1 Safe Drivers versus Reckless Drunk Drivers

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1 ECON 301: General Equilibrium IV (Externalitie) 1 Intermediate Microeconomic II, ECON 301 General Equilibrium IV: Externalitie In our dicuion thu far, we have implicitly aumed that all good can be traded in the economy, or ha value in term of price in the uual ene However, thi need not alway be true Conider your deire to have clean air in your local pub, but which i typically filled with moker Or your deire to have a road free of drunk driver, but i often difficult to enforce Or a fihery located right next to a power generation plant that ue coal In all the cae lited, the agent involved would like to have clean air, and afe road but their may not be market for them typically Or i there? The above example are what we term a externalitie 1 Conumption Externality: Occur when one conumer care directly about another agent production or conumption choice becaue it ha direct implication on her utility 2 Production Externality: Occur when the choice made by a firm ha implication on the production poibilitie of another When externalitie are preent, it turn out that the market typically will not achieve Pareto Efficiency in Allocation, which i not really urpriing However, there are ocial intitution, uch a the legal ytem, or ome form of government intervention that may be able to duplicate the market mechanim and thereby bring about Pareto Efficiency 1 Safe Driver veru Reckle Drunk Driver We will firt deal with the former, conumption externality Conider two agent or neighbor living in the ame town, A i a afe driver, who never drive if he drink, but B i a habitual town drunk, who love to drive while inebriated Let the two good on hand be income, y, and afe road, ay x Let x be ay a meaure between 0 to 1, where 0 implie road with no drunk driver, while 1 ha the road filled with them So you can think of A and B a two et of conumer rather than two conumer We can then depict their choice et in a Edgeworth Box a follow;

2 ECON 301: General Equilibrium IV (Externalitie) 2 Safe Road 0 Figure 1: Preference for Money & Safe Road Peron B 0 1 Peron A 1 Income Note the light difference in the orientation of the axi, where for afe road, A prefer the maller value, while B prefer the larger value Suppoe initially both individual (or group of individual) have an equal hare of income that i available a repreented by the dotted line The final equilibrium i dependent on who ha the legal right to afe road If the right are veted in A, then the initial endowment would be point E A, while if the legal right are veted in B, the initial endowment would be point E B It i clear then that the legal ytem determine the endowment In general, there i no reaon to believe that the initial endowment i Pareto efficient, that i it i poible that A i willing to trade ome of the afety in return for income, and the ame i true for B You can now ee that depending on whoe ide the Judicial Sytem take will determine who ha the greater level of utility at the final equilibrium How can uch trading be achieved, you could imagine an auctioneer calling the price for afe road, and income In that ene, it i almot equivalent to B bribing A

3 ECON 301: General Equilibrium IV (Externalitie) 3 to drive on the road drunk Thi eem like a imple enough analogy, where the pricing mechanim i like a fine levied on drunk driving hould B be caught Problem arie however when Property Right are not well defined The analyi thu highlight the often heard cry for clearer definition of Law 2 Quailinear Preference and the Coae Theorem Although it i true in general that Property Right determine the ultimate level of externalitie that exit in an economy There are ituation when it doe not Thi occur when the utility function of agent are of the Quailinear Form a illutrated in figure 2 Notice that each individual indifference curve are horizontal tranlation of the other In that cae, the optimal level of externality i independent of which ever initial allocation i choen or determined by Property Right, and thi i typically known a Coae Theorem, credited to Ronald Coae of the Univerity of Chicago The numeraire good here i income However, thi interpretation ha been conteted with ome economit aerting that Coae Theorem only aert that when bargaining between agent are cotle over externalitie Pareto Efficiency i derived, and not that it i independent of property right

