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Problem Set 3 Solutions

1. (i.)

The equilibrium (market clearing) production level, Q, is foundby settingM C=M R:

0.728Q= 120 2Q

Q = 43.98 44

The equilibrium price, P where M C= M R:

P(Q) = 120 2Q

P= 120 2(43.98)

P = $32.02

Total revenue is the product of the optimal price and quantity:

P Q = 32.02 43.98 = $1, 408.66

Total cost is found by plugging the optimal quantity into the costfunction:

T C(Q

) = 0.364(43.98)2

T C= $704.33

We can also derive consumer, producer, and social surplus (forsection iii):

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Producer Surplus: 12(43.98) 32.02 = $704.33

Consumer surplus: 12(43.98) (120 32.02) = $1934.97

Total social surplus: T SS= C S+ P S= $2, 639.30

Graphically:

(ii.)

Marginal damage (MD) is increase in damage from producing oneextra unit of output:

D(Q)/ Q= 0.2Q

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Total social cost (TSC) is the sum of total cost and total dam-age:

0.364Q2 + 0.1Q2 = 0.464Q2

The marginal social cost (MSC) is the sum of marginal cost andmarginal damage:

M C+ M D= 0.728Q + 0.2Q= 0.928Q

(iii.)

The Pigouvian tax is the difference between marginal social cost(MSC) and marginal cost at Q where M SC=Demand:

0.928Q= 120 2Q

Q= 40.39836 41

When Q= 41,M C= 0.728(41) = $29.84, and M SC= 0.928(41) =$38.04 the Pigouvian tax is :

38.04 29.84 = $8.20

Market clearing quantity is Q where M SC = Demand, so Q =41. The market clearing price is P where M SC = Demand, soP= $38.04.

This graph shows the total social surplus is greater with the taxthan without:

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Before the tax:

CS= A + B+ G + H

P S=C+ D+ J

Damage =F+ L, where L = I+ H+ J

Total Social Surplus=A + B+ G + H+ C+ D+ J F I H J

=A + B+ G + C+ D I F

After the tax:

CS=A

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P S= B + C

Tax Revenue =D + G

Damage =F

Total Social Surplus =A + B+ C+ D+ G F

By comparing the total social surplus before tax with the one aftertax, we find that the net gain with a tax is I.

(iv.)

This example illustrates the theory of the second best. Note thatthe monopolists production lies below the socially optimal level.So taxing the monopolist would reduce welfare! We should notrecommend a pollution tax given the structure of this market. Ifanything, we should subsidize production so as to improve socialwelfare. However, it is unlikely that a regulatory agency in chargeof regulating pollution would subsidize an industry to correct adistortion associated with the exercise of market power. But youcould calculate the optimal subsidy taking into account the mo-

nopolists profit maximization problem and the marginal damagefunction.

2. (i)

(i) The two firms are identical, so we can solve the optimizationproblem for one firm and then use symmetry to find the solution:

maxq1= p1(q1, q2)q1 c1(q1) =q1 q21+ zq1q2

Giving the FOC (interior solution):

1 2q1+ zq2= 0

q1 = 1+zq2

2

This is firm 1s best response function (BRF). By symmetry, firm2s BRF is:

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q

2 = 1+zq12

You can now plug one BRF into the other to solve. Or, you can usethis often-quicker symmetry strategy: using the fact thatq1 =q

2,so you can change theq2 in firm 1s BRF to a q

1 and quickly solvefor the optimal quantities (note that you can only do this aftertaking the first-order conditions).

q1 = 1+zq

1

2

2q1 = 1 + zq1

q1 = 12z =q

2

Now plug these quantities into the inverse demand functions tofind equilibrium prices:

p1= 1 12z +

z2z =

2z2z

12z +

z2z

p1= 12z =p

2

[Note: whenever you use any shortcut, just make sure you haveprovided everything the question asked for. In this case, you couldhave solved for equilibrium quantities and prices without ever writ-ing down firm 2s best response function, but the question explic-itly asks for this.]

Convert the two inverse demand functions into demand functions.Rearrange each equation and then plug one into the other:

q1= 1 + zq2 p1 and q2= 1 + zq1 p2

q1= 1 + z(1 + zq1 p2) p1

q1 = 1 + z+ z2q1 zp2 p1

(1 z2)q1= 1 + z zp2 p1

q1(p1, p2) = 1+z1z2

11z2

p1 z1z2

p2

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And of course, by symmetry:

q2(p1, p2) = 1+z1z2

11z2

p2 z1z2

p1

(ii.)

The strategy looks very similar to part (a), except now we willwrite profits as q(p)pinstead ofp(q)q, and we will maximize withrespect top. Consider firm 1:

maxp1= q1(p1, p2)p1 = 1+z1z2

p1 11z2

p21 z1z2

p2p1

The FOC (interior solution) is:

1+z1z2

21z2

p1 z1z2

p2= 0

This gives firm 1s best response function.

p1= 1+zzp2

2

By symmetry, firm 2s BRF is:

p2= 1+zzp1

2

Now lets use the fact that p1= p

2 in equilibrium (again by sym-

metry) to solve firm 1s BRF for p

1:

p1 = 1+zzp

1

2

2p1 = 1 + z zp

1

p1= 1+z1z =p

2

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Now substitute in to the demand functions to find equilibriumquantities. Before doing that, lets simplify the demand functionsby recognizing that p1= p

2:

q1(p

1) = 1+z1z2

1+z1z2

p1= 1

(1z)(2+z)

And by symmetry:

q1(p

1) = 1+z1z2

1+z1z2

p1= 1

(1z)(2+z)

(iii.)

First lets compare prices. Prices under quantity competition willbe higher than prices under price competition if:

12z >

1+z2+z

2 + z >2 + 2z z z2

0> z2

This will always be true for values of z between -1 and 1 (Note: ifyou do not restrict the values of z, you have to be careful. Ifz >2or 2< z < 1, then when you try to cross multiply, the sign flips.(But also note, ifz >1, then you will get a negative quantity underquantity competition, which does not make sense.) Ifz = 0, ie thetwo goods are completely unrelated or heterogenous, then bothsides of this equation are equal. If z = 1, we get the Cournotand Bertrand results for the case where the firms produce the same

exact product.

Now lets compare quantities. Quantities under quantity compe-tition will be lower than quantities under price competition if:

12z