1. A two-person partnership has revenue az + az. The effort cost for partner i is ļa? (thus the first-best effort levels, which maximize the joint payoff of the two partners, is a, = az = 1). Effort is not contractible; Partner 1 gets a share s (a1 + a2) and Partner 2 gets (1 – s) · (a1 + a2), where s can vary between 0 and 1. a) Calculate the effort level each partner chooses as a function of s. Use the answers to calculate the total surplus (sum of the partners' payoffs) as a function of s. What value of s maximizes this total surplus? b) Suppose the partners are not symmetric; instead revenue is aa, + Baz, where a and B are positive but not in general equal; costs remain as before. Answer the same questions as in (a). Provide some intuition for your result. c) The symmetric partnership (as in part (a)) contemplates hiring a "boss" who will monitor the partners' efforts, making them contractible. What is the most the boss could earn in his role? (Assume the partners would choose the optimal s you found in (a) without a boss, and argue from there.) 2. A large partnership has n identical partners; each partner i has effort cost a?. The revenue is E a4. a) Calculate the first-best effort, revenue and total surplus as a function of n. b) Suppose effort is not contractible, and the partners share the revenue equally (i.e., cach gets E a,). What level of effort will each partner choose? What is the resulting total revenue and total surplus? c) Compute a "performance ratio" (ratio of per-partner surplus using your answer from (b) to the first-best surplus using (a)) as a function of n. What happens as n gets large? 3. A study group with two members receives a single grade on a term project (so both students get the same same grade). The grade is an in- creasing function of each of their efforts. Specifically the grade is G(a1, a2) = 1
1. A two-person partnership has revenue az + az. The effort cost for partner i is ļa? (thus the first-best effort levels, which maximize the joint payoff of the two partners, is a, = az = 1). Effort is not contractible; Partner 1 gets a share s (a1 + a2) and Partner 2 gets (1 – s) · (a1 + a2), where s can vary between 0 and 1. a) Calculate the effort level each partner chooses as a function of s. Use the answers to calculate the total surplus (sum of the partners' payoffs) as a function of s. What value of s maximizes this total surplus? b) Suppose the partners are not symmetric; instead revenue is aa, + Baz, where a and B are positive but not in general equal; costs remain as before. Answer the same questions as in (a). Provide some intuition for your result. c) The symmetric partnership (as in part (a)) contemplates hiring a "boss" who will monitor the partners' efforts, making them contractible. What is the most the boss could earn in his role? (Assume the partners would choose the optimal s you found in (a) without a boss, and argue from there.) 2. A large partnership has n identical partners; each partner i has effort cost a?. The revenue is E a4. a) Calculate the first-best effort, revenue and total surplus as a function of n. b) Suppose effort is not contractible, and the partners share the revenue equally (i.e., cach gets E a,). What level of effort will each partner choose? What is the resulting total revenue and total surplus? c) Compute a "performance ratio" (ratio of per-partner surplus using your answer from (b) to the first-best surplus using (a)) as a function of n. What happens as n gets large? 3. A study group with two members receives a single grade on a term project (so both students get the same same grade). The grade is an in- creasing function of each of their efforts. Specifically the grade is G(a1, a2) = 1
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Transcribed Image Text:1. A two-person partnership has revenue az + az. The effort cost for partner
i is ļa? (thus the first-best effort levels, which maximize the joint payoff of
the two partners, is a, = az = 1). Effort is not contractible; Partner 1 gets
a share s (a1 + a2) and Partner 2 gets (1 – s) · (a1 + a2), where s can vary
between 0 and 1.
a) Calculate the effort level each partner chooses as a function of s. Use
the answers to calculate the total surplus (sum of the partners' payoffs) as a
function of s. What value of s maximizes this total surplus?
b) Suppose the partners are not symmetric; instead revenue is aa, + Baz,
where a and B are positive but not in general equal; costs remain as before.
Answer the same questions as in (a). Provide some intuition for your result.
c) The symmetric partnership (as in part (a)) contemplates hiring a
"boss" who will monitor the partners' efforts, making them contractible.
What is the most the boss could earn in his role? (Assume the partners
would choose the optimal s you found in (a) without a boss, and argue from
there.)
2. A large partnership has n identical partners; each partner i has effort
cost a?. The revenue is E a4.
a) Calculate the first-best effort, revenue and total surplus as a function
of n.
b) Suppose effort is not contractible, and the partners share the revenue
equally (i.e., cach gets E a,). What level of effort will each partner
choose? What is the resulting total revenue and total surplus?
c) Compute a "performance ratio" (ratio of per-partner surplus using your
answer from (b) to the first-best surplus using (a)) as a function of n. What
happens as n gets large?
3. A study group with two members receives a single grade on a term
project (so both students get the same same grade). The grade is an in-
creasing function of each of their efforts. Specifically the grade is G(a1, a2) =
1
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