Question

Question 6: Logan has preferences over olives ) and ice creams (x2). He prefers to eat them separately but not together, which is represented by: Also suppose his income is $100 and the prices of olives and ice creames arepx$1 and Pr2 $2,respectively. 1. Is this convex preferences or concave preferences? 2. Solve for the bundle that satisfis the tangency condition. 3. What's the level of utility using the bundle you just solved in Q6.2? 4. Consider spending all of your income on olives (xi). What's the utility from this bundle? 5. Which bundle does Logan prefer between the bundle in Q6.2 and Q6.4? Does the tangency condition lead to the optimal bundle? 6. What's the optimal bundle when price of olives is $3.

Answer 1

U (x_1, x_2) = x^2_1 + x^2_2 M = $ 100, P_x_1 = $1, P_x_2 == $ 2 The preferences are concave as the question states that he prefers to eat the goods separately but not together. For tangency condition: Slope of IC = MRS = Partialdifferential v/Partialdifferential x_1/Partialdifferential u/Partialdifferential x_2 = 2 x_1/2 x_2 = x_1/x_2 slope of budget line = P_1/P_2 = 1/2 Equating both we get x_1/x_2 = 1/2 2 x_1 = x-2 rightarrow (1) P_1 x_1 + P_2 x_2 = 100 (Budget line) rightarrow (2) x_1 + 2x_2 = 100 Substituting (1) in (2), x_1 + 2 times 2 x_1 = 100 5 x_1 = 100 x_1 = 20 units x_2 = 2 times 20 = 40 units v = (20)^2 + (40)^2 = 400 + 1600 = 2000 units)^d = 100/1 = M/p x_1 = 100 v = (100)^2 = 100 times 100 = 10000 units He will choose the bundle