Question

Consider my utility (U) functions for Q1and Q2:U1=30Q1–Q12and U2=20Q2–1.5Q22. The market prices for Q1and Q2respectively are PM1= 1and PM2= 3. My budget is $100.(a) How much Q1and Q2should I purchase to maximize my total utility (TU)?(b) Compute my TU at the optimum quantities.

Answer 1

Budget constraint:

100 = Q1 + 3 Q2

Total utility (TU) = 30 Q1 - Q1² + 20 Q2 - 1.5 Q2²

Lagrangian:

L = 30 Q1 - Q1² + 20 Q2 - 1.5 Q2² + b ( 100 - Q1 - 3Q2)

dL/dQ1 = 30 - 2Q1 - b = 0

dL/dQ2 = 20 - 3 Q2 - 3b = 0

dL/db = 100 - Q1 - 2Q2 = 0

Solving the above equations for Q1 and Q2, we get,

Q1 = 54.77

Q2 = 35.33

TU = 30 × 54.77 - (54.77)² + 20 × 35.33 -1.5 × (35.33)²

= -2522.366

(a) The quantity purchased to maximise utility is Q1 = 54.77 and Q2 = 35.33

b) The total utility is -2522.366