OPMT 5701 Term Project 2013

Transcription

1 OPMT 570 Term Project 03 Selected Answers. Willingness to Pa versus Equivalent Compensation Skipp and Mrtle are friends who consume the same goods: oga classes (X) and Timbits (Y ). Skipp has the utilit function u = u(, ) and faces the budget constraint: M = p + p. where M is income, and p, p are prices. Mrtle has the same budget constraint as Skipp but her utilit function is v = v(, ). NOTE: Skipp s Utilit is u = which can be transformed to u = 3 3 or u = or u = / / or u = ln + ln. All of these are MONOTONIC tansformations. You can use calculus to find the MRCS for each MRCS = u/ u/ = which means the all have the same demand functions: = M P and = M P Therefore, the answers for all four versions will be the same number for,, M onl the utilit numbers will differ Similarl, for Mrtle v = or v = /3 /3 or ma problem = M 3P and = M 3P. v = ln + ln will give the same solutions to the (a) Use the Lagrange Method to show that Skipp and Mrtles demand functions for and, which will be in the form: L = + λ (M p p ) L = λp = 0 L = λp = 0 L λ = M p p = 0 Substitute into budget to find = λp λp = p p s = M p s = M p u = M 4p p The ependiture function is Then, for Mrtle, M = 4p p u L = + λ (M p p ) L = λp = 0 L = λp = 0 L λ = M p p = 0 = λp λp = p p

2 and Mrtle s ependiture function is m = M m = M 3p 3p ( ) ( ) M M v = = 4M 3 3p 3p 7p p M = 3 7p p v 4 (b) Suppose M = 0, P = and P = 4. What is Skipp and Mrtles optimal, and utilit number? If the price of was lowered to $ what would be their, and utilit numbers? Using the formula s in part (a) At Then P = 4 Skipp Mrtle U P = Skipp Mrtle U (c) If the price remains as $4, how much additional income would each one require to get the same utilit as the did when the price was $ Use each person s ependiture function ((M ) shown in part (a) Skipp M rtle u = 800 v = 8000 M = 4p p u M = 3 7pp v 4 M = 69.7 M = 5. (d) Draw a graph for Skipp and another for Mrtle. In each graph show the budget constraints, indifference curves and equilibrium values of and for both p = 4 and p =. Be ACCURATE and NEAT. (minimum / page for each graph) (e) Suppose that the Yoga Studio offers two options: a drop-fee of $4, or a membership of $30 that lets the member take Yoga classes for $. What will be the Utilit of Skipp and Mrtle if the each bu a membership? Given the options (drop-in or member) which will each one choose? Carefull add the new budget constraint, indifference curve and equilibrium and to each girl s graph. Skipp will join, Mrtle will not join Skipp M rtle Drop in M ember (f) What is the membership fee that would make Skipp indifferent between drop at $4 and membership that charges $ per oga class? Add this to Skipp s graph. Membership = = 35.5 (g) What is the membership fee that would make Mrtle indifferent between drop at $4 and membership that charges $ per oga class? Add this to Mrtle s graph. : Membership = = 4. 6

3 69.7 Skipp Question Mrtle

4 . Labour Suppl OR Epected Utilit problem LABOUR SUPPLY PROBLEM (a) There are 68 hours in one week. Subtracting 48 hours for the weekend and 8 hours per night for sleep leaves 80 hours for work and leisure. Spark works at a job that pas w per hour for the first 40 hours, then.5w for each hour after that. Let h be the hours of work such that h = 80 L. Spark consumes a composite commodit C which costs p. Spark maimizes her utilit u(c, L) subject to the following budget constraints: L = 80 h 80w = pc + wl if 0 h 40 00w = pc +.5wL if h > 40 (b) If p = and w = 4, graph both constraints. Verif that the two constraints cross at 40 hours of work. Given the restrictions on l, use a hi-lite pen to show Spark s "Effective Budget Constraint". (c) If Spark s utilit function is u = ln C +ln L, find her optimal L and C. (hint: do the Lagrangian twice, once with each constraint; then check which solution satisfies all the conditions). Z = ln C + ln L + λ(30 C 4L) Z L = L 4λ = 0 Z C = C λ = 0 Z λ = 30 C 4L = 0 L = 6.67, C = 3, U = 4 h = = 53.3 Z = ln C + ln L + λ(400 C 6L) Z L = C 6λ = 0 Z C = L λ = 0 Z λ = 400 C 6L = 0 L =., C = 66.7, U = 4.7 h = 80. = 57.8 Therefore Spark prefers overtime 4

