Question

On a given evening, J. P. enjoys the consumption of cigars () and brandy (c) according to the utility function U(c,b) 20c - c2 + 18h 32 Cost is no object to J.P Which of the following are the first order conditions for this unconstrained maximization problem? Check all that apply. dU-20 -2c0 20 c2 dll 186b0 du -20c -c20 dlJ The combination of cigars and brandy that maximizes P.J.'s utility is and b* . This leads to a total utility level of U - Lately, however, J. P. has been advised by his doctors that he should limit the sum of glasses of brandy and cigars consumed to 5. With this new constraint, the combination of cigars and brandy that maximizes P.J.'s utility is c utility level ofU* and b* . This leads to a total

Answer 1

Ans.1 ) 1st and 5th option are the correct answers which we get after differentiating the Utility function with respect to Cigars and Brandy.

Putting d(U)/d(c) = 0 gives,

20 - 2c = 0

C = 20/2

C = 10

Putting d(U)/d(B) = 0 gives,

18 - 6b = 0

b = 18/6

b = 3

U(c,b) = 20c - c^2 + 18b - 3b^2

Taking, C = 10 and b= 3 gives,

Utility = 100+(18×(3))−(3×(3)^(2))

Utility = 127

Ans2.)

Taking, C = 5 and b= 3 gives

Utility = 20×(5)−(5)^(2)+((18×(3))−(3×(3)^(2))

Utility = 102

Taking, C = 4 and b= 3 gives

Utility = 20×(4)−(4)^(2)+((18×(3))−(3×(3)^(2))

Utility = 91

Taking, C = 3 and b= 3 gives

Utility = 20×(3)−(3)^(2)+((18×(3))−(3×(3)^(2))

Utility = 78

Taking, C = 2 and b= 3 gives

Utility = 20×(2)−(2)^(2)+((18×(3))−(3×(3)^(2))

Utility = 63

Similarly , if c = 1

As we have seen in the 1st part of the question that Brandy = 3 units is the optimal choice for JP as it maximizes the utility function.

As far as Cigar consumption is concerned, it has been limited to 5 units.

We can also ses that as we decrease the consumption of cigar, the total utility is also decreasing.

So in 2nd scenario,

C = 5 ; b = 3

Total utility = 102