Final answer:
To find the effort level, we maximize the entrepreneur's profit function by setting the first derivative with respect to effort to zero. In scenarios where the entrepreneur retains all, none, or a portion of the equity, their effort level is adjusted based on the retained profits from their effort contribution. The value of the firm is determined by the profit given the chosen effort level.
Step-by-step explanation:
To determine the effort level an entrepreneur chooses, we maximize the profit function, which is the revenue function minus the cost function. We differentiate the profit function P(e) = R(e) - C(e) with respect to effort level e and set the derivative equal to zero to find the maximum.
(a) P(e) = R(e) - C(e) = e² + 800e - 2e². To find the optimum effort level, we take the first derivative of P(e) with respect to e, set it equal to zero, and solve for e.
So, P'(e) = 2e + 800 - 4e = -2e + 800. Setting P'(e) to zero gives us e = 400. The value (gross profit) of the firm is then calculated as P(400) = R(400) - C(400).
(b) If the entrepreneur sells 100% equity, their incentive to exert effort decreases since they no longer receive the benefits of their effort. Thus, the effort level would be e = 0, as the entrepreneur does not gain from any additional effort.
(c) When selling β% equity, the new profit function becomes (1-β)*P(e). The entrepreneur will choose an effort level based on this modified profit function. For β = 0.70, the entrepreneur retains 30% of the profit, which changes the effort level calculation. The value of the firm will be based on the new effort level e and the shared profit equation (1-β) * P(e).