5 (d) Calculate the utilit values for both solutions in (b) illustrate both equilibrium points and indifference curves in our graph drawn for (a). Does the incentive of an overtime premium induce Spark to work more than 40 hours? (e) Spark s cousin Otis gets hired for the same job. However Otis s utilit function is u = ln C+ ln L. Re-do parts (a)-(c) from above ecept using Otis s utilit function. Overtime: Z = ln C + ln L + λ(30 C 4L) Z L = L 4λ = 0 Z C = C λ = 0 Z λ = 30 C 4L = 0 L = 53.3, C = 06.6, U =.6 h = 6.7 Z = ln C + ln L + λ(400 C 6L) Z L = L 6λ = 0 Z C = C λ = 0 Z λ = 400 C 6L = 0 L = 44.4, C = 33.3, U =.4 h = 35.6 Therefore Otis will NOT work an overtime EXPECTED UTILITY PROBLEM Skipp has the following utilit function U = W, where W is her wealth. She has a probabilit P she has a bad da (State ) loses an amount L.Her epected utilit is: EU = ( P )W / + P (W L) / Suppose her initial wealth is $500, the loss, L = $00 and P = 30% 5

6 (a) (b) What is Skipp s epected utilit? What is the certaint equivalent wealth? Graph our answer. Skipp can bu units of insurance at a price of a on good das that will give her units of coverage a on bad das. If the insurance is offered at fair odds, then ( P )a P a = 0 0.7a 0.3a = 0 which is the "budget constraint" Skipp faces. If she bus insurance her epected utilit will be EU = ( P ) (W a ) / + P (W L + a ) / Set up the Lagrangian for this problem. Show that Skipp will full insure (i.e. her net wealth on both good and bad das will be equal). from equations () and () L = 0.7(500 a ) / + 0.3(400 + a) / + λ (0.7a 0.3a ) L = 0.7 (500 a ) / ( ) + 0.7λ = 0 L = 0.3 (400 + a ) / 0.3λ = 0 L λ = 0.7a 0.3a = (500 a ) / 0.3 (400 + a = 0.7λ ) / 0.3λ (500 a ) / = (400 + a ) / Use the budget constraint, re-written as a = 3 7 a (500 a ) / = (400 + a ) / 500 a = a 00 = a + a Skipp s wealth in each state will be 00 = 3 7 a + a = 0 7 a a = 7 (00) = a = 630 good W = = 870 Bad W = = 870 6

7 (c) 3. Skipp lives on an island where she produces two goods, and, according the the production possibilit frontier 400 +, and she consumes all the goods herself. Skipp also faces and environmental constraint on her total output of both goods. The environmental constraint is given b + 8: Her utilit function is u = = = (a) Write down the Kuhn Tucker rst order conditions. Z = = = + (400 ) + (8 ) The Kuhn-Tucker conditions are now Z = = = Z = = = Z = Z = 8 0 =0 =0 0 0 Z Z =0 =0 (b) Find Skipp s optimal and. Identif which constaints are binding.from the rst two equations, we get: = = = = or = = tr = 0. then we get: 7 = =

8 Sub into the PPF constraint (400 = 0) to get, 400 = = 00 = 4.4 = 4.4 But this violates the other constraint ( + 8) so tr λ = 0 Environmental Constraint is Binding (c) Graph our results. = λ = λ = = 4, = 4 (d) On the net island lives Spark who has all the same constraints as Skipp but Spark s utilit function is u = ln + 3 ln. Redo a, b, and c for Spark. The onl difference is that from the first order conditions 3 = λ + λ λ + λ tr λ = 0. then we get: 3 = λ λ = 3 Sub into the PPF constraint (400 = 0) to get, and 400 = 4 = 00 = 0 = 300 = 7.3 which does NOt violate the other constraint ( + 8) 8

