For the utility functions (given below) of two consumers (1 and 2) in a pure exchange model with two goods (x and y):

a) draw the Edgeworth box for initial endowments ω_1x = 10, ω_1y = 10, ω_2x = 20, ω_2y = 20 (standard notation);

b) in the box depict the area representing possible improvements for both consumers relative to their initial situation;

c) determine the formula of the contract curve and draw it in the diagram;

d) find the equilibrium price ratio and calculate the consumers’ demand for each of the two goods.

i. U₁(x,y) = xy, U₂(x,y) = x^_0.5 y^1.5

ii. U1(x,y) = 4x + 2y, U2(x,y) = x + y

iii. U1(x,y) = xy, U2(x,y) = x + 3y

iv. U1(x,y) = x + 2y, U2(x,y) = min{2x,y}

v. U1(x,y) = x^0.5y^0.5, U2(x,y) = min{x,2y}

vi. U1(x,y) = 2x + ln(y), U2(x,y) = x + 2ln(y)

vii. U1(x,y) = 2x^0.5 + 4y, U2(x,y) = x + y^0.5

viii. U1(x,y) = 2ln(x) + ln(y), U2(x,y) = ln(x) + ln(y)