1.  Clyde produces chemicals.  For any c litres of chemicals he produces, he can sell all c litres at a per-litre market price pc  = 700. Clyde’s cost of producing c litres of chemicals is Cc (c) = 5c2 + 100c. Clyde wants to maximise his profit, given by

π  (c) = p ccc − C  (c) = 700cc − 5c2 − 100c.

Bonnie is a baker, and can sell any b loaves of bread she bakes at a per-loaf market price of pb  = 10. Bonnie is adversely affected by the noxious fumes emitted by Clyde’s production of chemicals, so that her cost of baking b loaves of bread depends on the litres of chemicals c produced by Clyde, and is given by Cb (b,c) = b2 − 140b+bc. Bonnie wants to maximise her profit—which now also depends on Clyde’s output choice c—and is given by

π b (b,c) = pb b − Cb (b,c) = 10b − b2 + 140b − bc.

(i) Derive the output of chemicals c ◦  that maximises Clyde’s profit π c , by solving for c in the first-order condition   = 0.  Interpret Clyde’s revenue Rc (c) = pc c as his  “benefit” from production, and sketch a graph that shows both Clyde’s marginal  benefit/revenue   and his marginal cost  as a function of c.  Use your graph to briefly explain how the optimal output c ◦  is determined.

(ii) Given any arbitrary c litres of chemicals produced by Clyde, find Bonnie’s best-response output

of bread (c), by solving for b in the first-order condition    = 0.  Letting Rb (b) = pb b

denote Bonnie’s revenue/benefit from baking and selling bread, sketch a graph that shows both

her marginal revenue  and her marginal cost  as a function of b, for some arbitrary fixed value c. Use your graph to briefly explain how Bonnie’s best response (c) is determined.

(iii) Now consider a situation where Clyde and Bonnie must simultaneously choose their outputs c

and b, and solve for the Nash equilibrium (c◦ ,b◦ ) of the associated simultaneous-move game. Depict the best responses of both Clyde and Bonnie on a graph where c is measured along the horizontal axis and b is measured along the vertical axis, and use your graph to explain how the Nash equilibrium (c◦ ,b◦ ) is determined.

(iv) The mayor of the village in which Clyde and Bonnie reside believes that the output choices of

Clyde and Bonnie should be chosen at levels that maximise the sum of their profits

W(c,b) = πc (c) + πb (b,c).

Derive the mayor’s preferred output choices (c* ,b* ), by jointly solving for the two variables c and b in the first-order conditions  = 0 and  = 0.  Briefly explain how the first-order conditions that characterise (c* ,b* ) take into account marginal external effects that are ignored by the conditions that characterise the Nash equilibrium (c◦ ,b◦ ), and identify the type of the associated externalities.  Is the mayor’s preferred outcome (c* ,b* ) Pareto efficient with respect to the profit levels of Clyde and Bonnie? Why or why not?

(v) The mayor has studied economics, and has learned about Pigouvian taxes—if he has the authority to impose any taxes on output of either chemicals (a per-litre tax) or bread (a per-loaf tax), or both, what taxes should he impose that would yield his preferred outcome (c* ,b* ) from part (iv).  Derive the exact amount of the associated tax rates, and show why it would yield the desired outcome by solving the profit-maximisation problems of Clyde and Bonnie that include the relevant taxes.

(vi) Compute the profits of Clyde and Bonnie resulting from the Nash equilibrium outcome (c◦ ,b◦ ),

and from the mayor’s preferred outcome (c* ,b* ) from part (iv) (i.e.,  not  including the taxes from part (v)).  If the mayor cannot introduce a Pigouvian tax, and there are no laws that prevent Clyde from emitting noxious fumes (so that property rights in the village allow Clyde to pollute as much as he likes), briefly discuss whether Coasean bargaining between Bonnie and Clyde could allow them to reach the outcome (c* ,b* ).  If this is indeed possible, explain what agreements between Bonnie and Clyde would yield the desired outcome.

Hint : Would Bonnie be able to pay Clyde to reduce his output and therefore the pollution he causes, if any agreement they reach can be enforced through the legal system?

2.  Flo is a farmer and grows flowers on her farm, which is located right next to Beatrice’s property. Beatrice is a beekeeper. Beatrice’s beehives pollinate Flo’s flowers, and Flo’s flowers provide food for Beatrice’s bees.  If Flo grows f acres of flowers, she can sell each acre of flowers at a market price pf  = 50, and incurs a production cost Cf (f,b) = 5 (f − b)2 , where b denotes the number of beehives kept by Beatrice.  If Beatrice keeps b beehives, she can sell each beehive worth of honey at a market price pb  = 100, and incurs a production cost Cb (f,b) = 10 (b − f)2 , where f is the number of acres of flowers grown by Flo. Assume that Flo and Beatrice simultaneously choose their outputs f and g , and that they each want to maximise their respective profits

π f (f,b) = pf f − Cf (f,b) = 50f − 5 (f − b)2 ,

π b (f,b) = pb b − Cb (f,b) = 100b − 10 (b − f)2 .

(i) Given any arbitrary number of beehives b kept by Beatrice, find Flo’s best response number of acres of flowers f˜(b), by solving for f in the first-order condition of her profit maximisation problem.   Sketch a graph that shows both her marginal revenue and her marginal cost as a function of f, for some arbitrary fixed value b, and use your graph to briefly explain how Flo’s best response f˜(b) is determined.

(ii) Given any arbitrary number of acres of flowers f grown by Flo, find Beatrice’s best response number of beehives (f), by solving for b in the first-order condition of her profit maximisation problem.   Sketch a graph that shows both her marginal revenue and her marginal cost as a function of b, for some arbitrary fixed value f, and use your graph to briefly explain how Beatrice’s best response (f) is determined.

(iii) Derive the Nash equilibrium (f ◦ ,b◦ ) of the simultaneous-move game between Flo and Beatrice.

Depict the best responses of both Flo and Beatrice on a graph where b is measured along the horizontal axis and f is measured along the vertical axis, and use your graph to explain how the Nash equilibrium (f ◦ ,b◦ ) is determined.

(iv) The mayor of the village in which Flo and Beatrice reside believes that the output choices of Flo

and Beatrice should be chosen at levels that maximise the sum of their profits

W(f,b) = πf (f,b) + πb (f,b).

Derive the mayor’s preferred output choices (f * ,b* ), by jointly solving for the two variables f and b in the first-order conditions  = 0 and  = 0.  Briefly explain how the first-order conditions that characterise (f * ,b* ) take into account marginal external effects that are ignored by the conditions that characterise the Nash equilibrium (f ◦ ,b◦ ), and identify the type of the associated externalities.