25(10 − 5) + 75(150 − 75) = 125 + 5625 = 5750 X1o = X1 (25) = 25·75 = 1875 and X2o = 75(80−0, 5·75)−0, 4∗1875 = 2437, 5 Hence X1o < X ∗1 , X2o > X2∗ and W o > W ∗ . c) The external cost is proportional to output X1 , in particular the external cost firm 2 incurs is given by P2 · V (X1 ) = 2V (X1 ) = 0, 8X1 . Since the Pigouvian tax on output of firm 1 equals the marginal external cost we have t = 0, 8.

2

d) Benefits of firm1 with Pigouvian tax: B1 (N1 ) = P1 X1 (N1 )−wN1 −0, 8X1 (N1 ) = 0, 2(100N1 −N12)−10N1 = 10N1 −0, 2N12 ¯ 1 = 25 = N1o. Hence The first-order condition is given by 10− 0, 4N¯1 = 0 ⇔ N the tax induces the firm to choose the optimal level of production. 4.

a) cost firm 1: C1 (R1 ) =

Z

21

0

 2 400Ri dRi = 200Ri2 0 1 = 200 · (21)2 = 88, 200

total cost= 2 · 88, 200 = 176, 400. b) Each firm picks the technology with the lower marginal cost of abatement. Firm 1 picks the scrubber and firm 2 picks low-sulfur coal, i.e. MC1 (R1 ) = 400R1 and MC2 (R2 ) = 300R2 . R 21 R 21 Total cost= 0 400R1 dR1 + 0 300R2 dR2 = 88, 200 + 150 · 212 = 88, 200 + 66, 150 = 154, 350. c) Cost-effective measures require that marginal abatement cost is equalized. Solve the system of equations 400R1o = 300Ro2 Ro1 + R2o = 42

(1) (2)

This gives R2o = 24 and R1o = 18. Abatement cost Firm 1: C1 (18) = 200 · 182 = 64800 Firm 2: C2 (24) = 150 · 242 = 86400 Total Cost= 64800 + 86400 = 151200. d) The Pigouvian tax also equalizes MC across firms. For it to give 42 tons of abatement we must equate t = MC1 (18) = MC2 (24) = 18 · 400 = 24 · 300 = 7200. The reduction by each firm and the total abatement cost is identical to c). e) Since each firm receives 19 permits, it has to reduce 40-19=21 tons. Hence the problem is isomorphic to question c). 5. The table takes the form:

3

(1) Firms

A B C D E F G TOTAL

(2) Historial Emissions (Tons/Year)

(3) Marginal Abatement Cost (e/Ton)

(4) Allowances Bought

(5) Allowances Sold

(6) Total Abatement Cost(NTA)

(7) Total Abatement Cost (TA)

600 600 600 600 800 800 800 4800

100 200 300 400 500 600 700

0 0 0 0 400 400 400 1200

300 300 300 300 0 0 0 1200

$30 K $60 K $90 K $120 K $200 K $240 K $280 K $1.02 mil

$60 K $120 K $180 K $240 K 0 0 0 $600 K

(NTA)No Tradable Allowances.(TA) Tradable Allowances.

a) column 6 b) column 4 , column 5 , column 7 c) The permit price is between e400 and e500. 6.

a) Firm 1: max py − 3y12 Firm 2: max py − y22

⇒ 5 = 6y1∗ ⇒ 5 = 2y2∗

Total Production: y1∗ + y2∗ = b) Firm A, y1∗ (p) =

p 6

, y2∗(p) =

⇔ y ∗1 = ⇔ y2∗ =

20 = 3 13 6 p If we charge 2

5 6

5 2

a tax τ , production is given by

y1∗ (p − τ ) + y2∗ (p − τ ) We want to solve

p−τ 6

+

p−τ 2

= 1 for τ with p = 5

p − τ + 3p − 3τ = 6 ⇔ τ = 3.5 . Then y1∗(5 − 3.5) + y ∗2 (5 − 3.5) = c) Total Costs= C1 (21) + C2 ( 21 ) =

3 4

+

31 + 23 12 26 1 =1 4

=

1 4

+

3 4

= 1.

7 4 9 Π∗2 = 5 · 0.5 − .25 = 4

Π∗1 = 5 · 0.5 − .75 =

d) Firms prefer taxes over quotas depending on which one gives them more profits. We have just found profits under quotas now we need profits under taxes. y1∗(p − τ ) =

4

1 4

y2∗(p − τ ) =

3 4

Π1 = (5 − 3.5) ·

1 1 3 − 3 · ( )2 = 16 4 4

Π2 = (5 − 3.5) ·

3 9 3 − 3 · ( )2 = 16 4 4

Comparing with (C) firms prefer quotas. e) Under tradable quotas each firm chooses yi to maximize (revenue - cost + net value of quota purchase), i.e. 1 max pyi − Ci (yi ) + ( − yi )pq 2 1 (p − pq )yi − Ci (y) + pq 2 The first-order condition is given by: (p − pq ) = Ci′(yi ∗) N.B. This is exactly the same we got for A and B. Y1∗(p − pq ) = (p − pq )/6 Y2∗ (p − pq ) = (p − pq )/2 Since the firms need one unit of quota for each unit of steel, if the market for quota clears, Y1∗ (p − pq ) + Y2∗(p − pq ) = 1 This is exactly the expression we solved to get τ ⇒ pq = τ = 3.5. Profits are also exactly the same as under taxes, but, both firms are better off by the value of their original 21 unit of quota. ˆ 1 = 3/16 + 1 pq = 3/16 + 7/4 = 31/16 Π 2 ˆ 2 = 9/16 + 1 pq = 9/16 + 7/4 = 37/16 Π 2 f) Firms will choose non- tradable quota if it is more profitable than tradable ˆ 1 = 7/4 − quota. What is the most the govt. can charge? Firm 1: Π1∗ − Π 31/16 = −3/16 ˆ 2 = 9/4 − 37/16 = −1/16 Firm 2: Π∗ − Π 2

The govt. can charge no more than opts for non tradable.

5

1 16

for initial

1 2

unit of quota or Firm 2...