Fall Final Examination Solutions Thursday 10 January 2012

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1 EC 20.2 & 20. Fall 202 Deniz Selman Bo¼gaziçi University Final Examination Solutions Thursday 0 January 202. (9 pts) It is the heart of winter the isl of Ludos has been devastated by a violent snowstorm which has destroyed almost everything on the isl. In order to survive, the original inhabitants of Ludos have escaped, only two people remain. One is the isl s ruler, King Lodewijk. The other is Defne. Digging through the snowy destruction, Defne King Lodewijk each nd small amounts of cheese (c) yogurt (y), left over from when the isl was a major producer of both goods. To be exact, Defne nds! D c = units of cheese! D y = 2 units of yogurt, King Lodewijk nds! L c = units of cheese! L y = 5 units of yogurt. Defne s utility function for cheese yogurt is u D c D ; y D = ln c D + y D ; whereas King Lodewijk s utility function is u L c L ; y L = c L + y L : Let p = (p c ; p y ) be the price vector in this exchange economy assume throughout the question that p y = : (a) ( pts) Based on the welfare theorems, would you expect a competitive equilibrium allocation of this exchange economy to be Pareto e cient? If so, which welfare theorem says so? If not, which assumptions (explicit or implicit) of the relevant welfare theorem are violated in this economy? Soln: Yes the rst basic welfare theorem states that a competitive equilibrium of an exchange economy in which all preferences are strictly monotonic is Pareto e cient. In this example, neither the monotonicity assumption nor any of the implicit assumptions are violated. (b) (6 pts) Find the contract curve (the set of all Pareto e cient allocations) in this exchange economy. Draw the contract curve on a well-labeled Edgeworth Box with Defne s origin in the bottom left, King Lodewijk s origin on the top right, cheese on the horizontal axis yogurt on the vertical axis. Soln: Defne s marginal rate of substitution is given by MRS D cy c D ; y D D(c D ;y D D D (c D ;y D D while King Lodewijk s marginal rate of substitution is given by MRS L cy c L ; y L = c D = c D L(c L ;y L L (c L ;y L L = = (for all cl ; y L ). so, because the interior of the contract curve is given where MRScy D c D ; y D = MRScy L c L ; y L ; we have that = ; or c D c D = : The contract curve hence begins at Defne s origin, moves horizontally to c D = ; moves vertically to the upper boundary, then horizontally to King Lodewijk s origin. (c) (5 pts) What are Defne s gross dem functions bc D p;! D by D p;! D ; given that! D = (; 2) p y =? Soln: Defne s consumer problem with endowments is c D ;y D 0 ln c D + y D s u b j. t o p c c D + p y y D p c! D c + p y! D y ; which (plugging in known values using the fact that Defne s utility is strictly increasing hence her budget constraint will hold with equality) becomes c D ;y D 0 ln c D + y D s u b j. t o p c c D + y D = p c + 2:

2 Substituting y D = c D p c + 2, we have that Defne is choosing c D to imize the expression ln c D + c D p c + 2: The FOC is c D p c = 0; while the SOC holds, so we have given that! D = (; 2) p y = : bc D p;! D = p c ; (0.) by D p;! D = bc D p;! D p c + 2 = p c + 2 p c = p c + ; (0.2) (d) (5 pts) What are King Lodewijk s gross dem functions bc L p;! L by L p;! L ; given that! L = (; 5) p y =? Soln: King Lodewijk s consumer problem with endowments is c L ;y L 0 c L + y L s u b j. t o p c c L + p y y L p c! L c + p y! L y ; which (plugging in known values using the fact that King Lodewijk s utility is strictly increasing hence her budget constraint will hold with equality) becomes c L ;y L 0 c L + y L s u b j. t o p c c L + y L = p c + 5: Substituting y L = 5 c L p c + 5, we have that King Lodewijk is choosing c L to imize the expression c L + 5 c L p c + 5 = 5p c ( p c ) c L ; which is strictly increasing in c L when p c <, strictly decreasing in c L when p c > ; constant for all c L when p c = : This means King Lodewijk will dem only cheese no yogurt when p c <, dem only yogurt no cheese when p c > ; be indi erent among all bundles on his budget line when p c = : Therefore, given that! L = (; 5) p y = ; we have that 8 bc L p;! L < + 5 p c if p c < = (0.) : 0 if p c > 8 by L p;! L < 0 if p c < = : p c + 5 if p c > ; (0.4) while bc L p;! L ; by L p;! L can be anything as long as dems are non-negative satisfy by L p;! L = 5 bc L p;! L p c + 5 when p c = : (e) (2 pts) Find the competitive equilibrium in this exchange economy, c D ; y D ; c L ; y L ; (p c ; p y ), continuing to assume that p y = : Soln: We are looking for 6 unknowns c D ; y D ; c L ; y L ; p c ; p y have already that py = : First let s assume that p c <. We then have from (0:4) that y L = 0; so y D =! y y L = 7 0 = 7, which from (0:2) yields 7 = p c +, p c = 2; which contradicts our assumption that p c < : Next let s assume p c > : Then from (0:) we have c L = 0, so c D =! c c L = 4 0 = 4; which from (0:) yields 4 = p c, p c = 4 ; 2

3 which contradicts our assumption that p c > : Next let s assume p c = : Then from (0:) (0:2) we have c D = p c = = so The competitive equilibrium is given by y D = p c + = () + = 4; c L =! c c D = 4 = : y L =! y y D = 7 4 = : c D ; y D ; c L ; y L ; (p c ; p y ) = ((; 4) ; (; ) ; (; )) : (f) (2 pts) Is the competitive equilibrium allocation you found in part (e) Pareto e cient? Very brie y explain your answer. Soln: Yes. The allocation clearly lies in the interior portion of the contract curve, where c D = : (g) (6 pts) King Lodewijk would like the nal allocation in the economy to be c D ; y D ; c L ; y L = ((; ) ; (; 6)) : Because he is king, he can impose (pre-exchange) transfers. Suppose he would like to achieve his desired nal allocation as a competitive equilibrium by imposing a transfer of cheese only. That is, the initial endowments of yogurt will remain! D y = 2;! L y = 5; but King Lodewijk will be free to choose the initial endowments of cheese! D c ;! L c to be any feasible amounts. Is it possible for him to choose these cheese endowments such that the resulting competitive equilibrium allocation is c D ; y D ; c L ; y L = ((; ) ; (; 6))? If so, what endowments! D c ;! L c must he choose? If not, explain why not. Soln: Yes, this is possible. King Lodewijk must choose to endow all of the cheese to himself,! D c ;! L c = (0; 4) ; so that the competitive equilibrium will be c D ; y D ; c L ; y L ; (p c ; p y ) = ((; ) ; (; 6) ; (; )) : 2. (4 pts) The snow continues to fall in Ludos the inhabitants have still not returned. King Lodewijk Defne have consumed all of the remaining cheese yogurt on the isl are now desperate for food to survive. Thankfully, Shalini has started an exotic plants delivery business, occasionally comes by Ludos on her boat. She is willing to sell Defne King Lodewijk any amount of xanthorrhoea (x ) xeranthemum (x 2 ) that they wish to buy, at a price of w for each unit of xanthorrhoea a price of for each unit of xeranthemum. While neither xanthorrhoea nor xeranthemum are edible by themselves, Defne King Lodewijk develop technology which allows them to produce a new, rather interesting kind of yogurt (y) out of the two exotic plants. The amount of yogurt they can produce from xanthorrhoea xeranthemum is given by the production function f (x ; x 2 ) = (x ) = (x 2 ) = : They incur no costs when producing yogurt other than the cost of the xanthorrhoea the xeranthemum. (a) ( pts) Is the technology Defne King Lodewijk develop consistent with that of a decreasing, constant or increasing returns to scale industry? Soln: The production function exhibits decreasing returns to scale. The easiest way to see this is to note that it is a Cobb-Douglas production function in which the sum of the coe cients + = 2 is less than one. It is of course also possible to see directly that, for any > ; f (x ; x 2 ) = (x ) = (x 2 ) = = 2= (x ) = (x 2 ) = = 2= f (x ; x 2 ) < f (x ; x 2 ) :

