Question

following are the information used to solve this problem

Joy the abandoned restaurant can be found a little further down the road. With facilities to produce food and water using the same technology as Dilek's it's a great rest stop for hikers. Manu and Kapi get to the restaurant next day they finish their food and water and now they are their only customers and owners. They decide to split the profits 50/50 and devote both of their time to produce in the restaurant. Each of them have exactly 16 hours a day to work and the same utility functions from before 1) Find the equilibrium of this new economy Problem 1 (30 Points) Kapi and Manu bumped into each other while hiking the Appalachian trail. While both of them started the hike well prepared by the moment they met each other Kapi had 5 bottles of water (x) and 1 energy bar (y) (the only supplies they need) while Manu had 3 bottles of water and 3 energy bars. Their preferences can be represented by the following utility functions identified by the subscript of their initials: Up(Xk, Yk) = VIKYK Um m, Ym) = VxmYm with marginal utilities: MUm = MU MUyk 1) Find the algebraic expression and graph the contract curve 2) What are the prices and allo cations of equilibrium? Depict in an Edgeworth's box clearly the pre exchange and post exchange situation 3) Why was it beneficial for them to trade? Problem 2 (40 Points) Dilek is also hiking the Appalachian trail but was not lucky and didn't bump into any of the other hikers but she is very resourceful. She ran out of both water and energy bars. Still she realized she can get water by walking to a nearby river and can get food by picking berries. Both production activities require only time to be carried out which is where she gets lucky since time is all she has, more specifically she has 16 hours of working time. The technology that she has to produce water and food is described as follows: VLs = i y= 0.25/Ly 1

Answer 1

x= (L_x)^0.5

y= 0.25(L_y)^0.5

As L_x + L_y = 16

=> x² + 16y² = 16

utility for both = (xy)^0.5

=>U(x)= (x^0.5)*(1- x²/16)^0.25

at equilibrium

dU/dx = 0

=> 0.5(x)^-0.5 * (1- x²/16)^0.25 + (x^0.5) * 0.25(1- x²/16)^-0.75 * (-2x/16) = 0

=> using this, we get x = 4,2^1.5,-2^1.5, -4, 0 as critical points.

so, Utility has maximum value at x = 2^1.5 or  2.83

Value of y = (1-x² /16)^0.5 = 1/(2)^0.5 = 0.71

Maximum value of their utilities = (xy)^0.5 = 1.42