Question

How do you solve this without lagrangians?

2. Suppose that Romeo has the utility function U=SORS', and Juliet has the utility function U = S'RS,, where Sr is Romeo's spaghetti consumption and S, is Juliet's. = They have 36 units of spaghetti to divide between them. a. Romeo would want to give Juliet some spaghetti if he had more than 18 units of spaghetti. b. Juliet would want to give Romeo some spaghetti if she had more than 25 units. c. Romeo and Juliet would never disagree about how to divide the spaghetti. d. Romeo would want to give Juliet some spaghetti if he had more than 30 units of spaghetti. Juliet would want to give Romeo some spaghetti if she had more than 24 units of spaghetti. e.

Answer 1

Q2) The answer is (d)

We do not need to use the Lagrangean to solve this.

Sincw the utility fuctions for both are symmetric, we can maximize for 1 and the symmetric outvome amximizes the utiltiy fror other.

Let us maximize Juliet's utility = S_RS_J⁵

= Max (36-S_J)*S_j⁵ (Since S_R +S_J = 36)

Taking FOC wrt to S_j and equating to 0 we get

180S_J⁴ = 6S_J⁵

=> S_J = 30

Thus, the utiltiy is maximized for both when they have 30 units of spaghetti and will be willing to gove some to the other person if they have more than 30 units of spaghetti.