4 (b) ( pts) Find Defne King Lodewijk s conditional input dem functions for xanthorrhoea xeranthemum, bx (y; w) bx 2 (y; w) ; where y is the amount of yogurt produced w = (w ; ) is the input price vector. Soln: Defne King Lodewijk s cost minimization problem is given by min x ;x 20 w x + x 2 s u b j. t o (x ) = (x 2 ) = y: Given that the production function is strictly increasing, we know that the production constraint will hold with equality at the optimal solution, so (x ) = (x 2 ) = = y: (0.5) We also know that the marginal rate of technical substitution will be equal to the ratio of the input prices at the optimal solution, so Solving equations (0:5) (0:6) together yields, MRT S 2 (x) = w, x 2 = w x ) (x ) =, (x ) 2= =, x = 2 (x ) 2= (x 2 ) = (x 2) 2= (x ) = w = w2 w, x 2 x = w : (0.6) w = y =2 y =2 ) x 2 = w x = w x w2 w = = y =2 y =2 = w =2 y =2 ; so Defne King Lodewijk s conditional input dem functions for xanthorrhoea xeranthemum are =2 w2 bx (y; w) = y =2 respectively. bx 2 (y; w) = w w =2 y =2 ; (c) ( pts) Write down Defne King Lodewijk s cost function for producing yogurt, C (y; w) : Soln: The cost function is given by C (y; w) = w bx (y; w) + bx 2 (y; w) =2 =2 w2 = w y =2 w + y =2 w = 2 (w ) =2 y =2 4

5 (d) ( pts) (For the rest of this question, suppose the prices Shalini charges for xanthorrhoea xeranthemum are w = 5 = 20.) The snow has stopped, the inhabitants of Ludos have nally returned to the isl. Because Defne King Lodewijk are the only producers of yogurt, they can operate together as a monopoly. Suppose total industry dem for yogurt on Ludos is given by D (p) = p 2 : What is the pro t imizing monopoly output y M the monopoly price p M? Soln: Defne Lodewijk s cost function is C (y) = 20y =2 ; so their marginal cost is We solve for inverse dem as follows: y = C 0 (y) = 0y =2 : 465, y =2 = 465, P (y) = P (y) 2 P (y) 2y =2 ; so revenue is R (y) = P (y) y = 0y 2y =2 ; so marginal revenue is given by R 0 (y) = 0 y =2 : The pro t imizing monopoly output is the solution to C 0 (y) = R 0 (y) ; so C 0 (y) = R 0 (y), 0y =2 = 0 y =2, y =2 = 0, y M = 00: The monopoly price is Note that monopoly pro t is p M = P y M = P (00) = 0 = 0 = 90 : 2 (00) =2 20 M = R y M C y M = p M y M 20 y M =2 = 90 = 9000 = (00)=2 > 0:

