5. On any given evening, J.P. enjoys consuming cigars (C) and glasses of brandy

Question

5. On any given evening, J.P. enjoys consuming cigars (C) and glasses of brandy (B) according to his utility function U(C, B) 20C- C2 18B - 3B2. While cost is no object to J.P. (i.e. he does not face a binding budget constraint), his doctors have strongly advised him to limit the sum of brandy glasses and cigars consumed to no more than 5. Givern this constraint, how many glasses of brandy and cigars will he consume? You should find that the Lagrange multiplier on this problem is 12. By how much should J.P's utility rise if the doctor relaxes the constraint he faces from 5 to 6?

Explanation / Answer

Max U(C,B)=20C-C^2+18B-3B^2 such that C+B<=5

Setting up the Lagrangian problem: L=20C-C^2+18B-3B^2 +LAMDAH(5-C-B)

Taking partial derivatives with respect to C and B:

dL/dC=20-2C-LAMDAH

dL/dB=18-6B-LAMDAH

Putting dL/dC and dL/dB = 0 given that LAMDAH=12;

20-2C=12 and 18-6B =12

We get, C=4 and B=1 The equilibrium quantities C* and B* need to be fed into the Utility function to get its value,

U=20(4)-16+18-3=79 The Utility remains constant at 79 even when the sum of brandy and cigars is set to no more than 6 ( from 5 initially)

Related Questions

Navigate

• Browse (All)
• Browse 5
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.