4 ECON 301: General Equilibrium IV (Externalitie) 4 Figure 2: Quailinear Preference & Coae Theorem Safe Road Income Peron A Peron B U A U B Pareto Set Production Externalitie We will now conider production externalitie but with the aid of a little calculu Conider two firm, one a fihing village owned by a ingle clan, F, the other an off hore natural ga firm, G, that extract and pipe natural ga to the mainland of thi economy The tory i uch that the extraction will inevitably involve ome eepage that would harm the indigenou fih population the F harvet Let denote thi eepage a x, the externality Denote the output of the fihing village a f, and the ga company, g, with repective price, p F and p G Their repective cot function are c F (f, x) and c G (g, x) The more eepage, the more cotly would the production proce for the fihing village be, and we can denote imply thi a c F > 0 On the other hand, the more care in reducing the eepage, the greater the cot of the ga company, o that c G 0

5 ECON 301: General Equilibrium IV (Externalitie) 5 The objective of each firm can be written a; max g,x p Gg c G (g, x) for the ga company, while for the fihing village, max p F f c F (f, x) f which i nothing but you tandard profit function, and the problem i one of profit maximization Note that it i only the natural ga company that get to chooe the level of externality, x, and that in maximizing it profit, it will not be concerned with the externality which it impoe on the fihing village The firt order condition for the natural ga firm are then, For the fihing village, it i jut, p G = c G(g, x ) g 0 = c G(g, x ) (1) (2) p F = c F (f, x ) f (3) where g, f and x are the optimal value of natural ga, fih output, and eepage in uch a program where each firm maximize their own profit independently What i clear from the above i that in maximizing their profit, the natural ga company G diregard it choice effect may have on the fihing village In producing natural ga and fihe, both produce up to the point where marginal revenue equate with marginal cot However, eepage from natural ga extraction increae the fihing village cot of producing fihe, and yet in chooing the optimal level of externality, firm G will produce x up to the point where the marginal cot of producing the next unit of externality i zero, and given the cot of production i decreaing in the level of externality, thi likely mean that G would produce more than they hould (We do not know yet We have to prove it) Thi cot impoed on the village i the ocial cot of natural ga extraction How can we examine what would be the Pareto efficient level of externality? If the firm performing the extraction, were alo the ame firm fihing the ame water, it would be in their interet to moderate the level of eepage Given thi level, we could

6 ECON 301: General Equilibrium IV (Externalitie) 6 then examine if in the prior independent etup, whether there i an over production of externality Thi collective firm would maximize the following objective function; The firt order condition are now; max p Gg + g F f c G (g, x) c F (f, x) g,f,x p G = c G(ĝ, x) g (4) p F = c F ( f, x) f 0 = c F ( f, x) + c G(ĝ, x) (6) The firt two condition are a before, but evaluated at the new level of production and externalitie, ince ĝ g, f f, and x x The key condition i the lat, which can be rewritten a, c F ( f, x) = c G(ĝ, x) > 0 Since we know that both ide of the equality i poitive, thi term imply ay that in equilibrium, marginal cot of the externalitie mut be the ame in both type of production What i the ignificance of thi? Combining what we have here, with what we found before under independent production, c G (ĝ, x) < c G(g, x ) Since cot increae a externalitie fall (negative), thi then mean that the abolute value of the marginal cot i greater in the new etup than the old, which mean either g i higher, or x lower, or both are true Conequently, the new firm exercie a greater level of care, in other word, internalize the ocial cot the independent natural ga firm impoe on the fihing village Diagrammatically, thi i repreented below, = 0 (5) 4 Interpretation of the Condition 1 We can think of thi Pareto efficient level of eepage production a the point up to which the natural ga company price that they are willing to pay for permiion

7 ECON 301: General Equilibrium IV (Externalitie) 7 Figure 3: Social & Private Cot p G, p F MC F = MC G MC G MC F x x x to pollute i jut equal to the actual ocial cot The inefficiency arie in the original ituation becaue firm G faced the wrong price Thi can be rectified by impoing a tax on the pollution of the water Suppoe the tax rate impoed i t per unit of pollution, o that for any amount of eepage x, the tax firm G ha to pay i xt The new profit maximization problem now i intead, max p Gg c G (g, x) tx g,x Thi then give the following firt order condition, p G = c G(g, x) g (7) t = c G(g, x) (8) Comparing equation (8) with (6), we ee that the olution i nothing but etting t = c F (g, x)