6 (e) (6 pts) As time goes on, other inhabitants of Ludos get access to xanthorrhoea xeranthemum (at the same prices as Defne King Lodewijk), as well as the same yogurt producing technology, so the yogurt industry becomes competitive. What will be the long-run equilibrium industry output of yogurt, Q ; the long-run equilibrium price p the long-run industry pro t (per rm),? Soln: In the long-run we know that the supply curve becomes completely elastic the long-run equilibrium price is equal to p = p min ; where p min is the minimum of the average cost curve. But note that AC (y) = C (y) y = 20y=2 y = 20y =2 ; so y min = 0 p min = 0! So the long-run equilibrium price of yogurt is p = 0; while the long-run equilibrium industry output of yogurt is Q = D (p ) = p 2 = = : Because each rm produces nothing, the long-run pro t per rm is given by = 0: (Note: This problem is a very clear example of how the concept of perfect competition relies on the possibility of having an in nite number of rms. Notice that although each rm produces zero units of yogurt, a positive amount of yogurt gets produced! This of course is possible because there are, in fact, an in nite number of rms in the industry.). (27 pts) Now that making yogurt out of xanthorrhoea xeranthemum is no longer pro table, Shalini is no longer able to sell either plant to yogurt makers. However, the inhabitants of Ludos have just discovered that xanthorrhoea (x ) can be used as a beautiful smelling fancy cologne, so every inhabitant of Ludos is willing to pay 20 euros for a xanthorrhoea plant (because it gives them 20 euros worth of utility). Xeranthemum (x 2 ) ; on the other h, smells absolutely terrible when made into a cologne, so nobody is willing to pay anything for any amount of xeranthemum (because it provides zero utility). Shalini realizes that (because of their very confusing names rather similar appearances) the inhabitants of Ludos cannot tell the two exotic plants apart when she is trying to sell them! Suppose the inhabitants of Ludos know that a fraction of the total plants that Shalini has are xanthorrhoea (x ) the other are xeranthemum (x 2 ), that they are willing to pay their expected utility for a plant when they do not know what it is. Assume that the inhabitants of Ludos are rational can perfectly calculate the probability of a plant being xanthorrhoea conditional on Shalini being willing to sell it on Ludos. (That is, if Shalini chooses not to sell any of her xanthorrhoea on Ludos, the inhabitants of Ludos will be able to calculate that the probability of a plant being sold on Ludos being xanthorrhoea is zero.) Throughout the question, assume that Shalini has 00 plants total incurs zero costs. (a) ( pts) If Shalini were to sell all of her exotic plants on Ludos by romly choosing a plant from her boat every time she made a sale, how much would each inhabitant of Ludos be willing to pay for such a plant? Soln: They would be willing to pay ( ) = 20 euros for the plant. (b) (8 pts) Close to Ludos, there is another isl called Gnosis, where the very knowledgeable inhabitants can always tell if a plant is xanthorrhoea or xeranthemum. On Gnosis, Shalini can sell a unit of xanthorrhoea (x ) for 5 euros but cannot sell xeranthemum (x 2 ) at all. Gnosis is close enough to Ludos that Shalini can go back forth between the isls at no cost. i. (4 pts) What will Shalini s total pro t be if she chooses to try sell all of her plants of Ludos? (Hint: Your answer should be a function of.) Soln: Shalini s total pro t will be = 2000 euros. ii. (4 pts) What will Shalini s total pro t be if she chooses to try sell all of her xanthorrhoea (x ) on Gnosis all of her xeranthemum (x 2 ) on Ludos? (Hint: Your answer should be a function of.) Soln: Shalini s total pro t will be ( ) = 500 euros. (c) (0 pts) Suppose Shalini chooses to sell a fraction of her xanthorrhoea (x ) (i.e. a fraction of her total plants) on Gnosis the rest of all of her plants on Ludos. i. (5 pts) How much would inhabitants on Ludos be willing to pay for a plant in this case? Soln: They would be willing to pay 20 times the probability that a plant is a xanthorrhoea (x ), which is ; so they are willing to pay 20 euros. 6

7 ii. (5 pts) What will Shalini s total pro t be if she chooses to do this? Soln: Shalini s total pro t is ( ) = ( ) = : (d) (6 pts) Suppose = 2 : How much of her xanthorrhoea (x ) will Shalini try to sell on Gnosis how much will she try to sell on Ludos? Is the resulting allocation e cient? Explain why or why not. Soln: Shalini s pro t is strictly decreasing in no matter what is, so for = 2 or any other she will choose = 0; meaning she will never try to sell xanthorrhoea on Gnosis. She will always try to sell it all on Ludos. The resulting allocation will be one in which all of the xanthorrhoea is consumed by the inhabitants of Ludos, who value it at 20 euros. This means that none of the xanthorrhoea is wasted or consumed by the inhabitants of Gnosis, who value it at 5 euros. This is indeed an e cient allocation of xanthorrhoea. As for xeranthemum, all of it is also consumed by inhabitants of Ludos, but since the plant does not provide positive or negative utility to anybody this does not a ect e ciency. The resulting allocation is therefore e cient. 7