Question

The utility function of Jim is given by  u(w, b, e) = w + b – c (e)

c(e) = 1 if e < 10

c(e) = (1/2 × (e − 10)^2) if e ≥ 10. 2 

Michael is Jim’s boss, who is maximising DM’s profits which are given by: 

  π (w, b, e) = R(e) − w − b, where R(e) is the firm′s sales revenues and R(e) = 100e 

Michael cannot observe Jim’s effort (e) directly but observes R(e) and can set the bonus level dependent on sales revenues.

(a). If Michael sets w=100 and b=0, what would be Jim’s effort level (e) and DM’s profits?

Answer 1

(a) If Michael sets w=100 and b=0, then Jim’s effort level (e) and DM’s profits will be:

MR = MC,

$MC = c'(e) = (1/2 × (e − 10)^2)' = e - 10,$

MR = R'(e) = 100,

e - 10 = 100,

e = 110,

$π (w, b, e) = 100×110 − 100 − 0 = 10,900.$