8 ECON 301: General Equilibrium IV (Externalitie) 8 thereby giving rie to the Pareto Efficient outcome, or level of externality Thi kind of tax i known a Pigouvian Tax The problem with uch a cheme i that we need to know the optimal level of pollution to impoe the right level of tax, and yet if we did know, we needn t even alter the tax cheme, ince we can imply init that firm G produce the correct level of externality 2 We could alternately think of the problem a one of miing market for pollutant intead of facing the wrong price Conider the property right a being veted with the fihing village, who could ell the right to pollute to firm G, at the price of q The problem for firm G and fihing village F i a follow, and max p Gg c G (g, x) qx g,x max f,x p F + qx c F (f, x) repectively It i clear that thi program involve the tranfer between the natural ga company and the fihing village In thi manner, the fihing village now wretle away the deciion of pollutant in their water, ince optimally, the price they are willing to accept mut be equal to the price firm G i willing to pay Show yourelf that thi mut be true 3 We could ue a imilar program a the lat, but now give the property right to firm G We already know that if firm G doe not internalize the externality it impoe it will produce x level of externality Since the property right are veted with G, the fihing village could intead pay G not to pollute their water In which cae, the profit maximizing problem of G become, max g,x p Gg + q(x x) c G (g, x) while the fihing village problem i intead, max f,x p F f q(x x) c F (f, x) Show that in equilibrium, the ame equilibrium condition i derived a before, and conequently yield the ame level of Pareto Efficient level of externality Although the ame level of externalitie i obtained, note that the wherea in the former the tranfer i from G to F, while in the current, the tranfer i from F to G

9 ECON 301: General Equilibrium IV (Externalitie) 9 4 Market Signal: We have found that if the firm could merge a one, it would then have to internalize any externalitie, thereby achieving higher profit The idea i that a long a the joint profit of the merged entity i greater than if the firm remain eparate, there i alway an impetu to merge That i the market itelf hould have provided the ignal for firm to merge if it i viable A claic example i that of an apiary and apple orchard haring the ame ownerhip 5 The Tragedy of Common: The Cae of Overfihing in International Water We have noted that a long a Property Right are well defined, the Pareto Efficient level allocation will be produced But when exactly i Property Right ambiguou? One of the claic example i from Overfihing Who own the ocean of the world? No one in truth, and everyone alo in truth Can we ue what we have learned about General Equilibrium and Externalitie to undertand the problem? Let the fih production in the Ocean of thi planet be F (), where i the ize of the fihing fleet where each fleet can catch a fixed maximum capacity Let the cot of fihing by each country (which you can think of a individual for implicity) be C() Now imagine there i a central world authority exit and can decide how much to fih through determining the number of hip in the water in any period What would that choice be? The firt order condition i imply, max F () C() F () = C() Which i nothing but the marginal product of the additional boat being equal to the marginal cot of obtaining that boat on the open ea Note that I have effectively normalized the price of the output at 1 for implicity Now what i occurring when an individual country or individual chooe to the optimal ize of the fleet? A country would continue to add to their fleet a long a the per unit haul per boat yield more in value than the cot of adding an additional boat Taken on the

10 ECON 301: General Equilibrium IV (Externalitie) 10 aggregate, it i equivalent to the following: The former meaure i nothing but the average haul which i F (), while the latter i C() Taken together thi mean that; F () = C() F () C() = 0 That i the countrie will keep fihing with more boat until they earn zero profit You can think of thi a each country acting like a perfectly competitive firm o that they believe their choice will have no effect on the output of the world Thi equilibrium i nothing but equating the average product to average cot a in perfect competition Since the production function will uffer from diminihing marginal rate of production, ie downward loping, the marginal product i alway lower than the average product, thu, the deciion made by the authority would yield a lower level of hip, than when every country act on their own intinct Thi i commonly referred to a the Tragedy of Common, due to common ownerhip of a reource, ie property right i ambiguou Diagrammatically, thi i repreented below where for implicity, the cot per hip i fixed at c,

11 ECON 301: General Equilibrium IV (Externalitie) 11 Figure 4: The Tragedy of Common AP, MP MC c AP Efficient Output Equilibrium Output

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