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Unformatted text preview: 795 Externalities in Competitive Markets 21.10 PolicyApplication: Pollution that increases ﬁrm costs 7 Barney’s Solution: Consider the same
situation as the one described in exercise 21.9. A: Assume again that the only impact of pollution is that it increases ﬁrm ﬁxed costs by 6 for every
unit ofx that is produced in the industry.
(a) Suppose there are N ﬁrms in the equilibrium you described in exercise 21.9. What is the pol lution related cost ofﬁrm i producingone more unit ofx? Answer: The pollution related cost of one more unit of output is then 6 N — because the
ﬁxed cost of every one of the N ﬁrms is increased by 6. (b) How much ofthis pollution related cost does ﬁrm i not take into account? Ifﬁrm i is one of a large number of ﬁrms, is it a good approximation to say that ﬁrm i does not take any ofthe
pollution related cost into account? How is this similar to our “pricetaking” assumption ﬁ2r
competitive ﬁrms? Answer: Of the marginal pollution related cost 6N, the ﬁrm only takes into account the
impact on its own ﬁrm — not the (N— 1) other ﬁrms. Thus, the ﬁrm does not take into
account 6 (N — 1) of the 6 N pollution related costs it causes by producing one more output
unit. As N gets large, this implies that the percentage of the pollution related costs it causes
approaches zero. It is therefore reasonable to simply assume the competitive ﬁrm does not
take any of the pollution related costs it causes into account — and it simply takes the level
of pollution as given. This is similar to our pricetaking assumption in that both appeal to
the fact that each ﬁrm is so small that it makes no sense for it to behave strategically in its
price setting or its pollution production. (c) Suppose that our benevolentsocial planner Barney can tell ﬁrms what to count as costs. Il lustrate how Barney’s suggestion foreach ﬁrm’s marginal costcurve is related to the marginal
costcurve ﬁrms would otherwise use (given a ﬁxed number N of ﬁrms in the industry)? Answer: This is illustrated in panel (a) of Graph 21.10 — the SM C that Barney would want
the ﬁrm to use includes the (SN marginal cost of pollution that results from marginal in
creases in output. (M smc m
gm Graph 21.10: Private and Social Marginal Costs on the Lake (d) What does your answer imply about the relationship between the ﬁrm’s AC curve and Barney’s suggestion for what the ﬁrm’s AC curve should be? Answer: This is illustrated in panel (b) of Graph 21.10. The additional marginal cost that
Barney would like to include is like a perunit tax that shifts up the AC curve in a parallel
way — leaving the lowest point unchanged. Externalities in Competitive Markets 796 (e) True or False: 1 f ﬁrms used Barney’s suggested cost curves, the long run industry supply curve
would be upward sloping as you should have concluded in exercise 21.9 it is in the absence of
Barney— but now it would lie above where it was in exercise 21.9. Answer: As the industry expands, the AC curves for ﬁrms will still be shifting up because of
the increased pollution costs — but the “SAC” in panel (b) of the graph will always lie above
the private AC curve used by ﬁrms. Thus, if Barney caused ﬁrms to consider the full cost
of their production choices, the lowest point of the AC curve for any size of the industry
would be higher — implying that the long run industry supply curve would be higher (and
still upward sloping). (f) True or False: Under the efﬁcient outcome, the industry would produce less ata higher price. Answer: This is true — the efﬁcient outcome would have ﬁrms take into account the full
cost of pollution — which would lead to an increase in cost, an increase in price and a drop
in output (assuming downward sloping demand). (g) I f a single corporation acquired all the ﬁrms around the lake, would that corporation take
the costs of pollution into account more like Barney or more like the individual competitive
ﬁrms? (In exercise 23.1 1, yo u’ll be asked to revisit this in the context ofsuch a monopoly.) Answer: If a single corporation owned all the ﬁrms around the lake, then it would consider
the full cost of the pollution since the only impact of the pollution is on the costs of the ﬁrms
around the lake. Thus, the corporation would take the pollution costs into consideration
like Barney. If the corporation still behaved as a price taker, that would fully resolve the
externality problem. (But if the corporation establishes a monopoly as it acquires all the
ﬁrms around the lake, it will cause efﬁciency problems deriving from the existence of market
power, as we will see in Chapter 23). B: Consider the same setup as in part B of exercise 21.9. In the previous case where we derived the
marketequilibrium, we said thati in a model with many ﬁrms 7 it was reasonable to model each
individual ﬁrm as not taking its own impact of pollution into account and to simply model the cost
function as c(x) = ﬁxz + (SN? (where the latter entered as a ﬁxed cost). (a) Now consider the cost function that benevolent Barney would use for each ﬁrm: From the
social planner’s perspective, the ﬁrm’s variable costs (captured by ) would still matter; as
would the ﬁxed cost from pollution (captured by 5N7 where? is the amount produced by each
ﬁrm and N is the number of ﬁrms in the industry.) B utBarney also cares about theﬁ2llowing:
each unit of x produced by ﬁrm i causes an increase in costs of6 for each of the N ﬁrms 7
which implies that the pollution cost Barney would consider ﬁrm i as imposing on society
is 6Nx. This implies that Barney’s costfunctionfor each ﬁrm is c3 (x) = + (mm 6Nx.