9 (c) 9

10 The Gains From Trade D 0 A u = 000 u = 34 (No trade) 8.i B P = 0.5 u = 85 C E P = u* (No trade).5 4. International Trade: Solving the problem of Production and Echange. The following graph is found in most economics tets and it illustrates how an econom can get outside its PPF through free trade. Solving this problem is a straight forward application of Lagrange. Figure : (a) Skipp lives on an island full of resources that lets her produce two goods: and. Her production possibilit frontier (constraint) is given b + = 00 Skipp works alone to maimize her utilit which is given b u = (b) Set up the Lagrangian for Skipp and solve for,.use our solution to calculate u. L = + λ ( 00 ) =.5, = 8.6, u = 089 (c) Carefull graph and label the PPF and Indifference Curve at the optimum. (d) Now Skipp has the opportunit of to trade with Mrtle on a neighboring island. Skipp can trade whatever and she has at prices p and p. To maimize her gains from trade, Skipp needs to follow a -step process. L E = p + p + λ(00 ) L U = + λ (p X E + p Y E p p ) (e) If p = p =, solve the Lagrangian that maimizes her endowments. Then use the endowment values and prices given in the trading constraint and maimize her utilit b the Lagrangian 0

11 method. Calculate her utilit number. L = + + λ(00 ) = = 0 L = + λ(0 ) = 3.3, = 6.7, u = 85 (f) CAREFULLY construct a graph containing her PPF, Trading Line, and indifference curve at the optimum. Is her utilit number greater than in part one? (g) Now suppose the trading prices change such that p = 0.5. Using our endowment values (X E, Y E ) from before, calculate Skipp s new utilit maimizing bundle. Draw the new trading line and indifference curve on our graph in (b) above. = = 0 L = + λ(5 0.5 ) = 0, = 5, u = 000 (h) Since the trading prices have changed, Skipp needs to adjust her production decision. Re-do (a) and (b) with the new prices. L = λ(00 ) = 6.3, =.7 L = + λ( ) =., = 5.3, u = 34

12 5. There are two firms, each with the same total savings functions: T S = 00e 0.5(e ) and T S = 50e (e ), where e and e are the emissions from firm and firm respectivel. The two firms want to maimize their savings from emissions, subject to whatever regulator constraints the face. Currentl, both firms are constrained b a pollution standard limiting their emissions to e = e = 48. The government decides to implement a new "Cap and Trade" program that will allow the firms to trade emission credits from their initial allocation of 48 credits each. However, an trade of emissions must maintain the air qualit constraint at the two receptors that are set up in the region the firms operate. The national air qualit constraint limits particles in the air to P = 0. If particles rise (P > 0) means air qualit falls. Each receptor has it s own diffusion equation that links emissions to particles in the air. At receptor one, the formula is At receptor two the diffusion equation is The economic and environmental problem is to subject to P =.5e + e P = e +.5e Ma T S + T S P =.5e + e 0 P = e +.5e 0 (a) Set up the Lagrangian for this problem and solve for the optimal e and e. Z = 00e 0.5(e ) + 50e (e ) + λ (0.5e e ) + λ (0 e.5e ) Tr λ = 0, use onl constraint # Solving for e an e ields Test answer against the other constraint Z = 00 e.5λ = 0 Z = 50 4e λ = 0 Z λ = 0.5e e = 0 e = 54.5 e = 38.5 e +.5e (38.5) =.875 < 0 NOTE, if ou tr λ = 0 (use onl constraint ) ou will get e = 88.9, e = 0.9, which will violate constraint ANS: Constraint is the onl binding con- (b) Determine which constraint(s) are binding. straint. (c) Given that both firms begin with 48 emission credits each, determine the amounts traded and the trade ratio. e = = 6.5 e = = 9.75 ratio = =.5

13 (d) Graph our answer carefull, labelling all the relevant points. 3