Assume an exchange economy with two individuals, A and B. Both individuals consume good x and good y. The individuals utility functions are uA(xA, yA) = x1 4 Ay3 4 A and uB (xB , yB ) = x1 4 B y3 4 B . The initial endowment is A = (Ax , Ay ) = (2, 0) for individual A, i.e., individual A initially has two units of good x and no units of good y. Analogously, the initial endowment for individual B is B = (Bx , By ) = (0, 1). Good y is the numeraire, that is, py = 1. Calculate the Walrasian equilibrium as follows. (i) Determine the Lagrangian function for individual A that is used to derive the Walrasian equilibrium. (ii) Explain the different parts of the Lagrangian function (determined in (i)) verbally. (iii) Determine the first-order conditions and solve for individual As gross demand, i.e., (xA, yA) depending on px. (iv) Determine the gross demand for individual B. (v) Determine the price of good x, i.e., px.

Problem 2. Assume an exchange economy with two individuals, A and B. Both individuals consume good x and goody. The individuals' utility functions are uA(xA,yA)=xA41yA43anduB(xB,yB)=xB41yB43. The initial endowment is A=(xA,yA)=(2,0) for individual A, i.e., individual A initially has two units of good x and no units of good y. Analogously, the initial endowment for individual B is B=(xB,yB)=(0,1). Good y is the numeraire, that is, py=1. Calculate the Walrasian equilibrium as follows. (i) Determine the Lagrangian function for individual A that is used to derive the Walrasian equilibrium. (ii) Explain the different parts of the Lagrangian function (determined in (i)) verbally. (iii) Determine the first-order conditions and solve for individual A's gross demand, i.e., (xA,yA) depending on px. (iv) Determine the gross demand for individual B. (v) Determine the price of good x, i.e., px