Derive from this the marginal and average cost functions that Barney would use for each ﬁrm
(beingsure to not treat the last term as a ﬁxed cost.) Answer: We now get 603m =2ﬁx+6N and AC(x) = 030‘) = ﬁx+6N+ E. (21.80)
(ix x x
(b) Repeat parts (c) through (i) from exercise 21.9 using the cost functions Barney would use for eachﬁrm to arrive atN*, p* and X*. Answer: Setting MC equal to AC, we get the quantity at the lowest point of the AC curve — i.e. x = (5W/ﬁ)llz. Setting )6 equal to x (since every ﬁrm produces the same in equi
librium), we also again get x(N) = (SN/ﬂ. Aggregating across N ﬁrms, we still get X (N) =
6N2 lﬁ — and the inverse N (X) = (ﬁX /6)1/2. Thus far, nothing has changed from the anal
ysis in the previous problem. But when we calculate p(N) — the zero proﬁt price as a func
tion of how many ﬁrms are in the market — we now get a different answer. We derive this
by substituting x(N) into either the MC or the AC curves to get MC(x) = p(N) = 36N (21.81) rather than 26 N as in the previous problem. This should make sense: Since ﬁrms now take
into account their pollution costs for the industry, the zero proﬁt price has to be larger than
it was before. Substituting N (X) into p(N), we then get the long run industry supply curve 797 Externalities in Competitive Markets p(X) =3(ﬁ§X)1l2 (21.82) as opposed to 2(ﬁt5X)“2 as in the previous problem. Thus, the industry supply curve is
shifted up — again because producers now are taking into account the social cost of the
pollution they are producing. Setting demand equal to supply, solving for X and then com
pleting the remaining steps, we end up with 1/2 1I4
YEW, p* =(3A)1’2(ﬁ6)“4, and N*=(§) . (21.83) (c) Compare your answers to those from exercise 21.9. How do they differ? Answer: Comparing these to the solutions from the previous problem, we see that industry
output X * and the numb er of ﬁrms N * fall while the price p* increases. Put differently,
were ﬁrms to take into account the externality costs they impose on other ﬁrms, there would
be fewer of them producing less and selling at a higher price. (d) Suppose, as in part (i) of exercise 21.9 thatﬁ = 1, 6 = 0.1 and A = 10,580. What are X *, p* and N * ? How much does each individual ﬁrm produce?
Answer: Plugging these values into our equations above, we get p" = 56.34, X" = 35,267 and N" = 1,878 (21.84) with each ﬁrm producing approximately 18.78 units of output. (e) Compare these to your answers in exercise 21.9. Can you give an intuitive explanation for why these answers differ despite the fact that pollution only affects the ﬁrms in the indusry? Answer: As already suggested by our formulas above, price has increased while (ﬁrm and in
dustry) output as well as the number of ﬁrms have fallen. The fact that pollution only affects
the costs of the ﬁrms in the industry therefore does not mean that the industrywill internal
ize the pollution costs — because no single ﬁrm has an incentive to take into account the
cost of the pollution it produces. In Chapter 27 we will call this the freerider problem. (D What is the Pigouvian tax that is required in order ﬁ2r competitive ﬁrms to implement the equilibrium you just calculated in (d)? What price does this imply consumers would pay and
what price does it imply producers would receive? Answer: The tax would have to cause the MC as perceived by the ﬁrms to rise to the MC
as desired by our benevolent Barney. This implies a perunit tax of 6 N — which is simply
the marginal social cost of the pollution from a ﬁrm increasing output by one unit. But the
important part is to get N right — because Pigouvian taxes are determined at the optimum,
not at the equilibrium (in the absence of taxes). We just calculated that, at the optimum,
there would be 1,878 ﬁrms. Thus, the necessaryPigouvian tax is t = 0.01(1,878) = 18.78. This
implies a consumer price of pd = 56.34 and a supplier price of p; = 56.34 — 18.78 = 37.56. (g) Verijji that your Pigouvian tax in fact results in prices for consumers and the industry that lead them to demand and supply the output level you calculated in part (d). (Note: You will
need to refer back to your answers to exercise 21.9 to do this part.) Answer: We calculated in exercise 21.9 that the industry supply curve (in the absence of
Barney's intervention) is p = 2(ﬁ6x)1/2. Substituting [3 = 1 and 6 = 0.01, this means an
industry supply curve for our example of p = 0.02x1l2 — or an industry supply function of
x; = 25p2. The market demand curve is p = A/xllz. Substituting A = 10,580 and inverting
to get the demand function, we get xd = (10580/ p)2. Substituting the supplier price of 37.56
into the supply function, we get x3 = 35,269, while substituting the consumer price of 56.34
into the demand function gives us xd = 35,265. Thus, aside from some rounding error, we
get that the quantity demanded is equal to the quantity supplies — which in turn is equal to
the industry supply X * we calculated in (b) for the cost curves recommended by Barney. ...
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This note was uploaded on 11/17/2010 for the course ECON 100A taught by Professor Woroch during the Fall '08 term at University of California, Berkeley.
 Fall '08
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