Question

Consider the problem max 4x1 + 2x2 s.t. x1 + 3x2 ≤ 5 (K) 2x1 + 8x2 ≤ 12 (N) x1 ≥ 0, x2 ≥ 0 and the following possible market equilibria: i) x1 = 0, x2 = 3/2, pK = 0, pN = 1/4, ii) x1 = 1, x2 = 2, pK = 2, pN = 1, iii) x1 = 1, x2 = 2, pK = 4, pN = 0, iv) x1 = 5, x2 = 0, pK = 4, pN = 0. Select one: a. i) is a market equilibrium b. ii) is a market equilibrium c. iii) is a market equilibrium d. iv) is a market equilibrium

Answer 1

Here the market equilibrium is iv) x1 = 5, x2 = 0, pK = 4, pN = 0

Note that we are maximizing Z = 4x1 + 2x2 and the constraints are

• x1 + 3x2 ≤ 5 (K)
• 2x1 + 8x2 ≤ 12 (N)
• x1 ≥ 0
• x2 ≥ 0

Solve the first two constraints to get x1 = 5 and x2 = 0. In the graph, the value of Z at A is 4*0 + 2*5/3 = 3.33. At B, Z is 4*2 + 2*1 = 10. At C, Z is 4*5 + 2*0 = 20. Since Z is maximum at C, equilibrium is at  x1 = 5 and x2 = 0