Consider two agents, Amanda and Billy Bob, agents A and B who are having dinner together at a restaurant. We will formulate their respective consumption choices via our standard consumption bundle choice framework. Specifically, they are each choosing two-dimensional consumption bundles x = (x1, x2) where good 1 is food at the restaurant and good 2 is a choice of "other stuff", a composite bundle representation of other consumption choices in the economy. Assume preferences are the same across the two diners, u(x) = 21.02. Furthermore, assume they have an equal income of m= 150. Finally, assume that prices are P1 = 1 and P2 = 1.
1. Separate bills:
Suppose each diner simply pays just for his/her own food. That is, the budget constraint is x_1 + x_2 = 150 for each of them. Solve for the optimal choice of consumption bundle x* = (x_1*,x_2*) for each diner. Use the usual conditions for an optimal interior choice, that is MRS(x*) = P1/P2 and the budget line.
2. Share the bill:
Now, suppose Amanda and Billy Bob agree to share the restaurant bill. That is each of them will have a restaurant food expenditure of (x_1A +x_1B)/2, where x_1A is Amanda's food choice and x_1B is Billy Bob's food choice. In this case, their respective budget lines are,

(1/2) x_1A + x_2A = 150 - (1/2) x_1B
(1/2) x_1B + x_2B = 150 - (1/2) x_1A.

That is, for a given Billy Bob food choice, Amanda's budget line is one where prices are now P1 = 1/2 and P2 = 1, and Amanda's income is m_A = 150 - 10/2. Similarly, for a given Amanda food choice, Billy Bob's budget line is one where prices are p1 = 1/2 and P2 = 1, and Billy Bob's income is m_B = 150 - x_1A/2.

(a) Solve for Amanda's optimal consumption bundle x^_A = (x^_1A, x^_2A) understanding that these expressions will depend on Billy Bob's food choice, x_1B. To solve for x_A*, use the standard conditions for an optimal interior consumption bundle choice MRS(x^_A) = P1/P2 = 1/2, and Amanda's budget line. I chose to use the "^" notation for the optimal consumption bundle choice because I am anticipating that we have yet to resolve the Nash equilibrium which I will eventually refer to with "*" notation.
(b) Solve for Billy Bob's optimal consumption bundle x^_B = (x^_1B, x^_2B) understanding that these expressions will depend on Amanda's food choice, x_1A.
(c) Verify that the above solutions for x^_1A(x_1B) and x^_1B(x_1A) correspond to the best food response equations for diners A and B in the slides (where I have now made explicit that the optimal food consumption choices depend on the other diner's food choice).
(d) Denote by (x*_A, x*_B) the Nash equilibrium food bundle choices for Amanda and Billy Bob.
i. State the two equations involving the best food response functions that characterize the Nash equilibrium, (x*_A, x*_B). These equations only involve the food parts of the consumption bundles, but the x_2 parts directly follow once you know the Nash equilibrium food choices.
ii. Solve for the Nash equilibrium, and state the bundles (x*_A, x*_B).