There is a nest with birds: Mama bird, Papa bird, and two baby chicks named Chick 1 and Chick 2. The parent birds have a fixed stock of worms W and crickets C to divide between their children. The utility (evolutionary fitness) of each chick is a function how many worms and crickets they get: for Chick 1, u¹(w, c), and for Chick 2, u² (w, c). Mama and Papa want to allocate the worms and crickets between their chicks in such a way that maximizes the sum of utilities. Unfortunately, Mama and Papa do not know what u¹and u² are.

(a) Suppose that the chicks could tell their parents what their utility functions were. What problem would Mama and Papa solve, and what first order conditions would characterize their optimal division of worms and crickets between the chicks?

(b) Now suppose the chicks have no way to communicate their utility functions. Instead, Mama and Papa set up a market in order to allocate the resources. In addition to worms and crickets, there is also an 'avian evolutionary fitness' (AEF) currency; one unit of AEF always increases a chick's utility by one unit. The chicks can borrow as much AEF as they want. Suppose that Mama and Papa set the price of worms at pw and the price of crickets at p_c. How many worms and crickets will each of the chicks demand? (Characterize with FOCs; don't try to find explicit expressions.)

(c) It might be that both chicks are asking for too many worms or too many crickets, so Mama and Papa have to raise or lower their prices a bit. Finally, they reach prices p^*_wand p^*_c at which the total amount of worms demanded by the two chicks is W and the total amount of crickets demanded is C. Demonstrate (using the math!) that the chicks are asking for precisely the divisions that Mama and Papa would have chosen in part (a).

(d) Now suppose that Mama and Papa have a general utility function U^MP(u¹ , u² ) over the fitness of their children which favors Chick 1 over Chick 2. Write down the FOCs characterizing their ideal division of resources. Could the parents find a market-based solution to implement that division? Why or why not?